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Encyclopedia > Equipartition theorem
Figure 1. Thermal motion of an α-helical peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the average kinetic energy of each atom to be computed, as well as the average potential energies of many vibrational modes. The grey, red and blue spheres represent atoms of carbon, oxygen and nitrogen, respectively; the smaller white spheres represent atoms of hydrogen.

In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among its various forms; for example, the average kinetic energy in the translational motion of a molecule should equal the average kinetic energy in its rotational motion. Many introductory textbooks actually define temperature as nothing more than a measure of the mean thermal kinetic energy per particle, and it is the equipartition theorem that justifies this elementary definition. However, this definition doesn't always work, particularly when quantum mechanical effects are important. More advanced texts will use a more general definition of temperature and then show mathematically how equipartition comes up in situations governed by classical mechanics. Image File history File links Thermally_Agitated_Molecule. ... Image File history File links Thermally_Agitated_Molecule. ... Side view of an α-helix of alanine residues in atomic detail. ... Peptides (from the Greek πεπτος, digestible), are the family of short molecules formed from the linking, in a defined order, of various α-amino acids. ... Properties In chemistry and physics, an atom (Greek ἄτομος or átomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... General Name, Symbol, Number carbon, C, 6 Chemical series nonmetals Group, Period, Block 14, 2, p Appearance black (graphite) colorless (diamond) Standard atomic weight 12. ... General Name, Symbol, Number oxygen, O, 8 Chemical series nonmetals, chalcogens Group, Period, Block 16, 2, p Appearance colorless (gas) very pale blue (liquid) Standard atomic weight 15. ... General Name, Symbol, Number nitrogen, N, 7 Chemical series nonmetals Group, Period, Block 15, 2, p Appearance colorless gas Standard atomic weight 14. ... General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Fig. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... In physics, a translation is the operation changing the positions of all objects according to the formula where is a constant vector. ... See also rotation around a fixed axis. ...


The equipartition theorem makes quantitative predictions. Like the virial theorem, it gives the total average kinetic and potential energies for a system at a given temperature, from which the system's heat capacity can be computed. However, equipartition also gives the average values of individual components of the energy, such as the kinetic energy of a particular particle or the potential energy of a single spring. For example, it predicts that every molecule in an ideal gas has an average kinetic energy of (3/2)kBT in thermal equilibrium, where kB is the Boltzmann constant and T is the temperature. More generally, it can be applied to any classical system in thermal equilibrium, no matter how complicated. The equipartition theorem can be used to derive the classical ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. It can also be used to predict the properties of stars, even white dwarfs and neutron stars, since it holds even when relativistic effects are considered. In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Helical or coil springs designed for tension A spring is a flexible elastic object used to store mechanical energy. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ... The Dulong-Petit law is a chemical law proposed in 1819 by French chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... STAR is an acronym for: Organizations Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... This article or section does not adequately cite its references or sources. ... A neutron star is one of the few possible endpoints of stellar evolution. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...


Although the equipartition theorem makes very accurate predictions in certain conditions, this is no longer true when quantum effects are significant. Equipartition is valid only when the thermal energy kBT is much larger than the spacing between the quantum energy levels. When it is smaller than the quantum energy spacing in a particular degree of freedom, the average energy and heat capacity of this degree of freedom are less than the values predicted by equipartition. Such a degree of freedom is said to be "frozen out". For example, the specific heat of a solid decreases at low temperatures as various types of motion become frozen out, rather than remaining constant as predicted by equipartition. Such decreases in specific heat were the first sign to physicists of the 19th century that classical physics was incorrect and that new physics was needed. Equipartition's failure for electromagnetic radiation — also known as the ultraviolet catastrophe — led Albert Einstein to suggest that light itself was quantized into photons, a revolutionary hypothesis that spurred the development of quantum mechanics and quantum field theory. Fig. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... It has been suggested that this article or section be merged with light. ... The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... The word light is defined here as electromagnetic radiation of any wavelength; thus, X-rays, gamma rays, ultraviolet light, microwaves, radio waves, and visible light are all forms of light. ... Fig. ... Quantum field theory (QFT) is the quantum theory of fields. ...

Contents

Basic concept and simple examples

See also: Kinetic energy and Heat capacity
Figure 2. Probability density functions of the molecular speed for four noble gases at a temperature of 298.15 K (25 °C). The four gases are helium (4He), neon (20Ne), argon (40Ar) and xenon (132Xe); the superscripts indicate their mass numbers. These probability density functions have dimensions of probability times inverse speed; since probability is dimensionless, they can be expressed in units of seconds per meter.

The name "equipartition" means "share and share alike". The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, on the average, once the system has reached thermal equilibrium. Equipartition also makes quantitative predictions for these energies. For example, it predicts that every atom of a noble gas, in thermal equilibrium at temperature T, has an average translational kinetic energy of (3/2)kBT, where kB is the Boltzmann constant. As a consequence, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases. The kinetic energy of an object is the extra energy which it possesses due to its motion. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Plot of Maxwell Boltzmann distribution of speeds of noble gasses I made this image, and am placing it in the public domain. ... Plot of Maxwell Boltzmann distribution of speeds of noble gasses I made this image, and am placing it in the public domain. ... Neon, like all noble gases, has a full valence (outermost) electron shell. ... Fig. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... Celsius is, or relates to, the Celsius temperature scale (previously known as the centigrade scale). ... General Name, Symbol, Number helium, He, 2 Chemical series noble gases Group, Period, Block 18, 1, s Appearance colorless Standard atomic weight 4. ... General Name, Symbol, Number neon, Ne, 10 Chemical series noble gases Group, Period, Block 18, 2, p Appearance colorless Standard atomic weight 20. ... General Name, Symbol, Number argon, Ar, 18 Chemical series noble gases Group, Period, Block 18, 3, p Appearance colorless Atomic mass 39. ... General Name, Symbol, Number xenon, Xe, 54 Chemical series noble gases Group, Period, Block 18, 5, p Appearance colorless Atomic mass 131. ... The mass number (A), also called atomic mass number or nucleon number, is the number of protons plus the number of neutrons in an atomic nucleus. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... Neon, like all noble gases, has a full valence (outermost) electron shell. ... Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... General Name, Symbol, Number xenon, Xe, 54 Chemical series noble gases Group, Period, Block 18, 5, p Appearance colorless Atomic mass 131. ... General Name, Symbol, Number helium, He, 2 Chemical series noble gases Group, Period, Block 18, 1, s Appearance colorless Standard atomic weight 4. ... The Maxwell–Boltzmann distribution is a probability distribution with applications in physics and chemistry. ...


In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of ½kBT and therefore contributes ½kB to the system's heat capacity. This has many applications. Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


Translational energy and ideal gases

See also: Ideal gas

The (Newtonian) kinetic energy of a particle of mass m, velocity v is given by An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

H^{mathrm{kin}} = tfrac12 m |mathbf{v}|^2 = tfrac{1}{2} mleft( v_{x}^{2} + v_{y}^{2} + v_{z}^{2} right),

where vx, vy and vz are the cartesian components of the velocity v. Here, H is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...


Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute ½kBT to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is (3/2)kBT, as in the example of noble gases above.


More generally, in an ideal gas, the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the average total energy of an ideal gas of N particles is (3/2) N kBT. An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...


It follows that the heat capacity of the gas is (3/2) N kB and hence, in particular, the heat capacity of a mole of a such gas particles is (3/2)NAkB=(3/2)R, where NA is Avogadro's number and R is the gas constant. Since R ≈ 2 cal/(mole·K), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal/(mole·K). This prediction is confirmed by experiment.[1] To meet Wikipedias quality standards, this article or section may require cleanup. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... Heat capacity (abbreviated Cth or just C, also called thermal capacity) is the ability of matter to store heat. ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ...


The mean kinetic energy also allows the root mean square speed vrms of the gas particles to be calculated: Root mean square speed is a measure of the velocity of particles in a gas. ...

v_{mathrm{rms}} = sqrt{langle v^{2} rangle} = sqrt{frac{3 k_{B} T}{m}} = sqrt{frac{3 R T}{M}},

where M = NAm is the mass of a mole of gas particles. This result is useful for many applications such as Graham's law of effusion, which provides a method for enriching uranium.[2]. Grahams law, also known as Grahams law of effusion, was formulated by Scottish physical chemist, Thomluyfkuyfj,gfhuas Graham. ... Effusion can refer to: In literature, effusion is the process of opening the flood gates to ones emotions, so to speak. ... These pie-graphs showing the relative proportions of uranium-238 (blue) and uranium-235 (red) at different levels of enrichment. ... General Name, Symbol, Number uranium, U, 92 Chemical series actinides Group, Period, Block n/a, 7, f Appearance silvery gray metallic; corrodes to a spalling black oxide coat in air Standard atomic weight 238. ...


Rotational energy and molecular tumbling in solution

See also: Angular velocity and Rotational diffusion

A similar example is provided by a rotating molecule with principal moments of inertia I1, I2 and I3. The rotational energy of such a molecule is given by Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ... Rotational diffusion is a process by which the equilibrium statistical distribution of the overall orientation of particles or molecules is maintained or restored. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ...

H^{mathrm{rot}} = tfrac{1}{2} ( I_{1} omega_{1}^{2} + I_{2} omega_{2}^{2} + I_{3} omega_{3}^{2} ),

where ω1, ω2, and ω3 are the principal components of the angular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is (3/2)kBT. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated.[3] Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...


The tumbling of rigid molecules — that is, the random rotations of molecules in solution — plays a key role in the relaxations observed by nuclear magnetic resonance, particularly protein NMR and residual dipolar couplings.[4] Rotational diffusion can also be observed by other biophysical probes such as fluorescence anisotropy, flow birefringence and dielectric spectroscopy.[5] Relaxation in the topic of Nuclear magnetic resonance (NMR) and Magnetic resonance imaging (MRI) phenomenology, describes the evolution of magnetizations separately in two directions: longitudinal relaxation: The part of the magnetization vector M that is parallel to the main magnetic field B0 is usually called longitudinal magnetization, designated as Mz. ... Pacific Northwest National Laboratorys high magnetic field (800 MHz, 18. ... Pacific Northwest National Laboratorys high magnetic field (800 MHz) NMR spectrometer being loaded with a sample. ... The residual dipolar coupling between two spins in a molecule occurs if the molecules in solution exhibit a partial alignment leading to an incomplete averaging of spatially anisotropic dipolar couplings. ... In biochemistry, fluorescence anisotropy assays the rotational diffusion of a molecule from the decorrelation of polarization in fluorescence, i. ... In biochemistry, flow birefringence is a hydrodynamic technique for measuring the rotational diffusion constants (or, equivalently, the rotational drag coefficients]]. The birefringence of a solution sandwiched between two concentric cylinders is measured as a function of the difference in rotational speed between the iner and outer cylinders. ... A dielectric permittivity spectrum over a wide range of frequencies. ...


Potential energy and harmonic oscillators

Equipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as a spring, which has a quadratic potential energy Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ... In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ... Helical or coil springs designed for tension A spring is a flexible elastic object used to store mechanical energy. ...

H^{mathrm{pot}} = tfrac 12 a q^{2},,

where the constant a describes the stiffness of the spring and q is the deviation from equilibrium. If such a one dimensional system has mass m, then its kinetic energy Hkin is ½mv2 = p2/2m, where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy[6]

H = H^{mathrm{kin}} + H^{mathrm{pot}} = frac{p^{2}}{2m} + frac{1}{2} a q^{2}.

Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy

langle H rangle = langle H^{mathrm{kin}} rangle + langle H^{mathrm{pot}} rangle = tfrac{1}{2} k_{B} T + tfrac{1}{2} k_{B} T = k_{B} T,

where the angular brackets leftlangle ldots rightrangle denote the average of the enclosed quantity,[7]


This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy kBT and hence contributes kB to the system's heat capacity. This can be used to derive the formula for Johnson–Nyquist noise[8] and the Dulong–Petit law of solid molar heat capacities. The latter application was particularly significant in the history of equipartition. Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ... Cross coupled LC oscillator with output on top An electronic oscillator is an electronic circuit that produces a repetitive electronic signal, often a sine wave or a square wave. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the noise generated by the thermal agitation of the charge carriers (the electrons) inside an electrical conductor in equilibrium, which happens regardless of any applied voltage. ... The Dulong-Petit law is a chemical law proposed in 1819 by French chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ... Specific heat capacity, also known simply as specific heat (Symbol: C or c) is the measure of the heat energy required to raise the temperature of a given amount of a substance by one degree. ...

Figure 4. Atoms in a crystal can vibrate about their equilibrium positions in the lattice. Such vibrations account largely for the heat capacity of crystalline dielectrics; with metals, electrons also contribute to the heat capacity.
Figure 4. Atoms in a crystal can vibrate about their equilibrium positions in the lattice. Such vibrations account largely for the heat capacity of crystalline dielectrics; with metals, electrons also contribute to the heat capacity.

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Enargite crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... A dielectric, or electrical insulator, is a substance that is highly resistant to electric current. ... Hot metal work from a blacksmith In chemistry, a metal (Greek: Metallon) is an element that readily loses electrons to form positive ions (cations) and has metallic bonds between metal atoms. ... e- redirects here. ...

Specific heat capacity of solids

For more details on the molar specific heat of solids, see Einstein solid and Debye model.

An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of 3N independent simple harmonic oscillators, where N denotes the number of atoms in the lattice. Since each harmonic oscillator has average energy kBT, the average total energy of the solid is 3NkBT, and its heat capacity is 3NkB. The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... For other uses, see Solid (disambiguation). ... This article needs to be cleaned up to conform to a higher standard of quality. ... In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat (heat capacity) in a solid. ... A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x, i. ...


By taking N to be Avogadro's number NA, and using the relation R = NAkB between the gas constant R and the the Boltzmann constant kB, this provides an explanation for the Dulong–Petit law of molar heat capacities of solids, which states that the heat capacity per mole of atoms in the lattice is 3R ≈ 6 cal/(mole·K). Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... The Dulong-Petit law is a chemical law proposed in 1819 by French chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ... Heat capacity (abbreviated Cth or just C, also called thermal capacity) is the ability of matter to store heat. ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ...


However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero.[8] A more accurate theory, incorporating quantum effects, was developed by Albert Einstein (1907) and Peter Debye (1911).[9] The third law of thermodynamics (hereinafter Third Law) states that as a system approaches the zero absolute temperature (hereinafter ZAT), all processes cease and the entropy of the system approaches a minimum value. ... Heat capacity (abbreviated Cth or just C, also called thermal capacity) is the ability of matter to store heat. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Petrus Josephus Wilhelmus Debije (March 24, 1884 – November 2, 1966) was a Dutch physical chemist. ...


Many other physical systems can be modeled as sets of coupled oscillators. The motions of such oscillators can be decomposed into normal modes, like the vibrational modes of a piano string or the resonances of an organ pipe. On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ergodicity, is important for the law of equipartition to hold. Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... Various normal modes in a 1D-lattice. ... This article or section does not cite any references or sources. ... This article is about resonance in physics. ... The choir division of the organ at St. ...


Sedimentation of particles

See also: Sedimentation, Mason–Weaver equation, and Brewing

Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom x contributes only a multiple of xs (for a fixed real number s) to the energy, then in thermal equilibrium the average energy of that part is kBT/s. Sedimentation describes the motion of particles in solutions or suspensions in response to an external force such as gravity, centrifugal force or electric force. ... The Mason-Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. ... A 16th century brewer A 21st century brewer This article concerns the production of alcoholic beverages. ...


There is a simple application of this extension to the sedimentation of particles under gravity.[10] For example, the haze sometimes seen in beer can be caused by clumps of proteins that scatter light.[11] Over time, these clumps settle downwards under the influence of gravity, causing more haze near the bottom of a bottle than near its top. However, in a process working in the opposite direction, the particles also diffuse back up towards the top of the bottle. Once equilibrium has been reached, the equipartition theorem may be used to determine the average position of a particular clump of buoyant mass mb. For an infinitely tall bottle of beer, the gravitational potential energy is given by Sedimentation describes the motion of particles in solutions or suspensions in response to an external force such as gravity, centrifugal force or electric force. ... “Gravity” redirects here. ... A selection of bottled beers A selection of cask beers Beer is one of the worlds oldest[1] and most popular[2] alcoholic beverage, selling more than 133 billion liters (35 billion gallons) per year - producing total global revenues of $294. ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ... Rayleigh scattering causing a reddened sky at sunset Rayleigh scattering (named after Lord Rayleigh (RAY-lee)) is the scattering of light, or other electromagnetic radiation, by particles much smaller than the wavelength of the light. ... This article or section does not cite its references or sources. ... In physics, buoyancy is the upward force on an object produced by the surrounding fluid (i. ... Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ...

H^{mathrm{grav}} = m_{b} g z,,

where z is the height of the protein clump in the bottle and g is the acceleration caused by gravity. Since s=1, the average potential energy of a protein clump equals kBT. Hence, a protein clump with a buoyant mass of 10 MDa (roughly the size of a virus) would produce a haze with an average height of about 2 cm at equilibrium. The process of such sedimentation to equilibrium is described by the Mason–Weaver equation.[12] Precise values of g vary depending on the location on the Earths surface. ... Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point In physics or physical science, acceleration (symbol: a) is defined as the rate of change (or derivative with respect to... The unified atomic mass unit (u), or dalton (Da), is a small unit of mass used to express atomic masses and molecular masses. ... Groups I: dsDNA viruses II: ssDNA viruses III: dsRNA viruses IV: (+)ssRNA viruses V: (-)ssRNA viruses VI: ssRNA-RT viruses VII: dsDNA-RT viruses A virus (from the Latin noun virus, meaning toxin or poison) is a microscopic particle (ranging in size from 20 - 300 nm) that can infect the... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... The Mason-Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. ...


History

This article uses the non-SI unit of cal/(mole·K) for molar specific heat, because it offers greater accuracy for single digits.
For an approximate conversion to the corresponding SI unit of J/(mole·K), such values should be multiplied by 4.2 J/cal.

The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston.[13] In 1859, James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy.[14] In 1876, Ludwig Boltzmann expanded on this principle by showing that the average energy was divided equally among all the independent components of motion in a system.[15][16] Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong–Petit law for the specific heat capacities of solids. Cover of brochure The International System of Units. ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... A calorie is a unit of measurement for energy. ... John James Waterston (1811 - June 18, 1883) was a Scottish physicist, a neglected pioneer of the kinetic theory of gases. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... The Dulong-Petit law is a chemical law proposed in 1819 by French chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ...

Figure 3. Idealized plot of the molar specific heat of a diatomic gas against temperature. It agrees with the value (7/2)R predicted by equipartition at high temperatures (where R is the gas constant), but decreases to (5/2)R and then (3/2)R at lower temperatures, as the vibrational and rotational modes of motion are "frozen out". The failure of the equipartition theorem led to a paradox that was only resolved by quantum mechanics. For most molecules, the transitional temperature Trot is much less than room temperature, whereas Tvib can be ten times larger or more. A typical example is carbon monoxide, CO, for which Trot ≈ 2.8 K and Tvib ≈ 3103 K. For molecules with very large or weakly bound atoms, Tvib can be close to room temperature (about 300 K); for example, Tvib ≈ 308 K for iodine gas, I2.
Figure 3. Idealized plot of the molar specific heat of a diatomic gas against temperature. It agrees with the value (7/2)R predicted by equipartition at high temperatures (where R is the gas constant), but decreases to (5/2)R and then (3/2)R at lower temperatures, as the vibrational and rotational modes of motion are "frozen out". The failure of the equipartition theorem led to a paradox that was only resolved by quantum mechanics. For most molecules, the transitional temperature Trot is much less than room temperature, whereas Tvib can be ten times larger or more. A typical example is carbon monoxide, CO, for which Trot ≈ 2.8 K and Tvib ≈ 3103 K. For molecules with very large or weakly bound atoms, Tvib can be close to room temperature (about 300 K); for example, Tvib ≈ 308 K for iodine gas, I2.[17]

The history of the equipartition theorem is intertwined with that of molar heat capacity, both of which were studied in the 19th century. In 1819, the French physicists Pierre Louis Dulong and Alexis Thérèse Petit discovered that the molar-specific heats of solids at room temperature were almost all identical, roughly 6 cal/(mole·K).[18] Their law was used for many years as a technique for measuring atomic weights.[9] However, subsequent studies by James Dewar and Heinrich Friedrich Weber showed that this Dulong–Petit law holds only at high temperatures;[19] at lower temperatures, or for exceptionally hard solids such as diamond, the specific heat was lower.[20] Image File history File links Size of this preview: 799 × 599 pixelsFull resolution (2398 × 1799 pixel, file size: 46 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Equipartition theorem User:PAR/Work7 ... Image File history File links Size of this preview: 799 × 599 pixelsFull resolution (2398 × 1799 pixel, file size: 46 KB, MIME type: image/png) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Equipartition theorem User:PAR/Work7 ... Specific heat capacity, also known simply as specific heat (Symbol: C or c) is the measure of the heat energy required to raise the temperature of a given amount of a substance by one degree. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... Fig. ... Carbon monoxide, with the chemical formula CO, is a colorless, odorless, and tasteless gas. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... General Name, Symbol, Number iodine, I, 53 Chemical series halogens Group, Period, Block 17, 5, p Appearance violet-dark gray, lustrous Standard atomic weight 126. ... Specific heat capacity, also known simply as specific heat (Symbol: C or c) is the measure of the heat energy required to raise the temperature of a given amount of a substance by one degree. ... Pierre Louis Dulong (February 12, 1785 - July 19, 1838) was a French physicist and chemist. ... Alexis Thérèse Petit (October 2, 1791 - June 21, 1820) was a French physicist. ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... ... For other persons named James Dewar, see James Dewar (disambiguation). ... Heinrich Friedrich Weber (1843-1912) was born in the town of Magdala, near Weimar, son of a merchant. ... The Dulong-Petit law is a chemical law proposed in 1819 by French chemists Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ... Fig. ... This article is about the gemstone. ...


Experimental observations of the specific heat of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mole·K), whereas that of diatomic gases should be roughly 7 cal/(mole·K). Experiments confirmed the former prediction,[1] but found that molar heat capacities of diatomic gases were typically about 5 cal/(mole·K),[21] and fell to about 3 cal/(mole·K) at very low temperatures.[22] Maxwell noted in 1875 that the disagreement between experiment and the equipartition theorem was much worse than even these numbers suggest;[23] since atoms have internal parts, heat energy should go into the motion of these internal parts, making the predicted specific heats of monatomic and diatomic gases much higher than 3 cal/(mole·K)and 7 cal/(mole·K), respectively. A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ...


A third discrepancy concerned the specific heat of metals.[24] According to the classical Drude model, metallic electrons act as a nearly ideal gas, and so they should contribute (3/2) Ne kB to the heat capacity by the equipartition theorem, where Ne is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same.[24] The Drude model of electrical conduction was developed in the 1900s by Paul Drude to explain the transport properties of electrons in materials (especially metals). ...


Several explanations of equipartition's failure to account for molar heat capacities were proposed. Boltzmann defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in thermal equilibrium because of their interactions with the aether.[25] Lord Kelvin suggested that the derivation of the equipartition theorem must be incorrect, since it disagreed with experiment, but was unable to show how.[26] Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem.[27] Albert Einstein provided that escape, by showing in 1907 that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid.[28] Einstein used the failure of equipartition to argue for the need of a new quantum theory of matter.[9] Nernst's 1910 measurements of specific heats at low temperatures[29] supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists.[30] Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... The luminiferous aether: it was hypothesised that the Earth moves through a medium of aether that carries light In the late 19th century luminiferous aether (light-bearing aether) was the term used to describe a medium for the propagation of light. ... William Thomson, 1st Baron Kelvin, OM, GCVO, PC, PRS, FRSE, (26 June 1824 – 17 December 1907) was a mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century. ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Walther Nernst. ... Quantum theory is a theory of physics that uses Plancks constant. ...


General formulation of the equipartition theorem

See also: Generalized coordinates, Hamiltonian mechanics, Microcanonical ensemble, and Canonical ensemble

The most general form of the equipartition theorem[10][7][3] states that under suitable assumptions (discussed below), for a physical system with Hamiltonian energy function H and degrees of freedom xn, the following equipartition formula holds in thermal equilibrium for all indices m and n: Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... The microcanonical ensemble is the simplest of the ensembles of statistical mechanics. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In physics, Hamiltonian has distinct but closely related meanings. ...

! Bigllangle x_{m} frac{partial H}{partial x_{n}} Bigrrangle = delta_{mn} k_{B} T.

Here δmn is the Kronecker delta, which is equal to one if m=n and is zero otherwise. The averaging brackets leftlangle ldots rightrangle may refer either to the long time average of a single system, or, more commonly, the ensemble average over phase space. The ergodicity assumptions implicit in the theorem imply that these two averages agree, and both have been used to estimate internal energies of complex physical systems. In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In statistical mechanics, the ensemble average is defined as the weighted average of a molecular property of a system, over the set of states available to the system. ... The ergodic hypothesis is a postulate of thermodynamics. ...


The general equipartition theorem holds in both the microcanonical ensemble,[7] when the total energy of the system is constant, and also in the canonical ensemble,[31][3] when the system is coupled to a heat bath with which it can exchange energy. Derivations of the general formula are given later in the article. The microcanonical ensemble is the simplest of the ensembles of statistical mechanics. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... A heat bath is a large system that is in thermal contact with some other system of interest. ...


The general formula is equivalent to the following two.

  1. Bigllangle x_{n} frac{partial H}{partial x_{n}} Bigrrangle = k_{B} T   for all n.
  2. Bigllangle x_{m} frac{partial H}{partial x_{n}} Bigrrangle = 0   for all m≠n.

If a degree of freedom xn appears only as a quadratic term anxn2 in the Hamiltonian H, then the first of these formulae implies that

k_{B} T = Bigllangle x_{n} frac{partial H}{partial x_{n}}Bigrrangle = 2langle a_n x_n^2 rangle,

which is twice the contribution that this degree of freedom makes to the average energy langle Hrangle. Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by s, applies to energies of the form anxns.


The degrees of freedom xn are coordinates on the phase space of the system and are therefore commonly subdivided into generalized position coordinates qk and generalized momentum coordinates pk, where pk is the conjugate momentum to qk. In this situation, formula 1 means that for all k, For other senses of this term, see phase space (disambiguation). ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ...

Bigllangle p_{k} frac{partial H}{partial p_{k}} Bigrrangle = Bigllangle q_{k} frac{partial H}{partial q_{k}} Bigrrangle = k_{B} T.

Using the equations of Hamiltonian mechanics,[6] these formulae may also be written Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

Bigllangle p_{k} frac{dq_{k}}{dt} Bigrrangle = -Bigllangle q_{k} frac{dp_{k}}{dt} Bigrrangle = k_{B} T.

Formula 2 additionally states that the averages

Bigllangle q_{j} frac{partial H}{partial q_{k}} Bigrrangle, quad Bigllangle q_{j} frac{partial H}{partial p_{k}} Bigrrangle, quad Bigllangle p_{j} frac{partial H}{partial p_{k}} Bigrrangle, quad Bigllangle p_{j} frac{partial H}{partial q_{k}} Bigrrangle, quad Bigllangle q_{k} frac{partial H}{partial p_{k}} Bigrrangle,   and   Bigllangle p_{k} frac{partial H}{partial q_{k}} Bigrrangle

are all zero for j≠k.


Relation to the virial theorem

See also: Virial theorem, Generalized coordinates, and Hamiltonian mechanics

The general equipartition theorem is an extension of the virial theorem (proposed in 1870[32]), which states that In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ...

Bigllangle sum_{k} q_{k} frac{partial H}{partial q_{k}} Bigrrangle = Bigllangle sum_{k} p_{k} frac{partial H}{partial p_{k}} Bigrrangle = Bigllangle sum_{k} p_{k} frac{dq_{k}}{dt} Bigrrangle = -Bigllangle sum_{k} q_{k} frac{dp_{k}}{dt} Bigrrangle,

where t denotes time.[6] Two key differences are that the virial theorem relates summed rather than individual averages to each other, and it does not connect them to the temperature T. Another difference is that traditional derivations of the virial theorem use averages over time, whereas those of the equipartition theorem use averages over phase space. A pocket watch, a device used to tell time Look up time in Wiktionary, the free dictionary. ... Fig. ... For other senses of this term, see phase space (disambiguation). ...


Applications

Ideal gas law

See also: Ideal gas and Ideal gas law

Ideal gases provide an important application of the equipartition theorem. As well as providing the formula An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ...

langle H^{mathrm{kin}} rangle = frac{1}{2m} langle p_{x}^{2} + p_{y}^{2} + p_{z}^{2} rangle
= frac{1}{2} biggl( Bigllangle p_{x} frac{partial H^{mathrm{kin}}}{partial p_{x}} Bigrrangle + Bigllangle p_{y} frac{partial H^{mathrm{kin}}}{partial p_{y}} Bigrrangle + Bigllangle p_{z} frac{partial H^{mathrm{kin}}}{partial p_{z}} Bigrrangle biggr) = frac{3}{2} k_{B} T

for the average kinetic energy per particle, the equipartition theorem can be used to derive the ideal gas law from classical mechanics.[3] If q = (qx, qy, qz) and p = (px, py, pz) denote the position vector and momentum of a particle in the gas, and F is the net force on that particle, then Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ...

langle mathbf{q} cdot mathbf{F} rangle = Bigllangle q_{x} frac{dp_{x}}{dt} Bigrrangle + Bigllangle q_{y} frac{dp_{y}}{dt} Bigrrangle + Bigllangle q_{z} frac{dp_{z}}{dt} Bigrrangle
=-Bigllangle q_{x} frac{partial H}{partial q_x} Bigrrangle - Bigllangle q_{y} frac{partial H}{partial q_y} Bigrrangle - Bigllangle q_{z} frac{partial H}{partial q_z} Bigrrangle = -3k_{B} T,

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition formula. Summing over a system of N particles yields Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ...

3Nk_{B} T = - biggllangle sum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k} biggrrangle.
Figure 5. The kinetic energy of a particular molecule can fluctuate wildly, but the equipartition theorem allows its average energy to be calculated at any temperature. Equipartition also provides a derivation of the ideal gas law, an equation that relates the pressure, volume and temperature of the gas. (In this diagram five of the molecules have been colored red to track their motion; this coloration has no other significance.)
Figure 5. The kinetic energy of a particular molecule can fluctuate wildly, but the equipartition theorem allows its average energy to be calculated at any temperature. Equipartition also provides a derivation of the ideal gas law, an equation that relates the pressure, volume and temperature of the gas. (In this diagram five of the molecules have been colored red to track their motion; this coloration has no other significance.)

By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence Image File history File links Translational_motion. ... Image File history File links Translational_motion. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ... The use of water pressure - the Captain Cook Memorial Jet in Lake Burley Griffin, Canberra. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Fig. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

-biggllanglesum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k}biggrrangle = P oint_{mathrm{surface}} mathbf{q} cdot mathbf{dS},

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

boldsymbolnabla cdot mathbf{q} = frac{partial q_{x}}{partial q_{x}} + frac{partial q_{y}}{partial q_{y}} + frac{partial q_{z}}{partial q_{z}} = 3,

the divergence theorem implies that In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradsky–Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

P oint_{mathrm{surface}} mathbf{q} cdot mathbf{dS} = P int_{mathrm{volume}} left( boldsymbolnabla cdot mathbf{q} right) dV = 3PV,

where dV is an infinitesimal volume within the container and V is the total volume of the container.


Putting these equalities together yields

3Nk_{B} T = -biggllangle sum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k} biggrrangle = 3PV,

which immediately implies the ideal gas law for N particles: Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ...

PV = NkBT = nRT,

where n=N/NA is the number of moles of gas and R=NAkB is the gas constant. The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ...


Diatomic gases

See also: Two-body problem, Rigid rotor, and Harmonic oscillator

A diatomic gas can be modelled as two masses, m1 and m2, joined by a spring of stiffness a, which is called the rigid rotor-harmonic oscillator approximation.[17] The classical energy of this system is In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. ... The rigid rotor is a mechanical model that is used to explain rotating systems. ... In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ... Helical or coil springs designed for tension A spring is a flexible elastic object used to store mechanical energy. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...

H = frac{left| mathbf{p}_{1} right|^{2}}{2m_{1}} + frac{left| mathbf{p}_{2} right|^{2}}{2m_{2}} + frac{1}{2} a q^{2},

where p1 and p2 are the momenta of the two atoms, and q is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute ½kBT to the total average energy, and ½kB to the heat capacity. Therefore, the heat capacity of a gas of N diatomic molecules is predicted to be 7N · ½kB: the momenta p1 and p2 contribute three degrees of freedom each, and the extension q contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be (7/2)NAkB=(7/2)R and, thus, the predicted molar heat capacity should be roughly 7 cal/(mole·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5 cal/(mole·K)[21] and fall to 3 cal/(mole·K) at very low temperatures.[22] This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only increase the predicted specific heat, not decrease it.[23] This discrepancy was a key piece of evidence showing the need for a quantum theory of matter. A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... A calorie is a unit of measurement for energy. ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... Quantum theory is a theory of physics that uses Plancks constant. ...

Figure 6. A combined X-ray and optical image of the Crab Nebula. At the heart of this nebula there is a rapidly rotating neutron star which has about one and a half times the mass of the Sun but is only 25 km across (roughly the size of Madrid). The equipartition theorem is useful in predicting the properties of such neutron stars.
Figure 6. A combined X-ray and optical image of the Crab Nebula. At the heart of this nebula there is a rapidly rotating neutron star which has about one and a half times the mass of the Sun but is only 25 km across (roughly the size of Madrid). The equipartition theorem is useful in predicting the properties of such neutron stars.

Image File history File links Download high-resolution version (2400x2400, 314 KB) A composite image of the Crab Nebula showing the X-ray (blue), and optical (red) images superimposed. ... Image File history File links Download high-resolution version (2400x2400, 314 KB) A composite image of the Crab Nebula showing the X-ray (blue), and optical (red) images superimposed. ... The Crab Nebula (catalogue designations M 1, NGC 1952, Taurus A) is a supernova remnant in the constellation of Taurus. ... A neutron star is one of the few possible endpoints of stellar evolution. ... The Sun is the star at the center of the Solar System. ... Motto: De Madrid al Cielo (From Madrid to Heaven) Location Coordinates: Country Spain Autonomous Community Comunidad Autónoma de Madrid Province Madrid Administrative Divisions 21 Neighborhoods 127 Founded 9th century Government  - Mayor Alberto Ruiz-Gallardón (PP) Area  - Land 607 km² (234. ...

Extreme relativistic ideal gases

See also: Special relativity, White dwarf, and Neutron star

Equipartition was used above to derive the classical ideal gas law from Newtonian mechanics. However, relativistic effects become dominant in some systems, such as white dwarfs and neutron stars,[7] and the ideal gas equations must be modified. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativistic ideal gas.[3] In such cases, the kinetic energy of a single particle is given by the formula The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... This article or section does not adequately cite its references or sources. ... A neutron star is one of the few possible endpoints of stellar evolution. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. ... It has been suggested that this article or section be merged with Classical mechanics. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... This article or section does not adequately cite its references or sources. ... A neutron star is one of the few possible endpoints of stellar evolution. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. ... A relativistic particle is a particle moving with a speed close to the speed of light, such that effects of special relativity are important for the description of its behavior. ...

H^{mathrm{kin}} approx cp = c sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}.

Taking the derivative of H with respect to the px momentum component gives the formula

p_{x} frac{partial H^{mathrm{kin}}}{partial p_{x}} = c frac{p_{x}^{2}}{sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}}

and similarly for the py and pz components. Adding the three components together gives

langle H^{mathrm{kin}} rangle = biggllangle c frac{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}{sqrt{p_{x}^{2} + p_{y}^{2} + p_{z}^{2}}} biggrrangle = Bigllangle p_{x} frac{partial H^{mathrm{kin}}}{partial p_{x}} Bigrrangle + Bigllangle p_{y} frac{partial H^{mathrm{kin}}}{partial p_{y}} Bigrrangle + Bigllangle p_{z} frac{partial H^{mathrm{kin}}}{partial p_{z}} Bigrrangle = 3 k_{B} T

where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for N particles, it is 3 N kBT.


Anharmonic oscillators

See also: Anharmonic oscillator

An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension q (the generalized position which measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem.[33][34] Simple examples are provided by potential energy functions of the form Anharmonicity is the deviation of a system from being a harmonic oscillator. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

H^{mathrm{pot}} = C q^{s},,

where C and s are arbitrary real constants. In these cases, the law of equipartition predicts that In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

k_{B} T = Bigllangle q frac{partial H^{mathrm{pot}}}{partial q} Bigrrangle = langle q cdot s C q^{s-1} rangle = langle s C q^{s} rangle = s langle H^{mathrm{pot}} rangle.

Thus, the average potential energy equals kBT/s, not kBT/2 as for the quadratic harmonic oscillator (where s=2).


More generally, a typical energy function of a one-dimensional system has a Taylor expansion in the extension q: As the degree of the taylor series rises, it approaches the correct function. ...

H^{mathrm{pot}} = sum_{n=2}^{infty} C_{n} q^{n}

for non-negative integers n. There is no n=1 term, because at the equilibrium point, there is no net force and so the first derivative of the energy is zero. The n=0 term need not be included, since the energy at the equilibrium position may be set to zero by convention. In this case, the law of equipartition predicts that[33] The integers are commonly denoted by the above symbol. ...

k_{B} T = Bigllangle q frac{partial H^{mathrm{pot}}}{partial q} Bigrrangle = sum_{n=2}^{infty} langle q cdot n C_{n} q^{n-1} rangle = sum_{n=2}^{infty} n C_{n} langle q^{n} rangle.

In contrast to the other examples cited here, the equipartition formula

langle H^{mathrm{pot}} rangle = frac{1}{2} k_{B} T - sum_{n=3}^{infty} left( frac{n - 2}{2} right) C_{n} langle q^{n} rangle

does not allow the average potential energy to be written in terms of known constants.


Brownian motion

Figure 7. Typical Brownian motion of a particle in three dimensions.
See also: Brownian motion

The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation.[3] According to that equation, the motion of a particle of mass m with velocity v is governed by Newton's second law Image File history File links Download high-resolution version (904x883, 97 KB) A single sample path of a three-dimensional Brownian motion (Wiener process) Wt, as generated by Wolfram Mathematica with a time step of size 0. ... Image File history File links Download high-resolution version (904x883, 97 KB) A single sample path of a three-dimensional Brownian motion (Wiener process) Wt, as generated by Wolfram Mathematica with a time step of size 0. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors. ... In statistical physics, a Langevin equation is a stochastic differential equation describing Brownian motion in a potential. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

frac{dmathbf{v}}{dt} = frac{1}{m} mathbf{F} = -frac{mathbf{v}}{tau} + frac{1}{m} mathbf{F}^{mathrm{rnd}},

where Frnd is a random force representing the random collisions of the particle and the surrounding molecules, and where the time constant τ reflects the drag force that opposes the particle's motion through the solution. The drag force is often written Fdrag = - γv; therefore, the time constant τ equals m/γ. In physics and engineering, the time constant, usually denoted by the Greek letter Ï„ (tau), characterizes the frequency response of a first-order, linear time-invariant (LTI) system. ... An object falling through a gas or liquid experiences a force in direction opposite to its motion. ...


The dot product of this equation with the position vector r, after averaging, yields the equation

Bigllangle mathbf{r} cdot frac{dmathbf{v}}{dt} Bigrrangle + frac{1}{tau} langle mathbf{r} cdot mathbf{v} rangle = 0

for Brownian motion (since the random force Frnd is uncorrelated with the position r). Using the mathematical identities

frac{d}{dt} left( mathbf{r} cdot mathbf{r} right) = frac{d}{dt} left( r^{2} right) = 2 left( mathbf{r} cdot mathbf{v} right)

and

frac{d}{dt} left( mathbf{r} cdot mathbf{v} right) = v^{2} + mathbf{r} cdot frac{dmathbf{v}}{dt},

the basic equation for Brownian motion can be transformed into

frac{d^{2}}{dt^{2}} langle r^{2} rangle + frac{1}{tau} frac{d}{dt} langle r^{2} rangle = 2 langle v^{2} rangle = frac{6}{m} k_{B} T,

where the last equality follows from the equipartition theorem for translational kinetic energy:

langle H^{mathrm{kin}} rangle = Bigllangle frac{p^{2}}{2m} Bigrrangle = langle tfrac{1}{2} m v^{2} rangle = tfrac{3}{2} k_{B} T.

The above differential equation for langle r^2rangle (with suitable initial conditions) may be solved exactly: A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ...

langle r^{2} rangle = frac{6k_{B} T tau^{2}}{m} left( e^{-t/tau} - 1 + frac{t}{tau} right).

On small time scales, with t << τ, the particle acts as a freely moving particle: by the Taylor series of the exponential function, the squared distance grows approximately quadratically: As the degree of the Taylor series rises, it approaches the correct function. ... The exponential function is one of the most important functions in mathematics. ...

langle r^{2} rangle approx frac{3k_{B} T}{m} t^{2} = langle v^{2} rangle t^{2}.

However, on long time scales, with t >> τ, the exponential and constant terms are negligible, and the squared distance grows only linearly:

langle r^{2} rangle approx frac{6k_{B} Ttau}{m} t = 6gamma k_{B} T t.

This describes the diffusion of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way. This article or section does not cite its references or sources. ...

Figure 8. Equipartition gives an accurate estimate of the temperature at the Sun's core.

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

Stellar physics

See also: Astrophysics and Stellar structure

The equipartition theorem and the related virial theorem have long been used as a tool in astrophysics.[35] As examples, the virial theorem may be used to estimate stellar temperatures or the Chandrasekhar limit on the mass of white dwarf stars.[36][37] Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature, and chemical composition) of celestial objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... The simplest commonly used model of stellar structure is the spherically symmetric quasi-static model, which assumes that a star is very close to an equilibrium state, and that it is spherically symmetric. ... In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ... Spiral Galaxy ESO 269-57 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature, and chemical composition) of celestial objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... The Chandrasekhar limit, is the maximum mass possible for a white dwarf (one of the end stages of stars when they cool down) and is approximately 3 × 1030 kg, around 1. ... This article or section does not adequately cite its references or sources. ...


The average temperature of a star can be estimated from the equipartition theorem.[38] Since most stars are spherically symmetric, the total gravitational potential energy can be estimated by integration Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ... Potential energy is the energy that is by virtue of the relative positions (configurations) of the objects within a physical system. ...

H^{mathrm{grav}}_{mathrm{tot}} = -int_{0}^{R} frac{4pi r^{2} G}{r} M(r), rho(r), dr,

where M(r) is the mass within a radius r and ρ(r) is the stellar density at radius r; G represents the gravitational constant and R the total radius of the star. Assuming a constant density throughout the star, this integration yields the formula According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

H^{mathrm{grav}}_{mathrm{tot}} = - frac{3G M^{2}}{5R},

where M is the star's total mass. Hence, the average potential energy of a single particle is

langle H^{mathrm{grav}} rangle = frac{H^{mathrm{grav}}_{mathrm{tot}}}{N} = - frac{3G M^{2}}{5RN},

where N is the number of particles in the star. Since most stars are composed mainly of ionized hydrogen, N equals roughly (M/mp), where mp is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature STAR is an acronym for: Organizations Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... “Multivalent” redirects here. ... General Name, Symbol, Number hydrogen, H, 1 Chemical series nonmetals Group, Period, Block 1, 1, s Appearance colorless Atomic mass 1. ...

Bigllangle r frac{partial H^{mathrm{grav}}}{partial r} Bigrrangle = langle -H^{mathrm{grav}} rangle = k_{B} T = frac{3G M^{2}}{5RN}.

Substitution of the mass and radius of the Sun yields an estimated solar temperature of T = 14 million Kelvin, very close to its core temperature of 15 million Kelvin. However, the Sun is much more complex than assumed by this model — both its temperature and density vary strongly with radius — and such excellent agreement (≈7% relative error) is partly fortuitous.[39] The Sun is the star at the center of the Solar System. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... In the mathematical subfield of numerical analysis the approximation error in some data is the discrepancy between an exact value and some approximation to it. ...


Star formation

The same formulae may be applied to determining the conditions for star formation in giant molecular clouds.[40] A local fluctuation in the density of such a cloud can lead to a runaway condition in which the cloud collapses inwards under its own gravity. Such a collapse occurs when the equipartition theorem — or, equivalently, the virial theorem — is no longer valid, i.e., when the gravitational potential energy exceeds twice the kinetic energy Star formation is the process by which dense parts of molecular clouds collapse into a ball of plasma to form a star. ... A molecular cloud is a type of interstellar cloud whose density and size permits the formation of molecules, most commonly molecular hydrogen (H2). ... In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ...

frac{3G M^{2}}{5R} > 3 N k_{B} T

Assuming a constant density ρ for the cloud

M = frac{4}{3} pi R^{3} rho

yields a minimum mass for stellar contraction, the Jeans mass MJ

M_{J}^{2} = left( frac{5k_{B}T}{G m_{p}} right)^{3} left( frac{3}{4pi rho} right)

Substituting the values typically observed in such clouds (T=150 K, ρ = 2×10-16 g/cm3) gives an estimated minimum mass of 17 solar masses, which is consistent with observed star formation. This effect is also known as the Jeans instability, after the British physicist James Hopwood Jeans who published it in 1902.[41] The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zero—the lowest possible temperature where nothing could be colder and no heat energy remains in a substance—is defined as zero kelvin (0 K). ... It has been suggested that Jeans mass be merged into this article or section. ... Sir James Hopwood Jeans (September 11, 1877 in Ormskirk – September 16, 1946 in Dorking) was a British physicist, astronomer, and mathematician. ...


Derivations

Derivation for kinetic energies

The original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average kinetic energy, (3/2)kBT.[42] This may be shown using the Maxwell–Boltzmann distribution (see Figure 2), which is the probability distribution In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... The kinetic energy of an object is the extra energy which it possesses due to its motion. ... The Maxwell–Boltzmann distribution is a probability distribution with applications in physics and chemistry. ...

f (v) = 4 pi left( frac{m}{2 pi k_B T}right)^{3/2}!!v^2 exp Bigl( frac{-mv^2}{2k_B T} Bigr)

for the speed of a particle of mass m in the system, where the speed v is the magnitude sqrt{v_x^2 + v_y^2 + v_z^2} of the velocity vector mathbf{v} = (v_x,v_y,v_z). This article or section does not cite its references or sources. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...


The Maxwell–Boltzmann distribution applies to any system composed of atoms, and assumes only a canonical ensemble, specifically, that the kinetic energies are distributed according to their Boltzmann factor at a temperature T.[42] The average kinetic energy for a particle of mass m is then given by the integral formula A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ...

langle H^{mathrm{kin}} rangle = langle tfrac{1}{2} m v^{2} rangle = int _{0}^{infty} tfrac{1}{2} m v^{2} f(v) dv = tfrac{3}{2} k_{B} T,

as stated by the equipartition theorem.


Derivation for all quadratic energies

More generally, the equipartition theorem states that any degree of freedom x which appears in the total energy H only as a simple quadratic term Ax2, where A is a constant, has an average energy of ½kBT in thermal equilibrium. In this case the equipartition theorem may be derived from the partition function Z(β), where β=1/(kBT) is the canonical inverse temperature.[43] Integration over the variable x yields a factor Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise... The inverse temperature is given by where k is the Boltzmann constant and T is the temperature. ...

Z_{x} = int_{-infty}^{infty} dx e^{-beta A x^{2}} = sqrt{frac{pi}{beta A}},

in the formula for Z. The mean energy associated with this factor is given by

langle H_{x} rangle = - frac{partial log Z_{x}}{partial beta} = frac{1}{2beta} = frac{1}{2} k_{B} T

as stated by the equipartition theorem.


General derivations

General derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble[7][3] and for the canonical ensemble.[31][3] They involve taking averages over the phase space of the system, which is a symplectic manifold. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... The microcanonical ensemble is the simplest of the ensembles of statistical mechanics. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... For other senses of this term, see phase space (disambiguation). ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...


To explain these derivations, the following notation is introduced. First, the phase space is described in terms of generalized position coordinates qj together with their conjugate momenta pj. The quantities qj completely describe the configuration of the system, while the quantities (qj,pj) together completely describe its state. In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ...


Secondly, the infinitesimal volume

dΓ = ∏ dqidpi
i

of the phase space is introduced and used to define the volume Γ(E, ΔE) of the portion of phase space where the energy H of the system lies between two limits, E and E+ΔE:

Gamma (E, Delta E) = int_{H in left[E, E+Delta E right]} dGamma .

In this expression, ΔE is assumed to be very small, ΔE<<E. Similarly, Σ(E) is defined to be the total volume of phase space where the energy is less than E:

Σ(E) = ∫ dΓ.
H < E

Since ΔE is very small, the following integrations are equivalent

int_{H in left[ E, E+Delta E right]} ldots dGamma = Delta E frac{partial}{partial E} int_{H < E} ldots dGamma,

where the ellipses represent the integrand. From this, it follows that Γ is proportional to ΔE

Gamma = Delta E frac{partial Sigma}{partial E} = Delta E rho(E),

where ρ(E) is the density of states. By the usual definitions of statistical mechanics, the entropy S equals kB log Σ(E), and the temperature T is defined by Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Ice melting - classic example of entropy increasing[1] described in 1862 by Rudolf Clausius as an increase in the disgregation of the molecules of the body of ice. ... Fig. ...

frac{1}{T} = frac{partial S}{partial E} = k_{b} frac{partial log Sigma}{partial E} = k_{b} frac{1}{Sigma},frac{partial Sigma}{partial E} .

The canonical ensemble

In the canonical ensemble, the system is in thermal equilibrium with an infinite heat bath at temperature T (in Kelvin).[31][3] The probability of each state in phase space is given by its Boltzmann factor times a normalization factor mathcal{N}, which is chosen so that the probabilities sum to one A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... Fig. ... For other senses of this term, see phase space (disambiguation). ... In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ... The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...

mathcal{N} int e^{-beta H(p, q)} dGamma = 1,

where β = 1/kBT. Integration by parts for a phase-space variable xk (which could be either qk or pk) between two limits a and b yields the equation In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

mathcal{N} int left[ e^{-beta H(p, q)} x_{k} right]_{x_{k}=a}^{x_{k}=b} dGamma_{k}+ mathcal{N} int e^{-beta H(p, q)} x_{k} beta frac{partial H}{partial x_{k}} dGamma = 1,

where dΓk = dΓ/dxk, i.e., the first integration is not carried out over xk. The first term is usually zero, either because xk is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ensemble follows immediately

mathcal{N} int e^{-beta H(p, q)} x_{k} frac{partial H}{partial x_{k}} ,dGamma = Bigllangle x_{k} frac{partial H}{partial x_{k}} Bigrrangle = frac{1}{beta} = k_{B} T.

Here, the averaging symbolized by langle ldots rangle is the ensemble average taken over the canonical ensemble. In statistical mechanics, the ensemble average is defined as the weighted average of a molecular property of a system, over the set of states available to the system. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...


The microcanonical ensemble

In the microcanonical ensemble, the system is isolated from the rest of the world, or at least very weakly coupled to it.[7] Hence, its total energy is effectively constant; to be definite, we say that the total energy H is confined between E and E+ΔE. For a given energy E and spread ΔE, there is a region of phase space Γ in which the system has that energy, and the probability of each state in that region of phase space is equal, by the definition of the microcanonical ensemble. Given these definitions, the equipartition average of phase-space variables xm (which could be either qkor pk) and xn is given by For other senses of this term, see phase space (disambiguation). ... For other senses of this term, see phase space (disambiguation). ...

Bigllangle x_{m} frac{partial H}{partial x_{n}} Bigr rangle = frac{1}{Gamma} , int_{H in left[ E, E+Delta E right]} x_{m} frac{partial H}{partial x_{n}} ,dGamma
quad=frac{Delta E}{Gamma}, frac{partial}{partial E} int_{H < E} x_{m} frac{partial H}{partial x_{n}} ,dGamma = frac{1}{rho} ,frac{partial}{partial E} int_{H < E} x_{m} frac{partial left( H - E right)}{partial x_{n}} ,dGamma,

where the last equality follows because E is a constant that does not depend on xn. Integrating by parts yields the relation In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

int_{H < E} x_{m} frac{partial ( H - E )}{partial x_{n}} ,dGamma = int_{H < E} frac{partial}{partial x_{n}} bigl( x_{m} ( H - E ) bigr) ,dGamma - int_{H < E} delta_{mn} ( H - E ) dGamma
= delta_{mn} int_{H < E} ( E - H ) ,dGamma,

since the first term on the right hand side of the first line is zero (it can be rewritten as an integral of H - E on the hypersurface where H = E). In mathematics, a hypersurface is some kind of submanifold. ...


Substitution of this result into the previous equation yields

Bigllangle x_{m} frac{partial H}{partial x_{n}} Bigrrangle = delta_{mn} frac{1}{rho} , frac{partial}{partial E} int_{H < E}left( E - H right),dGamma = delta_{mn} frac{1}{rho} , int_{H < E} ,dGamma = delta_{mn} frac{Sigma}{rho}.

Since rho = frac{partial Sigma}{partial E} the equipartition theorem follows:

Bigllangle x_{m} frac{partial H}{partial x_{n}} Bigrrangle = delta_{mn} Bigl(frac{1}{Sigma} frac{partial Sigma}{partial E}Bigr)^{-1} = delta_{mn} Bigl(frac{partial log Sigma} {partial E}Bigr)^{-1} = delta_{mn} k_{B} T.

Thus, we have derived the general formulation of the equipartition theorem

! Bigllangle x_{m} frac{partial H}{partial x_{n}} Bigrrangle = delta_{mn} k_{B} T,

which was so useful in the applications described above.


Limitations

Figure 9. Energy is not shared among the various normal modes in an isolated system of ideal coupled oscillators; the energy in each mode is constant and independent of the energy in the other modes. Hence, the equipartition theorem does not hold for such a system in the microcanonical ensemble (when isolated), although it does hold in the canonical ensemble (when coupled to a heat bath). However, by adding a sufficiently strong nonlinear coupling between the modes, energy will be shared and equipartition holds in both ensembles.
Figure 9. Energy is not shared among the various normal modes in an isolated system of ideal coupled oscillators; the energy in each mode is constant and independent of the energy in the other modes. Hence, the equipartition theorem does not hold for such a system in the microcanonical ensemble (when isolated), although it does hold in the canonical ensemble (when coupled to a heat bath). However, by adding a sufficiently strong nonlinear coupling between the modes, energy will be shared and equipartition holds in both ensembles.

Image File history File links 1D_normal_modes_(280_kB). ... Image File history File links 1D_normal_modes_(280_kB). ... Normal modes in an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency (called normal frequencies or allowed frequencies). ... In classical mechanics, a Harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hookes law: where is a positive constant. ... The microcanonical ensemble is the simplest of the ensembles of statistical mechanics. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ...

Requirement of ergodicity

See also: Ergodicity, Chaos theory, and Kolmogorov–Arnold–Moser theorem

The law of equipartition holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally likely to be populated.[7] Consequently, it must be possible to exchange energy among all its various forms within the system, or with an external heat bath in the canonical ensemble. The number of physical systems that have been rigorously proven to be ergodic is small; a famous example is the hard-sphere system of Yakov Sinai.[44] The requirements for isolated systems to ensure ergodicity — and, thus equipartition — have been studied, and provided motivation for the modern chaos theory of dynamical systems. A chaotic Hamiltonian system need not be ergodic, although that is usually a good assumption.[45] In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... The Kolmogorov–Arnold–Moser theorem is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. ... In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... A heat bath is a large system that is in thermal contact with some other system of interest. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... The Bunimovich stadium is a chaotic dynamical billiard A billiard is a dynamical system where a particle alternates between motion in a straight line and specular reflections with a boundary. ... Yakov G. Sinai (1935-) is a Russian- American mathematician. ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... The Lorenz attractor is an example of a non-linear dynamical system. ... In classical mechanics, a Hamiltonian system is a physical system in which forces are velocity invariant. ...


A commonly cited counter-example where energy is not shared among its various forms and where equipartition does not hold in the microcanonical ensemble is a system of coupled harmonic oscillators.[45] If the system is isolated from the rest of the world, the energy in each normal mode is constant; energy is not transferred from one mode to another. Hence, equipartition does not hold for such a system; the amount of energy in each normal mode is fixed at its initial value. If sufficiently strong nonlinear terms are present in the energy function, energy may be transferred between the normal modes, leading to ergodicity and rendering the law of equipartition valid. However, the Kolmogorov–Arnold–Moser theorem states that energy will not be exchanged unless the nonlinear perturbations are strong enough; if they are too small, the energy will remain trapped in at least some of the modes. Various normal modes in a 1D-lattice. ... The Kolmogorov–Arnold–Moser theorem is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. ...


Failure due to quantum effects

See also: Ultraviolet catastrophe, History of quantum mechanics, and Identical particles

The law of equipartition breaks down when the thermal energy kBT is significantly smaller than the spacing between energy levels. Equipartition no longer holds because it is a poor approximation to assume that the energy levels form a smooth continuum, which is required in the derivations of the equipartition theorem above.[7][3] Historically, the failures of the classical equipartition theorem to explain specific heats and blackbody radiation were critical in showing the need for a new theory of matter and radiation, namely, quantum mechanics and quantum field theory.[9] Other, more subtle quantum effects can lead to corrections to equipartition, such as identical particles and continuous symmetries. The effects of identical particles can be dominant at very high densities and low temperatures. For example, the valence electrons in a metal, if thought of as a gas of quasi-free particles, have a mean kinetic energy per electron of a few electron volts. Such a large kinetic energy would normally correspond to a temperature of some tens of thousands of Kelvin. This behavior persists even as the temperature approaches absolute zero. Such a state is called a degenerate fermion gas, in which the density is high enough that the Pauli exclusion principle invalidates the classical approach. The analogous degenerate state for bosons, in which a large number of identical particles are sitting in the lowest-energy state, is known as a Bose-Einstein condensate. The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... Niels Bohr’s 1913 quantum model of the atom, which incorporated an explanation of Johannes Rydbergs 1888 formula, Max Planck’s 1900 quantum hypothesis, i. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... Look up continuum in Wiktionary, the free dictionary. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... Fig. ... Quantum field theory (QFT) is the quantum theory of fields. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... Sphere symmetry group o. ...

Figure 10. Log-log plot of the average energy of a quantum mechanical oscillator (shown in red) as a function of temperature. For comparison, the value predicted by the equipartition theorem is shown in black. At high temperatures, the two agree nearly perfectly, but at low temperatures when kBT << hν, the quantum mechanical value decreases much more rapidly. This resolves the problem of the ultraviolet catastrophe: for a given temperature, the energy in the high-frequency modes (where hν >> kBT) is almost zero.
Figure 10. Log-log plot of the average energy of a quantum mechanical oscillator (shown in red) as a function of temperature. For comparison, the value predicted by the equipartition theorem is shown in black. At high temperatures, the two agree nearly perfectly, but at low temperatures when kBT << hν, the quantum mechanical value decreases much more rapidly. This resolves the problem of the ultraviolet catastrophe: for a given temperature, the energy in the high-frequency modes (where hν >> kBT) is almost zero.

To illustrate this breakdown, consider the average energy in a single (quantum) harmonic oscillator, which was discussed above for the classical case. Its quantum energy levels are given by En = nhν, where h is Planck's constant, ν is the fundamental frequency of the oscillator, and n is an integer. The probability of a given energy level being populated in the canonical ensemble is given by its Boltzmann factor Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The fundamental tone, often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ... A canonical ensemble in statistical mechanics is an ensemble of dynamically similar systems, each of which can share its energy with a large heat reservoir, or heat bath. ... In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ...

P(E_{n}) = frac{e^{-nbeta hnu}}{Z},

where β = 1/kBT and the denominator Z is the partition function, here a geometric series In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

Z = sum_{n=0}^{infty} e^{-nbeta hnu} = frac{1}{1 - e^{-beta hnu}}.

Its average energy is given by

langle H rangle = sum_{n=0}^{infty} E_{n} P(E_{n}) = frac{1}{Z} sum_{n=0}^{infty} nhnu e^{-nbeta hnu} = -frac{1}{Z} frac{partial Z}{partial beta} = -frac{partial log Z}{partial beta}.

Substituting the formula for Z gives the final result[7]

langle H rangle = hnu frac{e^{-beta hnu}}{1 - e^{-beta hnu}}.

At high temperatures, when the thermal energy kBT is much greater than the spacing hν between energy levels, the exponential argument βhν is much less than one and the average energy becomes kBT, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when hν >> kBT, the average energy goes to zero — the higher-frequency energy levels are "frozen out" (Figure 10). As another example, the internal excited electronic states of a hydrogen atom do not contribute to its specific heat as a gas at room temperature, since the thermal energy kBT (roughly 0.025 eV) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10 eV). The electronvolt (symbol eV, or, rarely and incorrectly, ev) is a unit of energy. ... The electronvolt (symbol eV, or, rarely and incorrectly, ev) is a unit of energy. ...


Similar considerations apply whenever the energy level spacing is much larger than the thermal energy. For example, this reasoning was used by Albert Einstein to resolve the ultraviolet catastrophe of blackbody radiation.[46] The paradox arises because there are an infinite number of independent modes of the electromagnetic field in a closed container, each of which may be treated as a harmonic oscillator. If each electromagnetic mode were to have an average energy kBT, there would be an infinite amount of energy in the container.[46][47] However, by the reasoning above, the average energy in the higher-ω modes goes to zero as ω goes to infinity; moreover, Planck's law of black body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.[46] Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... Black body spectrum In physics, Plancks law of black body radiation predicts the spectral intensity of electromagnetic radiation at all wavelengths from a black body at temperature  : where the following table provides the definition and SI units of measure for each symbol: The wavelength is related to the frequency...


Symmetry in a quantum system introduces another, more subtle correction to the law of equipartition. Excited rotational states are impossible for systems with a continuous symmetry, such as the rotation of a diatomic gas about the axis connecting the centers of the atoms. Since such states do not exist, they are effectively frozen out; hence, the system has fewer effective degrees of freedom. Thus, diatomic gases typically have a molar specific heat of (5/2)R, instead of (7/2)R; the vibrational degree of freedom is frozen out at room temperature, and one rotational degree of freedom is frozen out because of symmetry. At even lower temperatures, the two remaining rotational modes are frozen out, giving a molar specific heat of (3/2)R, corresponding to the three degrees of translational freedom. Other symmetries, such as those involving identical particles, can also produce deviations from the specific heats predicted by the equipartition theorem. Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ...


See also

In mechanics, the virial theorem provides a general equation relating the average total kinetic energy of a system with its average total potential energy , where angle brackets represent the average of the enclosed quantity. ... Kinetic theory attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. ...

Notes and references

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  2. ^ Fact Sheet on Uranium Enrichment U.S. Nuclear Regulatory Commission. Accessed 30th April 2007
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  4. ^ Cavanagh, J; Fairbrother WJ, Palmer AG III, Skelton NJ, Rance M (2006). Protein NMR Spectroscopy: Principles and Practice, 2nd ed., Academic Press. ISBN 978-0121644918. 
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  6. ^ a b c Goldstein, H (1980). Classical Mechanics, 2nd. ed, Addison-Wesley. ISBN 0-201-02918-9. 
  7. ^ a b c d e f g h i Huang, K (1987). Statistical Mechanics, 2nd ed., John Wiley and Sons, pp. 136–138. ISBN 0-471-81518-7. 
  8. ^ a b Mandl, F (1971). Statistical Physics. John Wiley and Sons, pp. 213–219. ISBN 0-471-56658-6. 
  9. ^ a b c d Pais, A (1982). Subtle is the Lord. Oxford University Press. ISBN 0-19-853907-X. 
  10. ^ a b Tolman, RC (1918). "A General Theory of Energy Partition with Applications to Quantum Theory". Physical Review 11: 261–275. 
  11. ^ Miedl M, Garcia M, Bamforth C (2005). "Haze formation in model beer systems". J. Agric. Food Chem. 53 (26): 10161-5. PMID 16366710. 
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  13. ^ Brush, SG (1976). The Kind of Motion We Call Heat, Volume 1. Amsterdam: North Holland, 134–159. ISBN 978-0444870094. 
    Brush, SG (1976). The Kind of Motion We Call Heat, Volume 2. Amsterdam: North Holland, 336–339. ISBN 978-0444870094. 
    Waterston, JJ (1846/1893). "On the physics of media that are composed of free and elastic molecules in a state of motion". Roy. Soc. Proc. 5: 604. (abstract only). Not published in full until (1893) "". Philos. Trans. R. Soc. London A183: 1–79.  Reprinted (1928) in J.S. Haldane: The collected scientific papers of John James Waterston. Edinburgh: Oliver & Boyd. 
    Waterston, JJ (1843). Thoughts on the Mental Functions.  (reprinted in his Papers, 3, 167, 183.)
    Waterston, JJ (1851). "". British Association Reports 21: 6.  Waterston's key paper was written and submitted in 1845 to the Royal Society. After refusing to publish his work, the Society also refused to return his manuscript and stored it among its files. The manuscript was discovered in 1891 by Lord Rayleigh, who criticized the original reviewer for failing to recognize the significance of Waterston's work. Waterston managed to publish his ideas in 1851, and therefore has priority over Maxwell for enunciating the first version of the equipartition theorem.
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  19. ^ Dewar, J (1872). "The Specific Heat of Carbon at High Temperatures". Philosophical Magazine 44: 461. 
    Weber, HF (1872). "Die specifische Wärme des Kohlenstoffs (The specific heat of carbon)". Annalen der Physik 147: 311–319.  (German)
    Weber, HF (1875). "Die specifische Wärmen der Elemente Kohlenstoff, Bor und Silicium (The specific heats of elemental carbon, boron, and silicon)". Annalen der Physik 154: 367–423, 553–582.  (German)
  20. ^ de la Rive, A; Marcet F (1840). "Quelques recherches sur la chaleur spécifique (Some research on specific heat)". Annales de Chimie et de Physique 75: 113–144.  (French)
    Regnault, HV (1841). "Recherches sur la chaleur spécifique des corps simples et des corps composés (deuxième Mémoire) (Studies of the specific heats of simple and composite bodies)". Annales de Chimie et de Physique 1 (3me Série): 129–207.  (French) Read at l'Académie des Sciences on 11 January 1841.
    Wigand, A (1907). "Über Temperaturabhängigkeit der spezifischen Wärme fester Elemente (On the temperature dependence of the specific heats of solids)". Annalen der Physik 22: 99–106.  (German)
  21. ^ a b Wüller, A (1896). Lehrbuch der Experimentalphysik (Textbook of Experimental Physics). Leipzig: Teubner, Vol. 2, 507ff.  (German)
  22. ^ a b Eucken, A (1912). "Die Molekularwärme des Wasserstoffs bei tiefen Temperaturen (The molecular specific heat of hydrogen at low temperatures)". Sitzungsberichte der königlichen Preussischen Akademie der Wissenschaften 1912: 141–151.  (German)
  23. ^ a b Maxwell, JC (1890). "On the Dynamical Evidence of the Molecular Constitution of Bodies", in WD Niven: The Scientific Papers of James Clerk Maxwell. Cambridge: At the University Press, Vol.2, pp.418–438. ASIN B000GW7DXY.  A lecture delivered by Prof. Maxwell at the Chemical Society on 18 February 1875.
  24. ^ a b Kittel, C (1996). Introduction to Solid State Physics. New York: John Wiley and Sons, 151–156. ISBN 978-0471111818. 
  25. ^ Boltzmann, L (1895). "On certain Questions of the Theory of Gases". Nature 51: 413–415. 
  26. ^ Thomson, W (1904). Baltimore Lectures. Baltimore: Johns Hopkins University Press, Sec. 27.  Re-issued in 1987 by MIT Press as Kelvin's Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives (Robert Kargon and Peter Achinstein, editors). ISBN 978-0262111171
  27. ^ Rayleigh, JWS (1900). "The Law of Partition of Kinetic Energy". Philosophical Magazine 49: 98–118. 
  28. ^ Einstein, A (1907). "Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme (The Planck theory of radiation and the theory of specific heat)". Annalen der Physik 22: 180–190. (German)
    Einstein, A (1907). "Berichtigung zu meiner Arbeit: 'Die Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme' (Correction to previous article)". Annalen der Physik 22: 800.  (German)
    Einstein, A (1911). "Eine Beziehung zwischen dem elastischen Verhalten and der spezifischen Wärme bei festen Körpern mit einatomigem Molekül (A connection between the elastic behavior and the specific heat of solids with single-atom molecules)". Annalen der Physik 34: 170–174.  (German)
    Einstein, A (1911). "Bemerkung zu meiner Arbeit: 'Eine Beziehung zwischen dem elastischen Verhalten and der spezifischen Wärme bei festen Körpern mit einatomigem Molekül' (Comment on previous article)". Annalen der Physik 34: 590.  (German)
    Einstein, A (1911). "Elementare Betrachtungen über die thermische Molekularbewegung in festen Körpern (Elementary observations on the thermal movements of molecules in solids)". Annalen der Physik 35: 679–694.  (German)
  29. ^ Nernst, W (1910). "Untersuchungen über die spezifische Wärme bei tiefen Temperaturen. II. (Investigations into the specific heat at low temperatures)". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1910: 262–282.  (German)
  30. ^ Hermann, Armin (1971). The Genesis of Quantum Theory (1899-1913), original title: Frühgeschichte der Quantentheorie (1899-1913), translated by Claude W. Nash, Cambridge, MA: The MIT Press, pp. 124–145. ISBN 0-262-08047-8, LCCN 73-151106. 
  31. ^ a b c Tolman, RC (1938). The Principles of Statistical Mechanics. New York: Dover Publications, pp. 93–98. ISBN 0-486-63896-0. 
  32. ^ Clausius, R (1870). "Ueber einen auf die Wärme anwendbaren mechanischen Satz". Annalen der Physik 141: 124–130.  (German)
    Clausius, RJE (1870). "On a Mechanical Theorem Applicable to Heat". Philosophical Magazine, Ser. 4 40: 122–127. 
  33. ^ a b Tolman, RC (1927). Statistical Mechanics, with Applications to Physics and Chemistry. Chemical Catalog Company, pp. 76–77. 
  34. ^ Terletskii, YP (1971). Statistical Physics, translated: N. Fröman, Amsterdam: North-Holland, pp. 83–84. ISBN 0-7204-0221-2, LCCN 70-157006. 
  35. ^ Collins, GW (1978). The Virial Theorem in Stellar Astrophysics. Pachart Press. 
  36. ^ Chandrasekhar, S (1939). An Introduction to the Study of Stellar Structure. Chicago: University of Chicago Press, pp. 49–53. 
  37. ^ Kourganoff, V (1980). Introduction to Advanced Astrophysics. Dordrecht, Holland: D. Reidel, pp. 59–60, 134–140, 181–184. 
  38. ^ Chiu, H-Y (1968). Stellar Physics, volume I. Waltham, MA: Blaisdell Publishing. LCCN 67-17990. 
  39. ^ Noyes, RW (1982). The Sun, Our Star. Cambridge, MA: Harvard University Press. ISBN 0-674-85435-7. 
  40. ^ Ostlie, DA; Carroll BW (1996). An Introduction to Modern Stellar Astrophysics. Reading, MA: Addison-Wesley. ISBN 0-201-59880-9. 
  41. ^ Jeans, JH (1902). "The Stability of a Spherical Nebula". Phil.Trans. A 199: 1–53. 
  42. ^ a b McQuarrie, DA (2000). Statistical Mechanics, revised 2nd ed., University Science Books, pp. 121-128. ISBN 978-1891389153. 
  43. ^ Callen, HB (1985). Thermodynamics and an Introduction to Thermostatistics. New York: John Wiley and Sons, pp. 375–377. ISBN 0-471-86256-8. 
  44. ^ Arnold, VI; Avez A (1967). Théorie ergodique des systèms dynamiques. Gauthier-Villars, Paris. (French) (English edition: Benjamin-Cummings, Reading, Mass. 1968). 
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  46. ^ a b c Einstein, A (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (A Heuristic Model of the Creation and Transformation of Light)". Annalen der Physik 17: 132–148.  (German). An English translation is available from Wikisource.
  47. ^ Rayleigh, JWS (1900). "Remarks upon the Law of Complete Radiation". Philosophical Magazine 49: 539–540. 

August Adolf Eduard Eberhard Kundt (18 November 1839 – 21 May 1894) was a German physicist. ... Emil Gabriel Warburg (9th March,1846 (Altona) - 28th July,1931 (Bayreuth)) was a German phycist who during his career was professor of physics at the Universities of Straßburg, Freiburg and Berlin. ... Herbert Goldstein (June 26, 1922 – January 12, 2005) was an American physicist and the author of the standard graduate textbook Classical Mechanics, widely considered to be one of the best books on the subject. ... Kerson Huang (黃克孫 pinyin: Huang Kesun), who grew up in Canton, China, is Professor of Physics Emeritus at MIT. His name, however, is mostly familiar to Chinese readers as the translator of the Rubaiyat of Omar Khayyam. ... Abraham (Bram) Pais (May 19, 1918, Amsterdam, The Netherlands — July 28, 2000, Copenhagen, Denmark) was a Dutch-born American physicist and science historian. ... Richard Chace Tolman (March 4, 1881–September 5, 1948) was an American mathematical physicist and physical chemist who was an authority on statistical mechanics and made important contributions to the early development of theoretical cosmology. ... Physical Review is one of the oldest and most-respected scientific journals publishing research on all aspects of physics. ... Physical Review is one of the oldest and most-respected scientific journals publishing research on all aspects of physics. ... John James Waterston (1811 - June 18, 1883) was a Scottish physicist, a neglected pioneer of the kinetic theory of gases. ... John James Waterston (1811 - June 18, 1883) was a Scottish physicist, a neglected pioneer of the kinetic theory of gases. ... John James Waterston (1811 - June 18, 1883) was a Scottish physicist, a neglected pioneer of the kinetic theory of gases. ... The premises of The Royal Society in London (first four properties only). ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... September 21 is the 264th day of the year (265th in leap years). ... 1859 (MDCCCLIX) is a common year starting on Saturday of the Gregorian calendar (or a common year starting on Monday of the Julian calendar). ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... Alexis Thérèse Petit (October 2, 1791 - June 21, 1820) was a French physicist. ... Pierre Louis Dulong (February 12, 1785 - July 19, 1838) was a French physicist and chemist. ... For other persons named James Dewar, see James Dewar (disambiguation). ... Heinrich Friedrich Weber (1843-1912) was born in the town of Magdala, near Weimar, son of a merchant. ... Heinrich Friedrich Weber (1843-1912) was born in the town of Magdala, near Weimar, son of a merchant. ... Henri Victor Regnault (July 21, 1810 – January 19, 1878) was a French chemist and physicist best known for his careful measurements of the thermal properties of gases. ... January 11 is the 11th day of the year in the Gregorian calendar. ... 1841 is a common year starting on Friday (link will take you to calendar). ... James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and theoretical physicist. ... February 18 is the 49th day of the year in the Gregorian calendar. ... 1875 (MDCCCLXXV) was a common year starting on Friday (see link for calendar). ... Ludwig Eduard Boltzmann (Vienna, Austrian Empire, February 20, 1844 – Duino near Trieste, September 5, 1906) was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics. ... William Thomson, 1st Baron Kelvin, OM, GCVO, PC, PRS, FRSE, (26 June 1824 – 17 December 1907) was a mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century. ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Walther Nernst. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ... Richard Chace Tolman (March 4, 1881–September 5, 1948) was an American mathematical physicist and physical chemist who was an authority on statistical mechanics and made important contributions to the early development of theoretical cosmology. ... Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 – August 24, 1888), was a German physicist and mathematician. ... Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 – August 24, 1888), was a German physicist and mathematician. ... Richard Chace Tolman (March 4, 1881–September 5, 1948) was an American mathematical physicist and physical chemist who was an authority on statistical mechanics and made important contributions to the early development of theoretical cosmology. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ... Chandrasekhar redirects here. ... The Library of Congress Control Number or LCCN is a serially based system of numbering books in the Library of Congress in the United States. ... Sir James Hopwood Jeans (September 11, 1877 in Ormskirk – September 16, 1946 in Dorking) was a British physicist, astronomer, and mathematician. ... Vladimir I. Arnold (Moscow, December 2001). ... Albert Einstein( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is widely considered to have been one of the greatest physicists of all time. ... Annalen der Physik is one of the best-known and oldest (it was founded in 1799) physics journals worldwide. ... The original Wikisource logo. ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ...

Further reading

  • Pathria, RK (1972). Statistical Mechanics. Pergamon Press, pp. 43–48, 73–74. ISBN 0-08-016747-0. 
  • Huang, K (1987). Statistical Mechanics, 2nd ed., John Wiley and Sons, pp. 136–138. ISBN 0-471-81518-7. 
  • Pauli, W (1973). Pauli Lectures on Physics: Volume 4. Statistical Mechanics. MIT Press, pp. 27–40. ISBN 0-262-16049-8. 
  • Khinchin, AI (1949). Mathematical Foundations of Statistical Mechanics (G. Gamow, translator). New York: Dover Publications, pp. 93–98. ISBN 0-486-63896-0. 
  • Mohling, F (1982). Statistical Mechanics: Methods and Applications. John Wiley and Sons, pp. 137–139, 270–273, 280, 285–292. ISBN 0-470-27340-2. 
  • Landau, LD; Lifshitz EM (1980). Statistical Physics, Part 1, 3rd ed., Pergamon Press, pp. 129–132. ISBN 0-08-023039-3. 
  • Mandl, F (1971). Statistical Physics. John Wiley and Sons, pp. 213–219. ISBN 0-471-56658-6. 
  • Tolman, RC (1938). The Principles of Statistical Mechanics. New York: Dover Publications, pp. 93–98. ISBN 0-486-63896-0. 
  • Tolman, RC (1927). Statistical Mechanics, with Applications to Physics and Chemistry. Chemical Catalog Company, pp. 72–81.  ASIN B00085D6OO

Kerson Huang (黃克孫 pinyin: Huang Kesun), who grew up in Canton, China, is Professor of Physics Emeritus at MIT. His name, however, is mostly familiar to Chinese readers as the translator of the Rubaiyat of Omar Khayyam. ... This article is about Austrian-Swiss physicist Wolfgang Pauli. ... Aleksandr Ya. ... George Gamow (pronounced GAM-off) (March 4, 1904 – August 19, 1968) , born Georgiy Antonovich Gamov (Георгий Антонович Гамов) was a Ukrainian born physicist and cosmologist. ... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... Evgeny Mikhailovich Lifshitz (Russian: ; February 21, 1915 – October 29, 1985) was a notable Soviet physicist. ... Richard Chace Tolman (March 4, 1881–September 5, 1948) was an American mathematical physicist and physical chemist who was an authority on statistical mechanics and made important contributions to the early development of theoretical cosmology. ... Richard Chace Tolman (March 4, 1881–September 5, 1948) was an American mathematical physicist and physical chemist who was an authority on statistical mechanics and made important contributions to the early development of theoretical cosmology. ...

External links

  • The equipartition theorem in stellar physics, written by Nir J. Shaviv, an associate professor at the Racah Institute of Physics in the Hebrew University of Jerusalem.

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Equipartition of Energy (346 words)
The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion and that the energy is
Equipartition of energy also has implication for electromagnetic radiation when it is in equilibrium with matter, each mode of radiation having kT of energy in the Rayleigh-Jeans law.
The average translational kinetic energy possessed by free particles given by equipartition of energy is sometimes called the thermal energy per particle.
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