The equipartition theorem is a principle of classical (non-quantum) statistical mechanics which states that the internal energy of a system composed of a large number of particles at thermal equilibrium will distribute itself evenly among each of the quadratic degrees of freedom allowed to the particles of the system. Fig. ... 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. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

History

The equipartition principle was proposed initially in 1867 by James Clerk Maxwell who stated that the energy of a gas is equally divided between linear and rotational energy. Then, in 1868 and 1872, Ludwig Boltzmann, an enthusiastic follower of Maxwell’s, expanded on this principle by showing that energy could not only be divided equally between linear and rotational movements but among all the independent components of motion in the system. James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematical physicist, born in Edinburgh. ... 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. ...

Overview

As an example, in thermodynamics, the equipartition theorem says that the mean internal energy associated with each degree of freedom of a monatomic ideal gas is the same. Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In physics and chemistry, monatomic is a combination of the words mono and atomic, and means single atom. ... An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of negligible volume, with no intermolecular forces. ...

For a molecule of gas, each component of velocity has an associated kinetic energy. This kinetic energy is, on average, Kinetic energy is the energy that a body possesses as a result of its motion. ...

where k_B is the Boltzmann constant, and T is the temperature of the molecule in kelvins. The components of velocity can be either linear or angular. Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... 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 general, for any system with a classical Hamiltonian of the form: Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...

where a_i and b_i are constant with respect to all q_{i < N} and p_{i < M},

q_j and p_i are spatial coordinates and their conjugate momenta,

each degree of freedom q_i and p_j will contribute a total of to the system's total energy, resulting in a total of equipartition energy. In classical mechanics, momentum (pl. ...

The equipartition theorem is valid only in the classical limit of an energy continuum. The equipartition theorem breaks down in the limit of large gaps between quantum energy levels, because it becomes more difficult to excite degrees of freedom which are highly quantized, such as electronic excitations in non-metals, vibrational modes with a large ratio of force constant to reduced mass, or rotational degrees of freedom about an axis with a low moment of inertia. Look up continuum in Wiktionary, the free dictionary. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... Generally, quantization is the state of being constrained to a set of discrete values, rather than varying continuously. ... It has been suggested that this article or section be merged with quantum state. ... The vibrational states of a molecule can be probed in a variety of ways. ... 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. ... Reduced mass is a concept that allows one to solve the two-body problem of mechanics as if it were a one body problem. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², English units lbs ft2) quantifies the rotational inertia of a rigid body, i. ...

See also

• Degrees of freedom (physics and chemistry)
• Kinetic theory
• Heat capacity
• Statistical mechanics
• Quantum statistical mechanics