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The microcanonical ensemble is the simplest of the ensembles of statistical mechanics. 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. ...
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 statistical mechanics, the grand canonical ensemble is a statistical ensemble, that means a set of identically prepared systems, each of which is in equilibrium with an external bath with respect to particle and energy exchange. ...
The isothermal–isobaric ensemble is a statistical mechanical ensemble where the system is allowed to exchange energy with a heat bath of temperature T and the volume can also change though its mean value is tuned by the pressure P applied. ...
In physics, a statistical ensemble is a very large set of similar systems, considered all at once. ...
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. ...
Assumptions of the ensemble A microcanonical ensemble is an ensemble consisting of copies of an isolated system. Its analysis is correspondingly simple. By the assumption that the system is isolated, each identical system in the ensemble has a common fixed energy E. The system may have many different microstates corresponding to the energy E. By the fundamental assumption of thermodynamics, each microstate corresponding to the same energy is equally probable. Therefore if Ω is the number of accessible microstates, the probability that a system chosen at random from the ensemble would be in a given microstate is simply . This leads to a formula for entropy (see below).
A microcanonical ensemble is a degenerate canonical ensemble in the sense that a canonical ensemble can be divided into sub-ensembles, each of which corresponds to a possible energy value and is itself a microcanonical ensemble. 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. ...
Thermodynamical systems that appear in physics are however sometimes constituted of extended objects, eg strings, and in this case the canonical and microcanonical ensembles are not equivalent. One must then resort to the microcanonical ensemble which is thought to be more fundamental. This in turn actually leads to a limiting maximum temperature called the Hagedorn temperature in string theory which has generated a lot of interest and is possibly relevant in the early universe which was, according to observations, much denser and hotter than it is today. We should emphasize that one can calculate with the canonical ensemble but to actually derive a physical quantity, such as the entropy or energy density, one need do so from the microcanonical ensemble, from Ω. (For more information see Deo et al.)
Entropy Entropy is defined by where kB is the Boltzmann constant. Or, equivalently, Ludwig Boltzmann The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ...
where Ω is the multiplicity of microstates in the ensemble, as before. Notice that, for the microcanonical ensemble, Ω plays the role of the partition function in the canonical and grand canonical ensembles. For this reason, it is also sometimes referred to as the microcanonical partition function. We should note here that the notion of multiplicity is valid for any thermodynamical system. Same can be said for partition functions and any ensemble. It is only for the microcanonical ensemble that they happen to be the same. In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ...
Ω is also called the characteristic state function of the microcanonical ensemble. The characteristic state function in statistical mechanics refers to a particular relationship between the partition function of an ensemble. ...
An application: residual entropy The expression for entropy above can be used to calculate the residual entropy. Residual entropy is physically significant entropy, which is present even after a substance is cooled arbitrarily close to absolute zero. ...
The third law of thermodynamics says that the entropy of a pure crystalline substance at 0 K is zero. However, in some solids, at temperatures close to 0 K, there may be many molecular orientations. For example, water molecules in ice crystal may arrange themselves in several different ways. In principle, there must be one molecular orientation with the lowest energy. But due the near randomness with which configurations occur, it is often impractical to attempt realization of the lowest energy configuration. This leads to the notion of residual entropy. Furthermore, there are often very little difference in the total energy of the system between the different molecular configurations. Therefore, as an approximation, the system can be viewed having fixed energy and the possible configurations as microstates, exactly a microcanonical ensemble. So it is sensible to estimate the residual entropy via the same expression for the microcanonical ensemble entropy): 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. ...
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). ...
where Ω is the number of possible molecular arrangements of the crystal, at some suitable temperature range close to 0 K.
Classical mechanical systems As with any ensemble of classical systems, we would like to find a corresponding probability measure on the phase space M. This constant energy assumption means that every system in the ensemble is confined to a submanifold of phase space of constant energy E. Call this submanifold ME. From the physical considerations given above, it is already clear what the probability measure on the constant energy surface (not the full phase space) should be, namely the trivial one that is constant everywhere. However, while only the submanifold ME is of interest for the microcanonical ensemble, in other more general ensembles, it is necessary to consider the full phase space. We now construct a measure on the full phase space that is suitable for the microcanonical ensemble. Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...
This is a glossary of terms specific to differential geometry and differential topology. ...
The Liouville measure dqdp on the full phase space induces a measure dA on ME in the following manner: In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...
The measure of an open subset R of ME is given by
Where Q is any open subset of M such that Q ∩ M = R, Q(E, E + ΔE) is part of Q with E < H < E + ΔE, and "vol" is the usual Liouville volume. Thus any sufficiently good (measurable) subset of ME can be characterized by its hyperarea(measure) with respect to dA. In mathematics, a measure is a function that assigns a number, e. ...
The density function on the full phase space ρ(q,p) is the generalized function , where H is the Hamiltonian and Ω is the hyperarea of ME. If Δ is a region of the phase space, the probability of a system being in a state within Δ is simply In mathematics, generalized functions are objects generalizing the notion of functions. ...
where ΔE is the intersection of ME and Δ.
Notice how one can either consider the whole phase space and use the measure whose density is a generalized function, or restrict to the constant energy surface in question and use the measure whose density is a constant function. For instance, consider a 1-dimensional harmonic oscillator. The phase space is (the position-momentum plane) and the constant energy hypersurface is the ellipse 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 later can be parametrized as where φ varies between 0 and 2π. The measure dA would then equal dφ up to a constant. On the other hand, if one considers the ellipse embedded in the plane, then it would have measure zero, which is why a generalized function is used as the density.
Connection with Liouville's theorem We have (the curly bracket is Poisson bracket) since ρ is a function of H. Therefore, according to Liouville's theorem we get In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...
Liouvilles theorem has various meanings: In complex analysis, see Liouvilles theorem (complex analysis). ...
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In particular, dA is time-invariant, that is, the ensemble is a stationary one.
Alternatively, one can say that since the Liouville measure is invariant under the Hamiltonian flow, so is the measure dA. In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ...
Physically speaking, this means the local density of a region of representative points in phase space is invariant, as viewed by an observer moving along with the systems.
Ergodic hypothesis A microcanonical ensemble of classical systems provides a natural setting to consider the ergodic hypothesis, that is, the long time average coincides with the ensemble average. More precisely put, an observable is a real valued function f on the phase space Γ that is integrable with respect to the microcanonical ensemble measure μ. Let x(0) denote a representative point in the phase space, and x(t) be its image under the Hamiltonian flow at time t. The time average of f is defined to be 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. ...
, provided that this limit exists μ-almost everywhere. The ensemble average is In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...
The system is said to be ergodic if they are equal.
Using the fact that μ is preserved by the Hamiltonian flow, we can show that indeed the time average exists for all observables. Whether classical mechanical flows on constant energy surfaces is in general ergodic is unknow at this time.
Remark The relationship between the microcanonical ensemble, Liouville's theorem, and ergodic hypothesis can be summarized as follows: The key assumption of a microcanonial ensemble is that all accessible microstates are equally probable. Therefore the density function on the relevant region of phase space is constant, say it is 1 everywhere, i.e. the phase space measure μ is just the Lebesgue measure. But, according to Liouville's theorem, this measure is invariant under the Hamiltonian time evolution. From this follows that the notion of time average makes sense for all observables. The ensemble average is defined using μ. The question of ergodicity is whether they coincide. It should perhaps be emphasized that while the microcanonical ensemble and Liouville's theorem are directly related, they should not be confused as being equivalent to the ergodic hypothesis.
Quantum mechanical systems
Semi-classical treatment So far, we have assumed the system in question is classical. Slight modification is required for quantum mechanical systems, although the results are essentially the same. For an ensemble consisting of quantum mechanical systems, it no longer make sense to speak of all members of the ensemble having the same definite energy E. So, instead of a level set H(q,p) = E in the phase space, one considers a small range of energies E < H < E + dE that a system in the ensemble may have and the corresponding region of the phase space. When classical states are replaced by quantum states, the degeneracy needs to be taken into account. Also, in the quantum mechanical case, due to the uncertainty principle, the states can no longer be viewed as continuously distributed in the phase space. Rather, one must find a "fundamental volume" ω0, which depends on the particulars of a given system. As we would expect, ω0 is usually related to in some way. Consequently, the multiplicity is not the total available volume of the phase space Ω but is replaced by , and entropy becomes In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...
Density operators The microcanonical ensemble can also be described by a density operator. Namely, if Ω is the total number of accessible microstates of the system, and are all states of the system (accessible and otherwise), then a microcanonical ensemble is the mixed state A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. ...
, where if is an accessible state and 0 otherwise.
We note here that, in this context, Ω is computed quantum-mechanically, taking into account indistinguishability of particles. The entropy is
When Ω = 1, the ensemble is said to be a pure ensemble. The fact that the entropy vanishes for pure states is essentially the third law of thermodynamics. 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. ...
References - R.K. Pathria, Statistical Mechanics, Elsevier 2001.
- Nivedita Deo, Sanjay Jain, Chung-I Tan, The Ideal Gas of Strings, Bombay Quant. Field Theory (1990) 112-148, http://www-spires.dur.ac.uk/cgi-bin/spiface/hep/www?rawcmd=FIND+T+%22IDEAL+GAS+OF+STRINGS%22+and+a+deo&FORMAT=www&SEQUENCE=
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