Canonical ensemble

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Statistical mechanics
Microcanonical ensemble
Canonical ensemble
Grand canonical ensemble
Isothermal-isobaric ensemble
Isoenthalpic-isobaric ensemble
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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. It is also referred to as an NVT ensemble: the number of particles (N), the volume (V), and the temperature (T) are constant in this ensemble. The distribution of the total energy amongst the possible dynamical states (i.e. the members of the ensemble) is given by the partition function. A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy. By contrast, in the microcanonical ensemble, the energy of each individual system is fixed.

In some derivations, the heat bath is considered to comprise a large number of copies of the original system, loosely coupled to the original and to each other, so as to share the same total energy - this then makes the combined (system+heat bath) describable by the statistics of a microcanonical ensemble.

This article derives the fundamental mathematical object in the canonical ensemble, the canonical partition function. Other related thermodynamic formulas are given in the partition function article. Mathematical treatments are given in the articles on the Potts model, where the canonical ensemble as a probability measure expressed in the language of measure theory, and quantum statistical mechanics.

The canonical ensemble is used in statistical thermodynamics to calculate various thermodynamic quantities.

1 A derivation
2 Quantum mechanical systems

[edit] A derivation

Illustration of a system of interest suspended in a heat bath. The system of interest is small compared to the heat bath.

Define the following:

S - the system of interest
S′ - the heat reservoir in which S resides; S is small compared to S′
S* - the system consisting of S and S′ combined together
m - an indexing variable which labels all the available energy states of the system
E_m - the energy of the state corresponding to the index m for the system S
E′ - the energy associated with the heat bath
E* - the energy associated with S*
Ω′(.) - denotes the number of microstates available at a particular energy for the heat reservoir. For example, Ω′(E) denotes the number of microstates available to the reservoir when the S has energy E.

It is assumed that the system S and the reservoir S′ are in thermal equilibrium. The objective is to calculate the set of probabilities p_m that S is in a particular energy state E_m.

From these definitions, the total energy of the system S* is given by

$E^* = E' + E_m \,$

Notice E* is constant, since the combined system S* is thought to be isolated. Suppose S is in a microstate indexed by m.

Now, arguably the key step in the derivation is that the probability of S being in the m-th state, $\; p_m$ , is proportional to the corresponding number of microstates available to the reservoir when S is in the m-th state. Therefore,

$p_m = C'\Omega'(E') \,$

for some constant $\; C'$ . Taking the logarithm gives

$\ln p_m = \ln C' + \ln \Omega' (E') = \ln C' + \ln \Omega' (E^* - E_m) \,$

Since E_m is small compared to E*, a Taylor series expansion can be performed on the latter logarithm around the energy E′. An appropriate approximation can be obtained by keeping the first two terms of the Taylor series expansion:

$\ln \Omega'(E') = \sum_{k=0}^\infty \frac{(E' - E^* )^k }{k!} \frac{d^k \ln \Omega' (E^*)}{dE'^k} \approx \ln \Omega'(E^*) - \frac{d}{dE'} \ln \Omega'(E^*) E_m$

The following quantity is a constant which is traditionally denoted by β, known as the thermodynamic beta.

$\beta = \frac{d}{dE'} \ln \Omega'(E^*) = \left . \frac{d}{dE'} \ln \Omega'(E') \right |_{E'=E^*}$

Finally,

$\ln p_m = \ln C' + \ln \Omega'(E^*) - \beta E_m . \,$

Exponentiating this expression gives

$p_m = C' \Omega'(E^*) e^{-\beta E_m}$

The factor in front of the exponential can be treated as a normalization constant C, where

$C = C' \Omega'(E^*). \,$

From this

$p_m = C e^{-\beta E_m}. \,$

[edit] Normalization to recover the partition function

Since probabilities must sum to 1, it must be the case that

$\sum_m p_m = 1 = \sum_m C e^{-\beta E_m} = C \sum_m e^{-\beta E_m} \iff C = \frac{1}{\sum_m e^{-\beta E_m}} \equiv \frac{1}{Z(\beta)}$

where $Z$ is known as the partition function for the canonical ensemble.

[edit] Note on derivation

As mentioned above, the derivation hinges on recognizing that the probability of the system being in a particular state is proportional to the corresponding multiplicities of the reservoir (the same can be said for the grand canonical ensemble). As long as one makes that observation, it is flexible as how one might proceed. In the derivation given, the logarithm is taken, then a linear approximation based on physical arguments is used. Alternatively, one can apply the thermodynamic identity for differential entropy:

$d S = {1 \over T} (d U + P d V - \mu d N)$

and obtain the same result. See the article on Maxwell-Boltzmann statistics where this approach is employed.

The canonical ensemble is also called the Gibbs ensemble, in honor of J.W. Gibbs, widely regarded with Boltzmann as being one of the two fathers of statistical mechanics. In his definitive original book "Elementary Principles in Statistical Mechanics", Gibbs viewed an ensemble as a list of the allowed states of the system (each state appearing once and only once in the list) and the associated statistical weights. The states do not interact with each other, or with a reservoir, until Gibbs treats what happens when two complete ensembles at two different temperatures are allowed to interact weakly (Gibbs, pp 160). Gibbs writes that "...the distribution in phase..." (the phase space density in modern language) "...[is] called canonical...[if] the index of probability" (the logarithm of the statistical weight of the phase space density) "...is a linear function of the energy..." (Gibbs, Ch. 4). In Gibbs' formulation, this requirement (his equation 91, in modern notation

$P = \exp \left ( \frac{E-A}{kT} \right )$

is taken to define the canonical ensemble and to be the fundamental postulate. Gibbs does show that a large collection of interacting microcanonical systems approaches the canonical ensemble, but this is part of his demonstration (Gibbs, pp 169-183) that the principle of equal a priori probabilities, therefore the microcanonical ensemble, are inferior to the canonical ensemble as an axiomatization of statistical mechanics, at every point where the two treatments differ.

Gibbs original formulation is still standard in modern mathematically rigorous treatments of statistical mechanics, where the canonical ensemble is defined as the probability measure exp( (E-A)/kT) dp dq, p and q being the canonical coordinates.

[edit] Characteristic state function

The characteristic state function of the canonical ensemble is the Helmholtz free energy function, as the following relationship holds:

$Z(T,V,N) = e^{- \beta A} \,\;$

[edit] Quantum mechanical systems

By applying the canonical partition function, one can easily obtain the corresponding results for a canonical ensemble of quantum mechanical systems. A quantum mechanical ensemble in general is described by a density matrix. Suppose the Hamiltonian H of interest is a self adjoint operator with only discrete spectrum. The energy levels ${E n}$ are then the eigenvalues of H, corresponding to eigenvector $| \psi _n \rangle$ . From the same considerations as in the classical case, the probability that a system from the ensemble will be in state $| \psi _n \rangle$ is $p_n = C e^{- \beta E_n}$ , for some constant $C$ . So the ensemble is described by the density matrix

$\rho = \sum p_n | \psi _n \rangle \langle \psi_n | = \sum C e^{- \beta E_n} | \psi _n \rangle \langle \psi_n|$

(Technical note: a density matrix must be trace-class, therefore we have also assumed that the sequence of energy eigenvalues diverges sufficiently fast.) A density operator is assumed to have trace 1, so

$\operatorname{Tr} (\rho) = Q = \sum C e^{- \beta E_n} = 1$

, which means

$C = \frac{1}{\sum e^{- \beta E_n} } = \frac{1}{Q}.$

Q is the quantum-mechanical version of the canonical partition function. Putting C back into the eqation for ρ gives

$\rho = \frac{1}{\sum e^{- \beta E_n}} \sum e^{- \beta E_n} | \psi _n \rangle \langle \psi_n| = \frac{1}{ \operatorname{Tr}( e^{- \beta H} ) } e^{- \beta H} .$

By the assumption that the energy eigenvalues diverge, the Hamiltonian H is an unbounded operator, therefore we have invoked the Borel functional calculus to exponentiate the Hamiltonian H. Alternatively, in non-rigorous fashion, one can consider that to be the exponential power series.

Notice the quantity

$\operatorname{Tr}( e^{- \beta H} )$

is the quantum mechanical counterpart of the canonical partition function, being the normalization factor for the mixed state of interest.

The density operator ρ obtained above therefore describes the (mixed) state of a canonical ensemble of quantum mechanical systems. As with any density operator, if A is a physical observable, then its expected value is

$\langle A \rangle = \operatorname{Tr}( \rho A ).$