Canonical ensemble - Citizendia

Statistical mechanics
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A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics The microcanonical ensemble is the simplest of the ensembles of Statistical mechanics. In Statistical mechanics, the grand canonical ensemble is a Statistical ensemble (a large collection of identically prepared systems where each system is in The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble is a statistical mechanical ensemble that maintains constant temperature T \ The isoenthalpic-isobaric ensemble (constant Enthalpy and constant Pressure ensemble is a statistical mechanical ensemble that maintains Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Mathematical physics, especially as introduced into Statistical mechanics and Thermodynamics by J In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable The probability distribution is characterised by the proportion p_i of members of the ensemble which exhibit a measurable macroscopic state i, where the proportion of microscopic states for each macroscopic state i is given by the Boltzmann distribution,

$p_i = \tfrac{1}{Z}e^{- E_i/(kT)} = e^{-(E_i -A)/(kT)}$

where E_i is the energy of state i. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one It can be shown that this is the distribution which is most likely, if each system in the ensemble can exchange energy with a heat bath, or alternatively with a large number of similar systems. A heat bath is a system whose Heat capacity is so large that when it is in Thermal contact with some other system of interest its temperature remains constant Equivalently, it is the distribution which has maximum entropy for a given average energy <E_i>. The principle of maximum entropy is a postulate about a universal feature of any Probability assignment on a given set of Propositions ( Events hypotheses

It is also referred to as an NVT ensemble: the number of particles (N), the volume (V), of each system in the ensemble are the same, and the ensemble has a well defined temperature (T), given by the temperature of the heat bath with which it would be in equilibrium.

The quantity k is Boltzmann's constant, which relates the units of temperature to units of energy. Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics It may be suppressed by expressing the absolute temperature using thermodynamic beta, $β = 1 / (k T)$ . In Statistical mechanics, the thermodynamic beta is a numerical quantity related to the Thermodynamic temperature of a system

The quantities A and Z are constants for a particular ensemble, which ensure that $Σ p i$ is normalised to 1. Z is therefore given by

$Z = \sum e^{- E_i/(kT)} = \sum e^{-\beta E_i}$ .

This is called the partition function of the canonical ensemble. In Statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in Thermodynamic Specifying this dependence of Z on the energies E_i conveys the same mathematical information as specifying the form of p_i above.

The canonical ensemble (and its partition function) is widely used as a tool to calculate thermodynamic quantites of a system under a fixed temperature. This article derives some basic elements of the canonical ensemble. Other related thermodynamic formulas are given in the partition function article. In Statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in Thermodynamic Mathematical treatments are given in the articles on the Potts model, where the canonical ensemble as a probability measure is expressed in the language of measure theory, and quantum statistical mechanics. In Statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a Crystalline lattice. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with Quantum statistical mechanics is the study of Statistical ensembles of quantum mechanical systems.

1 Deriving the Boltzmann factor from ensemble theory
2 A derivation from heat-bath viewpoint
3 Quantum mechanical systems
4 Relations with other ensembles
5 See also
6 References
7 External links

Deriving the Boltzmann factor from ensemble theory

Let $E_i\,$ be the energy of the microstate $i\,$ and suppose there are $n_i\,$ members of the ensemble residing in this state. In Statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system that the system visits in the course of its thermal fluctuations Further we assume the total number of systems in the ensemble, $\mathcal{N}\,$ , and the total energy of all systems of the ensemble, $\mathcal{E}\,$ , are fixed, i. e. ,

$\mathcal{N}= \sum_i n_i , \,$

$\mathcal{E}= \sum_i n_i E_i \,.$

Since systems in the ensemble are indistinguishable, for each set $\{n_i\} \,$ , the number of ways of shuffling systems is equal to

$W (\{n_i\}) = \mathcal{N}!/ \prod_{i} n_i! \, .$

So for a given $\{n_i\}\,$ , there are $W(\{n_i\})\,$ rearrangements that specify the same state of the ensemble.

The most probable distribution is the one that maximizes $W (\{n_i\})\,$ . The probability for any other distribution to occur is extremely small in the limit $\mathcal{N} \rightarrow \infty \,$ . To determine this distribution, one should maximize $W (\{n_i\})\,$ with respect to the $n_i\,$ 's, under two constraints specified above. This can be done by using two Lagrange multipliers $\alpha \,$ and $\beta\,$ . In mathematical optimization problems the method of Lagrange multipliers, named after Joseph Louis Lagrange, is a method for finding the extrema of (The assumption that $\mathcal{N} \rightarrow \infty \,$ would be invoked in such calculation, which allows one to apply Stirling's approximation. In Mathematics, Stirling's approximation (or Stirling's formula) is an approximation for large Factorials It is named in honour of James Stirling ) The result is

$n_i = e^{-\alpha -\beta E_i} \,$ .

This distribution is called the canonical distribution. To determine $\alpha \,$ and $\beta\,$ , it is useful to introduce the partition function as a sum over microscopic states

$Z(\beta) = \sum_j e^{-\beta E_j} .\,$

Comparing with thermodynamic formulae, it can be shown that $\beta\,$ , is related to the absolute temperature $T\,$ as, $\beta=1/k_B T\,$ . Moreover the expression

$- \ln Z(\beta) /\beta\,$

is identified as the Helmholtz free energy $F$ . In Thermodynamics, the Helmholtz free energy is a Thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic A derivation is given here. In Thermodynamics, the Helmholtz free energy is a Thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic Consequently, from the partition function we can obtain the average thermodynamic quantities for the ensemble. For example, the average energy among members of the ensemble is

$\langle E \rangle = \frac{ \mathcal{E}}{ \mathcal{N} } = - \frac{\partial}{\partial \beta } \ln Z(\beta) \,$ .

This relation can be used to determine $\beta\,$ . $\alpha\,$ is determined from

$e^{\alpha} = Z(\beta)/ \mathcal{N}\,$ .

A derivation from heat-bath viewpoint

Illustration of a system of interest suspended in a heat bath. The system of interest is taken to be 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 S
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′(E) - denotes the number of microstates available at a particular energy E for the heat reservoir.

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.

Suppose S is in a microstate indexed by m. From the above definitions, the total energy of the system S* is given by

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

Notice E* is constant, since the combined system S* is taken to be isolated.

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*. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives A good approximation can be obtained by keeping the first two terms of the Taylor series expansion:

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

The following quantity is a constant which is traditionally denoted by β, known as the thermodynamic beta. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Statistical mechanics, the thermodynamic beta is a numerical quantity related to the Thermodynamic temperature of a system

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

Finally,

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

Exponentiating this expression gives

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

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

$C = C' \Omega'(E^\ast) \,$

From this

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

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. In Statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in Thermodynamic

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. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy See the article on Maxwell-Boltzmann statistics where this approach is employed. In Statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in Thermal equilibrium

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

$P = e^{\frac{E-A}{kT} } \,$

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

$e^{ {E - A \over kT} } dp \, dq$

with p and q being the canonical coordinates. A probability space, in Probability theory, is the conventional Mathematical model of Randomness.

Characteristic state function

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

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

Quantum mechanical systems

By applying the canonical partition function, one can easily obtain the corresponding results for a canonical ensemble of quantum mechanical systems. The characteristic state function in Statistical mechanics refers to a particular relationship between the Partition function of an ensemble In Thermodynamics, the Helmholtz free energy is a Thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic 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. In Mathematics, on a finite-dimensional Inner product space, a self-adjoint operator is one that is its own adjoint, or equivalently one whose matrix In Physics, discrete spectrum is a Finite set or a Countable set of Eigenvalues of an Operator. 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. In Mathematics, a trace class operator is a Compact operator for which a trace may be defined such that the trace is finite and independent of the choice ) 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. In Mathematics, specifically in Functional analysis, closed linear operators are an important class of Linear operators on Banach spaces They In Functional analysis, a branch of Mathematics, the Borel functional calculus is a Functional calculus (that is an assignment of Operators 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

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

Relations with other ensembles

A generalization of this is the grand canonical ensemble, in which the systems may share particles as well as energy. In Statistical mechanics, the grand canonical ensemble is a Statistical ensemble (a large collection of identically prepared systems where each system is in By contrast, in the microcanonical ensemble, the energy of each individual system is fixed. The microcanonical ensemble is the simplest of the ensembles of Statistical mechanics.

References

External links

The Ornstein-Zernike equation in the canonical ensemble

Draft Chapters of Thermal and Statistical Physics Textbook

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