Canonical Ensemble

Article Index for Canonical

Website Links For Canonical

Information About Canonical Ensemble

	CATEGORIES ABOUT CANONICAL ENSEMBLE
	statistical mechanics

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME
	MORE SHOPPING...

A canonical ensemble in Statistical Mechanics is an Ensemble (a large number of mental copies of a system, representing in effect a Probability Distribution for the exact microscopic state of the system), that is characterised by the proportion ''p''i of members of the ensemble occupying the state ''i'' being given by the Boltzmann Distribution , :$p\_i\; =\; frac\{1\}\{Z\}e^\{-\; E\_i/kT\}\; =\; e^\{-(E\_i\; -A)/kT\}$ where ''E''''i'' is the energy of state ''i''. 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. Equivalently, it is the distribution which has Maximum Entropy for a given average energy <''E''i>. 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. It may be suppressed by expressing the absolute temperature using Thermodynamic Beta , β = 1/''kT''. 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 :$Z\; =\; \backslash sum\; \backslash exp(-\; E\_i/kT)\; =\; \backslash sum\; \backslash exp(-\backslash beta\; E\_i)$. This is called the Partition Function of the canonical ensemble. 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. 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 . DERIVING THE BOLTZMANN FACTOR FROM ENSEMBLE THEORY Let $E\_i\backslash ,$ be the energy of the Microstate $i\backslash ,$ and suppose there are $n\_i\backslash ,$ members of the ensemble residing in this state. Further we assume the total number of systems in the ensemble, $\backslash mathcal\{N\}\backslash ,$, and the total energy of all systems of the ensemble, $\backslash mathcal\{E\}\backslash ,$, are fixed, i.e., :$\backslash mathcal\{N\}=\; \backslash sum\_i\; n\_i\; ,\; \backslash ,$ :$\backslash mathcal\{E\}=\; \backslash sum\_i\; n\_i\; E\_i\; \backslash ,.$ Since systems in the ensemble are indistinguishable, for each set $\backslash \{n\_i\backslash \}\; \backslash ,$, the number of ways of shuffling systems is equal to :$W\; (\backslash \{n\_i\backslash \})\; =\; \backslash mathcal\{N\}!/\; \backslash prod\_\{i\}\; n\_i!\; \backslash ,\; .$ So for a given $\backslash \{n\_i\backslash \}\backslash ,$, there are $W(\backslash \{n\_i\backslash \})\backslash ,$ rearrangements that specify the same state of the ensemble. The most probable distribution is the one that maximizes $W\; (\backslash \{n\_i\backslash \})\backslash ,$. The probability for any other distribution to occur is extremely small in the limit $\backslash mathcal\{N\}\; ightarrow\; \backslash infty\; \backslash ,$. To determine this distribution, one should maximize $W\; (\backslash \{n\_i\backslash \})\backslash ,$ with respect to the $n\_i\backslash ,$'s, under two constraints specified above. This can be done by using two Lagrange Multipliers $\backslash alpha\; \backslash ,$ and $\backslash beta\backslash ,$. (The assumption that $\backslash mathcal\{N\}\; ightarrow\; \backslash infty\; \backslash ,$ would be invoked in such calculation, which allows one to apply Stirling's Approximation .) The result is :$n\_i\; =\; \backslash exp(-\backslash alpha\; -\backslash beta\; E\_i)\; \backslash ,$. This distribution is called the canonical distribution. To determine $\backslash alpha\; \backslash ,$ and $\backslash beta\backslash ,$, it is useful to introduce the Partition Function as a sum over microscopic states :$Z(\backslash beta)\; =\; \backslash sum\_j\; \backslash exp(\; -\backslash beta\; E\_j)\; .\backslash ,$ Comparing with thermodynamic formulae, it can be shown that $\backslash beta\backslash ,$, is related to the absolute temperature $T\backslash ,$ as, $\backslash beta=1/k\_B\; T\backslash ,$. Moreover the expression :$-\; \backslash ln\; Z(\backslash beta)\; /\backslash beta\backslash ,$ is identified as the Helmholtz Free Energy $F$. 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 :. This relation can be used to determine $\backslash beta\backslash ,$. $\backslash alpha\backslash ,$ is determined from :$\backslash exp(\backslash alpha)\; =\; Z(\backslash beta)/\; \backslash mathcal\{N\}\backslash ,$. A DERIVATION FROM HEAT-BATH VIEWPOINT 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''. is given by :$E^\backslash ast\; =\; E\text{'}\; +\; E\_m\; \backslash ,$ 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, $\backslash ;\; p\_m$, is proportional to the corresponding number of microstates available to the reservoir when ''S'' is in the m-th state''. Therefore, :$p\_m\; =\; C\text{'}\backslash Omega\text{'}(E\text{'})\; \backslash ,$ for some constant $\backslash ;\; C\text{'}$. Taking the logarithm gives - E_m) \, , a Taylor Series expansion can be performed on the latter logarithm around the energy ''E''---. A good approximation can be obtained by keeping the first two terms of the Taylor Series expansion: : \approx \ln \Omega'(E^\ast) - rac{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 .

\equiv rac{1}{Z(\beta)}

where $Z$ is known as the Partition Function for the canonical ensemble.


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 :

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

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) "... 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

:

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
:$\backslash exp\; \backslash left\; (\; \{E\; -\; A\; \backslash over\; kT\}\; ight\; )\; dp\; \backslash ,\; dq$
with ''p'' and ''q'' being the canonical coordinates.


Characteristic state function


The Characteristic State Function of the canonical ensemble is the Helmholtz Free Energy function, as the following relationship holds:

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


QUANTUM MECHANICAL SYSTEMS


  Ho \sum p_n \psi _n angle \langle \psi_n = \sum C e^{- \beta E_n} \psi _n angle \langle \psi_n