Canonical Ensemble - A Derivation From Heat-bath Viewpoint

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.

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

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, is proportional to the corresponding number of microstates available to the reservoir when S is in the m-th state. Therefore,

for some constant . Taking the logarithm gives

Since E_m is small compared to E*, 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:

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

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

Finally,

Exponentiating this expression gives

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

From this

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