Introduction to Mathematical Physics/Statistical physics/Canonical distribution in classical mechanics

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Consider a system for which only the energy is fixed. Probability for this system to be in a quantum state (l) of energy E_l is given (see previous section) by:

P_l=\frac{1}{Z}e^{-E_l/k_BT}

Consider a classical description of this same system. For instance, consider a system constituted by N particles whose position and momentum are noted q_i and p_i, described by the classical hamiltonian H(q_i,p_i). A classical probability density w^c is defined by:

eqdensiprobaclas

w^c(q_i,p_i)=\frac{1}{A}e^{-H(q_i,p_i)/k_BT}

Quantity w^c(q_i,p_i)dq_idr_i represents the probability for the system to be in the phase space volume between hyperplanes q_i,p_i and q_i+dq_i, p_i+dp_i. Normalization coefficients Z and A are proportional.

A=\int dq_1...dq_n\int dp_1...dp_n e^{-H(q_i,p_i)/k_BT}

One can show [ph:physt:Diu89] that

Z=\frac{1}{(2\pi\hbar)^{3N}}A

2\pi\hbar^N being a sort of quantum state volume.

Remark:

This quantum state volume corresponds to the minimal precision allowed in the phase space from the Heisenberg uncertainty principle:

\index{Heisenberg uncertainty principle}

\Delta x \Delta p > \hbar

Partition function provided by a classical approach becomes thus:

Z=\frac{1}{(2\pi\hbar)^N}\int dq_1...dq_n\int dp_1...dp_n e^{-H(q_i,p_i)/k_BT}

But this passage technique from quantum description to classical description creates some compatibility problems. For instance, in quantum mechanics, there exist a postulate allowing to treat the case of a set of identical particles. Direct application of formula of equation eqdensiprobaclas leads to wrong results (Gibbs paradox). In a classical treatment of set of identical particles, a postulate has to be artificially added to the other statistical mechanics postulates:

Postulate:

Two states that does not differ by permutations are not considered as different.

This leads to the classical partition function for a system of N identical particles:

Z=\frac{1}{N!}\frac{1}{(2\pi\hbar)^{3N}}\int\prod dp^{3}_i dq^{3}_i
e^{-H(q_i,p_i)/k_BT}