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

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:

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