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:

$P_{l}={\frac {1}{Z}}e^{-E_{l}/k_{B}T}$

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

$w^{c}(q_{i},p_{i})={\frac {1}{A}}e^{-H(q_{i},p_{i})/k_{B}T}$

Quantity $w^{c}(q_{i},p_{i})dq_{i}dr_{i}$ 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.

$A=\int dq_{1}...dq_{n}\int dp_{1}...dp_{n}e^{-H(q_{i},p_{i})/k_{B}T}$

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:

$Z={\frac {1}{(2\pi \hbar )^{N}}}\int dq_{1}...dq_{n}\int dp_{1}...dp_{n}e^{-H(q_{i},p_{i})/k_{B}T}$

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:

$Z={\frac {1}{N!}}{\frac {1}{(2\pi \hbar )^{3N}}}\int \prod dp_{i}^{3}dq_{i}^{3}e^{-H(q_{i},p_{i})/k_{B}T}$