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:

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:

Quantity $w^c(q_i,p_i)d{q}_{i}d{r}_i$ represents the probability for the system to be in the phase space volume between hyperplanes $q_i,p_i$ and $q_i+d{q}_i,p_i+d{p}_i$. Normalization coefficients $Z$ and $A$ are proportional.

One can show Template:IMP/cite that

$2\pi {\hslash }^N$ being a sort of quantum state volume.

Template:IMP/rem Partition function provided by a classical approach becomes thus:

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:

This leads to the classical partition function for a system of $N$ identical particles: Template:IMP/eq