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 $$w^c(q_i,p_i)dq_idr_i$$ 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.

One can show  that

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

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: