Introduction to Mathematical Physics/N body problems and statistical equilibrium/Thermodynamical perfect gas

In this section, a perfect gas model is presented: all the particles are independent, without any interaction.

Classical approximation (see section secdistclassi) allows to replace the sum over the quantum states by an integral of the exponential of the classical hamiltonian $$H(q_i,p_i)$$. The price to pay is just to take into account a proportionality factor $$\frac{1}{2\pi \hbar}$$. Partition function $$z$$ associated to one particle is:

Partition function $$z$$ is thus proportional to $$V$$ :

Because particles are independent, partition function $$Z$$ for the whole system can be written as:

It is known that pressure (proportional to the Lagrange multiplier associated to the internal variable "volume") is related to the natural logarithm of $$Z$$; more precisely if one sets:

then

This last equation and the expression of $$Z$$ leads to the famous perfect gas state equation: