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. Template:IMP/rem 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 $\frac{1}{2\pi \hslash }$. Partition function $z$ associated to one particle is: Template:IMP/eq Template:IMP/eq Partition function $z$ is thus proportional to $V$ : Template:IMP/eq Because particles are independent, partition function $Z$ for the whole system can be written as: Template:IMP/eq 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: Template:IMP/eq then Template:IMP/eq This last equation and the expression of $Z$ leads to the famous perfect gas state equation: Template:IMP/eq