Normal (probability) distribution and Partition function.

[Kevin_spencer2]

Let be the continuous partition function:

$$Z(\beta)=(N!)^{-1}\int_{V}dx_1 dx_2 dx_3 dx_4 ...dx_N exp(-\beta H(x_1, x_2 , x_3 , ... ,x_n,p_1 , p_2 , ..., p_N$$

if the Hamiltonian is 'quadratic' in p's are q's do this mean that the particles in the gas solid or whatever follow a Normal distribution (is Maxwell distribution under a quadratic potential or with U=0 potential a Normal distribution??)

 

[Physics Monkey]

Hi Kevin,

First a notational issue, you need to also integrate over the momenta, not just the positions, to get the partition function.

Now, to answer your question, a normal distribution is synonymous with a Gaussian distribution. A Gaussian or normal random variable has a probability distribution which is an exponential of an expression quadratic in the variable. Hence, since the probability distribution for your particles is proportional to $$\exp{(-\beta H)}$$, the positions and momenta of your particles are normal random variables if the Hamiltonian is quadratic in those positions and momenta.