Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) consisting in $3N$ particles moving in one dimension. I know that the partition function is given by

$$Q_{3N}=\frac{1}{(3N)!h^{3N}} \int e^{-\beta H(q,p)}d\omega,$$

where $d\omega$ denotes a volume element of the phase space. In this case

$$d\omega=dq_1dp_1\cdots dq_{3N}dp_{3N},$$

and

$$H(q,p)=\sum_{i=1}^{3N}cp_i.$$

Then, making the substitution I find that

$$Q_{3N}=\frac{L^{3N}}{(3N)!h^{3N}} \left[\int_{-\infty}^{+\infty} e^{-\beta c p_j}dp_j\right]^{3N}.$$

$L$ being the "length" of the space available. But I'm pretty sure that this integral does not converge.

Where am I wrong?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Energy has to be bounded below. The relationship between energy & momentum in this case is $E_i = c |p_i|$, not $E_i =c p_i$. So the integral you should be trying is $\int_{-\infty}^\infty e^{-\beta c |p_i|} dp_i$, which converges just fine.

share|improve this answer
    
Yes, you are right. Because for each particle $E=cp$ where $p=\sqrt{\mathbf{p}\cdot\mathbf{p}}=\sqrt{p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}=|p_{x}|‌​$ if we consider that the motion is along the x-axis. Thanks! –  Anuar Dec 1 '12 at 20:22

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.