This shouldn't be too hard but I'm struggling.(adsbygoogle = window.adsbygoogle || []).push({});

Consider N identical particles in a box of volume V. The relation between their linear momentum and kinetic energy is given by [tex]E = c\left|\overrightarrow{p}\right|[/tex], where c is the speed of light. So, the Hamiltonian of the system is

[tex]H_N\left(\overrightarrow{q_1},\,\cdots,\overrightarrow{q_N},\overrightarrow{p_1},\,\cdots,\overrightarrow{p_N}\right) = c \sum_{i = 1}^N \left|p_i\right|[/tex]

Now, I'm trying to find the average energy of the system. So, I need to get Z:

[tex]Z = \int \exp\left(-\beta \cdot c \cdot \sum_i \left|p_i\right|\right) d\overrightarrow{q_1}\cdots d\overrightarrow{q_N} d\overrightarrow{p_1}\cdots d\overrightarrow{p_N} = V^{N} \cdot \left(\int \exp\left(-\beta \cdot c \cdot \left|\overrightarrow{p}\right|\right) d\overrightarrow{p}\right)^N[/tex]

Now, I've got two questions:

1) Is there any mistake in the calculations for Z above?

2) How to evaluate the integral??

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# Statistical mechanics

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