Turning summations into integrals in the thermodynamic limit

[Gabriel Maia]

Hi. I'm reading a solution to a problem concerning a gas of photons. In the solution, the energy of the gas is given as

[itex]E=2\sum_{\vec{k}} \frac{\displaystyle \epsilon_{\vec{k}}}{\displaystyle \exp[\beta\epsilon_{\vec{k}}]+1}[/itex]

where [itex]\epsilon_{\vec{k}}[/itex] is one photon's energy. It is said then that in the thermodynamic limit we have

[itex]\sum_{\vec{k}} \rightarrow \frac{\displaystyle V}{\displaystyle (2\pi)}\int_{0}^{\infty}\,4\pi\,k^{2}dk[/itex]Could you explain how is this change from the summation to the integral is done?

Thank you very much.

 

[king vitamin]

I'll just do the 1D case for simplicity. In a finite 1D box with periodic BCs, the momentum [itex]k[/itex] is really defined as
[tex]
k = k(n) = \frac{2 \pi n}{L}
[/tex]
where [itex]n[/itex] runs over all of the integers. Then the sum is really over these integers
[tex]
\sum_{k} f(k) = \sum_{n = \infty}^{\infty} f(k(n))
[/tex]
The thermodynamic limit can be though of as approximating this sum as an integral
[tex]
\sum_{n = \infty}^{\infty} f(k(n)) \rightarrow \int_{-\infty}^{\infty} dn f(k(n))
[/tex]
However, since [itex]k[/itex] has a more natural physical interpretation, we prefer to change variables. Then using the fact that [itex]dn = \frac{L}{2 \pi} dk[/itex], we obtain
[tex]
\int_{-\infty}^{\infty} dn f(k(n)) = \frac{L}{2 \pi} \int_{-\infty}^{\infty} dk f(k)
[/tex]
In [itex]d[/itex] dimensions, the factor is [itex]V/(2 \pi)^d[/itex] instead.