- #1
wphysics
- 29
- 0
I am following up Weinberg Cosmology book, but I have one question.
In chapter 3.1, we have Eq (3.1.3) and (3.1.4)
[itex] s(T) = \frac{\rho(T) + p(T)}{T} [/itex]
[itex] T\frac{dp(T)}{dT} = \rho(T) + p(T) [/itex]
In Eq (3.1.5), we have the Fermi-Dirac or Bose-Einstein distributions.
[itex] n(p, T) = \frac{4 \pi g p^2}{(2 \pi \hbar)^3} \frac{1}{exp(\sqrt{p^2 + m^2} / k_B T) \pm 1} [/itex].
From using this number distribution, the author said we have the energy density and pressure of a particle mass m are given by Eq (3.1.6) and (3.1.7).
[itex] \rho(T) = \int n(p, T) dp \sqrt{p^2 + m^2} [/itex]
[itex] p(T) = \int n(p, T) dp \frac{p^2}{3\sqrt{p^2 + m^2}} [/itex]
Here, energy density is straightforward by the definition of number density.
But, for pressure, the author said it can be derived from Eq(3.1.4), the second equation on this post.
However, I cannot derive this pressure equation using Eq(3.1.4). Can somebody help me do this?
Thank you.
In chapter 3.1, we have Eq (3.1.3) and (3.1.4)
[itex] s(T) = \frac{\rho(T) + p(T)}{T} [/itex]
[itex] T\frac{dp(T)}{dT} = \rho(T) + p(T) [/itex]
In Eq (3.1.5), we have the Fermi-Dirac or Bose-Einstein distributions.
[itex] n(p, T) = \frac{4 \pi g p^2}{(2 \pi \hbar)^3} \frac{1}{exp(\sqrt{p^2 + m^2} / k_B T) \pm 1} [/itex].
From using this number distribution, the author said we have the energy density and pressure of a particle mass m are given by Eq (3.1.6) and (3.1.7).
[itex] \rho(T) = \int n(p, T) dp \sqrt{p^2 + m^2} [/itex]
[itex] p(T) = \int n(p, T) dp \frac{p^2}{3\sqrt{p^2 + m^2}} [/itex]
Here, energy density is straightforward by the definition of number density.
But, for pressure, the author said it can be derived from Eq(3.1.4), the second equation on this post.
However, I cannot derive this pressure equation using Eq(3.1.4). Can somebody help me do this?
Thank you.