Return to Question

I've been trying to follow the procedure that some books give in order to prove that the entropy of the universe is conserved (S is constant). It usually goes like this:

Consider the second law of thermodynamics: $$ TdS=dU+PdV. $$ Now, using $U=\rho V$, we get:

$$dS=\frac{1}{T} \left( d[(\rho + P)V]-VdP \right) $$

From this moment on, to continue proving the conservation of entropy, I need to use:

$$\frac{\partial P}{\partial T}=\frac{\rho+P}{T},$$ which is the equation I'm struggling to get.

In some books, like Kolb and Turner's Early Universe, they argue that this comes from the following Maxwell relation: $$ \frac{\partial S}{\partial T\partial V}=\frac{\partial S}{\partial V\partial T}. $$ This, for me, seems a little harder to grasp (although I'm open to hear responses that are related to this way of dealing with the problem), therefore I tried looking into other books which may have another ways of going through this. In the book Modern Cosmology by Scott Dodelson, we are told to assume $f=f(E/T)$, where $f$ is the distribution function, and subsitute this into the integral expression of $P$, $$P=g\int \frac{d^3p}{(2\pi)^3}f(p)\frac{p^2}{3E(p)}.$$ Then in order to find $dP/dT$, we rewrite $df/dT$ under the integral sign as $-(E/T)df/dE$ (I don't get this), and by integrating by parts we should get our result. I've been trying to work it out but I don't know how to procced.

I've been trying to follow the procedure that some books give in order to prove that the entropy of the universe is conserved (S is constant). It usually goes like this:

Consider the second law of thermodynamics: $$ TdS=dU+PdV. $$ Now, using $U=\rho V$, we get:

$$dS=\frac{1}{T} \left( d[(\rho + P)V]-VdP \right) $$

From this moment on, to continue proving the conservation of entropy, I need to use:

$$\frac{\partial P}{\partial T}=\frac{\rho+P}{T},$$ which is the equation I'm struggling to get.

In some books, like Kolb and Turner's Early Universe, they argue that this comes from the following Maxwell relation: $$ \frac{\partial S}{\partial T\partial V}=\frac{\partial S}{\partial V\partial T}. $$ This, for me, seems a little harder to grasp (although I'm open to hear responses that are related to this way of dealing with the problem), therefore I tried looking into other books which may have another ways of going through this. In the book Modern Cosmology by Scott Dodelson, we are told to assume $f=f(E/T)$, where $f$ is the distribution function, and subsitute this into the integral expression of $P$, $$P=g\int \frac{d^3p}{(2\pi)^3}f(p)\frac{p^2}{3E(p)}.$$ Then in order to find $dP/dT$, we rewrite $df/dT$ under the integral sign as $-(E/T)df/dE$ (I don't get this), and by integrating by parts we should get our result. I've been trying to work it out but I don't know how to procced.

References: Cosmology by Baumann pages 55-56, "The Early Universe" by E. Kolb and M. Turner pages 65-66, Modern Cosmology by Scott Dodelson pages 44-45, page 56 exercise 14.

I've been trying to follow the procedure that some books give in order to prove that the entropy of the universe is conserved (S is constant). The part that I struggle the most with is deriving $\partial P/\partial T = (\rho + P)/T$.

In the book Modern Cosmology by Scott Dodelson, we are told to assume $f=f(E/T)$ and subsitute this into the integral expression of $P$, $$P=g\int \frac{d^3p}{(2\pi)^3}f(p)\frac{p^2}{3E(p)},$$ then in order to find $dP/dT$, we rewrite $df/dT$ under the integral sign as $-(E/T)df/dE$ (why??), and by integrating by parts we should get our result. I've been trying to work it out but I don't know how to procced.

I've been trying to follow the procedure that some books give in order to prove that the entropy of the universe is conserved (S is constant). It usually goes like this:

Consider the second law of thermodynamics: $$ TdS=dU+PdV. $$ Now, using $U=\rho V$, we get:

$$dS=\frac{1}{T} \left( d[(\rho + P)V]-VdP \right) $$

From this moment on, to continue proving the conservation of entropy, I need to use:

$$\frac{\partial P}{\partial T}=\frac{\rho+P}{T},$$ which is the equation I'm struggling to get.

In some books, like Kolb and Turner's Early Universe, they argue that this comes from the following Maxwell relation: $$ \frac{\partial S}{\partial T\partial V}=\frac{\partial S}{\partial V\partial T}. $$ This, for me, seems a little harder to grasp (although I'm open to hear responses that are related to this way of dealing with the problem), therefore I tried looking into other books which may have another ways of going through this. In the book Modern Cosmology by Scott Dodelson, we are told to assume $f=f(E/T)$, where $f$ is the distribution function, and subsitute this into the integral expression of $P$, $$P=g\int \frac{d^3p}{(2\pi)^3}f(p)\frac{p^2}{3E(p)}.$$ Then in order to find $dP/dT$, we rewrite $df/dT$ under the integral sign as $-(E/T)df/dE$ (I don't get this), and by integrating by parts we should get our result. I've been trying to work it out but I don't know how to procced.