Thermodynamics, a model of the symmetry restoration in the universe

[molkee]

Homework Statement

The energy density (u=E/V) of a thermodynamic system (used as a model of symmetry restoration in the early universe) is given by:

$$u(T)=aT^4 + \Lambda(T)$$ where $$\Lambda =0, T \leq T_0$$ or $$\Lambda =\Lambda_0, T>T_0$$

$$k_B T_0 = 10^{14} GeV$$

a) Calculate the Helmholtz free energy for the system

b) Calculate the pressure and entropy from a). To fix any constants of integration, use the condition $$p=aT^4/3$$ at *very low *temperatures

c) find the factor by which the volume changes if a container of this stiff is adiabatically and reversibly cooled from T just above T_0 to T just below T_0

d) suppose that the system cools reversible to zero in a metastable phase in which $$\Lambda$$ remains stuck at $$\Lambda_0$$ (instead of going to zero below T_0). What are the values of energy and entropy in this limit?

e) the system then spontaneously and rapidly decays to the stable state in which $$\Lambda=0$$. Find the final temperature of the system and the entropy change of the transition.

Homework Equations

will be used in the next section

The Attempt at a Solution

a) The equation $$\left (\frac{\partial (F/T)}{\partial T}\right )_V=-\frac{E}{T^2}$$ should be used. After the integration, F/T is defined up to a constant F_0.

b) F(T,V) from a) should be differentiated with respect to T and V.

c) The equation $$dE(T,V)+p(T)dV=0$$ for adiabatic process should be used. It will give us the connection between T and V in this process.

We know p(T) (from b)) and E(T,V).

d)no idea

e)no idea

Am I doing something wrong or not?

 

[mark726]

Your approach seems correct so far. Here are some additional steps for parts d) and e):

d) For the metastable phase where \Lambda remains at \Lambda_0, the Helmholtz free energy is still given by F(T,V) from part a). To find the energy and entropy in this limit, you can use the equations E(T,V) = -T^2 \frac{\partial (F/T)}{\partial T} and S(T,V) = -\frac{\partial F}{\partial T}.

e) To find the final temperature and entropy change of the transition, you can use the condition that the system is at equilibrium at both \Lambda = \Lambda_0 and \Lambda = 0. This means that the Helmholtz free energy is the same at both points, so F(T_f,V_f) = F(T_i,V_i), where T_f is the final temperature and T_i is the initial temperature. You can also use the fact that the change in entropy is given by \Delta S = \int_{T_i}^{T_f} \frac{C_V}{T} dT, where C_V is the specific heat at constant volume. You can use the expression for C_V given in the problem statement to evaluate this integral and find the final temperature.