# Thermodynamics, a model of the symmetry restoration in the universe

• molkee
In summary, thermodynamics is a branch of physics that deals with the relationship between heat, energy, and work. It plays a crucial role in the symmetry restoration model of the universe, explaining how the laws of physics and fundamental forces were unified in the early universe. Entropy, which is a measure of disorder in a system, is closely related to thermodynamics and helps explain the evolution of the universe from a state of low entropy to high entropy. Thermodynamics can be applied to the entire universe, but its properties may be difficult to measure and understand due to its immense size and complexity. The symmetry restoration model also accounts for the second law of thermodynamics, which states that the total entropy of a closed system will always increase over time.
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?

Last edited:

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.

## 1. What is thermodynamics?

Thermodynamics is a branch of physics that deals with the relationship between heat, energy, and work. It is based on the laws of thermodynamics, which describe how energy can be transferred and transformed in a system.

## 2. What is the role of thermodynamics in the symmetry restoration model of the universe?

In the symmetry restoration model, thermodynamics plays a crucial role in understanding how the universe evolved from a state of high symmetry to its current state of lower symmetry. It helps explain how the laws of physics and the fundamental forces of nature were unified in the early universe.

## 3. How does thermodynamics relate to the concept of entropy?

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it is closely related to the concept of heat, as heat transfer increases the disorder or entropy of a system. This relationship is important in understanding how the universe evolved from a state of low entropy to high entropy.

## 4. Can thermodynamics be applied to the entire universe?

Yes, thermodynamics can be applied to the entire universe. The laws of thermodynamics are universal and apply to all systems, including the universe as a whole. However, due to the immense size and complexity of the universe, some of its properties may be difficult to measure and understand.

## 5. How does the symmetry restoration model of the universe account for the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system will always increase over time. In the symmetry restoration model of the universe, this law is accounted for by the continuous expansion of the universe, which leads to an increase in entropy. This expansion also allows for the eventual breaking of symmetries and the formation of distinct structures in the universe.

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