Maximum entropy principle from minimum energy principle

[Tomas Molina]

Homework Statement

Formulate a proof that the energy minimum principle implies the entropy maximum principle. That is, show that if the entropy were not maximum at constant energy then the enrgy could not be minimum at constant entropy. HINT: First show that the permissible increase in entropy in the system can be exploited to extract heat from a reversible heat source (initially at the same temperature as the system) and to deposit it in a reversible work source. The reversible heat source is thereby cooled. Continue the argument.

Reference https://www.physicsforums.com/threa...he-energy-minimum-and-entropy-maximum.915479/

Homework Equations

we want to prove ds/dx = 0 and d/dx (ds/dx) < 0

The Attempt at a Solution

has something to do with cycles the argument? :c

 

[mark726]

Hello,

Thank you for your forum post. I am a scientist and I would be happy to help you formulate a proof for the energy minimum principle implying the entropy maximum principle.

First, let's define the terms used in the problem. The energy minimum principle states that at constant entropy, a system will tend towards a state of minimum energy. On the other hand, the entropy maximum principle states that at constant energy, a system will tend towards a state of maximum entropy. In other words, a system will naturally move towards a state of minimum energy and maximum entropy.

To prove that the energy minimum principle implies the entropy maximum principle, we can use the following steps:

1. Assume that the entropy is not maximum at constant energy, meaning that the system is not in a state of maximum entropy.

2. This means that there is still room for the system to increase its entropy without changing its energy.

3. Now, let's consider a reversible process in which the system increases its entropy by a small amount, while keeping its energy constant.

4. According to the energy minimum principle, the system will tend towards a state of minimum energy. Therefore, the system will try to decrease its energy in order to reach a state of minimum energy.

5. However, since the energy is constant, the system cannot decrease its energy without decreasing its entropy.

6. This creates a contradiction, as we assumed that the system is not in a state of maximum entropy, but decreasing its energy would decrease its entropy.

7. Therefore, our initial assumption that the entropy is not maximum at constant energy must be false. This means that the entropy must be maximum at constant energy.

8. This proves that the energy minimum principle implies the entropy maximum principle.

Now, let's look at the hint provided in the forum post. The hint suggests that we can exploit the permissible increase in entropy in the system to extract heat from a reversible heat source and deposit it in a reversible work source. This process would cool the reversible heat source.

Using this hint, we can continue the argument by considering a cycle in which the system increases its entropy by extracting heat from the reversible heat source and depositing it in the reversible work source, thus cooling the reversible heat source.

This cycle can be repeated indefinitely, allowing the system to continuously increase its entropy while the energy remains constant. This process is in line with the entropy maximum principle, as the system is continuously moving towards a state of maximum entropy at constant energy.

In conclusion, the