# Why is Entropy a concave function of internal energy?

• Wentu
In summary, according to the PDF, entropy is a concave function of internal energy, meaning that as the temperature rises, the entropy of the system increases until it reaches a maximum and then decreases. The behaviour of entropy is not linear when the amount of energy available to the system increases, but instead becomes less-than-linear.
Wentu
Hello

I may well be all wrong about this but I am trying to understand from a microscopic point of view why Entropy is a concave function of internal energy. I found this in the following .pdf:

http://physics.technion.ac.il/ckfinder/userfiles/files/avron/thermodynamics_potentials.pdf

I started from this wikipedia article and i understand why, if the particles composing the system have a limited number of available energy levels, then S(E) first increases and then decreases.

But saying that S(E) is concave should mean:
- when the temperature is T1, if i give a dE to the system its entropy increases of dS1
- when the tempereture is T2>T1, if I give the same dE to the system, its Entropy increases only of dS2 < dS1

I cannot see this with single particles.
If I have N particles in their lowest energy state there is only one microstate: all the particles are still.
If I give to this system the tiniest possible amount of energy, it will be taken by just one of the particle, so the possible microstates are N.
If I add another dE, the possible microstates should be N + N(N-1) = N^2 ... that is or one particle gets both dE or two different particles get it. Every time I add a dE I should increase the power of N.
Now, if the entropy is somehow proportional to the logarithm of the number of microstates, I should get S proportional to K ln(N^E), that is, something that is proportianl to E... taht is, no concavity

I am sure I am getting all this wrong... could you please help me understand this?

Thank You

Wentu

Wentu said:
If I add another dE, the possible microstates should be N + N(N-1) = N^2 .

Where do you get the first N on the left hand side from?

The first N is for a single particle having 2*dE energy and all other particles ground energy

Ah, ok. Shouldn't the second term should rather read N(N-1)/2?

You are right... I was considering distinguishble particles but this isn't enough, so yes, the term should be divided by 2. I wonder if this is enough to change the behaviour from linear to less-than-linear... I think the number of microstates still increases as a power with the increasing of E... but again, i could be all wrong

W.

## 1. What is Entropy and internal energy?

Entropy is a measure of the disorder or randomness in a system, while internal energy is the total energy contained within a system.

## 2. Why is Entropy a concave function of internal energy?

This is because as the internal energy of a system increases, the number of microstates or possible arrangements of the particles also increases. This means that the system becomes more disordered and the entropy increases at a decreasing rate, leading to a concave function.

## 3. How does the concavity of the entropy function relate to the Second Law of Thermodynamics?

The Second Law of Thermodynamics states that the total entropy of a closed system will always increase over time. The concavity of the entropy function reflects this by showing that the rate of entropy increase decreases as the internal energy increases.

## 4. Can the entropy function of a system ever be convex?

No, the entropy function will always be concave. This is because a convex function would imply that the rate of entropy increase increases as the internal energy increases, which goes against the Second Law of Thermodynamics.

## 5. How is the concavity of the entropy function used in thermodynamic calculations?

The concavity of the entropy function is used to determine the stability of a system. A concave function indicates a stable system, while a convex function would indicate an unstable system. This information is useful in analyzing and predicting the behavior of thermodynamic systems.

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