Why is Entropy a concave function of internal energy?

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.
  • #1
Wentu
14
2
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
 
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  • #2
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?
 
  • #3
The first N is for a single particle having 2*dE energy and all other particles ground energy
 
  • #4
Ah, ok. Shouldn't the second term should rather read N(N-1)/2?
 
  • #5
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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