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Consider the Wikipedia article "https://en.wikipedia.org/wiki/Principle_of_minimum_energy". It clearly says

  1. For an isolated system with fixed energy, the entropy is maximized.
  2. For a closed system with fixed entropy, the energy is minimized.

The problem comes when they provide a mathematical explanation under the header "Mathematical explanation" (in the same article). It first states that $$\bigg(\dfrac{\partial S}{\partial X}\bigg)_U=0,~~\bigg(\dfrac{\partial^2 S}{\partial X^2}\bigg)_U<0,$$ at equilibrium for an isolated system with fixed internal energy. Then the article uses some algebra to connect these terms to $$-\dfrac{1}{T}\bigg(\dfrac{\partial U}{\partial X}\bigg)_S~~\&~~-\dfrac{1}{T}\bigg(\dfrac{\partial^2 U}{\partial X^2}\bigg)_S~~\text{respectively}.$$ This is then used to claim the minimization of energy.

My confusion is that the entropy maximization requires internal energy constant (point 1 above). Thus $$\dfrac{\partial U}{\partial X}=\dfrac{\partial U^2}{\partial X^2}=0.$$

The first equality is okay, but for the minimization of the internal energy, we require that the second one be positive and not $0$.

I was not aware of a connection, mathematical in nature, between the minimization of internal energy and the maximization of entropy, till I came across this Wikipedia document. This then leads to all of this confusion. I don't see where I am going wrong. Also, this is the proof given in the thermodynamics books by R.H. Swendsen. Any help is highly appreciated.

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I will attempt an answer to this, although I'm not 100% sure I understand the question, I see that there is some confusion with respect to the derivation of the minimum energy principle, and I think I know where the confusion comes from.

What the article shows is that if the function $S$ has an extrema at a point $(U_0, X_0)$, for which it takes the value $S_0 = S(U_0, X_0)$ then the function $U$ has an extrema at the point $(S_0, X_0)$, for which it takes the value $U_0$. This extrema is a maximum for the function $S$ but a minimum for the function $U$ it is a minimum.

I think your confusion is related with the fact that writing down $$\frac{\partial S}{\partial X} \bigg|_U$$ seems to imply somehow that now the other function, $U$ is a constant, and when you take its derivative it should vanish, but this is not the case, using this notation for the derivative only means that $S$ is being considered as a function of $X$ and $U$, and not other variables. A different way to write this would be:

$$\frac{\partial S_{U,X}}{\partial X}$$ where the subindices just indicate as a function of which variables you are considering your function to depend on. Using this notation then, the derivation reads something like this, for the first derivative:

$$\frac{\partial S_{U,X}}{\partial X} = -\frac{\partial S_{U,X}}{\partial U} \frac{\partial U_{S, X}}{\partial X} = - T \frac{\partial U_{S, X}}{\partial X} $$ Where I've used the cyclic chain rule. From this equality you can see that the gradients of both functions are related, so that if one has a critical point then the other one does too, and at no point I have considered $U$ to be just a constant.

Edit after comment:

It seems that what is unclear is that the derivative $$\frac{\partial S_{U, X}}{\partial U}$$ is not zero for a fixed value of energy. Maybe this way of thinking about it would help:

If you have a function $F(x,y) = yx + x^2$ for example, and you take the derivative with respect to x then the result is $$\frac{\partial F_{x,y}}{\partial x}(x, y) = y + 2x$$. Lets say now that we want to study this function for a fixed value of x, namely $x_0$, then our function will be $$F(x_0, y) = yx_0 + x^2_0$$ and our derivative will be $$\frac{\partial F_{x,y}}{\partial x}(x_0, y) = y + 2x_0$$ which is not necessarily equal to zero.

A similar thing happens when you want to study the function $S(U_0, X)$ and its derivative with respect to $U$; $$\frac{\partial S_{U,X}}{\partial U}(U_0, X)$$ is not necessarily equal to zero.

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  • $\begingroup$ Well, Entropy maximization is when you are dealing with a system at constant energy. Thus when you write down $\dfrac{\partial S(U, X)}{\partial U}$, this term should be 0, as S is not changing with U.\\ PS- I have modified the question a bit. Please let me know if its still unclear, also what I can do to change it. $\endgroup$
    – Shoham Sen
    Mar 3 '20 at 20:16
  • $\begingroup$ @Shoham Sen I added some explanation to try to clarify why that derivative is not zero $\endgroup$
    – Ignacio
    Mar 3 '20 at 22:10
  • $\begingroup$ appreciate the example, though I don't really think that it's an accurate description of whats going on. I guess my doubt can be boiled down to the following. If I look at the math, it would claim that whenever you have Entropy maximized, you also have Internal energy minimized since they are related as such, isn't that what the math says? However, the entropy Maximization principle claims that Entropy is maximized at fixed internal energy. Thus energy cannot be minimized. PS- thanks for the help, really appreciate it. $\endgroup$
    – Shoham Sen
    Mar 3 '20 at 23:48

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