# Thermodynamics – Why Does the Minimum Energy Principle Work?

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The principle of minimum energy states that in a thermodynamic system the equilibrium state corresponds to the minimum energy state among a set of states of constant entropy. I believe I understand the mathematical derivation of this, however, my immediate intuition is that this should not be the case.

People sometimes handwave something like "Thermodynamics should agree with mechanics when entropy is constant" or similar. Other arguments imply some sort of "interaction" with the environment, which increases entropy when one reaches a minimum value of energy (I'm not sure where I have read this, I wish I had a source) but I would prefer to steer away from those kinds of arguments. It is clear to me from the mathematical derivation that this principle does not rely on dynamics, mechanics, or other auxiliary systems to be true, only on the fact that the entropy is a concave function of its variables and that its hessian is negative definite at equilibrium.

My intuition, however, says that if a system has a bunch of states available to it, and all states have the same entropy, then it should not prefer one state over the other and all of them should be equally good "equilibrium states". This is for sure valid when the energy is constant; I know this has to be wrong when the states have different energies, I just don't see how.

Edit for clarity: As an example of the application of the principle of maximum entropy, consider a system composed of two ideal gasses with fixed numbers of particles in different compartments. The total energy and volume of the system are held constant but the entropies and volumes of both gasses are allowed to change subject to constraints, so that $$U(S_1, S_2, V_1, V_2)$$ has to be a constant, $$V_1 + V_2 = V$$ has to be a constant but $$S_1$$ and $$S_2$$ can freely change. There are many possible states for this system, but the principle of maximum entropy says that the state that corresponds to thermodynamic equilibrium is the state with maximum entropy $$S_1 + S_2$$. The principle of minimum energy is analogous but the roles of $$S$$ and $$U$$ are reversed, and the energy is actually a minimum at thermodynamic equilibrium instead of a maximum.

The principle of minimum energy states that in a thermodynamic system the equilibrium state corresponds to the minimum energy state among a set of states of constant entropy.

which is very close to the statement in the introductory part of wikipedia page you cited. However, this is not a consistent way to express the minimum energy principle in thermodynamics. The reason for inconsistency should become clear by looking at formulas. In the case a thermodynamic state is fixed by the value of entropy, volume, and number of particles, the fundamental function from which the whole thermodynamic behavior can be obtained is the internal energy $$U(S,V,N)$$. Now, it is clear that once the independent variables are fixed, a unique value for $$U$$ is possible. There is one thermodynamic state and it is not clear which should be the states "among which energy should be minimum".

Actually, the correct statement of the minimum principle for energy is the following: in an equilibrium system at fixed entropy, volume and number of particles, and subject to internal constraints controlled by a set of parameters $$X_{\alpha}$$, the internal energy is a function $$U(S,V,N;\{X_{\alpha}\})$$ and the final equilibrium state, obtained after removal of the constraints, corresponds to the minimum of the energy among the all the possible values of the constraint variables $$X_{\alpha}$$ (see Callen's textbook on Thermodynamics for a reference).

Starting from the correct statement of the minimum principle, a first observation is that it is more general than just the convexity property of the function $$U(S,V,N)$$. Indeed, from the minimum principle, one can derive convexity of $$U(S,V,N)$$. But there are cases where the minimum principle provides results which are not derivable from convexity. For example, if one can determine different functions of energy at fixed $$S,N$$, as a function of $$V$$, minimum energy allows to determine for each $$V$$ the equilibrium state.

What about intuition? Frankly, I think that in the case of the minimum energy principle, is far from being intuitive. The main reason is that the underlying condition of constant entropy is difficult to manage both from the experimental and from the conceptual point of view. However, since from the minimum of energy $$U(S,V,N;\{X_{\alpha}\})$$ one can easily obtain similar minimum principles for the Legendre transforms of energy (Helmholtz free energy, Gibbs free energy), the difficult condition of fixed volume and entropy can be transformed into the conceptually and experimentally easier conditions of minimum at fixed temperature and volume or temperature and pressure.

Edit after a few comments and the editing of the question.

Notwithstanding the previous words of caution about the non-intuitive condition of constant entropy, an example with a fluid system could help to get a better understanding. Let me start recasting in a correct way the situation, if it should be analyzed in term of minimum energy principle.

There is a composite system made by two compartments such that initially the first compartment contains a fluid (the same in both compartments for simplicity) described by the thermodynamic variables $$S_1,V_1,N_1$$, and the second by $$S_2,V_2,N_2$$. $$V_1,N_1$$ and $$V_2,N_2$$ remain always fixed.

The energy of this composite system is the sum of the energies of the two subsystems and, being filled with the same fluid (for example both Neon gas), the same function $$U$$ of entropy, volume and number of particles describes both. By introducing the subscript $$tot$$ for the extensive quantities describing the composite system we have $$S_{tot}=S_1+S_2$$, $$V_{tot}=V_1+V_2$$ and $$N_{tot}=N_1+N_2$$. For a given partition of the total entropy into a value $$S_1$$ and $$S_2=S_{tot}-S_1$$ (this is the constraint on our composite system) we have $$U_{tot}(S_{tot},V_{tot},N_{tot};S_1)=U(S_1,V_1,N_1)+U(S_{tot}-S_1,V_2,N_2).$$ The minimum energy principle applied to the present case says that if we eliminate the constraint that system $$1$$ should have entropy $$S_1$$, but always keeping fixed $$S_{tot}$$, the final equilibrium state of the composite system will correspond to the value of $$S_1$$ which minimize $$U_{tot}$$.

That there should be a minimum can be seen by noting that $$U(S,V,N)$$, at fixed $$V$$ and $$N$$ must be an increasing function of $$S$$ (let's recall that $$\left.\frac{\partial{U}}{\partial{S}}\right|_{V,N}=T\gt 0$$). So, $$U_{tot}$$ is the sum of an increasing and a decreasing (convex) function in the interval $$0 and therefore there should have a minimum.

It is possible to check everything explicitly in the case of a perfect gas in two equal volume containers with the same density. The total energy is $$U_{tot} \propto \left( e^{\frac{2S_1}{3N_1k_B}} + e^{\frac{2(S_{tot}-S_1)}{3N_1k_B}} \right),$$ which has a minimum at $$S_1=S_{tot}/2$$.

In a less formal way, one could say that the reason for the minimum is directly connected to the constraint of keeping fixed the total entropy. Since entropy is proportional to the logarithm of the number of states, a fixed total entropy in our composite system is equivalent to keep fixed the product of the number of states of system $$1$$ and system $$2$$. The way the number of states varies with energy provides the mechanism on which the minimum principle is based.