# Does the basic postulate about equilibrium violate the minimum energy principle?

For the sake of simplicity, suppose that the volume V of the system is its only relevant external parameter. An isolated system of this kind consists then of a given number N of particles in a specified volume V, the constant energy of the system being known to lie in some range between E and E + delta E. Probability statements are then made with reference to an ensemble which consists of many such systems, all consisting of this number N of particles in this volume V, and all with their energy lying in the range between E and E + delta E. The fundamental statistical postulate asserts that in an equilibrium situation the system is equally likely to be found in any one of its accessible states.

My qustion is whether this fundamental statistical postulate violate the minimum energy principle for it assumes that the higher energy case will be equally likely to occur as the lower energy case.

## 2 Answers

The minimum energy principle is in fact a consequence of the fundamental postulate. The fundamental postulate can be rephrased as "entropy is maximised in equilibrium". For a system in contact with an environment, the total entropy of the system plus the environment is maximised when the internal energy of the system is minimised. See for example here.

The fundamental statistical postulate asserts that in an equilibrium situation the system is equally likely to be found in any one of its accessible states.

The term accessible is important. The states that are accessible by the system are the ones that have the macroscopic variables fixed to the values taken at equilibrium. This includes energy. So what this says is that all the states who have volume $$V$$, number if particles $$N$$ and energy $$E$$ are equally likely.

• But it doen not say with energy E. It say that energy lying in the range between E and E + delta E are equally likely. That's why I am puzzled. – Yuan Fang Mar 1 '20 at 5:32
• Yes. But $\Delta E\to 0$ – Superfast Jellyfish Mar 1 '20 at 5:42
• Then why it does not directly say energy E but say energy in the range E and E+delta E? – Yuan Fang Mar 1 '20 at 5:49
• That’s an artefact of going from discrete to continuum. – Superfast Jellyfish Mar 1 '20 at 5:58
• I don't think you can include temperature here. Temperature is not a property of a single state, it is a parameter of the distribution of states. A fixed temperature implies that energy is not fixed. – Elias Riedel Gårding Mar 2 '20 at 13:11