Principle of Minimum Energy - Mathematical Explanation

Mathematical Explanation

The total energy of the system is where S is entropy, and the are the other extensive parameters of the system (e.g. volume, particle number, etc.). The entropy of the system may likewise be written as a function of the other extensive parameters as . Suppose that X is one of the which varies as a system approaches equilibrium, and that it is the only such parameter which is varying. The principle of maximum entropy may then be stated as:

and at equilibrium.

The first condition states that entropy is at an extremum, and the second condition states that entropy is at a maximum. Note that for the partial derivatives, all extensive parameters are assumed constant except for the variables contained in the partial derivative, but only U, S, or X are shown. It follows from the properties of an exact differential (see equation 7 in the exact differential article) and from the energy/entropy equation of state that, for a closed system:

It is seen that the energy is at an extremum at equilibrium. By similar but somewhat more lengthy argument it can be shown that

which is greater than zero, showing that the energy is, in fact, at a minimum. (See Callen (1985) chapter 5).