Principle of minimum energy

The principle of minimum energy is essentially a restatement of the second law of thermodynamics. It states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. External parameters generally means the volume, but may include other parameters which are specified externally, such as a constant magnetic field.

In contrast, for isolated systems (and fixed external parameters), the second law states that the entropy will increase to a maximum value at equilibrium. An isolated system has a fixed total energy and mass. A closed system, on the other hand, is a system which is connected to another, and cannot exchange matter (i.e. particles), but other forms of energy (e.g. heat), with the other system. If, rather than an isolated system, we have a closed system, in which the entropy rather than the energy remains constant, then it follows from the first and second laws of thermodynamics that the energy of that system will drop to a minimum value at equilibrium, transferring its energy to the other system. To restate:

The total energy of the system is ${\displaystyle U(S,X_{1},X_{2},\dots )}$ where S is entropy, and the ${\displaystyle X_{i}}$ 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 ${\displaystyle S(U,X_{1},X_{2},...)}$. Suppose that X is one of the ${\displaystyle X_{i}}$ 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:

...
Wikipedia