Principle of minimum energy Totally Explained

Everything about The Principle Of Minimum Energy totally explained

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, the second law states that for isolated systems, (and fixed external parameters) 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 system, and may exchange energy, but not mass, with the other system. If, rather than an isolated system, we've 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 maximum entropy principle: For a closed system with fixed internal energy (for example an isolated system), the entropy is maximised at equilibrium.

The minimum energy principle: For a closed system with fixed entropy, the total energy is minimized at equilibrium. This shouldn't be confused with the minimum total potential energy principle which states that, at equilibrium, the total potential energy of a system with dissipation will be at a minimum, which is a special case of the maximum entropy principle.
As an example, consider the familiar example of a marble on the edge of a bowl. If we consider the marble and bowl to be an isolated system, then when the marble drops, the potential energy will be converted to the kinetic energy of motion of the marble. Frictional forces will convert this kinetic energy to heat, and at equilbrium, the marble will be at rest at the bottom of the bowl, and the marble and the bowl will be at a slightly higher temperature. The total energy of the marble-bowl system will be unchanged. What was previously the potential energy of the marble, will now reside in the increased heat energy of the marble-bowl system. This will be an application of the maximum entropy principle as set forth in the principle of minimum potential energy, since due to the heating effects, the entropy has increased to the maximum value possible given the fixed energy of the system.
If, on the other hand, the marble is lowered very slowly to the bottom of the bowl, so slowly that no heating effects occur (for example reversibly), then the entropy of the marble and bowl will remain constant, and the potential energy of the marble will be transferred as work energy to the apparatus that's lowering the marble. Since the potential energy is now at a minimum with no increase in the energy due to heat of either the marble or the bowl, the total energy of the system is at a minimum. This is an application of the minimum energy principle.

Mathematical explanation

The total energy of the system is

U(S,X_1,X_2,dots)

where S is entropy, and the

X_i

are the other extensive parameters of the system (for example volume, particle numbers, etc.). The entropy of the system may likewise be written as a function of the other extensive parameters as

S(U,X_1,X_2,...)

. Suppose that X is one of the

X_i

which varies as a system approaches equilibrium, and that it's the only such parameter which is varying. The principle of maximum entropy may then be stated as:
ť

left(frac_x(A_o(T_o,x))

showing that the Helmholtz free energy is minimized at equilibrium.
The Enthalpy and Gibbs free energy, are similarly derived.

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