Minimum Energy Principle

The minimum energy principle, introduced by Eq. (251), is a consequence of the maximum entropy principle. This can be shown by considering the system at thermal equilibrium described by the following diagram:
 

 

Consider a small displacement of heat $\delta E$ from compartment (2) to compartment (1). Since the system was originally at equilibrium, such a contraint in the distribution of thermal energy produces a constrained system whose entropy is smaller than the entropy of the system at equilibrium. Mathematically,

$$S(E^{(1)}+\delta E, {\bf X}) + S(E^{(2)}-\delta E, {\bf X}) < S(E^{(1)}, {\bf X}) + S(E^{(2)}, {\bf X}).$$ (256)

Now consider the system at equilibrium (i.e., without any constraints) with entropy $S(E, {\bf X})$ such that

$$S(E, {\bf X}) = S(E^{(1)}+\delta E, {\bf X}) + S(E^{(2)}-\delta E, {\bf X}).$$ (257)

Since, according to Eqs. (257) and (256),

$$S(E, {\bf X}) < S(E^{(1)}, {\bf X}) + S(E^{(2)}, {\bf X}),$$ (258)

and according to Eq. (39),

$$\frac{\partial S}{\partial E} \Bigg)_{V,N} = \frac{1}{T} >0,$$ (259)

then

$$E < E^{(1)} + E^{(2)}.$$ (260)

Eq. (257) thus establishes that by imposing internal constraints at constant entropy the system that was initially at equilibrium with entropy $S(E, {\bf X})$ moves away from such equilibrium and its internal energy increases from $E$ to $E^{(1)}+E^{(2)}$. Mathematically,

$$dE \Bigg)_{S,V} \geq 0,$$ (261)

which is the minimum energy principle.