Principle of the Shortest Way to the Equilibrium State

Abstract

Abstract We investigate the temporal evolution of the entropy S(t) of an isolated system from an initial non-equilibrium state i of entropy Si to the final equilibrium state of maximum entropy Smax > Si . The equations of classical thermodynamics and of phenomenological irreversible thermodynamics are unsuited for handling this problem. Hence, in order to determine S(t), we make use of Hamilton´s principle of mechanics and of the Euler-Lagrange equation of motion, which were adapted to thermodynamics. Now the Lagrangian F(S, P, t) depends on the entropy S, the entropy production P = dS/dt, and the time t. Application of variational calculus leads us straightforward to the time-dependent equations S(t) and P(t), which represent the geodesic line and the geodesic slope on a S,t diagram. Both equations agree with results of former calculations on the basis of quantum thermodynamics. The geodesic line S(t) describes the shortest way of the system from its initial state i to the final equilibrium state. This way is irreversible.