Variational Functions in Open Systems

This paper looks for an entropy-like quantity having a monotonic time development. In the case of spontaneous emission, the final state usually consists of a single ground state assigning zero to the ordinary expressions for entropy. Thus entropy ceases to be a monotonic measure of the direction of time. The point is illustrated by a simple test case consisting of three levels coupled by spontaneous emission. It is shown how this case allows the definition of a monotonic function. Using the theory of non-Hermitian operators, the paper shows how such a function may be constructed in the general case, and it explores the main consequences of the expressions suggested. The generalization of the entropy concept is found to relate to time-reversal properties of the dynamics. The paper concludes by discussing open questions and possible further explorations.