On the Construction of a Time-Reversed Markoff Process

Abstract

For a given Markoff process characterized by a set of transition probability densities there exists another process with time reversed (the retrodictive vs predictive process in the theory of measurements) such that any one of them multiplied by a single-event density may be symmetric with respect to an interchange of the events expressed as space-time variables, yielding a joint probability density. It is shown how this time-reversed process can be constructed by means of the generating operator of the associated evolution equation, and the basic properties with explicit applications to master equations and Fokker-Planck equations. Onsager's microscopic reversibility is reformulated on this basis. Possible sym. metries concerning time-correlation functions under the Markoffian law is summarized in comparion with the Kubo formula. In the application to Fokker-Planck equations, the Onsager-Machlup most probable paths are extended to general type of diffusion processes, and it is shown that the present method corresponds to a gauge transformation in dynamics of a charged particle which leaves its paths invariant. Reciprocity has been a fundamental subject in the theory of irreversible proces­ ses, since Onsager initiated the approach to the problem based on the consideration of microscopic reversibility.!) The Onsager relations for a linear dissipative sys­ tem in an external magnetic field @, given by L~,( -8) =L,~(@), have since been discussed in a number of papers. Kubo's general linear response theory•> among them provided an accurate statistical-mechanical foundation of these relations, ex­ pressing the microscopic reversibility in the time correlation functions between Hamiltonian-driven dynamical variables. Recently, the interests have been revised in connection with the statistical mechanics for non-equilibrium or open systems far from equilibrium conditions. Van Kampen discussed a possibility of extending the relations straightforwardly to the nonlinear regime of flux-force equations. 3> Another systematic approach has been developed by using equations of evolutions for probability densities based on the theory of Markoff processes. 4> An interesting finding in the latter approach has been that the microscopic reversibility as represented by a form of detailed balance condition (or its equivalent) is a situation rather restrictive for such non­ equilibrium states: There exist a number of important examples of violation such as complex optical systems and chemical reactions. A typical non-trivial system, a single-mode laser, is a special example for which the potential condition equiva­ lent to the reversibility is well satisfied, as discussed by Graham and Haken.