Stochastic processes, slaves and supersymmetry (Q2879924)

The authors present an extension of works of Tănase-Nicola and Kurchan on the structure of diffusion processes and the associated supersymmetry algebra by the analysis of responses of a simple statistical system to external disturbances of various kinds. The authors consider both the stochastic differential equations (SDEs) for the process and the associated diffusion equation. They introduce slave variables the evolution of which describes the behavior of line elements carried along in the stochastic flow. These elements together with the associated surface and volume elements constructed from them provide the basis of the supersymmetry properties of the theory. NEWLINENEWLINENEWLINENEWLINEThe authors work in three dimensions obtaining results that can be generalized to higher dimensions and can be specialized to one and two dimensions. They also use electromagnetic analogy as a useful starting point for calculating asymptotic results at low temperature that can be compared with direct numerical evaluations. They also examine the problems that arise in a direct numerical simulation of the stochastic equation together with the slave equations. They pay a special attention to the dependence of the slave variable statistics on the temperature. NEWLINENEWLINENEWLINE NEWLINEThis paper is divided into thirteen sections. After the introduction, the authors give (Section 2) a description of the studied model, and then (Section 3) the derivation of the slave equations, both for an infinitesimal change in the initial position of the particle and also for a weak external field. Section 4 contains a brief exposition of the numerical method used for the simulation of the stochastic differential and slave equations. In Section 5, a brief account of the state space on which the Hamiltonian acts is presented, whereas the next section develops an electromagnetic analogy that helps to understand the structure of the eigenstates of the supersymmetry. The analogy is developed in Section 7, where the authors give a presentation of the evolution of the eigenstates with a principle of minimum dissipation. In Sections 8 an 9, the low temperature limit in which it is possible to compute low lying eigenvalues of the supersymmetry is discussed. The application of the supersymmetry theory to the case of an external electric field together with a derivation of the associated Einstein relation comprises Section 10. Then, in the next two sections, they give explicit examples of calculations in one and two dimensions. Section 13 presents the conclusions. There are two appendices.