Relaxation and Correlations in Time in a Finite Volume

Abstract

To describe relaxation towards equilibrium and correlating in time in stochastic numerical simulations, it is necessary to examine dynamical stochastic evolution equations from the renormalization group (R. G.) point of view, and then finite volume effects since all numerical simulations take place of course in a finite volume. The stochastic evolution equation which describes in the continuum space and time limit the time behavior of numerical simulations is the Langevin equation and in a first part we shall discuss briefly its algebraic properties. We shall show how it is possible to associate with the Langevin equation a conventional effective action in such a way that its renormalization and R.G. properties can be discussed in the language of standard field theory. However in this discussion it is essential. to recognize that this effective action has a B. R. S. ^[1] type symmetry of the form encountered in the quantization of gauge theories.