On Irreversibility and Stochastic Systems; Part Two

TL;DR

This paper explores how irreversibility in dynamical systems can be characterized through different forward and backward stochastic representations, linking deterministic Hamiltonian systems with stochastic diffusions via heat baths.

Contribution

It demonstrates that heat baths induce stochastic diffusions in deterministic systems and connects these stochastic models to Hamiltonian frameworks, extending understanding of irreversibility.

Findings

Heat baths induce dissipative stochastic diffusions.

Forward-backward representations can describe deterministic systems with heat baths.

Stationary processes with rational spectra relate to lossless deterministic systems.

Abstract

We attempt to characterize irreversibility of a dynamical system from the existence of different forward and backward mathematical representations depending on the direction of the time arrow. Such different representations have been studied intensively and are shown to exist for stochastic diffusion models. In this setting one has however to face the preliminary justification of the existence of a stochastic description for physical systems which are traditionally described by classical mechanics as inherently deterministic and conservative. In part one of this paper we have addressed this modeling problem from a deterministic viewpoint for linear systems. We have shown that there are forward-backward representations which can describe conservative finite dimensional deterministic systems when they are coupled to an infinite-dimensional conservative heat bath. A key observation is that the heat bath acts on the finite-dimensional system by {\em state-feedback} which can shift its eigenvalues to make the system dissipative, but also may generate a totally unstable system which naturally evolves backward in time. In this second part, we address the stochastic description of these two representations. Under a natural family of invariant measures it is shown that the heat bath induces a white noise input acting on both the forward-backward representations making them true dissipative diffusions. We also consider how to relate the Stochastics to the Hamiltonian deterministic picture discussed in Part one. In our current context, we show that a continuous stationary process with a rational spectrum can always be thought of, or represented as, the output of a lossless deterministic system coupled to an infinite dimensional heat bath.