Information symmetries in irreversible processes

Abstract

We study dynamical reversibility in stationary stochastic processes from an information theoretic perspective. Extending earlier work on the reversibility of Markov chains, we focus on finitary processes with arbitrarily long conditional correlations. In particular, we examine stationary processes represented or generated by edge-emitting, finite-state hidden Markov models. Surprisingly, we find pervasive temporal asymmetries in the statistics of such stationary processes with the consequence that the computational resources necessary to generate a process in the forward and reverse temporal directions are generally not the same. In fact, an exhaustive survey indicates that most stationary processes are irreversible. We study the ensuing relations between model topology in different representations, the process's statistical properties, and its reversibility in detail. A process's temporal asymmetry is efficiently captured using two canonical unifilar representations of the generating model, the forward-time and reverse-time -machines. We analyze example irreversible processes whose -machine presentations change size under time reversal, including one which has a finite number of recurrent causal states in one direction, but an infinite number in the opposite. From the forward-time and reverse-time -machines, we are able to construct a symmetrized, but nonunifilar, generator of a process-the bidirectional machine. Using the bidirectional machine, we show how to directly calculate a process's fundamental information properties, many of which are otherwise only poorly approximated via process samples. The tools we introduce and the insights we offer provide a better understanding of the many facets of reversibility and irreversibility in stochastic processes. One of the principal early mysteries of thermodynamics was the origin of irreversibility: While microscopic equations of motion describe behaviors that are the same in both time directions, why do large-scale systems exhibit temporal asymmetries? Many thermodynamic processes go in one direction: Closed systems devolve from order to disorder, heat flows from high temperature to low temperature, and shattered glass does not reassemble itself spontaneously. These are described as transient relaxation processes in which a system moves from one macroscopic state to another with high probability since there is an * cellison@cse.ucdavis. Here, we analyze a generalized notion of irreversibility: Behavior in reverse time gives rise to a different stochastic process than that in forward time. This dynamical irreversibility subsumes relaxation, but is not so constrained, since it can occur in a nonequilibrium steady state. A dynamical parallel to the shattered glass example of transient relaxation is found in a "continuousflow" glass grinder: Continuously fed whole glass, the grinder eventually produces glass pieces that are sufficiently small to pass out via a sieve. After a transient start-up time, the distribution of glass sizes settles down to a steady state. The gl...

Informational and Causal Architecture of Continuous-time Renewal Processes

“…Pr(σ + = (τ, g, x)|σ − = (g , x , τ )) = Pr(τ |g, x, τ )δ x,x Pr(g|g , x , x, τ ) = φ g,x (τ + τ ) Pr(g|g , x)δ x,x , where we obtain p(g|g , x) from standard methods [12,26] applied to (only) the dynamic on G. Note that p(g|g , x , x, τ ) reduces to p(g|g , x ) as g and x uniquely specify the distribution from which τ is drawn and since x = x . We leave the the final steps to E as an exercise.…”

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“…A corollary is that the predictive information bottleneck-compression of semi-infinite pasts to retain information about semi-infinite futures-can be recast as compression of forward-time causal states to retain information about reverse-time causal states. The joint probability distribution of forward-and reverse-time causal states may seem somewhat elusive, but previous work has shown that this joint probability distribution can be obtained given the process' model [23].…”

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