Expansion Exponents for Nonequilibrium Systems

TL;DR

This paper introduces local expansion exponents for nonequilibrium systems described by PDEs, discusses their calculation, and explores principles related to phase volume expansion, pattern selection, and time irreversibility.

Contribution

It defines expansion exponents for both deterministic and stochastic systems and proposes principles for understanding phase volume behavior and time irreversibility in nonequilibrium dynamics.

Findings

Exponents indicate whether phase volume expands, contracts, or is conserved.

Principle of minimal expansion aids pattern selection.

Asymptotic expansion principle shows phase volumes expand at large times.

Abstract

Local expansion exponents for nonequilibrium dynamical systems, described by partial differential equations, are introduced. These exponents show whether the system phase volume expands, contracts, or is conserved in time. The ways of calculating the exponents are discussed. The {\it principle of minimal expansion} provides the basis for treating the problem of pattern selection. The exponents are also defined for stochastic dynamical systems. The analysis of the expansion-exponent behaviour for quasi-isolated systems results in the formulation of two other principles: The {\it principle of asymptotic expansion} tells that the phase volumes of quasi-isolated systems expand at asymptotically large times. The {\it principle of time irreversibility} follows from the asymptotic phase expansion, since the direction of time arrow can be defined by the asymptotic expansion of phase volume.