Reversal symmetries for cyclic paths away from thermodynamic equilibrium

TL;DR

This paper demonstrates that even in far-from-equilibrium systems, cyclic paths exhibit local reversibility properties, with traversal frequency ratios determined solely by the thermodynamic force on the cycle, independent of the broader system complexity.

Contribution

It reveals that cyclic paths maintain local symmetry in non-equilibrium steady states, with ratio of traversal frequencies governed only by the cycle's thermodynamic force.

Findings

Cycle traversal ratios depend only on the thermodynamic force.

Reversibility of cycles persists regardless of system complexity.

No net energy expenditure implies equal traversal frequencies.

Abstract

If a system is at thermodynamic equilibrium, an observer cannot tell whether a film of it is being played forward or in reverse: any transition will occur with the same frequency in the forward as in the reverse direction. However, if expenditure of energy changes the rate of even a single transition to yield a non-equilibrium steady state, such time-reversal symmetry undergoes a widespread breakdown, far beyond the point at which the energy is expended. An explosion of interdependency also arises, with steady-state probabilities of system states depending in a complicated manner on the rate of every transition in the system. Nevertheless, in the midst of this global non-equilibrium complexity, we find that cyclic paths have reversibility properties that remain local, and which can exhibit symmetry, no matter how far the system is from thermodynamic equilibrium. Specifically, given any cycle of reversible transitions, the ratio of the frequencies with which the cycle is traversed in one direction versus the other is determined, in the long-time limit, only by the thermodynamic force on the cycle itself, without requiring knowledge of transition rates elsewhere in the system. In particular, if there is no net energy expenditure on the cycle, then, over long times, the cycle traversal frequencies are the same in either direction.