Network representations of nonequilibrium steady states: Cycle decompositions, symmetries, and dominant paths

Nonequilibrium steady states of Markov processes give rise to nontrivial cyclic probability fluxes. Cycle decompositions of the steady state offer an effective description of such fluxes. Here we present an iterative cycle decomposition exhibiting a natural dynamics on the space of cycles that satisfies detailed balance. Expectation values of observables can be expressed as cycle “averages,” resembling the cycle representation of expectation values in dynamical systems. We illustrate our approach in terms of an analogy to a simple model of mass transit dynamics. Symmetries are reflected in our approach by a reduction of the minimal number of cycles needed in the decomposition. These features are demonstrated by discussing a variant of an asymmetric exclusion process. Intriguingly, a continuous change of dominant flow paths in the network results in a change of the structure of cycles as well as in discontinuous jumps in cycle weights.

Figure 1

Representation of a NESS in terms of linear superpositions of cycle fluxes. The numbers on the arrows (representing the directed transitions) are the values of the fluxes. The steady-state fluxes between states 1 and 6 can be decomposed into cycle fluxes (labeled by Greek letters) with positive weights. Two different decompositions are possible.

Figure 2

Transformed graph $H$ obtained for the original graph $G$ for the two decomposition of the flux field introduced in Fig. 1.

Figure 3

The network of states for the TASEP example. The rates shown in (a) lead to the steady state shown in (b). For $x=2$ the fluxes are proportional to the ones shown in the original example (Fig. 1). (a) Transition rates of the TASEP example with a variable jump rate $x$ for transitions involving a particle jump over the boundary. (b) Dependence of the steady-state fluxes on $x$. For clarity, the edge fluxes shown are divided by the common factor $x/C\left(x\right)$; cf. Eq. (30).