A circulation bound on the frenetic component of branch selection

Abstract

Plain-language summary Driven systems, from reacting chemical mixtures and active materials to living systems maintained away from equilibrium, can settle into more than one stable state. The maximum entropy production principle is often invoked to predict which state will be selected, but it is not a universal rule: a system may instead favor a less dissipative state. This study asks what controls such departures from a dissipation-based ordering. In the minimum-action description of rare transitions, the cost of switching between stable states has two components. One changes sign under time reversal and is associated with entropy production. The other is unchanged by time reversal and represents dynamical activity, or frenesy. If the dominant forward and backward transitions follow different routes, their frenetic contributions may differ. That difference can either reinforce the dissipation ordering or overturn it. The central result is the structural inequality |ΔC| ≤ Σcyc / 2. Here, ΔC is the difference in frenetic cost between the dominant forward and backward transition tubes, while Σcyc is the entropy production around the closed cycle formed by those tubes. In simple terms, the time-symmetric route asymmetry cannot exceed half of the dissipation circulating around the transition cycle. The proof requires only that each tube minimize the action in its own direction and that the time reversal of the competing tube be an admissible comparison. The associated frenetic polarization index, η = 2ΔC / Σcyc, lies between -1 and +1 whenever Σcyc is positive. Its sign shows whether frenesy reinforces or opposes the dissipation ordering. As the system approaches a route-handover point, where one route becomes as favorable as the time reversal of its competitor, |η| approaches 1. At the handover itself, several routes may be equally optimal, so η need not be uniquely defined. The result also gives a detailed-balance no-go: at equilibrium, ΔC = 0. Circulation is therefore necessary, although not sufficient, for frenetic branch selection. The bound is not proposed as a new universal thermodynamic law. It is a consistency inequality within the minimum-action framework and applies to a specified pair of forward and backward transition tubes, including pairs embedded in larger networks. It does not extend directly to arbitrary spanning-tree comparisons that determine occupations in systems with many attractors. Boundary or blowtorch effects can also alter selection through mechanisms not constrained by this bound. The result is tested in finite-state networks, chemical reaction networks, irreversible diffusions, a rotational Maier-Stein model, a two-channel continuous benchmark, and spatial field models. The two-channel benchmark reaches η ≈ 0.97 near a genuine route handover, showing that the bound is sharp. By contrast, the single-channel examples studied remain below about |η| = 0.37, indicating that near-saturation is possible but not typical. The paper also develops a trajectory-based diagnostic for distinguishing frenetic, dissipation-dominated, and boundary-conditioned selection, provided that the relevant routes, contacts, and action calibration are known. In a blind numerical test, the trajectory-based estimate gives η = 0.72, compared with an independent minimum-action value of 0.725. The diagnostic works only within a finite operating window: weak driving is dominated by sampling noise, large noise weakens the rare-transition approximation, near saturation the competing routes cease to be cleanly separated, and coarse graining can erase the transition cycle altogether. Why it matters Predicting which state a driven system will select remains an open problem in nonequilibrium physics, chemistry, active matter, biology, and related fields. Dissipation-based principles account only for the time-antisymmetric part of the transition cost. This work shows that a time-symmetric route contribution can also influence or reverse the outcome and places a quantitative upper bound on that contribution. Rather than proving or disproving maximum entropy production as a universal principle, the framework identifies when frenesy reinforces or opposes a dissipation ordering and separates that effect from boundary-conditioned selection. Reproduction package The numerical code and data are deposited separately at Mendeley Data: doi: 10.17632/3dy4nv92r8.5. The package regenerates all fourteen figures and reproduces the numerical results underlying Tables I-IV. Preprint; not peer reviewed. This upload contains the extended manuscript (38 pages) and its LaTeX source archive.