A circulation bound on the frenetic component of branch selection

Abstract

Plain-language summary Driven systems — a chemical mixture kept reacting, a fluid continuously stirred, a living cell burning fuel — typically settle into one of several possible stable states or repeating patterns. A long-standing rule of thumb, the *maximum entropy production* (MEP) principle, guesses that such a system will choose whichever option dissipates energy fastest. The guess often works, but not always: sometimes the system settles instead on a lower-dissipation option. This paper asks what governs those failures. It splits the "cost" of a rare switch between states into two distinct parts: one tied to how much energy is dissipated (the quantity MEP cares about), and a separate, time-symmetric part that measures how much restless back-and-forth activity — called *frenesy* — the switch involves. When this second, activity-based part is what tips the balance, the system selects against the MEP guess. The central result is a clean inequality: the activity imbalance between the forward and backward switching routes can never exceed half of the dissipation circulating around the loop those two routes form. Equivalently, a single number η between −1 and +1 measures how strongly activity, rather than dissipation, is steering the choice; it reaches its extreme values exactly where the system hands off from one preferred route to another. The result also implies a strict no-go: at equilibrium, where nothing circulates, this activity imbalance is exactly zero. Sustained circulation — a genuinely non-equilibrium condition — is therefore required for activity-driven, anti-MEP selection to occur at all. The bound is not a new physical law but an exact identity of the standard least-cost-path (large-deviation) description of rare events. What makes the proof work is a single structural condition: the two competing routes must be comparable under time reversal. Where that condition fails — for instance when three or more states compete and the comparison is no longer between a route and the reversal of its rival — the inequality genuinely breaks, and the accompanying code exhibits the breakdown explicitly. That is not a caveat but the sharpest evidence for what the theorem actually rests on. The reproduction package (Mendeley Data, doi:10.17632/3dy4nv92r8) The code is not part of this upload: it is deposited at Mendeley Data and linked from this record under "Is supplemented by". The package confirms the bound across random networks, chemical reaction networks, a rotating model system, and a spatial field model, and turns it into a practical diagnostic: from a single recorded trajectory — once the competing switching routes are identified — one can tell whether an observed choice was driven by activity, by dissipation, or by boundary effects. The diagnostic is not merely proposed but demonstrated. It is run *blind* on simulated trajectories: the routes are resolved from the data alone, the circulating dissipation is estimated without any knowledge of the escape prefactor, and only afterwards is the answer compared with the exact computation. At a fresh operating point the blind prediction recovers η = 0.72 against a true value of 0.725. The diagnostic also has honest limits, and the code maps them: it works inside a window of intermediate driving, and fails outside it — at weak driving the estimate is swamped by noise, and near the extreme value of η the two competing routes become indistinguishable in the data, exactly where the theory predicts they must. Why it matters Predicting which state a driven system will select is a basic, still-open problem across physics, chemistry, biology, and climate science, and several proposed selection principles — maximum entropy production, and related ideas such as dissipative adaptation — try to answer it by appealing to dissipation alone. This work shows that dissipation is only part of the story: a time-symmetric activity channel, invisible to those principles, can override them, and it does so specifically under non-equilibrium driving. Rather than refuting MEP, the result places it. MEP-like alignment holds only when the activity channel is quiet, and the inequality pins the size of the activity imbalance — the part that can reverse the outcome — to the circulating dissipation, with equality exactly at the hand-off between competing routes. A caution the paper makes explicit: reaching that extreme value is *not* generic. It requires a genuine hand-off between two distinct escape routes; in ordinary driven bistable systems the activity imbalance stays well below its ceiling, so the bound is loose rather than tight. Where the effect is strongest is precisely where a single dominant route ceases to exist — a tension the paper states rather than hides. The framework is operational. Because its key quantities can be estimated from a single observed steady-state trajectory, the bound doubles as a diagnostic that classifies the mechanism behind an observed selection — activity-, dissipation-, or boundary-driven — once the relevant routes are known. That makes the ideas testable in simulation and, in principle, in experiments on active matter, chemical reaction networks, and other driven systems where competing stable states are the rule rather than the exception. Preprint; not peer reviewed. The upload contains the manuscript (37 pp.).