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LIT-1890: Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

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Properties5
StatusActive — read in full 2026-10-31 (NOTE-1474) from the PMC copy; worth reading as the clearest worked case of a maintained chemical state that is metastable at finite size: bistability needs the chemostats out of equilibrium, long-run occupancy does not depend on the starting state, the slow eigenvalue is the sum of the escape rates (THEORY-1273), and entropy production does not pick the stable state (THEORY-1270). Its large-volume rate formula carries a spurious 1/V, so its reported breakdown of asymptotics is mostly that factor.
Version2
Tagsthermodynamics, natural-sciences
Cited as “paper” byNOTE-1474 — Vellela & Qian — Schlögl model, stochastic and thermodynamic
Cited as “source” byTHEORY-1265 — Approximating a chemical master equation by a Fokker–Planck or nonlinear Langevin equation gets the quasi-potential right only at the fixed points: between them it mis-estimates the barriers, so escape rates come out wrong by a factor exponential in system size and the approximation can pick the wrong one of two states as the long-run favourite, and its apparent entropy production vanishes at every steady state, missing the dissipation that sustains it, THEORY-1270 — Entropy production does not select which of several maintained steady states is the stable one: in the bistable Schlögl model the state the system occupies in the long run can carry either the higher or the lower deterministic entropy production rate, and the mean stochastic entropy production follows whichever state is more probable; in general the most dissipative attractor is the longest-lived only under special conditions, and lifetime tracks relaxation entropy only near detailed balance, THEORY-1273 — In a driven chemical system with two stable states and a finite number of molecules, each maintained state is metastable: it is left by fluctuation after a time exponential in system size, set by a barrier in a nonequilibrium quasi-potential that closes at the saddle-node, and the long-run share of time in each state is fixed by the parameters and system size, not by the initial condition; the deterministic rate equation, where the initial condition decides, holds only on times short of that escape time

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Vellela, M., & Qian, H. (2009). Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. Journal of the Royal Society Interface, 6, 925–940.

Read in full in NOTE-1474.

Why this is here

A13’s worked example of a bistable dissipative system whose maintained state has a finite stochastic lifetime. It is the chemical counterpart of the saddle-node constructions in CASE-189, and of CASE-111. The reading bears that out and says what the deterministic cases leave out: at finite size a maintained regime is metastable, it is lost by fluctuation on a time exponential in size, and the barrier setting that time closes at the fold, as (distance)^{3/2} in this model (THEORY-1273). It is also a clean counterexample to ranking steady states by entropy production (THEORY-1270), against THEORY-671.

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