A Thermodynamic Constraint on Persistence

In open systems maintained far from equilibrium, continual exchange with an environment produces irreversible entropy and tends to erode coarse-grained distinguishability. We derive a necessary persistence condition that follows from (i) nonnegativity and (ii) additivity of irreversible entropy production, together with (iii) the fact that any fixed coarse-grained identity criterion has finite resolution and therefore a finite admissible tolerance Ω to accumulated degradation. For a persistence interval [0,τ], admissibility requires that cumulative irreversible ¯ entropy production Σirr(τ) remain below Ω; equivalently τ ≤Ω/ Φτ, where¯ Φτ is the time- averaged entropy production rate. In approximately stationary regimes this reduces to τ ≤Ω/Φ. An energetic form, Pirr = kBTsinkΦ, and a mass-normalized form, Γ = Pirr/(Mc2), yield τ ≤κ/Γ with κ= (kBTsinkΩ)/(Mc2). Spohn’s theorem gives an information-theoretic interpretation for quantum Markovian semigroups via monotone decay of relative-entropy distinguishability. A single diagnostic coordinate χ= τΓ makes boundary dependence explicit and enables cross- domain bookkeeping comparisons of persistence architectures.