Conjectured dissipation–stimulus formula in the fully adaptive Langevin chemotaxis model

Prove the conjectured piecewise expression for the steady-state dissipation rate as a function of stimulus magnitude s in the fully adaptive (β = 1) two-variable Langevin adaptation model with reflecting boundaries and equal effective temperatures, namely: for s < s0, W_diss(s) = W_diss^a(s); for s ≥ s0, W_diss(s) = K0 exp(−λ(s − s0)) + ε0 (1 − exp(−λ(s − s0))), where K0 = W_diss^a(s0) and ε0 = min_s W_diss(s).

Background

The standard steepest-descent analytical approximation for dissipation, W_diss^a, breaks down when the probability distribution is significantly influenced by phase-space boundaries at large stimuli, motivating an empirical description.

To capture this boundary-dominated regime, the authors introduce a threshold s0 and conjecture a piecewise dissipation formula that matches simulations: the analytical form W_diss^a(s) holds below s0, while an exponential relaxation from K0 to ε0 with rate λ describes dissipation above s0.