Cosh gradient systems and tilting

TL;DR

This paper explores cosh-gradient systems, their emergence in large deviations and multi-scale limits, and examines the concept of tilt-independence, showing how it is generally lost in certain limits like the Kramers high-activation-energy scenario.

Contribution

It provides a comprehensive analysis of cosh-type gradient systems, their tilt-dependence, and the implications for modeling and convergence in stochastic processes and chemical reaction theories.

Findings

Cosh dissipation potentials arise from large deviations and multi-scale limits.

Tilt-independence is generally lost in the Kramers limit and similar scenarios.

Discrete network analysis reveals tilt-dependence akin to electrical networks.

Abstract

We review a class of gradient systems with dissipation potentials of hyperbolic-cosine type. We show how such dissipation potentials emerge in large deviations of jump processes, multi-scale limits of diffusion processes, and more. We show how the exponential nature of the cosh derives from the exponential scaling of large deviations and arises implicitly in cell problems in multi-scale limits. We discuss in-depth the role of "tilting" of gradient systems. Certain classes of gradient systems are "tilt-independent", which means that changing the driving functional does not lead to changes of the dissipation potential. Such tilt-independence separates the driving functional from the dissipation potential, guarantees a clear modelling interpretation, and gives rise to strong notions of gradient-system convergence. We show that although in general many gradient systems are tilt-independent, certain cosh-type systems are not. We also show that this is inevitable, by studying in detail the classical example of the Kramers high-activation-energy limit, in which a diffusion converges to a jump process and the Wasserstein gradient system converges to a cosh-type system. We show and explain how the tilt-independence of the pre-limit system is lost in the limit system. This same lack of independence can be recognized in classical theories of chemical reaction rates in the chemical-engineering literature. We illustrate a similar lack of tilt-independence in a discrete setting. For a class of "two-terminal" fast subnetworks, we give a complete characterization of the dependence on the tilting, which strongly resembles the classical theory of equivalent electrical networks.