Exponent Inequalities in Dynamical Systems

TL;DR

This paper derives general inequalities linking the dynamic exponent to the steady state exponent in stochastic dynamical systems, providing insights for critical phenomena and surface growth models.

Contribution

It introduces a unified framework of exponent inequalities for a broad class of stochastic systems, distinguishing between two system classes.

Findings

Derived a general exact inequality relating response and correlation functions.

Established different inequalities for two classes of dynamical systems.

Discussed implications for critical dynamics and KPZ-like problems.

Abstract

In this letter we derive exponent inequalities relating the dynamic exponent $z$ to the steady state exponent $\Gamma$ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating $z$ and $\Gamma$. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems are discussed.