Diffusion from Convection

TL;DR

This paper develops a formalism to understand diffusion in many-body systems by analyzing quadratic fluctuations of convective modes, revealing conditions for super-diffusive behavior and matching exact diffusion constants in integrable systems.

Contribution

It introduces a new method to derive diffusion constants from quadratic fluctuations of convective modes, applicable to both quantum and classical integrable systems.

Findings

Second-order terms in current expansion contribute to diffusion.

Degenerate group velocities lead to super-diffusive behavior.

Expression matches exact diffusion constants in integrable models.

Abstract

We introduce non-trivial contributions to diffusion constant in generic many-body systems arising from quadratic fluctuations of ballistically propagating, i.e. convective, modes. Our result is obtained by expanding the current operator in the vicinity of equilibrium states in terms of powers of local and quasi-local conserved quantities. We show that only the second-order terms in this expansion carry a finite contribution to diffusive spreading. Our formalism implies that whenever there are at least two coupled modes with degenerate group velocities, the system behaves super-diffusively, in accordance with the non-linear fluctuating hydrodynamics theory. Finally, we show that our expression saturates the exact diffusion constants in quantum and classical interacting integrable systems, providing a general framework to derive these expressions.