Cross-Diffusion Waves as a Mesoscopic Uncertainty Relationship for Multi-Physics Instabilities

TL;DR

This paper introduces a mesoscopic uncertainty relationship for cross-diffusion waves in multi-physics systems, revealing their unique dispersion, particle-like behavior, and dependence on material parameters within coupled THMC equations.

Contribution

It proposes a novel uncertainty framework for cross-diffusion waves in multi-physics instabilities, linking wave properties to material parameters in coupled thermodynamic equations.

Findings

Cross-diffusion waves exhibit unusual dispersion patterns.

Waves have a solitary, particle-like state but are not true solitons.

Wave speed and wavenumber are determined by material parameters.

Abstract

We propose a generic uncertainty relationship for cross-diffusion (quasi-soliton) waves triggered by local instabilities through Thermo-Hydro-Mechano-Chemical (THMC) coupling and cross-scale feedbacks. Cross-diffusion waves nucleate when the overall stress field is incompatible with accelerations from local feedbacks of generalized THMC thermodynamic forces with generalized thermodynamic fluxes of another kind. Cross-diffusion terms in the 4 x 4 THMC diffusion matrix are shown to lead to multiple diffusional $P$- and $S$-wave solutions of the coupled THMC equations. Uncertainties in the location of local material instabilities are captured by wave scale correlation of probability amplitudes. Cross-diffusional waves have unusual dispersion patterns and, although they assume a solitary state, do not behave like solitons but have a quasi-elastic particle-like state. Their characteristic wavenumber and constant speed defines mesoscopic internal material time-space relations entirely defined by the coefficients of the coupled THMC reaction-cross-diffusion equations. These coefficients are identified here as material parameters underpinning the criterion for nucleation and speed of diffusional waves. Interpreting patterns in nature as features of standing or propagating diffusional waves offers a simple mathematical framework for analysis of multi-physics instabilities and evaluation of their uncertainties similar to their quantum-mechanical analogues.