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We propose a generic uncertainty relationship for cross-diffusion(quasi-soliton) waves triggered by local instabilities throughThermo-Hydro-Mechano-Chemical (THMC) coupling and cross-scale feedbacks.Cross-diffusion waves nucleate when the overall stress field is incompatiblewith accelerations from local feedbacks of generalized THMC thermodynamicforces with generalized thermodynamic fluxes of another kind. Cross-diffusionterms in the 4 x 4 THMC diffusion matrix are shown to lead to multiplediffusional $P$- and $S$-wave solutions of the coupled THMC equations.Uncertainties in the location of local material instabilities are captured bywave scale correlation of probability amplitudes. Cross-diffusional waves haveunusual dispersion patterns and, although they assume a solitary state, do notbehave like solitons but have a quasi-elastic particle-like state. Theircharacteristic wavenumber and constant speed defines mesoscopic internalmaterial time-space relations entirely defined by the coefficients of thecoupled THMC reaction-cross-diffusion equations. These coefficients areidentified here as material parameters underpinning the criterion fornucleation and speed of diffusional waves. Interpreting patterns in nature asfeatures of standing or propagating diffusional waves offers a simplemathematical framework for analysis of multi-physics instabilities andevaluation of their uncertainties similar to their quantum-mechanicalanalogues.

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