Stochastic Mechanics and Fractals

Stochastic mechanics is not simply the addition of randomness to continuum mechanics; rather, it provides a framework linking continuum mechanics, statistical physics, homogenization, random fields, fractals, and multiscale phenomena. Stochastic Mechanics and Fractals: Within or Beyond the Entropy Inequality develops this perspective for complex media in which randomness, multiscale geometry, and nonclassical continuum effects are intrinsic rather than perturbative. The book addresses porous materials, fractured solids, composites, turbulent flows, and odd passive and active media, showing how statistical variability, scale dependence, and generalized continuum descriptions can be incorporated consistently into balance laws, constitutive relations, and variational principles. Part I develops the foundations of stochastic mechanics, classical and micropolar continua, and scale-dependent homogenization. Part II treats random processes and scalar and tensor-valued random fields, including fractal scaling, persistence, and long-range dependence. Part III develops stochastic and multiscale descriptions of cracks, damage, and elastic–inelastic transitions. Part IV uses fluctuation theorems to examine finite-scale departures from the classical entropy inequality and their disappearance under appropriate averaging, including molecular and macroscopic granular systems and extensions to odd matter. Part V develops continuum models for fractal and random porous media, including non-integer-dimensional spaces and fractal and fractional operators. Three appendices provide concise introductions to random-media models, fractals, and random processes for graduate coursework and self-study. The book is intended for researchers and graduate students in continuum mechanics, applied mathematics, statistical physics, materials science, and related areas.