Fractional calculus modifications to the kinetic equations of statistical mechanics

Abstract Statistical mechanics describes the behavior of the particles (atoms and molecules) that comprise solids, liquids, and gases. The dynamics of particle motion is encoded in kinetic equations (e.g., Liouville, Boltzmann, Fokker-Planck, and Langevin) that predict the hydrodynamic, magnetic, and dielectric properties of materials, the rates of chemical reactions, and the stability of plasmas. However, interest in multiscale, composite, heterogeneous, and fractal materials has stimulated the development of generalized, and in some cases, fractional calculus models in order to describe the emergence of anomalous transport of mass, momentum, and energy. Extending integer time and space derivatives to fractional order provides a concise way to incorporate memory and non-locality into the force, flow, and flux terms of each equation. This approach recasts the fundamental equations and complements the nonlinear and perturbation models commonly used to address complex dynamics and stochastic processes. In this review, we compare fractional calculus methods with other techniques, such as the assumption of time- or position-dependent diffusion coefficients, and highlight the connections between fractional-order operators, stochastic differential equations, and the asymptotic behavior of physical systems. We focus on fractional-calculus modifications of the Boltzmann, Fokker–Planck, and Langevin equations and distinguish time-fractional (memory) from space-fractional (spatial nonlocality) generalizations and their main stochastic-process mechanisms. The goal is to identify situations where mathematics and physics combine in models that not only provide better fits to data, but also animate the mathematics, allowing it to be tailored to new applications.