We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (Sect. 1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (Sect. 2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis induces us to introduce a hydrodynamic model suitable to revisit in Sect. 3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In Sect. 4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order $0< \beta <2$. Led by our analysis we express the fundamental solutions (the Green functions) in terms of two interrelated auxiliary functions in the similarity variable, which turn out to be of Wright type (see Appendix), and to distinguish slow-diffusion processes ($0 < \beta < 1$) from intermediate processes ($1 < \beta < 2$).

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Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

Time dependent correlation functions and mode-mode coupling theories

The Langevin equation describing Brownian motion is considered as a contraction from the more fundamental, but still phenomenological, description of an incompressible fluid governed by fluctuating hydrodynamics in which a Brownian particle with stick boundary condition is immersed. First, the derivation of fluctuating hydrodynamics is reconsidered to clarify certain ambiguities as to the treatment of boundaries. Subsequently the contraction is carried out. Since Brownian particles of arbitrary shape are considered, rotations and translations are in general coupled. The symmetry of the 6×6 friction tensorγ

 ij
 
(t) is proved for arbitrary shape without appeal to microscopic arguments. This symmetry is then used to prove that the fluctuation-dissipation theorem on the contracted level (nonwhite noise in general) follows from the corresponding statement on the level of fluctuating hydrodynamics (white noise). The condition under which the contracted description reduces to the classical Langevin equation is given, and the connection between our theory and related work is discussed.

Fluctuating hydrodynamics and Brownian motion

Exact generalized Fokker–Planck equations are derived from the linear Mori–Kubo generalized Langevin equation for the case of Gaussian but non‐Markovian noise. Fokker–Planck equations which generate the momentum and phase space probability distribution functions (pdf’s) for free Brownian particles and the phase space pdf for Brownian oscillators are presented. Also given is the generalized diffusion equation for the free Brownian particle pdf in the zero inertia limit. The generalized Fokker–Planck equations are similar in structure to the corresponding phenomenological equations. They, however, involve time‐dependent friction and frequency functions rather than phenomenological constants. Explicit results for the frequency and friction functions are given for the Debye solid model. These functions enter as simple multiplicative factors rather than as retarded kernels. Further the phase space Fokker–Planck equations contain an extra diffusive term, a mixed phase space second partial derivative, not occurr...

Fokker-Planck equations for simple non-Markovian systems

Brownian motion of particles suspended in a fluid is studied, and expressions derived for the particle diffusivity and velocity autocorrelation function. The theory of thermal noise in a general linear system is applied to both the particles and the fluid in a novel formulation. This enables the recent modification of the Langevin equation to include the effect of fluid inertia to be seen as just a necessary but simple reinterpretation of the original analysis, without introducing the theory of non-Markovian processes.

Application of the Langevin equation to fluid suspensions

From Boussinesq's work it is known that the frictional resistance of a particle in a viscous, inert fluid, depends on its history. This plays an important part in Brownian motion. The general theory of fluctuations in fluid dynamics leads to explicit expressions for the autocorrelation function G(t) for the random force acting on a Brownian particle and the autocorrelation function φ(t) for its velocity. It is demonstrated that G(t) contains an inertial term which depends on the velocity distribution in the liquid surrounding the particle. The results obtained lead to the correct value of the diffusivity.

J. J. Hermans

Papers

Effect of Inertia on the Brownian Motion of Rigid Particles in a Viscous Fluid