Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Membrane deformation is necessary to generate endocytic vesicles, but the molecular mechanisms proposed to drive membrane bending are controversial. Stachowiak and Schmid et al. report that crowding of proteins at the membrane is sufficient to induce curvature in vitro.

Membrane bending by protein-protein crowding

We study the inducement of passive nonlinear sinks in linear vibrating systems. These are substructures that absorb vibrational energy in a one-way, irreversible fashion. The systems considered are composed of strongly coupled, grounded damped linear oscillators with a strongly nonlinear attachment at the end. Applying a complex averaging technique we derive a set of modulation equations that is directly amenable to physical interpretation, and provides insight into the energy pumping phenomenon. For the case of a two DOF system we show that nonlinear energy pumping occurs when a certain frequency of envelope modulation crosses through zero; then the dynamics of the envelope modulation of the motion resemble the dynamics of a forced rigid body. For the case of an impulsively loaded multi-DOF chain with a nonlinear attachment at the end, we show that after some initial transients the response of the nonlinear attachment sets to a motion dominated by a fast frequency identical to the lower bound of the propagation zone of the linear chain. This feature reduces the study of energy pumping in the chain to a two DOF equivalent problem. The applications of the energy pumping phenomenon to practical engineering problems are discussed.

Inducing Passive Nonlinear Energy Sinks in Vibrating Systems

In this paper, the homotopy–perturbation method proposed by J.-H. He is adopted for solving pure strong nonlinear second-order differential equation. For the oscillatory differential equation the initial approximate solution is assumed in the form of Jacobi elliptic function and the forementioned method is used for obtaining of the approximate analytic solution. Two types of differential equations are considered: with strong cubic and strong quadratic nonlinearity. The obtained solution is compared with exact numerical one. The difference between these solutions is negligible for a long time period. The method is found to work extremely well in the examples, but the theoretical reasons are not yet clear.

Homotopy–perturbation method for pure nonlinear differential equation

http://server.ihim.uran.ru/files/info/Rusanov.pdf

Surface thermodynamics revisited

An equation of state for a fluid of hard disks is proposed: Z=[1−2η+(2η0−1)(η/η0)2]−1. The exact fit of the second virial coefficient and the existence of a single pole singularity at the close‐packing fraction η0 are the only requirements imposed on its construction. A comparison of the prediction of virial coefficients and of the values of the compressibility factor Z with those stemming out of other known equations of state is made. The overall performance of this very simple equation of state is quite satisfactory.

An accurate and simple equation of state for hard disks

The radial distribution function g(r) provided by the solution of the Percus-Yevick (PY) equation for hard spheres is rederived in terms of the simplest Pad\'e approximant of a function defined in the Laplace space that is consistent with the following physical requirements: g(r) is continuous for rg1, the isothermal compressibility is finite, and the zeroth- and first-order coefficients in the density expansion of g(r) must be exact. An explicit expression for the solution of the generalized mean-spherical approximation (GMSA) is obtained as a simple extension involving two new parameters, which are determined by imposing two conditions: (i) the virial and the compressibility routes to the equation of state agree consistently, and (ii) this equation of state coincides with that of Carnahan and Starling [J. Chem. Phys. 51, 635 (1969)]. The second- and third-order coefficients in the density expansion of g(r) given by the GMSA are compared with the exact ones and with those given by the PY equation.

Radial distribution function for hard spheres.

Construction of approximate analytical solutions to a new class of non-linear oscillator equations

Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

A rational-function approximation method to analytically derive the radial distribution function g(r ) and the structure factorS(q) of a hard-sphere fluid is revisited. The method provides a fluid structure thermodynamically consistent with a given equation of state. The Carnahan-Starling and a recently derived Pade ́ ~4,3! equations of state are considered. While the Carnahan-Starling equation leads to functions g(r ) andS(q) even for densities larger than the crystalline close-packing density, their physical value is questionable already in the metastable fluid region. In the case of the much more reliable Pade ́ equation of state, the method shows the existence of a threshold density, beyond which no meaningful fluid structure can be derived. This threshold value, whose signature is a diverging slope of g(r ) at the contact distance, coincides with the one associated in the literature with the glass transition. @S1063-651X~96!01805-3#

S. Bravo Yuste

Papers

An accurate and simple equation of state for hard disks

Radial distribution function for hard spheres.

Construction of approximate analytical solutions to a new class of non-linear oscillator equations

Improvement of a Krylov-Bogoliubov method that uses Jacobi elliptic functions

Structure of hard-sphere metastable fluids.