Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included.

/pdf/the-kuramoto-model-a-simple-paradigm-for-synchronization-2akwf0t24m.pdf

The Kuramoto model: A simple paradigm for synchronization phenomena

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.

/pdf/the-restaurant-at-the-end-of-the-random-walk-recent-1w7yr3snyb.pdf

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

Magnetic fluid hyperthermia: focus on superparamagnetic iron oxide nanoparticles.

Artificial Molecular Rotors

TOPICAL REVIEW: Supermagnetism

The Langevin Equation

The concept of a random process the Brownian motion of a free particle - the velocity distribution the Brownian motion of a free particle - the distribution of the displacements one-dimensional Brownian motion in a potential excluding inertia effects the multi-dimensional Langevin equation rotational Brownian motion application to relaxation and loss processes in electric and magnetic fields.

The Langevin equation : with applications in physics, chemistry, and electrical engineering

The relaxation behavior of an assembly of noninteracting single-domain ferromagnetic particles in the presence of a constant magnetic field is studied by solving the corresponding Fokker-Planck equation. The analysis is performed by first converting that equation into a hierarchy of differential-recurrence relations by expanding the solution in Legendre polynomials. The spectrum of eigenvalues and their associated amplitudes is then determined by matrix methods where all the desired physical quantities such as the magnetization correlation time and complex magnetic susceptibility may be computed numerically. In order to ensure the accuracy of the results obtained this solution is compared with an exact solution derived in terms of matrix continued fractions. It is shown that the conventional assumption in the theory of superparamagnetism, that except in the very early stages of relaxation to equilibrium the only appreciable time constant is the one associated with the smallest nonvanishing eigenvalue, is no longer true when an applied constant magnetic field exceeds a certain critical value. The breakdown of this assumption manifests itself in (a) a dramatically large deviation of the magnetization correlation time (area under the curve of the decay of the magnetization) from the inverse of the lowest eigenvalue, and (b) in the presence of relatively strong high-frequency modes superimposed on the N\'eel one usually assigned to the lowest eigenvalue. The results are compared with available experimental data.

/pdf/constant-magnetic-field-effect-in-neel-relaxation-of-single-2hypgbpl70.pdf

Constant-magnetic-field effect in Néel relaxation of single-domain ferromagnetic particles.

The effect of a constant magnetic field, applied at an angle $\ensuremath{\psi}$ to the easy axis of magnetization, on the N\'eel relaxation time $\ensuremath{\tau}$ of a single domain ferromagnetic particle (with uniaxial anisotropy) is studied by calculating the lowest nonvanishing eigenvalue ${\ensuremath{\lambda}}_{1}$ (the escape rate) of the appropriate Fokker-Planck equation using matrix methods. The effect is investigated by plotting ${\ensuremath{\lambda}}_{1}$ versus the anisotropy parameter $\ensuremath{\alpha}$ for various values of $\ensuremath{\psi}$, and the ratio $h=\frac{\ensuremath{\xi}}{2\ensuremath{\alpha}}$, where $\ensuremath{\xi}$ is the external field parameter and ${\ensuremath{\lambda}}_{1}$ versus $\ensuremath{\psi}$ for various $h$ values (for rotation of the magnetization vector M both in a plane and in three dimensions). If M rotates in a plane the curve of ${\ensuremath{\lambda}}_{1}$ versus $\ensuremath{\psi}$ is symmetric about $\ensuremath{\psi}=\frac{\ensuremath{\pi}}{4}$ in the range $0l\ensuremath{\psi}l\frac{\ensuremath{\pi}}{2}$ and significant decrease in $\ensuremath{\tau}$ with increasing $\ensuremath{\psi}$ is predicted for large $\ensuremath{\xi}$ and $\ensuremath{\alpha}$. The maximum decrease in $\ensuremath{\tau}$ occurs at $\ensuremath{\psi}=\frac{\ensuremath{\pi}}{4}$ whereupon $\ensuremath{\tau}$ increases again to the $\ensuremath{\psi}=0$ value at $\ensuremath{\psi}=\frac{\ensuremath{\pi}}{2}$. For rotation of M in three dimensions, the curve of ${\ensuremath{\lambda}}_{1}$ versus $\ensuremath{\psi}$ ($0l\ensuremath{\psi}l\ensuremath{\pi}$) is symmetric about $\ensuremath{\psi}=\frac{\ensuremath{\pi}}{2}$. Thus the maximum decrease in $\ensuremath{\tau}$ again occurs at $\ensuremath{\psi}=\frac{\ensuremath{\pi}}{4}$ with maximum increase to a value exceeding that at $\ensuremath{\psi}=0$ (i.e., with the field applied along the polar axis with that axis taken as the easy axis), at $\ensuremath{\psi}=\frac{\ensuremath{\pi}}{2}$ (field applied along the equator), the $\ensuremath{\psi}=0$ value being again attained at $\ensuremath{\psi}=\ensuremath{\pi}$. The results are shown to be consistent with the behavior predicted by the Kramers theory of the rate of escape of particles over potential barriers. This theory when applied to the potential barriers for the equatorial orientation of the field for rotation in three dimensions yields a simple approximate formula for the escape rate which is in reasonable agreement with the exact ${\ensuremath{\lambda}}_{1}$ calculated from the Fokker-Planck equation. Pfeiffer's approximate formula for the barrier height as a function of $\ensuremath{\alpha}$ [H. Pfeiffer, Phys. Status Solidi 122, 377 (1990)] is shown to be in reasonable agreement with our results.

/pdf/effect-of-an-oblique-magnetic-field-on-the-superparamagnetic-4cs6dc24sj.pdf

Effect of an oblique magnetic field on the superparamagnetic relaxation time

/pdf/exact-analytic-formula-for-the-correlation-time-of-a-single-1sb2kcyvnz.pdf

Yu. P. Kalmykov

Papers

The Langevin Equation

The Langevin equation : with applications in physics, chemistry, and electrical engineering

Constant-magnetic-field effect in Néel relaxation of single-domain ferromagnetic particles.

Effect of an oblique magnetic field on the superparamagnetic relaxation time

Exact analytic formula for the correlation time of a single-domain ferromagnetic particle.