Yu. P. Kalmykov

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

Historical Background and Introductory Concepts Methods for Solving Langevin and Fokker-Planck Equations Matrix Continued Fractions Escape Rate Theory Linear and Nonlinear Response Theory Brownian Motion of a Free Particle and a Harmonic Oscillator Rotational Brownian Motion about a Fixed Axis in a Periodic Potential Brownian Motion in a Tilted Periodic Potential: Application to the Josephson Tunnelling Junction and Ring Lasers Brownian Motion in a Double-Well Potential Isotropic and Anisotropic Rotational Brownian Motion in Space in the Presence of an External Potential with Applications to Dielectric and Kerr Effect Relaxation in Fluids and Liquid Crystals Brownian Motion of Classical Spins with Applications to Superparamagnetism Magnetic Stochastic Resonance Dynamic Hysteresis Switching Field Surfaces Inertial Langevin Equations with Applications to Orientational Relaxation in Liquids Itinerant Oscillator Model Anomalous Diffusion Continuous Time Random Walks Methods for the Solution of Fractional Fokker-Planck Equations.

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

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.

Anomalous dielectric relaxation in the context of the Debye model of noninertial rotational diffusion

The Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker–Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. The fractional Fourier–Planck equation in question is a generalization of the Smoluchowski equation for the dynamics of Brownian particles in configuration space to include anomalous relaxation. It is shown that this model can reproduce nonexponential Cole–Cole-type anomalous dielectric relaxation behavior and that it reduces to the classical Debye model of rotational diffusion when the anomalous exponent is unity.

Exact solution for the extended Debye theory of dielectric relaxation of nematic liquid crystals

The exact solution for the transverse (i.e. in the direction perpendicular to the director axis) component α⊥(ω) of a nematic liquid crystal and the corresponding correlation time T⊥ is presented for the uniaxial potential of Martin et al. [Symp. Faraday Soc. 5 (1971) 119]. The corresponding longitudinal (i.e. parallel to the director axis) quantities α⊥(ω), T⊥ may be determined by simply replacing magnetic quantities by the corresponding electric ones in our previous study of the magnetic relaxation of single domain ferromagnetic particles Coffey et al. [Phys. Rev. E 49 (1994) 1869]. The calculation of α⊥(ω) is accomplished by expanding the spatial part of the distribution function of permanent dipole moment orientations on the unit sphere in the Fokker-Planck equation in normalised spherical harmonics. This leads to a three term recurrence relation for the Laplace transform of the transverse decay functions. The recurrence relation is solved exactly in terms of continued fractions. The zero frequency limit of the solution yields an analytic formula for the transverse correlation time T⊥ which is easily tabulated for all nematic potential barrier heights σ. A simple analytic expression for T∥ which consists of the well known Meier-Saupe formula [Mol. Cryst. 1 (1966) 515] with a substantial correction term which yields a close approximation to the exact solution for all σ, and the correct asymptotic behaviour, is also given. The effective eigenvalue method is shown to yield a simple formula for T⊥ which is valid for all σ. It appears that the low frequency relaxation process for both orientations of the applied field is accurately described in each case by a single Debye type mechanism with corresponding relaxation times (T∥, T⊥).

Effect of a dc bias field on the dynamic hysteresis of single-domain ferromagnetic particles

Dynamic magnetic hysteresis in uniaxial superparamagnetic nanoparticles in superimposed ac and dc magnetic fields of arbitrary amplitude is considered using Brown’s model of coherent rotation of the magnetization. The dependence of the area of the dynamic hysteresis loop on the temperature, frequency, and ac and dc bias fields is analyzed. In particular, the dynamic hysteresis loop of a single-domain ferromagnetic particle is substantially altered by applying a relatively weak dc field. Furthermore, it is shown that at intermediate to low ac field amplitudes, the dc bias field permits tuning of the magnetic power absorption of the particles, while for strong ac field amplitudes the effect becomes entirely analogous to that produced by the exchange biased anisotropy. Simple analytical formulas are provided in the linear response regime for the steady-state magnetization and loop area, exhibiting perfect agreement with the numerical solution of Brown’s Fokker–Planck equation. Comparison with previous result...

Exact analytic formulae for the correlation times for single domain ferromagnetic particles

Abstract Exact solutions for the longitudinal relaxation time T ∥ and the complex susceptibility χ ∥ (ω) of a thermally agitated single domain ferromagnetic particle are presented for the simple uniaxial (Maier-Saupe) potential of the crystalline anisotropy considered by Brown [Phys. Rev. 130 (1963) 1677].

Semiclassical Klein–Kramers and Smoluchowski equations for the Brownian motion of a particle in an external potential

The quantum Brownian motion of a particle in an external potential V(x) is treated using the master equation for the Wigner distribution function W(x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(4) and in the classical limit, → 0, reduces to the Klein–Kramers equation. For a quantum oscillator, the method yields an evolution equation for W(x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.

On the calculation of field-dependent relaxation times from the noninertial Langevin equation

The problem of the dielectric relaxation of an assembly of noninteracting rigid dipoles subjected to a strong direct current field superimposed on which is a weak alternating field is treated by averaging of the noninertial Langevin equation regarded as a Stratonovich stochastic differential equation. The hierarchy of differential‐difference equations so obtained is linearized in the alternating current (ac) field. It is then closed by assuming that the ratio of the Fourier–Laplace transforms of the ensemble averages of the first and second spherical harmonics may be replaced by its zero‐frequency value. This allows one to obtain closed‐form expressions in terms of the Langevin function for the relaxation times, complex susceptibility and loss tangent induced by the coupling between the direct current (dc) field and the weak ac field. The relaxation times produced by this closure procedure for ac fields applied parallel and perpendicular to the dc field are in good agreement with the results of a numerica...

Rotational Brownian motion in an external potential: The Langevin equation approach

Abstract The theory of orientation relaxation is developed without recourse to the Fokker-Planck equation by direct averaging of the underlying Langevin equation for the noninertial rotational Brownian motion of a polar particle in an arbitrary external potential. Two problems are considered: (i) orientation relaxation of a polar molecule governed by the vector Euler-Langevin equation and (ii) magnetisation relaxation of a single domain ferromagnetic particle described by Gilberts equation augmented by a random field term. Those equations being regarded as stochastic non-linear differential equations of the Stratonovich type.

Exact analytic solution for the correlation time of a Brownian particle in a double-well potential from the Langevin equation

The correlation time of the positional autocorrelation function is calculated exactly for one‐dimensional translational Brownian motion of a particle in a 2–4 double‐well potential in the noninertial limit. The calculations are carried out using the method of direct conversion (by averaging) of the Langevin equation for a nonlinear stochastic system to a set of differential–recurrence relations. These, in the present problem, reduce on taking the Laplace transform, to a three‐term recurrence relation. Thus the correlation time Tc of the positional autocorrelation function may be formally expressed as a sum of products of infinite continued fractions which may be represented in series form as a sum of two term products of Whittaker’s parabolic cylinder functions. The sum of this series may be expressed as an integral using the integral representation of the parabolic cylinder functions and subsequently the Taylor expansion of the error function, thus yielding the exact solution for Tc. This solution is in ...