ABSTRACT
The solution of the Langevin equation describing the dynamics of a Brownian particle in a tilted periodic potential in the overdamped limit is obtained in terms of a matrix continued fraction, allowing us to evaluate statistical averages governing the nonlinear response to a strong ac force. Pronounced nonlinear effects are observed for large values of the ac force. For a weak ac force and low noise strength, the results obtained agree closely with previously available linear response and noiseless solutions, respectively.
ABSTRACT
An equation for the smallest nonvanishing eigenvalue lambda(1) of the Fokker-Planck equation (FPE) for the Brownian motion of a particle in a potential is derived in terms of continued fractions. This equation is directly applicable to the calculation of lambda(1) if the solution of the FPE can be reduced to the solution of a scalar three-term recurrence relation for the moments (the expectation values of the dynamic quantities of interest) describing the dynamical behavior of the system under consideration. In contrast to the previously available continued fraction solution for lambda(1) [for example, H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer, Berlin, 1989)], this equation does not require one to solve numerically a high order polynomial equation, as it is shown that lambda(1) may be represented as a sum of products of infinite continued fractions. Besides its advantage for the numerical calculation, the equation so obtained is also very useful for analytical purposes, e.g., for certain problems it may be expressed in terms of known mathematical (special) functions. Another advantage of such an approach is that it can now be applied to systems whose relaxation dynamics is governed by divergent three-term recurrence equations. To test the theory, the smallest eigenvalue lambda(1) is evaluated for several double-well potentials, which appear in various applications of the theory of rotational and translational Brownian motion. It is shown that for all ranges of the barrier height parameters the results predicted by the analytical equation so obtained are in agreement with those obtained by independent numerical methods. Moreover, the asymptotic results for lambda(1) previously derived for these particular problems by solving the FPE in the high barrier limit are readily recovered from the analytical equations.
ABSTRACT
An equation for the smallest nonvanishing eigenvalue lambda(1) of the Fokker-Planck equation (FPE) for the Brownian motion of a particle in a potential is derived in terms of matrix-continued fractions. This equation is applicable to the calculation lambda(1) if the solution of the FPE can be reduced (by expanding the probability distribution function in terms of a complete set of appropriate functions) to the solution of a multiterm recurrence relation for the moments describing the dynamics of the Brownian particle. In contrast to the available continued fraction solution for lambda(1) [H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1989)], this equation does not require one to solve numerically a high order polynomial equation. To test the theory, the smallest eigenvalue lambda(1) is evaluated for the FPE, which appears in the theory of magnetic relaxation of single domain (superparamagnetic) particles. Various regimes of relaxation of the magnetization in superparamagnetic particles are governed by a damping parameter alpha, the limiting values of which correspond to the high damping (alpha-->infinity) and the low damping (alpha<<1) limits in the theory of the escape rate over potential barriers. It is shown that for all ranges of the barrier height and damping parameters the smallest eigenvalue lambda(1) predicted by the continued fraction equation is in agreement with those gained by independent numerical methods and the asymptotic estimates for lambda(1) (in the high barrier limit) and, moreover, the matrix continued fraction approach may be successfully applied to the evaluation of lambda(1) in those ranges of parameters where traditional methods fail or are not applicable.
ABSTRACT
The nonlinear dielectric relaxation ac stationary response of an assembly of rigid polar molecules acted on by strong superimposed external dc E0 and ac E1(t)=E(1) cos omegat electric fields is evaluated in the context of the noninertial rotational diffusion model. The calculation proceeds by expanding the relaxation functions f(n)(t) (the expectation value of the Legendre polynomials P(n)), which describe the nonlinear relaxation of the system, as a Fourier series in the time domain. Hence, an infinite hierarchy of recurrence equations for the Fourier components of f(n)(t) is obtained. The exact solution of this hierarchy can be obtained in terms of a matrix continued fraction, so allowing us to evaluate the ac nonlinear response. For a weak ac field our results are in complete agreement with previous solutions obtained by perturbation methods. Diagrams showing the behavior of the in-phase and out-of-phase components of the electric polarization are presented.
