Approximate and numerically exact solutions of the Fokker-Planck equation with bistable potentials (original) (raw)
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The Fokker–Planck equation for a bistable potential
Phys.Rev.E Fokker-Planck equation for bistable potential in the optimized expansion, 2014
The Fokker–Planck equation is studied through its relation to a Schrodinger-type equation. The advantage of this combination is that we can construct the probability distribution of the Fokker–Planck equation by using well-known solutions of the Schrodinger equation. By making use of such a combination, we present the solution of the Fokker–Planck equation for a bistable potential related to a double oscillator. Thus, we can observe the temporal evolution of the system describing its dynamic properties such as the time τ to overcome the barrier. By calculating the rates k=1/τ as a function of the inverse scaled temperature 1/D, where D is the diffusion coefficient, we compare the aspect of the curve k×1/D, with the ones obtained from other studies related to four different kinds of activated process. We notice that there are similarities in some ranges of the scaled temperatures, where the different processes follow the Arrhenius behavior. We propose that the type of bistable potential used in this study may be used, qualitatively, as a simple model, whose rates share common features with the rates of some single rate-limited thermally activated processes.
Physica A: Statistical Mechanics and its Applications, 1978
The motion of a Brownian particle in an external field can be described on two levels: by a Fokker-Planck equation for the joint probability distribution of position and velocity, and by a Smoluchowski equation for the distribution in position space only. We derive the second description, with corrections, from the first by means of a systematic expansion procedure of the Chapman-Enskog type in terms of the inverse friction coefficient. We also derive equations describing the initial period, when the Smoluchowski description is not yet valid; in particular we find formulae connecting the initial value to be used for the Smoluchowski equation with that of the full Fokker-Planck equation. The special case of an harmonically bound Brownian particle can be solved exactly; the results are used to check and to illustrate our expressions for general potential.
Escape rates in bistable systems with position-dependent friction coefficients
Chemical Physics, 1993
In this paper we consider the generalization of the Kramers' model of chemical reactions to the case that the friction coefficient y(x) depends on the reaction coordinate x. Extending previous efforts the escape rate is exactly evaluated in the high-friction limit imposing on y(x) much milder conditions than used so far in the literature. The ensuing rate retains the Kramers' form and reproduces the renormalization effect of the damping coefficient which has been reported in laboratory experiments on chemical reactions. The origin of variable friction coefficients is then discussed within the framework of a multidimensional Markovian model and ascribed to the nonlinear coupling of the reaction coordinate with faster auxiliary variables. Finally, the implications of our results for the rate of ligands migration in proteins are briefly considered.
Surface diffusion in the low-friction limit: Occurrence of long jumps
Physical Review B, 1996
We present a molecular dynamics ͑MD͒ study of a Brownian particle in a two-dimensional periodic potential. For a separable potential, the study of the diffusion constant along the symmetry directions reduces to two one-dimensional problems. In this case, our MD study agrees with the existing analytical results on the temperature and the friction () dependence of the diffusion constant (D). For a nonseparable and anisotropic potential such as the adsorption potential on a bcc͑110͒ surface, the present study predicts an alternative Dϳ1/ 0.5 dependence in the low friction regime as opposed to the Dϳ1/ dependence found in previous studies of one-dimensional or separable potentials. We find that the dependence of D on in the low friction regime is directly related to the occurrence of long jumps. The probability for the long jumps depends not only sensitively on the value of the friction but also on the geometry of the surface. On the bcc͑110͒ surface, the path connecting adjoining adsorption sites does not coincide with the direction of easy crossing at the saddle point. Consequently, the probability of deactivation is enhanced, leading to the reduction of long jumps and the different dependence of D on .
Influence of the periodic potential shape on the Fokker–Planck dynamics
Physica A: Statistical Mechanics and its Applications, 2004
The in uence of the periodic potential structure on the di usion mechanism of a Brownian particle is studied using the Fokker-Planck equation. The equation is solved numerically by the matrix continued fraction method in order to calculate relevant correlation functions. In particular, jump length probability, di usion coe cient and the half-width of the quasi-elastic peak of the dynamical structure factor S(q; !) are fully studied in a wide range of physical parameters for two forms of periodic potential (bistable and metastable potential). There is some di erence between results provided by these two potential models, indicating that dynamical properties are very sensitive to the structure of the periodic potential, especially in the low friction regime.
Numerical Resolution of Fokker-Planck Type Kinetic Equations
2017
espanolLa tesis se centra en la resolucion numerica , mediante diferencias finitas y tecnicas de Fourier de dos ecuaciones de tipo Fokker-Planck. EnglishThe Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation is often used to approximate the description of particle transport processes with highly forward-peaked scattering, then the Fokker-Planck equation is an asymptotic approximation to the linear Boltzmann equation. In this thesis it is considered a new finite difference method and an iterative method to solve the Fokker-Planck equation when the angular flux depends on spatial, polar and azimuthal variables. Fourier technique is applied to split the problem into a set of azimuthal angle-independent problem.
Physical Review E, 2000
An equation for the smallest nonvanishing eigenvalue 1 of the Fokker-Planck equation ͑FPE͒ 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 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 interest͒ describing the dynamical behavior of the system under consideration. In contrast to the previously available continued fraction solution for 1 ͓for example, H. Risken, The Fokker-Planck Equation, 2nd ed. ͑Springer, Berlin, 1989͔͒, this equation does not require one to solve numerically a high order polynomial equation, as it is shown that 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 ͑special͒ 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 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 1 previously derived for these particular problems by solving the FPE in the high barrier limit are readily recovered from the analytical equations.
Derivation and solution of a low-friction Fokker-Planck equation for a bound Brownian particle
Zeitschrift f�r Physik B Condensed Matter, 1985
The low-friction region of an anharmonically bound Brownian particle is examined using systematic elimination procedures. We obtain an asymptotic expression for the spectrum of the Fokker-Planck operator. Asymptotic means both small anharmonicities and small friction constants y compared to the oscillatory frequency co. We conclude that Kramers' low-friction equation is generally valid only for 0<7<0.01 co and has to be modified for 7~>0.01 co by including phase-dependent terms. From these the nonlinear part of the force field in connection with a finite temperature is shown to shorten the correlation time of the equilibrium velocity autocorrelation function and to renormalize the frequency of the corresponding spectral density.
Diffusion in a bistable potential: A systematic WKB treatment
Journal of Statistical Physics, 1979
We study the distribution P of a single stochastic variable, the evolution of which is described by a Fokker-Planck equation with a first moment deriving from a bistable potential, in the limit of constant and small diffusion coefficient. A systematic WKB analysis of the lowest eigenmodes of the equivalent Schr6dinger-like equation yields the following results: the final approach to equilibrium is governed by the Kramers high-viscosity rate, which is shown to be exact in this limit; for intermediate times, we show that Suzuki's scaling statement does give the correct behavior for the transition between the one-peak and the two-peak structure for P. However, the intermediate time domain also contains a second "half," where P enters the diffusive equilibrium regions, characterized by a time scale of the same order as Suzuki's time.
Physical Review E, 2002
Based on a true phase space probability distribution function and an ensemble averaging procedure we have recently developed [Phys. Rev. E 65, 021109 (2002)] a non-Markovian quantum Kramers' equation to derive the quantum rate coefficient for barrier crossing due to thermal activation and tunneling in the intermediate to strong friction regime. We complement and extend this approach to weak friction regime to derive quantum Kramers' equation in energy space and the rate of decay from a metastable well. The theory is valid for arbitrary temperature and noise correlation. We show that depending on the nature of the potential there may be a net reduction of the total quantum rate below its corresponding classical value which is in conformity with earlier observation. The method is independent of path integral approaches and takes care of quantum effects to all orders.