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A shortcut way to the Fokker-Planck equation for the non-Markovian dynamics
2020
Using a shortcut way we have derived the Fokker-Planck equation for the Langevin dynamics with a generalized frictional memory kernel and time-dependent force field. Then we have shown that this method is applicable for the non-Markovian dynamics with additional force from harmonic potential or magnetic field or both of them. The simplicity of the method in these complex cases is highly noticeable and it is applicable to derive the Fokker-Planck equation for any kind of linear Langevin equation of motion which describes additive colored noise driven non Markovian dynamics with or without frictional memory kernel. For example, one may apply the method even for the linear system with an additional colored Gaussian noise which is not related to the damping strength. With these the present study may get strong attention in the field of stochastic thermodynamics which is now at early stage to consider the non-Markovian dynamics.
2021
During the last years with the studies of stochastic processes: neurons networks, molecular motors, dynamics models, anomalous diffusion, disordered media, etc, several methods have evolved to apply the Focker-Planck equation (FPE) to these phenomena. We present here the solution of the Fokker-Planck equation by the Crank-Nicholson formalism. The von Neumann amplification factor, ξ (k), is independent of dt, so the method is stable for any size dt. The method is suitable for modeling molecular motors because the great amount of interactions in these systems, vectors and matrices oriented methods are needed, suited to work with Matlab. In the Appendix of this chapter is given some notions of Stochastic Dynamics 2.1 The Methods The method of Fractional Focker-Planck equation (FFPE) [1] was derived from a generalized continuous time random walk, which includes space dependent jump probabilities which are the result of an external field. In [2] was presented the solution of the FFPE in terms of an integral transformation. In [3] a half order FFPE was derived from the generalized scheme of random walks on the comlike structure. In [4, 5] the FFPE was used to study and describe also the anomalous diffusion in external fields. In [6] the FFPE was used to study ultraslow kinetics. In [7] was introduced a heterogeneous FFPE involving external force fields describing systems on heterogeneous fractal structure medium. In [8] was studied the dynamical properties of bistable systems described by the one-dimensional sub-diffusive FFPE, for the natural boundary conditions as well as the absorbing boundary conditions. In [9] was shown that the subordinated Brownian process is a stochastic solution of the FFPE. In [10] was proven that the asymptotic shape of the solution of the FFPE is a stretched Gaussian and that its solution can be expressed in the form of a function of
Journal of Statistical Mechanics-theory and Experiment, 2008
A harmonic oscillator that evolves under the action of both a systematic time-dependent force and a random time-correlated force can do work w. This work is a random quantity, and Mai and Dhar have recently shown, using the generalized Langevin equation (GLE) for the oscillator's position x, that it satisfies a fluctuation theorem. In principle, the same result could have been derived from the Fokker-Planck equation (FPE) for the probability density function, P (x, w, t), for the oscillator being at x at time t, having done work w. Although the FPE equivalent to the above GLE is easily constructed and solved, one finds, unexpectedly, that its predictions for the mean and variance of w do not agree with the fluctuation theorem. We show that to resolve this contradiction, it is necessary to construct an FPE that includes the velocity of the oscillator, v, as an additional variable. The FPE for P (x, v, w, t) does indeed yield expressions for the mean and variance of w that agree with the fluctuation theorem.
Physica A: Statistical Mechanics and its Applications
We study the Quantum Brownian motion of a charged particle moving in a harmonic potential in the presence of an uniform external magnetic field and linearly coupled to an Ohmic bath through momentum variables. We analyse the growth of the mean square displacement of the particle in the classical high temperature domain and in the quantum low temperature domain dominated by zero point fluctuations. We also analyse the Position Response Function and the long time tails of various correlation functions. We notice some distinctive features, different from the usual case of a charged quantum Brownian particle in a magnetic field and linearly coupled to an Ohmic bath via position variables.
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.
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.
Quantum Brownian motion in a magnetic field: Transition from monotonic to oscillatory behaviour
Physica A: Statistical Mechanics and its Applications
We investigate the Brownian motion of a charged particle in a magnetic field. We study this in the high temperature classical and low temperature quantum domains. In both domains, we observe a transition of the mean square displacement from a monotonic behaviour to a damped oscillatory behaviour as one increases the strength of the magnetic field. When the strength of the magnetic field is negligible, the mean square displacement grows linearly with time in the classical domain and logarithmically with time in the quantum domain. We notice that these features of the mean square displacement are robust and remain essentially the same for an Ohmic dissipation model and a single relaxation time model for the memory kernel. The predictions stemming from our analysis can be tested against experiments in trapped cold ions.
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.
Fokker-Planck equation for a periodic potential
Physica A: Statistical Mechanics and its Applications, 1977
The Fokker-Planck equation in one dimension has been solved for a system in a periodic potential and linearly perturbed by a time-and space-varying external electric field. The solution is not exact, but we believe it is less approximate than any attempted previously on this problem. We have calculated the frequency and wave-vector dependent mobility and discussed its behaviour in some limiting cases. The application of our model to various physical situations is discussed.