A shortcut way to the Fokker-Planck equation for the non-Markovian dynamics (original) (raw)
Related papers
Relationship between a non-Markovian process and Fokker–Planck equation
Physics Letters A, 2006
We demonstrate the equivalence of a non-Markovian evolution equation with a linear memory-coupling and a Fokker–Planck equation (FPE). In case the feedback term offers a direct and permanent coupling of the current probability density to an initial distribution, the corresponding FPE offers a non-trivial drift term depending itself on the diffusion parameter. As the consequence the deterministic part of the underlying Langevin equation is likewise determined by the noise strength of the stochastic part. This memory induced stochastic behavior is discussed for different, but representative initial distributions. The analytical calculations are supported by numerical results.
Stochastic Langevin equations: Markovian and non-Markovian dynamics
Physical Review E, 2009
Non-Markovian stochastic Langevin-like equations of motion are compared to their corresponding Markovian (local) approximations. The validity of the local approximation for these equations, when contrasted with the fully nonlocal ones, is analyzed in details. The conditions for when the equation in a local form can be considered a good approximation are then explicitly specified. We study both the cases of additive and multiplicative noises, including system dependent dissipation terms, according to the Fluctuation-Dissipation theorem.
Nonlinear effects in the dynamics governed by non-Markovian stochastic Langevin-like equations
Journal of Physics: Conference Series, 2010
The influence of nonlinear effects in stochastic equations of motion with both additive and multiplicative noises is studied. Non-Markovian stochastic dynamics are compared with their corresponding Markovian (local approximations). Non-Markovian effects are implemented through Ornstein-Uhlenbeck and exponential damped harmonic dissipative kernels.
Nonergodic Brownian Dynamics and the Fluctuation-Dissipation Theorem
2006
Nonergodic Brownian motion is elucidated within the framework of the generalized Langevin equation. For thermal noise yielding either a vanishing or a divergent zero-frequency friction strength, the non-Markovian Browninan dynamics exhibits a riveting, anomalous diffusion behavior being characterized by a ballistic or possibly also a localized dynamics. As a consequence, such tailored thermal noise may cause a net acceleration of directed transport in a rocking Brownian motor. Two notable conditions for the thermal noise are identified in order to guarantee the fluctuation-dissipation theorem of first kind.
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.
Derivation of a Fokker–Planck equation for generalized Langevin dynamics
Physica A: Statistical Mechanics and its Applications, 2005
A Fokker-Planck equation describing the statistical properties of Brownian particles acted upon by long-range stochastic forces with power-law correlations is derived. In contrast with previous approaches (Wang, Phys. Rev. A 45 (1992) 2), it is shown that the distribution of Brownian particles after release from a point source is broader than Gaussian and described by a Fox function. Transport is shown to be ballistic at short times and either sub-diffusive or super-diffusive at large times. The imposition of occasional trapping events onto the Brownian dynamics can result in confined diffusion (d=dthx 2 i ! 0) at long times when the mean trapping time is divergent. It is suggested that such dynamics describe protein motions in cell membranes.
Intermediate dynamics between Newton and Langevin
Physical Review E, 2006
A dynamics between Newton and Langevin formalisms is elucidated within the framework of the generalized Langevin equation. For thermal noise yielding a vanishing zero-frequency friction the corresponding non-Markovian Brownian dynamics exhibits anomalous behavior which is characterized by ballistic diffusion and accelerated transport. We also investigate the role of a possible initial correlation between the system degrees of freedom and the heat-bath degrees of freedom for the asymptotic long-time behavior of the system dynamics. As two test beds we investigate ͑i͒ the anomalous energy relaxation of free non-Markovian Brownian motion that is driven by a harmonic velocity noise and ͑ii͒ the phenomenon of a net directed acceleration in noise-induced transport of an inertial rocking Brownian motor.
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.
Langevin-Vladimirsky approach to Brownian motion with memory
2015
A number of random processes in various fields of science is described by phenomenological equations of motion containing stochastic forces, the best known example being the Langevin equation (LE) for the Brownian motion (BM) of particles. Long ago Vladimirsky (1942) in a little known paper proposed a simple method for solving such equations. The method, based on the classical Gibbs statistics, consists in converting the stochastic LE into a deterministic equation for the mean square displacement of the particle, and is applicable to linear equations with any kind of memory in the dynamics of the system. This approach can be effectively used in solving many of the problems currently considered in the literature. We apply it to the description of the BM when the noise driving the particle is exponentially correlated in time. The problem of the hydrodynamic BM of a charged particle in an external magnetic field is also solved.
A generalized Langevin equation for dealing with nonadditive fluctuations
Journal of Statistical Physics, 1982
A suitable extension of the Mori memory-function formalism to the non-Hermitian case allows a "multiplicative" process to be described by a Langevin equation of non-Markoffian nature. This generalized Langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well-known theoretical approach developed by Kubo, the stochastic Liouville equation (SLE) theory. It is shown, furthermore, that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath. The stochastic process under study can also be described by a Fokker-Planck-like equation, which results in a Gaussian equilibrium distribution for the variable of interest. The main flaw of the SLE theory, that resulting in an uncorrect equilibrium distribution, is therefore completely eliminated.