Fokker-Planck equation evolution times

In references (Santamaria Holek, 2005 2009 2001), the Smoluchowski equation was obtained by calculating the evolution equations for the first moments of the distribution function. These equations constitute the hydrodynamic level of description and can be obtained through the Fokker-Planck equation. The time evolution of the moments include relaxation equations for the diffusion current and the pressure tensor, whose form permits to elucidate the existence of inertial (short-time) and diffusion (long-time) regimes. As already mentioned, in the diffusion regime the mesoscopic description is carried out by means of a Smoluchowski equation and the equations for the moments coincide with the differential equations of nonequilibrium thermodynamics. 

This equation is the familiar Fokker-Planck equation for the time evolution of the distribution function for the number density of the nuclei with different sizes N. 

The perturbation reduction of the corresponding Markovian Fokker-Planck equation for the two-variable process (x t), ((t)) to an approximate one in x(t) has been carried out in Section V.A of Chapter II. For brevity we report only the approximate time-evolution equation for a x,t) up to order D-,... 

Figure 8. Time evolution of the Levy flight-PDF in the presence of the superharmonic external potential [Eq. (26)] with c — 4 (quartic Levy oscillator) and Levy index a = 1.2, obtained from the numerical solution of the fractional Fokker-Planck equation, using the Griinwald-Letnikov representation of the fractional Riesz derivative (full line). The initial condition is a 8-function at the origin. The dashed lines indicate the corresponding Boltzmann distribution. The transition from one to two maxima is clearly seen. This picture of the time evolution is typical for 2 < c <4 (see below).

This concept, which is based on a random walk with a well-defined characteristic time and which applies when collisions are frequent but weak [13], leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 71, due to Fiirth), we obtain the Fokker Planck equation for the evolution of the distribution function in phase space which describes normal diffusion. 

The essence of the calculation that leads to the desired Fokker-Planck equation, known in this limit as the Smoluchowski equation, is a coarse-grained average of the time evolution (8.131) over the fast variation of 7 (Z). This procedure, described in Appendix 8B, leads to... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

In principle, the presence of slow stochastic torques directly affecting the solute reorientational motion can be dealt with in the framework of generalized stochastic Fokker-Planck equations including frequency-dependent frictional terms. However, the non-Markovian nature of the time evolution operator does not allow an easy treatment of this kind of model. Also, it may be difficult to justify the choice of frequency dependent terms on the basis of a sound physical model. One would like to take advantage of some knowledge of the physical system under... 

Suppose the particle moves in the one-dimensional potential shown schematically in Fig. 3.2. Kramers assumes that the time evolution of the phase-space distribution function F q, v, t) is given by the Fokker-Planck equation... 

The (time) discretized evolution of the Fokker-Planck equation may be written in terms of R, and this gives the following Langevin-type equation... 

For a two state, a stochastic description of the dynamics is described by the possibility density Pi(x,i) for the motor to be at position x at time t in state i. This system progresses with period 1. The evolution of the system can be described by two Fokker—Planck equations with source terms ... 

The three-pulse EOM-PMA can be formulated not only in terms of density matrices and master equations but also in terms of wavefunctions and Schrodinger equations [29]. The EOM-PMA can therefore be straightforwardly incorporated into computer programs which provide the time evolution of the density matrix or the wavefunction of material systems. Besides the multilevel Redlield theory, the EOM-PMA can be combined with the Lindblad master equation [49], the surrogate Hamiltonian approach [49], the stochastic Liouville equation [18], the quantum Fokker-Planck equation [18], and the density matrix [50] or the wavefunction [14] multiconfigurational time-dependent Hartree (MCTDH) methods. When using the... 

We now summarize the equations that govern the dynamical evolution of the probability that the system is in state S at time t, which we denote by P S, t). Typically, such equations are written in one of three formalisms the master equation (ME), the Fokker-Planck equation (FPE), or the stochastic differential equation (SDE) each is summarized below in turn. Details of the derivations can be found in Refs. 2-4. 

The mapping of the time-evolution equation for the translocation kinetics to the Fokker-Planck equation allows immediate deduction of the various properties of polymer translocation, directly from the equations presented in Chapter 6. The inputs in obtaining the results are the free energy landscapes derived in Chapter 5 and the diffusion constants km. We give below the key results for polymer translocation by copying the general solutions presented in Chapter 6. We shall take the diffusion coefficient of the monomer km to be uniform (ko) in the following sections. 

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