Probability distribution Fokker-Planck equation

The second way is to obtain the solution of Eq. (2.6) for one-dimensional probability density with the initial distribution (2.9). Indeed, multiplying (2.6) by VT(xo, to) and integrating by xo while taking into account (2.4), we get the same Fokker-Planck equation (2.6). 

Let us now consider a NESS where the trap is moved at constant velocity, x t) = vt. It is not possible to solve the Fokker-Planck equation to find the probability distribution in the steady state for arbitrary potentials. Only for harmonic potentials, U x) = ioc /2, can the Fokker-Planck equation be solved exactly. The result is... 

If the velocity dependence of the rate of a reaction could be assumed to be constant and equal to k for velocities in excess of u0 and zero below 0, then reaction could be regarded as bleeding-off those reactant (Brownian) particles which have an energy in excess lmti02. This perturbs the velocity distribution of reactants and hence of solvent molecules [446]. Under such circumstances, the Fokker—Planck equation should be used to describe the chemical reaction. If this simple form of representing reaction is incorrect, there is little that can be done currently. The Fokker—Planck equation contains too much information about Brownian motion. In particular, the velocity dependence of the Brownian particles distribution is relatively unimportant. Davies [447] reduced the probability... 

An alternative treatment of the fluctuations is provided by the Fokker-Planck equation for the probability distribution of q... 

If A < 0 the stationary solution (1.4) is Gaussian. In fact, in that case it is possible by shifting y and rescaling, to reduce (1.5) to (IV.3.20), so that one may conclude the stationary Markov process determined by the linear Fokker-Planck equation is the Ornstein-Uhlenbeck process. For Al 0 there is no stationary probability distribution. 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

The probability distribution belonging to Eq. (18) obeys the diffusion-type magnetic Fokker-Planck equation [137, 140, 145]... 

The Brownian motion of a particle under the influence of an external force field, and its consequent escape over a potential barrier has to be treated, in general, using the Fokker-Planck equation. This equation gives the distribution function W governing the probability that a particle will be after time t at a point x with velocity u (Chandrasekhar, 1943). In one dimension it has the form ... 

It is important to notice that both the original and the modified Fokker-Planck equations give the probability distribution of a particle as a function of time, position and velocity. However, if we are interested in time intervals large enough compared to jS 1, the Fokker-Planck equation, equation (3), can be reduced to a diffusional equation for the distribution function w, frequently called the Smoluchowski equation (Chandrasekhar, 1943) ... 

The Langevin equation (3a, b) is equivalent to the following Fokker-Planck equation which drives the probability distribution in phase space ... 

The three relevant parameters y, f/a, and have been suitably defined above. The result of our perturbative approach is a Fokker-Planck equation for the reduced probability distribution P x t) of the form... 

We perform concrete calculations in the complex P-representation [Drummond 1980 McNeil 1983] in the frame of both probability distribution functions and stochastic equations for the complex c-number variables. We follow the standard procedures of quantum optics to eliminate the reservoir operators and to obtain a master equation for the density operator of the modes. The master equation is then transformed into a Fokker-Planck equation for the P-quasiprobability distribution function. In particular, for an ordinary NOPO and in the case of high cavity losses for the pump mode (73 7), if in the operational regime the pump depletion effects are involved, this approach yields... 

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

We have mentioned that the question posed above was answered in part by Shliomis and Stepanov [9]. They showed that for uniaxial particles, for weak applied magnetic fields, and in the noninertial limit, the equations of motion of the ferrofluid particle incorporating both the internal and the Brownian relaxation processes decouple from each other. Thus the reciprocal of the greatest relaxation time is the sum of the reciprocals of the Neel and Brownian relaxation times of both processes considered independently that is, those of a frozen Neel and a frozen Brownian mechanism In this instance the joint probability of the orientations of the magnetic moment and the particle in the fluid (i.e., the crystallographic axes) is the product of the individual probability distributions of the orientations of the axes and the particle so that the underlying Fokker Planck equation for the joint probability distribution also... 

In many practical situations the random process under observation is continuous in the sense that (1) the space of possible states is continuous (or it can be transformed to a continuous-like representation by a coarse-graining procedure), and (2) the change in the system state during a small time interval is small, that is, if the system is found in state x at time t then the probability to find it in state y x at time t + St vanishes when St 0. When these, and some other conditions detailed below, are satisfied, we can derive a partial differential equation for the probability distribution, the Fokker-Planck equation, which is discussed in this Section. 

In Appendix 8A we show that when these conditions are satisfied, the Chapman-Kolmogorov integral equation (8.118) leads to two partial differential equations. The Fokker-Planck equation describes the future evolution of the probability distribution... 

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. 

Equation (8.8.12) may be solved by passing to a Fokker-Planck equation (see Appendix 8. A) for Fic(iik,t) the probability distribution function for the normal coordinate... 

Most basic to the theoretical formulation of wonnlike chains is the distribution function G(r, u uo s), which represents the probability density that the tangent u(s) and the position r(s) of a contour point s have given values u and r, respectively, when the chain end s = 0 is at a laboratory-fixed coordinate origin and the tangent u(0) at that time has a given value uq. Since wormlike chains are Markoffian, this function satisfies a Fokker-Planck equation, which reads... 

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