Fokker-Planck equation stationary solution

In particular let us take for the stationary solution of (1.10) = s = 1. Then (1.11) reduces to a time-independent Fokker-Planck equation whose solution is the Ornstein-Uhlenbeck process. More directly one finds from (1.12b)... 

The probability density of the response state vector of a nonlinear system under the excitation of Gaussian white noises is governed by a parabolic partial differential equation, called the Fokker-Planck equation. Exact solutions to such equations are difficult especially when both parametric (multiplicative) and external (additive) random excitations are present. In this paper, methods of solution for response vectors at the stationary state are discussed under two schemes based on the concept of detailed balance and the concept of generalized stationary potential, respectively. It is shown that the second scheme is more general and includes the first scheme as a special case. Examples are given to illustrate their applications. 

The Fokker-Planck equation is a partial differential equation. In most cases, its time-dependent solution is not known analytically. Also, if the Fokker-Planck equation has more than one state variable, exact stationary solutions are... 

Homogeneous, linear Fokker-Planck equations are known to admit a multi-variate Gaussian PDF as a solution.33 Thus, this closure scheme ensures that a joint Gaussian velocity PDF will result for statistically stationary, homogeneous turbulent flow. 

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. 

This is a linear Fokker-Planck equation. Apart from constants which can be scaled away, it is identical with the equation (IV.3.20) obeyed by the transition probability of the Ornstein-Uhlenbeck process. The stationary solution of (4.6) is the same as the Pl given in (IV.3.10). Thus, in equilibrium V(t) is the Ornstein-Uhlenbeck process. 

In the case of the quasilinear Fokker-Planck equation (2.4), the free energy U defined in terms of the stationary solution by (2.6) is identical with the potential in the deterministic equation (5.2). That identity is often taken for granted when time-dependent solutions have to be constructed for systems of which only the equilibrium distribution is known. We shall now show, however, that it holds only for systems of diffusion type whose Fokker-Planck equation is quasilinear, i.e., of the form (2.4). 

Condition (c) requires that the stationary solution of the Fokker-Planck equation should be the Maxwellian distribution function. Substitution leads to... 

We will now search for a non-trivial solution to the stationary Fokker-Planck equation. 

Formally, the H-theorem valid for general Fokker-Planck equations states that the solution of (3) becomes unique at long times [47], Yet, because colloidal particles have a non-penetrable core and exhibit excluded volume interactions, corresponding to regions where the potential is infinite, and the proof of the H-theorem requires fluctuations to overcome all barriers, the formal H-theorem may not hold for nondilute colloidal dispersions. Nevertheless, we assume that the system relaxes into a unique stationary state at long times, so that f (f 1 holds. This assumption... 

In case when the KM operator is limited to the adjoint FP operator, this one, on its turn, can be rewritten in function of the stationary solution (5.269) of the direct Fokker-Planck equation ... 

In order to write the eigen-functions associated to Fokker-Planck equation the stationary solution for the harmonic external potential should be firstly evaluated the stationary solution equation (5.298), with Eq. (5.302b), yields immediately ... 

The external enharmonic potential case will be also the next fiamewoik in which, by means of the path integrals formalism, the non-stationary solutions of the Fokker-Planck equation will be searched for. 

Worth to observe that by recovering the harmonic case (g O) there is automatically regain also the harmonic stationary solution (5.319). Besides, the general stationary solution can be cross-checked by considering the stationary condition in Fokker-Planck equation (5.268) and (5.269) ... 

Adding a term due to Gaussian white noise to the second equation of (5.176) and adopting the centre-manifold approach it can be calculated that the stationary solution of the reduced Fokker-Planck equation is... 

If the deterministic system is subject to multiplicative noise the stationary solution of the reduced Fokker-Planck equation is... 

In this chapter we formulate the thermodynamic and stochastic theory of the simple transport phenomena diffusion, thermal conduction and viscous ffow (1) to present results parallel to those listed in points 1-7, Sect. 8.1, for chemical kinetics. We still assume local equilibrium with respect to translational and internal degrees of freedom. We do not assume conditions close to chemical or hydrodynamic equilibrium. For chemical reactions and diffusion the macroscopic equations for a given reaction mechanism provide sufficient detail, the fluxes in the forward and reverse direction, to write a birth-death master equation with a stationary solution given in terms of For thermal conduction and viscous flow we derive the excess work and then find Fokker-Planck equations with stationary solutions given in terms of that excess work. 

Stratonovich has treated a simple case referred to as the case of stationary potential for which solution to the reduced Fokker-Planck equation is easily obtainable. Let the Fokker-Planck equation (5) be rewritten as... 

A reduced Fokker-Planck equation Is said to possess a solution of the type of generalized stationary potential if a consistent function can be found which satisfies Eqs. (26) and (27). 

The concept of detailed balance has the origin from thermodynamics. It describes a state not necessarily in thermal equilibrium where microscopic reversibility is permissible. On the other hand, the concept of generalized stationary potential is based on the pattern of probability flow. The two concepts are unrelated, and it is remarkable that the procedures developed from the two to obtain exact solutions for the Fokker-Planck equations are essentially the same. However, since one of the conditions for detailBd balance, which places a restriction on the type of diffusion coefficients, is not required in the method of generalized stationary potential, the latter method is more general. 

Here, / is the probability density of the velocity of the Brownian particle. The stationary solution to the Fokker-Planck equation given by (86) is given as... 

This type of equation is known as a Fokker-Planck equation. The stationary solution of (11) is given by... 

The stationary solution of the Fokker-Planck equation, which includes the friction force F=— /3v, and the momentum diffusion coefficient (eqn 5.22), is a 3D Gaussian distribution... 

Solutions to the master equation, the Fokker-Planck equation and the mean value equations in the stationary and in the time dependent cases will now be derived. 

Corresponding to the exact stationary solution (2.54 or 55) of the master equation the stationary solution M of the Fokker-Planck equation can also be determined. 

Case c) Fluctuation Initiated Motion. If motion is started from a distribution concentrated around an unstable stationary point Xq where K(xo) = 0 and K (xq) = y > 0, the expansion of (2.82, 84,85) fails since the initial fluctuations are enhanced exponentially, see (2.92) before the drift dominated motion sets in. The appropriate approximation here, developed by Haake [2.2, 3] and by Suzuki [2.4] essentially consists of two steps 1) Solve the Fokker-Planck equation for the first fluctuation dominated stage and 2) Find a smoothly fitting solution for the second drift dominated stage. 

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