Equation of Fokker-Planck

To seek the answer of a system given to an excitation is a problem running in physics and in various techniques. This answer is generally defined by a differential equation or more generally by a differential connection to several variables. For the simplicity of exposed one will stick to a traditional equation of the second order, this equation corresponding in particular to the oscillations of a mechanical system provided with a force of recall and a damping (see oscillating Systèmes with a degree of freedom).

The problem more the current relates to a sinewave excitation. If the equation is linear the answer is itself sinusoidal. On the contrary, when it contains nonlinear terms one can in general find only solutions approximate: numerical solutions, search for a sinusoidal approximation by the equivalent Linearization or development in a succession of sinusoids by the method of the disturbances.

One meets sometimes also stochastic differential equations in which the excitation is represented by a random process which is wholesale a whole of functions having the same statistical properties. Under these conditions the solution is also a random process characterized by a density of probability which relates to, for the second order, the excursions and speeds.

If one limits oneself to a Gaussian excitation, the answer given by a linear equation is also Gaussian and can be determined by means of the techniques of spectral description.

In the case of a nonlinear equation, the united probability of the movements and speeds is given by a partial derivative equation named equation of Fokker-Planck. Like the majority of the partial derivative equations, it gives explicit solutions only in quite particular cases relating at the same time on the form of the equation and the nature of the excitation.

General expression of the equation

For a probability distribution depending on NR variables:

\ frac {\ partial F} {\ partial T} = - \ sum_ {i=1} ^ {NR} \ frac {\ partial} {\ partial x_i} \ left D_i^1 (x_1, \ ldots, x_N) F \ right + \ sum_ {i=1} ^ {NR} \ sum_ {j=1} ^ {NR} \ frac {\ partial^2} {\ partial x_i \, \ partial x_j} \ left D_ {ij} ^2 (x_1, \ ldots, x_N) F \ right,

where $D^1$ is the vector of tendency (drift transistor) and $D^2$ the tensor of diffusion

Example

Thus, for the equation of the second order, it is convenient to suppose that the excitation is roughly a white Bruit, which is suitable for a little deadened system. The method applies in particular to the equation having a nonlinear force of recall:

M \ ddot {X} + B \ dowry {X} + H (X) = F (T)

in which F (T) represents a Gaussian white vibration of spectral concentration S₀ (density expressed in units squared by radian a second).

The density of united probability of the excursion and speed is written

p_ {X \ dowry {X}} (X, \ dowry {X}) = C e^ {- {B \ mathcal {E}}/{\ pi S_0}}

In this formula, $\ mathcal {E} = {1 \ over 2} \, M \, {\ dowry {X} ^2} \, + \, \ int_0^x H (\ xi) D \ xi$ is the total energy of the system not deadened. The density of probability speed remains Gaussian while that of the excursion is not it any more. It becomes again it obviously when the function H is linear.

It is remarkable that this solution, relatively simple, is exact whereas there does not exist nothing like it for a sinewave excitation.

In addition, such an exact solution does not exist if it is the damping which is nonlinear; in this case there exists an exact solution for a more abstract equation which provides a nonlinear approximation better than the linear approximation.

Case of the Brownian Movement

In the case of a movement of a particle and within the framework of the equation of Smoluchowski which relates to the particles such as $\ gamma v \ gg m has$ , typically molecules, or objects of mass " négligeable" (athmospheric molecules, proteins in biology…):

$\ dowry {X} + \ frac {F (X)}{\ gamma} = \ sigma B$

where B and a white vibration, $\ textstyle \ gamma$ the viscosity coefficient and F (X) a field of forces. If p (X, T) is the probability of finding the particle as in point X at the moment T, by application of the Lemme of Itô one has then:

$\ frac {\ partial p (X, T)}{\ partial T} = \ frac {\ sigma^2} {2 \ gamma^2} \ triangle p (X, T) + \ frac {F (X)}{\ gamma}. \ nabla p (X, T)$

where $\ textstyle \ frac {\ sigma^2} {2 \ gamma^2} =D$ the coefficient of diffusion.

This particular equation of Fokker Planck allows then, with conditions at the adequate edges and the beginning, to study the Brownian movement of a particle in a field of forces.

References

Flax, Y.K.: " Probabilistic Theory off Structural Dynamics" , Robert E. Krieger Publishing Company, New York (1967)
I. Kosztin, Non-Equilibrium Statistical Mechanics chap 4: Smoluchowski Diffusion Equation, http://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/LectureNotes/chp4.pdf
H. Risken, " The Fokker-Planck Equation: Methods off Solutions and Applications" , 2nd edition, Springer Series in Synergetics, Springer, ISBN 3-540-61530-X.
S. Redner: With guide to first passage processes. Cambridge University Close, (2001)

See too

Formula of Itô
Brownian Movement

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