# Einstein-Smoluchowski equation

An integral equation for the probability density of the transition function $P ( t _ {0} , x _ {0} \mid t , x )$ from a state $x _ {0}$ at a moment $t _ {0}$ to a point $x$ at a moment $t$:

$$P ( t _ {0} , x _ {0} \mid t , x ) = \ \int\limits P ( t _ {0} , x _ {0} \mid t ^ \prime , x ^ \prime ) P ( t ^ \prime , x ^ \prime \mid t , x ) dx ^ \prime ,$$

$$t _ {0} < t ^ \prime < t ,\ \int\limits P ( t _ {0} , x _ {0} \mid t , x ) dx = 1 .$$

The function $P$ describes a stochastic process without after-effects (a Markov process), one characteristic feature of which is the independence of the evolution of the system from $t _ {0}$ to $t$ of its possible states preceding the moment $t _ {0}$. The equation was formulated by M. von Smoluchowski (1906) in connection with the representation of Brownian motion as a stochastic process, and was developed simultaneously by him and A. Einstein. In the literature the Einstein–Smoluchowski equation is called the Kolmogorov–Chapman equation.

The physical analysis of a process of Brownian-motion type shows that it can be described by means of the function $P$ on intervals $\Delta t = t - t _ {0}$ considerably larger than the correlation time of the stochastic process (even if $\Delta t \rightarrow 0$ formally), and that the moments

$$\overline{ {( x - x _ {0} ) ^ {k} }}\; = M _ {k}$$

computed by means of this function must satisfy

$$\lim\limits _ {\Delta t \rightarrow 0 } \frac{M _ {k} }{\Delta t } = 0 ,\ \ k \geq 3 ; \ \lim\limits _ {\Delta t \rightarrow 0 } \ \frac{M _ {2} }{\Delta t } \neq 0 .$$

In this case the Einstein–Smoluchowski equation reduces to a linear differential equation of parabolic type, called the Fokker–Planck equation (see Kolmogorov equation; Diffusion process), for which the initial and boundary conditions are chosen in accordance with the specific problem considered.

How to Cite This Entry:
Einstein–Smoluchowski equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Einstein%E2%80%93Smoluchowski_equation&oldid=22378