# Einstein-Smoluchowski equation

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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.

#### References

 [1] A. Einstein, M. von Smoluchowski, "Brownian motion" , Moscow-Leningrad (1936) (In Russian; translated from German) [2] S. Chandrasekhar, "Stochastic problems in physics and astronomy" Rev. Modern Physics , 15 (1943) pp. 1–89 [3] M. Kac, "Probability and related topics in physical sciences" , Proc. summer sem. Boulder, Col., 1957 , 1 , Interscience (1959) pp. Chapt. 4