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.

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

Comments

The chain equation for the transition density of a Markov process is usually called the Chapman–Kolmogorov equation in the English literature. It was already introduced in 1900 by L. Bachelier, see [a1]. For references and discussion of the original work by Einstein and (von) Smoluchowski see the collection of papers reproduced in [a2]. The Fokker–Planck equation corresponds to Kolmogorov's forward differential equation [a3], Sect. 5.26. There exist non-Markovian processes satisfying the Chapman–Kolmogorov equation [a4], Chapt. XV.13.

References

[a1]	P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965)
[a2]	N. Wax (ed.) , Selected papers on noise and stochastic processes , Dover, reprint (1954)
[a3]	E.B. Dynkin, "Markov processes" , 1 , Springer (1965) pp. Sect. 5.26 (Translated from Russian)
[a4]	W. Feller, "An introduction to probability theory and its applications", 1 , Wiley (1966) pp. Chapt. XV.13
[a5]	I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , II , Springer (1975) (Translated from Russian)