View source for 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|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|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|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|Fokker–Planck equation]] (see [[Kolmogorov equation|Kolmogorov equation]]; [[Diffusion process|Diffusion process]]), for which the initial and boundary conditions are chosen in accordance with the specific problem considered.

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

====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 [[#References|[a1]]]. For references and discussion of the original work by Einstein and (von) Smoluchowski see the collection of papers reproduced in [[#References|[a2]]]. The Fokker–Planck equation corresponds to Kolmogorov's forward differential equation [[#References|[a3]]], Sect. 5.26. There exist non-Markovian processes satisfying the Chapman–Kolmogorov equation [[#References|[a4]]], Chapt. XV.13.

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