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An integral equation for the probability density of the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352301.png" /> from a state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352302.png" /> at a moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352303.png" /> to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352304.png" /> at a moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352305.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352306.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352307.png" /></td> </tr></table>
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An integral equation for the probability density of the transition function  $  P ( t _ {0} , x _ {0} \mid  t , x ) $
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from a state  $  x _ {0} $
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at a moment  $  t _ {0} $
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to a point  $  x $
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at a moment  $  t $:
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352308.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e0352309.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523010.png" /> of its possible states preceding the moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523011.png" />. 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]].
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$$
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P ( t _ {0} , x _ {0} \mid  t , x )  = \
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\int\limits P ( t _ {0} , x _ {0} \mid  t  ^  \prime  , x  ^  \prime  ) P
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( t  ^  \prime  , x  ^  \prime  \mid  t , x ) dx  ^  \prime  ,
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$$
  
The physical analysis of a process of Brownian-motion type shows that it can be described by means of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523012.png" /> on intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523013.png" /> considerably larger than the correlation time of the stochastic process (even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523014.png" /> formally), and that the moments
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$$
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t _ {0}  < t  ^  \prime  < t ,\  \int\limits P ( t _ {0} , x _ {0} \mid  t , x )  dx  = 1 .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523015.png" /></td> </tr></table>
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The function  $  P $
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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} $
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to  $  t $
 +
of its possible states preceding the moment  $  t _ {0} $.
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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]].
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The physical analysis of a process of Brownian-motion type shows that it can be described by means of the function  $  P $
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on intervals  $  \Delta t = t - t _ {0} $
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considerably larger than the correlation time of the stochastic process (even if  $  \Delta t \rightarrow 0 $
 +
formally), and that the moments
 +
 
 +
$$
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\overline{ {( x - x _ {0} )  ^ {k} }}\;  =  M _ {k}  $$
  
 
computed by means of this function must satisfy
 
computed by means of this function must satisfy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035230/e03523016.png" /></td> </tr></table>
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$$
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\lim\limits _ {\Delta t \rightarrow 0 } 
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\frac{M _ {k} }{\Delta t }
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  = 0 ,\ \
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k \geq  3 ; \  \lim\limits _ {\Delta t \rightarrow 0 } \
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 +
\frac{M _ {2} }{\Delta t }
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  \neq  0 .
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$$
  
 
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.
 
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.
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====References====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 19:37, 5 June 2020


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)
How to Cite This Entry:
Einstein-Smoluchowski equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Einstein-Smoluchowski_equation&oldid=46796
This article was adapted from an original article by I.A. Kvasnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article