Difference between revisions of "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 | 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. | 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==== | ||
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====References==== | ====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, | + | <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> |
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) |
Einstein-Smoluchowski equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Einstein-Smoluchowski_equation&oldid=11374