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''Machlup–Onsager function''
 
''Machlup–Onsager function''
  
 
A function having its origin in physics and arising in a particular description of the dynamics of macroscopic systems. In this description the starting point is the calculation of a probability density for observing a complete path of a system in phase space spanned by the macroscopic variables. This approach was pioneered by L. Onsager and S. Machlup in [[#References|[a1]]], who used this to develop a theory of fluctuations in (non-) equilibrium thermodynamics. Their work was restricted to the linear Gaussian case, which was subsequently extended to non-linear equations. This probability density can be expressed, apart from a normalizing factor, by means of a functional integral over paths of the process. The corresponding integrand has the form of the Lagrangian and has been called the Onsager–Machlup function by physicists.
 
A function having its origin in physics and arising in a particular description of the dynamics of macroscopic systems. In this description the starting point is the calculation of a probability density for observing a complete path of a system in phase space spanned by the macroscopic variables. This approach was pioneered by L. Onsager and S. Machlup in [[#References|[a1]]], who used this to develop a theory of fluctuations in (non-) equilibrium thermodynamics. Their work was restricted to the linear Gaussian case, which was subsequently extended to non-linear equations. This probability density can be expressed, apart from a normalizing factor, by means of a functional integral over paths of the process. The corresponding integrand has the form of the Lagrangian and has been called the Onsager–Machlup function by physicists.
  
R.L. Stratonovich [[#References|[a2]]] first calculated this Onsager–Machlup function from a probabilistic viewpoint. The idea is to fix a smooth path in the state space, form a tube of small radius around this path and calculate asymptotically the probability of the sample paths of a diffusion lying within this tube. The most general result in this direction may be found in [[#References|[a3]]]. Consider a [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300401.png" /> and suppose that a non-singular [[Diffusion process|diffusion process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300402.png" /> is generated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300403.png" /> by
+
R.L. Stratonovich [[#References|[a2]]] first calculated this Onsager–Machlup function from a probabilistic viewpoint. The idea is to fix a smooth path in the state space, form a tube of small radius around this path and calculate asymptotically the probability of the sample paths of a diffusion lying within this tube. The most general result in this direction may be found in [[#References|[a3]]]. Consider a [[Riemannian manifold|Riemannian manifold]] $M$ and suppose that a non-singular [[Diffusion process|diffusion process]] $X ( . )$ is generated on $M$ by
  
<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/o/o130/o130040/o1300404.png" /></td> </tr></table>
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\begin{equation*} A = \frac { 1 } { 2 } \Delta + b, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300405.png" /> is the Laplace–Beltrami operator (cf. [[Laplace–Beltrami equation|Laplace–Beltrami equation]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300406.png" /> is a [[Vector field|vector field]]. Let
+
where $\Delta$ is the Laplace–Beltrami operator (cf. [[Laplace–Beltrami equation|Laplace–Beltrami equation]]) and $b$ is a [[Vector field|vector field]]. Let
  
<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/o/o130/o130040/o1300407.png" /></td> </tr></table>
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\begin{equation*} \mu _ { \varepsilon } ^ { x } : = \mathcal{P} _ { x } \{ \omega : \rho ( X _ { t } ( \omega ) , \phi ( t ) ) \leq \varepsilon \text { for every }t \in [ 0 , T ] \}, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o1300409.png" />, is a smooth curve on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004011.png" /> is the Riemannian distance. For any two smooth curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004013.png" />,
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where $\phi : [ 0 , T ] \rightarrow M$, $\phi ( 0 ) = x$, is a smooth curve on $M$ and $\rho ( x , y )$ is the Riemannian distance. For any two smooth curves $\phi$ and $\psi$,
  
<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/o/o130/o130040/o13004014.png" /></td> </tr></table>
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\begin{equation*} \operatorname { lim } _ { \varepsilon \downarrow 0 } \frac { \mu _ { \varepsilon } ^ { x } ( \phi ) } { \mu _ { \varepsilon } ^ { x } ( \psi ) } \end{equation*}
  
 
exists and can be expressed as
 
exists and can be expressed as
  
<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/o/o130/o130040/o13004015.png" /></td> </tr></table>
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\begin{equation*} \operatorname { exp } \left[ \int _ { 0 } ^ { T } L ( \dot { \phi } ( s ) , \phi ( s ) ) d s - \int _ { 0 } ^ { T } L ( \dot { \psi } ( s ) , \psi ( s ) ) d s \right] \end{equation*}
  
for some functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004016.png" /> on the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004017.png" />. This function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004018.png" />, which has the form of a [[Lagrangian|Lagrangian]] (cf. also [[Lagrange function|Lagrange function]]), is the Onsager–Machlup function. A detailed calculation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004019.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o130/o130040/o13004020.png" /> may be founded in [[#References|[a4]]].
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for some functional $L ( \dot { x } , x )$ on the tangent bundle $T M$. This function $L$, which has the form of a [[Lagrangian|Lagrangian]] (cf. also [[Lagrange function|Lagrange function]]), is the Onsager–Machlup function. A detailed calculation for $L ( \dot { x } , x )$ when $M = \mathbf{R} ^ { d }$ may be founded in [[#References|[a4]]].
  
 
The Onsager–Machlup function has recently (1990s) been applied to the non-linear filtering problem. Along with the development of the usual theory using conditional mean, various attempts have been made in the 1960s to heuristically calculate the conditional mode in the non-linear filtering problem (maximum a posteriori estimator). The mathematical difficulty arose due to the lack of a translation-invariant measure in function spaces. The first correct probabilistic interpretation of this filter has been given by A. Dembo and D. Zeitouni [[#References|[a5]]] using the Onsager–Machlup function. Maximizing this function may be interpreted as giving the most probable trajectory, which is a natural generalization of the maximum-likelihood principle in statistics (cf. also [[Maximum-likelihood method|Maximum-likelihood method]]). An extension of this approach to the estimation (smoothing) problem for random fields (cf. also [[Random field|Random field]]) may be found in [[#References|[a6]]].
 
The Onsager–Machlup function has recently (1990s) been applied to the non-linear filtering problem. Along with the development of the usual theory using conditional mean, various attempts have been made in the 1960s to heuristically calculate the conditional mode in the non-linear filtering problem (maximum a posteriori estimator). The mathematical difficulty arose due to the lack of a translation-invariant measure in function spaces. The first correct probabilistic interpretation of this filter has been given by A. Dembo and D. Zeitouni [[#References|[a5]]] using the Onsager–Machlup function. Maximizing this function may be interpreted as giving the most probable trajectory, which is a natural generalization of the maximum-likelihood principle in statistics (cf. also [[Maximum-likelihood method|Maximum-likelihood method]]). An extension of this approach to the estimation (smoothing) problem for random fields (cf. also [[Random field|Random field]]) may be found in [[#References|[a6]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Onsager,  S. Machlup,  "Fluctuations and irreversible proceses, I-II"  ''Phys. Rev.'' , '''91'''  (1953)  pp. 1505–1512; 1512–1515</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.L. Stratonovich,  "On the probability functional of diffusion processes"  ''Selected Transl. Math. Statist. Prob.'' , '''10'''  (1971)  pp. 273–286</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Fujita,  S. Kotani,  "The Onsager–Machlup function for diffusion processes"  ''J. Math. Kyoto Univ.'' , '''22'''  (1982)  pp. 115–130</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Ikeda,  S. Watanabe,  "Stochastic differential equations and diffusion processes" , North-Holland  (1989)  (Edition: Second)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Dembo,  O. Zeitouni,  "A maximum a posteriori estimator for trajectories of diffusion processes"  ''Stochastics'' , '''20'''  (1987)  pp. 221–246</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S.I. Aihara,  A. Bagchi,  "Nonlinear smoothing for random fields"  ''Stochastic Processes Appl.'' , '''55'''  (1995)  pp. 143–158</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  L. Onsager,  S. Machlup,  "Fluctuations and irreversible proceses, I-II"  ''Phys. Rev.'' , '''91'''  (1953)  pp. 1505–1512; 1512–1515</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.L. Stratonovich,  "On the probability functional of diffusion processes"  ''Selected Transl. Math. Statist. Prob.'' , '''10'''  (1971)  pp. 273–286</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  T. Fujita,  S. Kotani,  "The Onsager–Machlup function for diffusion processes"  ''J. Math. Kyoto Univ.'' , '''22'''  (1982)  pp. 115–130</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Ikeda,  S. Watanabe,  "Stochastic differential equations and diffusion processes" , North-Holland  (1989)  (Edition: Second)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Dembo,  O. Zeitouni,  "A maximum a posteriori estimator for trajectories of diffusion processes"  ''Stochastics'' , '''20'''  (1987)  pp. 221–246</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  S.I. Aihara,  A. Bagchi,  "Nonlinear smoothing for random fields"  ''Stochastic Processes Appl.'' , '''55'''  (1995)  pp. 143–158</td></tr></table>

Latest revision as of 16:52, 1 July 2020

Machlup–Onsager function

A function having its origin in physics and arising in a particular description of the dynamics of macroscopic systems. In this description the starting point is the calculation of a probability density for observing a complete path of a system in phase space spanned by the macroscopic variables. This approach was pioneered by L. Onsager and S. Machlup in [a1], who used this to develop a theory of fluctuations in (non-) equilibrium thermodynamics. Their work was restricted to the linear Gaussian case, which was subsequently extended to non-linear equations. This probability density can be expressed, apart from a normalizing factor, by means of a functional integral over paths of the process. The corresponding integrand has the form of the Lagrangian and has been called the Onsager–Machlup function by physicists.

R.L. Stratonovich [a2] first calculated this Onsager–Machlup function from a probabilistic viewpoint. The idea is to fix a smooth path in the state space, form a tube of small radius around this path and calculate asymptotically the probability of the sample paths of a diffusion lying within this tube. The most general result in this direction may be found in [a3]. Consider a Riemannian manifold $M$ and suppose that a non-singular diffusion process $X ( . )$ is generated on $M$ by

\begin{equation*} A = \frac { 1 } { 2 } \Delta + b, \end{equation*}

where $\Delta$ is the Laplace–Beltrami operator (cf. Laplace–Beltrami equation) and $b$ is a vector field. Let

\begin{equation*} \mu _ { \varepsilon } ^ { x } : = \mathcal{P} _ { x } \{ \omega : \rho ( X _ { t } ( \omega ) , \phi ( t ) ) \leq \varepsilon \text { for every }t \in [ 0 , T ] \}, \end{equation*}

where $\phi : [ 0 , T ] \rightarrow M$, $\phi ( 0 ) = x$, is a smooth curve on $M$ and $\rho ( x , y )$ is the Riemannian distance. For any two smooth curves $\phi$ and $\psi$,

\begin{equation*} \operatorname { lim } _ { \varepsilon \downarrow 0 } \frac { \mu _ { \varepsilon } ^ { x } ( \phi ) } { \mu _ { \varepsilon } ^ { x } ( \psi ) } \end{equation*}

exists and can be expressed as

\begin{equation*} \operatorname { exp } \left[ \int _ { 0 } ^ { T } L ( \dot { \phi } ( s ) , \phi ( s ) ) d s - \int _ { 0 } ^ { T } L ( \dot { \psi } ( s ) , \psi ( s ) ) d s \right] \end{equation*}

for some functional $L ( \dot { x } , x )$ on the tangent bundle $T M$. This function $L$, which has the form of a Lagrangian (cf. also Lagrange function), is the Onsager–Machlup function. A detailed calculation for $L ( \dot { x } , x )$ when $M = \mathbf{R} ^ { d }$ may be founded in [a4].

The Onsager–Machlup function has recently (1990s) been applied to the non-linear filtering problem. Along with the development of the usual theory using conditional mean, various attempts have been made in the 1960s to heuristically calculate the conditional mode in the non-linear filtering problem (maximum a posteriori estimator). The mathematical difficulty arose due to the lack of a translation-invariant measure in function spaces. The first correct probabilistic interpretation of this filter has been given by A. Dembo and D. Zeitouni [a5] using the Onsager–Machlup function. Maximizing this function may be interpreted as giving the most probable trajectory, which is a natural generalization of the maximum-likelihood principle in statistics (cf. also Maximum-likelihood method). An extension of this approach to the estimation (smoothing) problem for random fields (cf. also Random field) may be found in [a6].

References

[a1] L. Onsager, S. Machlup, "Fluctuations and irreversible proceses, I-II" Phys. Rev. , 91 (1953) pp. 1505–1512; 1512–1515
[a2] R.L. Stratonovich, "On the probability functional of diffusion processes" Selected Transl. Math. Statist. Prob. , 10 (1971) pp. 273–286
[a3] T. Fujita, S. Kotani, "The Onsager–Machlup function for diffusion processes" J. Math. Kyoto Univ. , 22 (1982) pp. 115–130
[a4] N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland (1989) (Edition: Second)
[a5] A. Dembo, O. Zeitouni, "A maximum a posteriori estimator for trajectories of diffusion processes" Stochastics , 20 (1987) pp. 221–246
[a6] S.I. Aihara, A. Bagchi, "Nonlinear smoothing for random fields" Stochastic Processes Appl. , 55 (1995) pp. 143–158
How to Cite This Entry:
Onsager-Machlup function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Onsager-Machlup_function&oldid=16546
This article was adapted from an original article by A. Bagchi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article