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Published: 10 December 2022
Last updated: 28 January 2023

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1

Koide, J. "Microscopic Basis for Onsager-Machlup Theory." Progress of Theoretical Physics 102, no. 6 (December 1, 1999): 1065–84. http://dx.doi.org/10.1143/ptp.102.1065.

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2

Ayanbayev, Birzhan, Ilja Klebanov, Han Cheng Lie, and T. J. Sullivan. "Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces." Inverse Problems 38, no. 2 (December 28, 2021): 025006. http://dx.doi.org/10.1088/1361-6420/ac3f82.

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Abstract We derive Onsager–Machlup functionals for countable product measures on weighted ℓ p subspaces of the sequence space R N . Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Γ-convergence of sequences of Onsager–Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 ⩽ p ⩽ 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.
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3

Sugiura, Nozomi. "The Onsager–Machlup functional for data assimilation." Nonlinear Processes in Geophysics 24, no. 4 (December 1, 2017): 701–12. http://dx.doi.org/10.5194/npg-24-701-2017.

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Abstract. When taking the model error into account in data assimilation, one needs to evaluate the prior distribution represented by the Onsager–Machlup functional. Through numerical experiments, this study clarifies how the prior distribution should be incorporated into cost functions for discrete-time estimation problems. Consistent with previous theoretical studies, the divergence of the drift term is essential in weak-constraint 4D-Var (w4D-Var), but it is not necessary in Markov chain Monte Carlo with the Euler scheme. Although the former property may cause difficulties when implementing w4D-Var in large systems, this paper proposes a new technique for estimating the divergence term and its derivative.
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4

Moret, Sílvia, and David Nualart. "Onsager-Machlup functional for the fractional Brownian motion." Probability Theory and Related Fields 124, no. 2 (October 1, 2002): 227–60. http://dx.doi.org/10.1007/s004400200211.

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5

Wolf, Eddy Mayer, and Ofer Zeitouni. "Onsager Machlup functionals for non trace class SPDE's." Probability Theory and Related Fields 95, no. 2 (June 1993): 199–216. http://dx.doi.org/10.1007/bf01192270.

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6

Tanaka, Shigenori. "Information geometrical characterization of the Onsager-Machlup process." Chemical Physics Letters 689 (December 2017): 152–55. http://dx.doi.org/10.1016/j.cplett.2017.10.005.

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7

BONACCORSI, STEFANO. "ONSAGER–MACHLUP FUNCTIONAL FOR VOLTERRA EQUATIONS PERTURBED BY NOISE." Stochastics and Dynamics 02, no. 04 (December 2002): 587–98. http://dx.doi.org/10.1142/s0219493702000534.

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The Onsager–Machlup operator is a useful tool in order to study the regularity of the trajectories of the solution to a stochastic differential equation. In this paper, we prove the existence of this operator for the solution of a stochastic Volterra equation in bounded domain. This kind of equation has relevant interest in the applications, as discussed in [6] or [18].
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8

Hu, Jianyu, Xiaoli Chen, and Jinqiao Duan. "An Onsager–Machlup approach to the most probable transition pathway for a genetic regulatory network." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 4 (April 2022): 041103. http://dx.doi.org/10.1063/5.0088397.

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We investigate a quantitative network of gene expression dynamics describing the competence development in Bacillus subtilis. First, we introduce an Onsager–Machlup approach to quantify the most probable transition pathway for both excitable and bistable dynamics. Then, we apply a machine learning method to calculate the most probable transition pathway via the Euler–Lagrangian equation. Finally, we analyze how the noise intensity affects the transition phenomena.
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9

Ayryan, Edik, Alexander Egorov, Dmitri Kulyabov, Victor Malyutin, and Leonid Sevastianov. "Functional Integral Approach to the Solution of a System of Stochastic Differential Equations." EPJ Web of Conferences 173 (2018): 02003. http://dx.doi.org/10.1051/epjconf/201817302003.

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A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature.
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10

Shepp, Larry A., and Ofer Zeitouni. "A Note on Conditional Exponential Moments and Onsager-Machlup Functionals." Annals of Probability 20, no. 2 (April 1992): 652–54. http://dx.doi.org/10.1214/aop/1176989796.

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11

Taniguchi, Tooru, and E. G. D. Cohen. "Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems." Journal of Statistical Physics 126, no. 1 (December 22, 2006): 1–41. http://dx.doi.org/10.1007/s10955-006-9252-2.

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12

Capitaine, Mireille. "Onsager-Machlup functional for some smooth norms on Wiener space." Probability Theory and Related Fields 102, no. 2 (June 1995): 189–201. http://dx.doi.org/10.1007/bf01213388.

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13

Lee, Juyong, and Bernard R. Brooks. "Direct global optimization of Onsager-Machlup action using Action-CSA." Chemical Physics 535 (July 2020): 110768. http://dx.doi.org/10.1016/j.chemphys.2020.110768.

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14

Chaleyat-Maurel, Mireille, and David Nualart. "The Onsager-Machlup functional for a class of anticipating processes." Probability Theory and Related Fields 94, no. 2 (June 1992): 247–70. http://dx.doi.org/10.1007/bf01192445.

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15

Fujisaki, Hiroshi, Akinori Kidera, and Motoyuki Shiga. "Sampling Path Ensembles using the Onsager-Machlup Action with Replica Exchange." Biophysical Journal 98, no. 3 (January 2010): 573a. http://dx.doi.org/10.1016/j.bpj.2009.12.3111.

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16

Hasegawa, Hiroshi. "Self-contained framework of stochastic mechanics for reconstructing the Onsager-Machlup theory." Physical Review D 33, no. 8 (April 15, 1986): 2508–11. http://dx.doi.org/10.1103/physrevd.33.2508.

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17

Carmona, RenéA, and David Nualart. "Traces of random variables on Wiener space and the Onsager-Machlup functional." Journal of Functional Analysis 107, no. 2 (August 1992): 402–38. http://dx.doi.org/10.1016/0022-1236(92)90116-z.

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18

Li, Tiejun, and Xiaoguang Li. "Gamma-Limit of the Onsager--Machlup Functional on the Space of Curves." SIAM Journal on Mathematical Analysis 53, no. 1 (January 2021): 1–31. http://dx.doi.org/10.1137/20m1310539.

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19

Serdyukov, S. I., and V. K. Bel’nov. "Extension of the variational formulation of the Onsager-Machlup theory of fluctuations." Physical Review E 51, no. 5 (May 1, 1995): 4190–95. http://dx.doi.org/10.1103/physreve.51.4190.

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20

Bardina, Xavier, Carles Rovira, and Samy Tindel. "Onsager Machlup Functional for Stochastic Evolution Equations in a Class of Norms." Stochastic Analysis and Applications 21, no. 6 (January 11, 2003): 1231–53. http://dx.doi.org/10.1081/sap-120026105.

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21

Zeitouni, Ofer. "On the Onsager-Machlup Functional of Diffusion Processes Around Non $C^2$ Curves." Annals of Probability 17, no. 3 (July 1989): 1037–54. http://dx.doi.org/10.1214/aop/1176991255.

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22

Singh, Navinder. "Onsager-Machlup Theory and Work Fluctuation Theorem for a Harmonically Driven Brownian Particle." Journal of Statistical Physics 131, no. 3 (February 23, 2008): 405–14. http://dx.doi.org/10.1007/s10955-008-9503-5.

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23

Hu, Yaozhong. "MULTI-DIMENSIONAL GEOMETRIC BROWNIAN MOTIONS, ONSAGER-MACHLUP FUNCTIONS, AND APPLICATIONS TO MATHEMATICAL FINANCE." Acta Mathematica Scientia 20, no. 3 (July 2000): 341–58. http://dx.doi.org/10.1016/s0252-9602(17)30641-0.

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24

SEWELL, GEOFFREY L. "QUANTUM MACROSTATISTICAL THEORY OF NONEQUILIBRIUM STEADY STATES." Reviews in Mathematical Physics 17, no. 09 (October 2005): 977–1020. http://dx.doi.org/10.1142/s0129055x05002492.

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We provide a general macrostatistical formulation of nonequilibrium steady states of reservoir driven quantum systems. This formulation is centered on the large scale properties of the locally conserved hydrodynamical observables, and our basic physical assumptions comprise (a) a chaoticity hypothesis for the nonconserved currents carried by these observables, (b) an extension of Onsager's regression hypothesis to fluctuations about nonequilibrium states, and (c) a certain mesoscopic local equilibrium hypothesis. On this basis, we obtain a picture wherein the fluctuations of the hydrodynamical variables about a nonequilibrium steady state execute a Gaussian Markov process of a generalized Onsager–Machlup type, which is completely determined by the position dependent transport coefficients and the equilibrium entropy function of the system. This picture reveals that the transport coefficients satisfy a generalized form of the Onsager reciprocity relations in the nonequilibrium situation and that the spatial correlations of the hydrodynamical observables are generically of long range. This last result constitutes a model-independent quantum mechanical generalization of that obtained for special classical stochastic systems and marks a striking difference between the steady nonequilibrium and equilibrium states, since it is only at critical points that the latter carry long range correlations.
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25

Ayanbayev, Birzhan, Ilja Klebanov, Han Cheng Lie, and T. J. Sullivan. "Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems." Inverse Problems 38, no. 2 (December 28, 2021): 025005. http://dx.doi.org/10.1088/1361-6420/ac3f81.

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Abstract The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager–Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Γ-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.
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26

Fujisaki, Hiroshi, Motoyuki Shiga, Kei Moritsugu, and Akinori Kidera. "Multiscale enhanced path sampling based on the Onsager-Machlup action: Application to a model polymer." Journal of Chemical Physics 139, no. 5 (August 7, 2013): 054117. http://dx.doi.org/10.1063/1.4817209.

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27

Chao, Ying, and Jinqiao Duan. "The Onsager–Machlup function as Lagrangian for the most probable path of a jump-diffusion process." Nonlinearity 32, no. 10 (September 5, 2019): 3715–41. http://dx.doi.org/10.1088/1361-6544/ab248b.

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28

Dembo, Amir, and Ofer Zeitouni. "Onsager-Machlup functionals and maximum a posteriori estimation for a class of non-gaussian random fields." Journal of Multivariate Analysis 36, no. 2 (February 1991): 243–62. http://dx.doi.org/10.1016/0047-259x(91)90060-f.

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29

Koehl, Patrice. "Minimum action principle and shape dynamics." Journal of The Royal Society Interface 14, no. 130 (May 2017): 20170031. http://dx.doi.org/10.1098/rsif.2017.0031.

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In this paper, we propose a new method for computing a distance between two shapes embedded in three-dimensional space. Instead of comparing directly the geometric properties of the two shapes, we measure the cost of deforming one of the two shapes into the other. The deformation is computed as the geodesic between the two shapes in the space of shapes. The geodesic is found as a minimizer of the Onsager–Machlup action, based on an elastic energy for shapes that we define. Its length is set to be the integral of the action along that path; it defines an intrinsic quasi-metric on the space of shapes. We illustrate applications of our method to geometric morphometrics using three datasets representing bones and teeth of primates. Experiments on these datasets show that the variational quasi-metric we have introduced performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.
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30

Renger, D. "Gradient and GENERIC Systems in the Space of Fluxes, Applied to Reacting Particle Systems." Entropy 20, no. 8 (August 9, 2018): 596. http://dx.doi.org/10.3390/e20080596.

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In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager–Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or GENERIC system in the space of fluxes. In a general setting we study how flux gradient or GENERIC systems are related to gradient systems of concentrations. This shows that many gradient or GENERIC systems arise from an underlying gradient or GENERIC system where fluxes rather than densities are being driven by (free) energies. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.
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31

Chen, Jianyu, Jianyu Hu, Wei Wei, and Jinqiao Duan. "A data-driven approach for discovering the most probable transition pathway for a stochastic carbon cycle system." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 11 (November 2022): 113140. http://dx.doi.org/10.1063/5.0116643.

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Many natural systems exhibit tipping points where changing environmental conditions spark a sudden shift to a new and sometimes quite different state. Global climate change is often associated with the stability of marine carbon stocks. We consider a stochastic carbonate system of the upper ocean to capture such transition phenomena. Based on the Onsager–Machlup action functional theory, we calculate the most probable transition pathway between the metastable and oscillatory states via a neural shooting method. Furthermore, we explore the effects of external random carbon input rates on the most probable transition pathway, which provides a basis to recognize naturally occurring tipping points. Particularly, we investigate the transition pathway’s dependence on the transition time and further compute the optimal transition time using a physics-informed neural network, toward the maximum carbonate concentration state in the oscillatory regimes. This work may offer some insights into the effects of noise-affected carbon input rates on transition phenomena in stochastic models.
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32

Fujisaki, Hiroshi, Motoyuki Shiga, and Akinori Kidera. "Onsager–Machlup action-based path sampling and its combination with replica exchange for diffusive and multiple pathways." Journal of Chemical Physics 132, no. 13 (April 7, 2010): 134101. http://dx.doi.org/10.1063/1.3372802.

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33

Cugliandolo, Leticia F., and Vivien Lecomte. "Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach." Journal of Physics A: Mathematical and Theoretical 50, no. 34 (July 26, 2017): 345001. http://dx.doi.org/10.1088/1751-8121/aa7dd6.

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34

Fujisaki, Hiroshi, Kei Moritsugu, and Yasuhiro Matsunaga. "Exploring Configuration Space and Path Space of Biomolecules Using Enhanced Sampling Techniques—Searching for Mechanism and Kinetics of Biomolecular Functions." International Journal of Molecular Sciences 19, no. 10 (October 15, 2018): 3177. http://dx.doi.org/10.3390/ijms19103177.

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To understand functions of biomolecules such as proteins, not only structures but their conformational change and kinetics need to be characterized, but its atomistic details are hard to obtain both experimentally and computationally. Here, we review our recent computational studies using novel enhanced sampling techniques for conformational sampling of biomolecules and calculations of their kinetics. For efficiently characterizing the free energy landscape of a biomolecule, we introduce the multiscale enhanced sampling method, which uses a combined system of atomistic and coarse-grained models. Based on the idea of Hamiltonian replica exchange, we can recover the statistical properties of the atomistic model without any biases. We next introduce the string method as a path search method to calculate the minimum free energy pathways along a multidimensional curve in high dimensional space. Finally we introduce novel methods to calculate kinetics of biomolecules based on the ideas of path sampling: one is the Onsager–Machlup action method, and the other is the weighted ensemble method. Some applications of the above methods to biomolecular systems are also discussed and illustrated.
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35

Fujisaki, Hiroshi, Motoyuki Shiga, and Akinori Kidera. "2P311 Path sampling for a model polymer using the Onsager-Machlup action(The 48th Annual Meeting of the Biophysical Society of Japan)." Seibutsu Butsuri 50, supplement2 (2010): S137. http://dx.doi.org/10.2142/biophys.50.s137_4.

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36

Fujisaki, Hiroshi, Motoyuki Shiga, and Akinori Kidera. "3P-033 Transition path sampling using the Onsager-Machlup action with replica exchange : Model calculations(Protein:Structure & Function,The 47th Annual Meeting of the Biophysical Society of Japan)." Seibutsu Butsuri 49, supplement (2009): S156. http://dx.doi.org/10.2142/biophys.49.s156_3.

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37

Fujisaki, Hiroshi, Yasuhiro Matsunaga, and Akinori Kidera. "2B1412 Path sampling for small peptide systems using the Onsager-Machlup action method(Proteins:Structure & Function II:Theory, Aggregation,Oral Presentation,The 50th Annual Meeting of the Biophysical Society of Japan)." Seibutsu Butsuri 52, supplement (2012): S40. http://dx.doi.org/10.2142/biophys.52.s40_2.

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38

Serdyukov, Sergey I. "The Onsager–Machlup theory of fluctuations and time-dependent generalized normal distribution." Journal of Non-Equilibrium Thermodynamics, January 5, 2023. http://dx.doi.org/10.1515/jnet-2022-0071.

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Abstract Generalization of the variational formulation of the Onsager–Machlup thermodynamic theory of fluctuation is considered. Within the framework of variational theory, we introduce the time-dependent generalized normal distribution and Hamilton–Jacobi equation. The family of higher-order partial differential equations, which generalize classical Fokker–Planck equation, is considered. It is shown that proposed theory can be used for describing anomalous diffusion.
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39

Yasuda, Kento, Akira Kobayashi, Li-Shing Lin, Yuto Hosaka, Isamu Sou, and Shigeyuki Komura. "The Onsager–Machlup Integral for Non-Reciprocal Systems with Odd Elasticity." Journal of the Physical Society of Japan 91, no. 1 (January 15, 2022). http://dx.doi.org/10.7566/jpsj.91.015001.

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40

Doi, Masao, Jiajia Zhou, Yana Di, and Xianmin Xu. "Application of the Onsager-Machlup integral in solving dynamic equations in nonequilibrium systems." Physical Review E 99, no. 6 (June 10, 2019). http://dx.doi.org/10.1103/physreve.99.063303.

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41

Du, Qiang, Tiejun Li, Xiaoguang Li, and Weiqing Ren. "The graph limit of the minimizer of the Onsager-Machlup functional and its computation." Science China Mathematics, March 25, 2020. http://dx.doi.org/10.1007/s11425-019-1650-7.

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42

Jiang, Jinlian, Wei Xu, Ping Han, and Lizhi Niu. "Most probable transition paths in eutrophicated lake ecosystem under Gaussian white noise and periodic force." Chinese Physics B, February 17, 2022. http://dx.doi.org/10.1088/1674-1056/ac5616.

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Abstract The effects of stochastic perturbations and periodic excitations on the eutrophicated lake ecosystem are explored. Unlike the existing work in detecting early warning signals, the literature presents most probable transition paths to characterize the regime shifts. Most probable transition paths are obtained by minimizing the Freidlin-Wentzell (FW) action functional and Onsager–Machlup (OM) action functional, respectively. The most probable path shows the movement trend of lake eutrophication system under noise excitation, and describes the global transition behavior of the system. Under the excitation of Gaussian noise, the results show that the stability of the eutrophic state and the oligotrophic state has different results from two perspectives of potential well and most probable transition paths. Under the excitation of Gaussian white noise and periodic force, we find that the transition occurs near the nearest distance between the stable periodic solution and the unstable periodic solution.
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43

Deza, Roberto, Gonzalo Izús, and Horacio Wio. "Fluctuation theorems from non-equilibrium Onsager-Machlup theory for a Brownian particle in a time-dependent harmonic potential." Open Physics 7, no. 3 (January 1, 2009). http://dx.doi.org/10.2478/s11534-009-0038-4.

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AbstractWe discuss the case of a Brownian particle which is harmonically bound and multiplicatively forced-namely bound by V(x,t)=1/2 a(t)x 2 where a(t)is externally controlled-as another instance that provides a generalization of Onsager-Machlup’s theory to non-equilibrium states, thus allowing establishment of several fluctuation theorems. In particular, we outline the derivation of a fluctuation theorem for work, through the calculation of the work probability distribution as a functional integral over stochastic trajectories.
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44

Aibara, Noriaki, Naoaki Fujimoto, So Katagiri, Mayumi Saitou, Akio Sugamoto, Takashi Yamamoto, and Tsukasa Yumibayashi. "Gravity analog model of non-equilibrium thermodynamics." Progress of Theoretical and Experimental Physics 2019, no. 7 (July 1, 2019). http://dx.doi.org/10.1093/ptep/ptz068.

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Abstract The non-equilibrium thermodynamics of Onsager and Machlup and of Hashitsume is reformulated as a gravity analog model, in which thermodynamic variables, kinetic coefficients, and generalized forces form, respectively, coordinates and metric tensor and vector fields in a space of thermodynamic variables. The relevant symmetry of the model is the general coordinate transformation. Then, the entropy production is classified into three categories, when a closed path is depicted as a thermodynamic cycle. One category is time-reversal odd, and is attributed to the number of lines of magnetic flux passing through the closed path, having the monopole as a source. There are two time-reversal-even categories, one of which is attributed to the space curvature around the path, having the gravitational instanton as a source, which dominates for a rapid operation of the cycle. The last category is the usual one, which remains even for the quasi-equilibrium operation. It is possible to extend the model to include non-linear responses. In introducing new terms, dimensional counting is important, using two parameters, the temperature and the relaxation time. The effective action, being induced by the non-equilibrium thermodynamics, is derived. This is a candidate for the action that controls the dynamics of kinetic coefficients and thermodynamic forces. An example is given in a chemical oscillatory reaction in a solvent of van der Waals type. The fluctuation–dissipation theorem is examined à la Onsager, and a derivation of the gravity analog thermodynamic model from quantum mechanics is sketched, based on an analogy to the resonance problem.
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