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Path Probability and an Extension of Least Action Principle to Random Motion Tongling Lin Path probability and an extension of least action principle to random motion Tongling Lin To cite this version: Tongling Lin. Path probability and an extension of least action principle to random motion. Other [cond-mat.other]. Université du Maine, 2013. English. NNT : 2013LEMA1001. tel-00795600 HAL Id: tel-00795600 https://tel.archives-ouvertes.fr/tel-00795600 Submitted on 28 Feb 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LUNAM Universit´e,Universit´edu Maine Th`esede Doctorat Sp´ecialit´e:Physique R´elis´eeau Laboratoire de Physique Statistique et Syst`emes Complexes de l’ISMANS par Tongling Lin pour l’obtention du grade de Docteur en Sciences Path Probability and An Extension of Least Action Principle to Random Motion Pr´esent´eele 19 F´evrier2013 Accept´eesur proposition du jury: Prof. Arto Annila, University of Helsinki, Finland (Pr´esident du jury) Prof. Q. Alexandre Wang, ISMANS et Universit´edu Maine, France (Directeur de th`ese) Prof. Florent Calvayrac, Universit´edu Maine, France (Co-Directeur de th`ese) Prof. Sumiyoshi Abe, Mie University, Japan (Rapporteur) Prof. Eduard Vakarin, Universit´ePierre et Marie Curie, France (Rapporteur) Prof. Umberto Lucia, Politecnico di Torino, Italy (Examinateur) Prof. Fran¸cois Tsobnang, ISMANS, France (Examinateur) Abstract The present thesis is devoted to the study of path probability of random motion on the basis of an extension of Hamiltonian/Lagrangian mechanics to stochastic dynamics. The path probability is first investigated by numerical simulation for Gaussian stochas- tic motion of non dissipative systems. This ideal dynamical model implies that, apart from the Gaussian random forces, the system is only subject to conservative forces. This model can be applied to underdamped real random motion in the presence of friction force when the dissipated energy is negligible with respect to the variation of the potential energy. We find that the path probability decreases exponentially with γA increasing action, i.e., P (A) e− , where γ is a constant characterizing the sensitivity ∼ T of the action dependence of the path probability, the action is given by A = 0 Ldt , a time integral of the Lagrangian L = K V over a fixed time period T , K is the kinetic − R energy and V is the potential energy. This result is a confirmation of the existence of a classical analogue of the Feynman factor eiA/ ~ for the path integral formalism of quantum mechanics of Hamiltonian systems. The above result is then extended to real random motion with dissipation. For this purpose, the least action principle has to be generalized to damped motion of mechanical systems with a unique well defined Lagrangian function which must have the usual simple connection to Hamiltonian. This has been done with the help of the following Lagrangian L = K V Ed, where Ed is the dissipated energy. By − − T variational calculus and numerical simulation, we proved that the action A = 0 Ldt is stationary for the optimal paths determined by Newtonian equation. More precisely,R the stationarity is a minimum for underdamped motion, a maximum for overdamped motion and an inflexion for the intermediate case. On this basis, we studied the path probability of Gaussian stochastic motion of dissipative systems. It is found that the path probability still depends exponentially on Lagrangian action for the underdamped T motion, but depnends exponentially on kinetic action A = 0 Kdt for the overdamped motion. R i Keywords: Path probability, Gaussian stochastic motion, Action, Dissipative systems, Classical mechanics, Variational principle ii R´esum´e La pr´esente th`eseest consacr´ee`al’´etude de la probabilit´edu chemin d’un mouvement al´eatoiresur la base d’une extension de la m´ecanique Hamiltonienne/Lagrangienne `a la dynamique stochastique. La probabilit´ed’un chemin est d’abord ´etudi´eepar simu- lation num´erique dans le cas du mouvement stochastique Gaussien des syst`emes non dissipatifs. Ce mod`eledynamique id´ealimplique que, outre les forces al´eatoiresGaussi- ennes, le syst`eme est seulement soumis `ades forces conservatrices. Ce mod`elepeut ˆetre appliqu´e`aun mouvement al´eatoirer´eelde r´egime pseudo-p´eriodique en pr´esence d’une force de frottement lorsque l’´energie dissip´eeest n´egligeable par rapport `ala variation de l’´energie potentielle. Nous constatons que la probabilit´ede chemin d´ecroˆıtexpo- γA nentiellement lorsque le son action augmente, c’est `adire, P (A) e− , o`u γ est une ∼ constante caract´erisant la sensibilit´ede la d´ependance de l’action `ala probabilit´ede T chemin, l’action est calcul´eepar la formule A = 0 Ldt , int´egraletemporelle du La- grangien. L = K V sur une p´eriode de temps fixe T , K est l’´energie cin´etique et V − R est l’´energie potentielle. Ce r´esultat est une confirmation de l’existence d’un analogue classique du facteur de Feynman eiA/ ~ pour le formalisme int´egralde chemin de la m´ecanique quantique des syst`emes Hamiltoniens. Le r´esultat ci-dessus est ensuite ´etendu au mouvement al´eatoirer´eelavec dissipa- tion. A cet effet, le principe de moindre action doit ˆetreg´en´eralis´eau mouvement amorti de syst`emes m´ecaniques ayant une fonction unique de Lagrange bien d´efinie qui doit avoir la simple connexion habituelle au Hamiltonien. Cela a ´et´efait avec l’aide du Lagrangien suivant L = K V Ed, o`u Ed est l’´energie dissip´ee. Par le calcul − − T variationnel et la simulation num´erique, nous avons prouv´eque l’action A = 0 Ldt est stationnaire pour les chemins optimaux d´etermin´espar l’´equation newtonienne.R Plus pr´ecis´ement, la stationnarit´eest un minimum pour les mouvements de r´egime pseudo-p´eriodique, un maximum pour les mouvements d’amortissement ap´eriodique et une inflexion dans le cas interm´ediaire. Sur cette base, nous avons ´etudi´ela probabilit´e du chemin du mouvement stochastique Gaussien des syst`emes dissipatifs. On constate que la probabilit´edu chemin d´epend toujours de fa¸conexponentielle de l’action La- grangien pour les mouvements de r´egime pseudo-p´eriodique, mais d´epend toujours de T fa¸conexponentielle de l’action cin´etique A = 0 Kdt pour r´egime ap´eriodique. R iii Mots-clefs: Probabilit´echemin, Mouvement stochastique Gaussien, Action, Syst`emes dissipatifs, M´ecanique classique, Principe variationnel iv Contents Abstract, Keywords, R´esum´e,Mots-clefs . i Contents . ...............v Acknowledgements . ix 1 Introduction . ............1 1.1 Background . ..............1 1.2 Research purpose . ...........4 1.3 Overview of the thesis . 5 2 Stochastic action principle and path probability distribution . 9 2.1 Stochastic action principle . 9 2.1.1 Principle of least action . 9 2.1.1.1 Statements of action pinciples. .9 2.1.1.2 Euler-Lagrange equation . 10 2.1.2 Principle of virtual work . 12 2.1.2.1 Basic definitions . 12 2.1.2.2 Static equilibrium . 14 2.1.3 Stochastic action principle. .15 2.1.3.1 Random motion . 15 2.1.3.2 Stochastic least action principle. .17 2.2 A path probability distribution . 20 2.2.1 Directed schema . 20 2.2.2 Panoramic shcema . 20 2.2.3 Initial condition schema . 21 3 Path probability for stochastic motion of non dissipative systems . 23 v 3.1 Introduction. ..............23 3.2 Technical details of numerical computation . 26 3.3 View path probability `ala Wiener . 28 3.4 Path probability distribution by numerical simulation . 29 3.4.1 Free paticles . 29 3.4.2 Particles under constant force . 29 3.4.3 Particles under harmonic force . 31 3.4.4 Particles in cubic potential . 31 3.4.5 Particles in quartic potential . 32 3.5 Correlation between path probability and action . 32 3.6 Sensitivity of path probability to action. .34 3.7 Conclusions . ..............36 4 Extended least action principle to dissipative mechanical systems . 39 4.1 Introduction. ..............39 4.2 Least action principle for dissipative systems . 42 4.2.1 The conservative Hamiltonian . 42 4.2.2 A dissipative Lagrangian function . 44 4.3 Variational formulation . 45 4.3.1 The “global” variational calculus . 45 4.3.2 The “forward” variational calculus . 45 4.3.3 Derivation from virtual work principle . 47 4.3.4 Application of Maupertuis’ principle. .48 4.3.5 The variational calculus with local Lagrangian. .49 4.4 The optimal path and action with constant force and Stokes’ drag . 50 4.5 Transition of extrema of action . 51 4.6 Other forces . .............56 4.7 Concluding remarks . .........57 vi 5 Path probability for stochastic motion of dissipative systems . 61 5.1 Introduction. ..............61 5.2 Numerical simulation . 62 5.3 Determination of path probability distributions . 64 5.3.1 Particles with constant force and Stokes’ drag . 64 5.3.2 Particles with harmonic force and Stokes’ drag . 66 5.4 Discussion. .............66 5.5 Conclusions . ..............68 6 Conclusions . ..........71 Appendix A ............................................. ..............75 Appendix B ............................................. ..............77 Bibliography . ...........79 Publications . ............ 89 vii viii Acknowledgements During three years of study and work in ISMANS and Universit´edu Maine, I was greatly supported both materially and spiritually by all of my professors, colleagues, friends and family. I could not finish this thesis without their continuous assistance. First of all, I would like to warmly thank my PhD supervisor, Prof.
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