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Path probability and an extension of least action principle to random motion

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 stochastic 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 increasing action, i.e., P(A) ~ eˉγA, where γ is a constant characterizing the sensitivity of the action dependence of the path probability, the action is given by A = ∫T0 Ldt, a time integral of the Lagrangian L = K–V over a fixed time period T, K is the kinetic energy and V is the potential energy. This result is a confirmation of the existence of a classical analogue of the Feynman factor eiA/ħ 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 variational calculus and numerical simulation, we proved that the action A = ∫T0 Ldt is stationary for the optimal paths determined by Newtonian equation. More precisely, 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 motion, but depends exponentially on kinetic action A = ∫T0 Kdt for the overdamped motion.
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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⟩

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