Random Perturbations of Hamiltonian Systems

2012 ◽  
pp. 258-354 ◽  
Author(s):  
Mark I. Freidlin ◽  
Alexander D. Wentzell
Keyword(s):  
Hamiltonian Systems ◽  
Random Perturbations

scholarly journals A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems

2020 ◽  
pp. 1-35
Author(s):  
Yichun Zhu
Keyword(s):  
Probability Theory ◽  
Weak Convergence ◽  
Hamiltonian Systems ◽  
Random Perturbations ◽  
Identity Matrix ◽  
Compact Sets ◽  
Original Result ◽  
Drift Term ◽  
State Dependent ◽  
Auxiliary Process

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

scholarly journals Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems

2002 ◽  
Vol 30 (1) ◽  
pp. 101-134 ◽  
Author(s):  
Peter H. Baxendale ◽  
Levon Goukasian
Keyword(s):  
Lyapunov Exponents ◽  
Hamiltonian Systems ◽  
Random Perturbations ◽  
Small Random Perturbations

Random perturbations of Hamiltonian systems

1994 ◽  
Vol 109 (523) ◽  
pp. 0-0 ◽  
Author(s):  
Mark I. Freidlin ◽  
Alexander D. Wentzell
Keyword(s):  
Hamiltonian Systems ◽  
Random Perturbations

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

Bifurcations into Pathology for Hamiltonian Systems,

1986 ◽  
Author(s):  
Konstantin Mischaikow
Keyword(s):  
Hamiltonian Systems
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