Stochastic Lagrangians for noisy dynamics. (May 2020) - Materassi, Massimo
Highlights: Criticism on the "Deterministic Principle" in the presence of stochastic terms in the equations, and formulation of the Stochastic Cauchy Problem. Review of the functional formalism for non-quantum stochastic systems. Introduction of the concept of Stochastic Lagrangian. Construction of the stochastic lagrangian in few interesting examples: Newton's mechanics point particle, Leibniz, Hamiltonian and metriplectic dynamics, Lotka-Volterra system. Explicit and complete calculation of the functional measure in a particular case. Inclusion of discussion about benefits of the presented formalism, its ma- turity, expected developments and field of experimental applicability. Abstract: The dynamical variables ψ of a classical system, undergoing stochastic stirring forces, satisfy equations of motion with noise terms . Hence, these ψ show a stochastic evolution themselves. The probability of each possible realization of ψ within a given time interval, arises from the interplay between the deterministic parts of dynamics and the statistics of noise terms. In this work, we discuss the construction of the stochastic Lagrangian out of the dynamical equations, that is a tool to calculate the realization probabilities of the variables ψ as path integrals . In this formulation, the study of classical statistical dynamics can benefit from all the techniques developed in Quantum Mechanics of path integrals; moreover, as the path integral is expressed in terms of a Lagrangian, the invariance properties of the system become transparent. After a coincise review of the stochastic Lagrangian formalism, some applications of it to physically relevant cases are illustrated. Then, the advantages and maturity of this approach, and its expected future developments, are outlined.