Abstract
A permanent challenge in physics and other disciplines is to solve Euler–Lagrange equations. Thereby, a beneficial investigation is to continue searching for new procedures to perform this task. A novel Monte Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the path space with a probability functional obtained by using the maximum caliber principle. Free particle and harmonic oscillator problems are numerically implemented by sampling the path space for a given action by using Monte Carlo simulations. The average path converges to the solution of the equation of motion from classical mechanics, analogously as a canonical system is sampled for a given energy by computing the average state, finding the least energy state. Thus, this procedure can be general enough to solve other differential equations in physics and a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.
Highlights
The main objective of this work is to show a new computational framework based on MonteCarlo Metropolis, for the study of dynamical systems that are described by a Lagrangian, being a first approach for the understanding of non-equilibrium statistical mechanics (NESM) by using constraints, as performed in statistical mechanics.Here, we propose a technique capable of simulating deterministic, dynamical systems through a stochastic formulation
This procedure can be general enough to solve other differential equations in physics and a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems
We described a technique for implementing Monte Carlo sampling of dynamical trajectories in classical Lagrangian systems under the maximum caliber formalism
Summary
We propose a technique capable of simulating deterministic, dynamical systems through a stochastic formulation. This technique is based on sampling a statistical ensemble of paths defined as having the maximum path entropy ( known as the caliber) available under imposed time-dependent constraints. The principle of maximum entropy (MaxEnt) is a systematic method for constructing the simplest, most unbiased probability distribution function under given constraints, a conceptual generalization of Gibbs’ method of ensembles in statistical mechanics. The complete generality of the principle makes it widely used in several areas of science, such as astronomy, ecology, biology, quantitative finance, image processing, electronics, and physics, among others. According to Jaynes, choosing a candidate probability distribution by maximizing its entropy is a rule of inferential reasoning far beyond its original application in physics, which makes this rule a powerful tool for creating models in any context
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