A derivation of the diffusion equation is presented using the maximum caliber principle and the continuity equation for a system composed of paths traveled by a free particle in a time interval. By identifying the diffusion coefficient in the obtained diffusion equation, it is shown that there is an inverse proportionality relationship concerning the particle’s mass so that a higher mass is related to lower diffusion, and the lower mass is connected to the higher diffusion. This relationship is also shown using Monte Carlo simulations to sample the path space for a free particle system and then using the time slicing equation to obtain the probability of the particle position for each time showing the diffusion behavior for different masses.
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.
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