Teaching the Principles of Statistical Dynamics

TL;DR

This paper introduces a unified framework based on the principle of Maximum Caliber for teaching fundamental transport laws and dynamical distributions, linking microtrajectories to macroscopic behavior.

Contribution

It presents a simple, principled approach to derive transport laws from the maximization of Caliber, connecting microscopic trajectories to macroscopic transport phenomena.

Findings

Derivation of Fick's, Fourier's, and viscosity laws from Maximum Caliber.

Introduction of dynamical distribution functions for microtrajectories.

Application to single-particle and single-molecule experimental data.

Abstract

We describe a simple framework for teaching the principles that underlie the dynamical laws of transport: Fick's law of diffusion, Fourier's law of heat flow, the Newtonian viscosity law, and mass-action laws of chemical kinetics. In analogy with the way that the maximization of entropy over microstates leads to the Boltzmann law and predictions about equilibria, maximizing a quantity that E. T. Jaynes called "Caliber" over all the possible microtrajectories leads to these dynamical laws. The principle of Maximum Caliber also leads to dynamical distribution functions which characterize the relative probabilities of different microtrajectories. A great source of recent interest in statistical dynamics has resulted from a new generation of single-particle and single-molecule experiments which make it possible to observe dynamics one trajectory at a time.