In this chapter we present dynamical systems and their probabilistic description. We distinguish between system descriptions with discrete and continuous state-spaces as well as discrete and continuous time. We formulate examples of statistical models including Markov models, Markov jump processes, and stochastic differential equations. In doing so, we describe fundamental equations governing the evolution of the probability of dynamical systems. These equations include the master equation, Langevin equation, and Fokker–Plank equation. We also present sampling methods to simulate realizations of a stochastic dynamical process such as the Gillespie algorithm. We end with case studies relevant to chemistry and physics.

Kinetic theory is a framework for calculating macroscopic physical properties of systems from their microscopic degrees of freedom.This idea is applied to an ideal gas to derive the Maxwell--Boltzmann velocity distribution, which is demonstrated to be compatible with the ideal gas law and is used to calculate the rate of effusion of an ideal gas.When molecular collisions are important, the mean free path and collision time are quantities that can characterize these collisions.Situations in which collisions are important, such as Brownian motion and diffusion, are presented, along with relevant equations: the Langevin equation and Fick's Law.

This chapter presents microscopic models of diffusion (Brownian motion). The discussed diffusion models explicitly describe the dynamics of solvent molecules. Such molecular dynamics models provide many more details than the models discussed in Chapter 4 (which simply postulate that the diffusing molecule is subject to a random force) and can be used to assess the accuracy of the stochastic diffusion models from Chapter 4. The analysis starts with theoretical solvent models, including a simple “one-particle” description of the solvent (heat bath), which is used to introduce the generalized Langevin equation and the generalized fluctuation–dissipation theorem. Analytical insights are provided by theoretical models with short- and long-range interactions. The chapter concludes with less analytically tractable, but more realistic, computational models, introducing molecular dynamics (molecular mechanics) and applying it to the Lennard-Jones fluid and to simulations of ions in aquatic solutions.