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1: Stochastic Processes and Brownian Motion

  • Page ID
    364807
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    Equilibrium thermodynamics and statistical mechanics are widely considered to be core subject matter for any practicing chemist [1]. There are plenty of reasons for this:

    • A great many chemical phenomena encountered in the laboratory are well described by equilibrium thermodynamics.
    • The physics of chemical systems at equilibrium is generally well understood and mathematically tractable.
    • Equilibrium thermodynamics motivates our thinking and understanding about chemistry away from equilibrium.

    This last point, however, raises a serious question: how well does equilibrium thermodynamics really motivate our understanding of nonequilibrium phenomena? Is it reasonable for an organometallic chemist to analyze a catalytic cycle in terms of rate-law kinetics, or for a biochemist to treat the concentration of a solute in an organelle as a bulk mixture of compounds? Under many circumstances, equilibrium thermodynamics suffices, but a growing number of outstanding problems in chemistry - from electron transfer in light-harvesting complexes to the chemical mechanisms behind immune system response- concern processes that are fundamentally out of equilibrium.

    This course endeavors to introduce the key ideas that have been developed over the last century to describe nonequilibrium phenomena. These ideas are almost invariably founded upon a statistical description of matter, as in the equilibrium case. However, since nonequilibrium phenomena contain a more explicit time-dependence than their equilibrium counterparts (consider, for example, the decay of an NMR signal or the progress of a reaction), the probabilistic tools we develop will require some time-dependence as well.

    In this chapter, we consider systems whose behavior is inherently nondeterministic, or stochastic, and we establish methods for describing the probability of finding the system in a particular state at a specified time.

    • 1.1: Markov Processes
    • 1.2: Master Equations
      The techniques developed in the basic theory of Markov processes are widely applicable, but there are of course many instances in which the discretization of time is either inconvenient or completely unphysical. In such instances, a master equation (more humbly referred to as a rate equation) may provide a continuous-time description of the system that is in keeping with all of our results about stochastic processes.
    • 1.3: Fokker-Planck Equations
      We have already generalized the equations governing Markov processes to account for systems that evolve continuously in time, which resulted in the master equations. In this section, we adapt these equations further so that they may be suitable for the description of systems with a continuum of states, rather than a discrete, countable number of states.
    • 1.4: The Langevin Equation
      A variety of interesting and important phenomena are subject to combinations of deterministic and stochastic processes. We concern ourselves now with a particular class of such phenomena which are described by Langevin equations. In its simplest form, a Langevin equation is an equation of motion for a system that experiences a particular type of random force. The archetypal system governed by a Langevin equation is a Brownian particle, that is, a particle undergoing Brownian motion.
    • 1.5: Appendix: Applications to Brownian Motion

    Thumbnail: This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions. (CC BY-SA 3.0; Lookang via Wikipedia)


    This page titled 1: Stochastic Processes and Brownian Motion is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jianshu Cao (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.