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1 results

  • 4 - Stochastic Processes

  • Pierre Gaspard, Université Libre de Bruxelles
  • Book:
    The Statistical Mechanics of Irreversible Phenomena
    Published online:
    14 July 2022
    Print publication:
    28 July 2022, pp 147-201

  • Summary

    At the mesoscale, the fluctuating phenomena are described using the theory of stochastic processes. Depending on the random variables, different stochastic processes can be defined. The properties of stationarity, reversibility, and Markovianity are defined and discussed. The classes of discrete- and continuous-state Markov processes are presented including their master equation, their spectral theory, and their reversibility condition. For discrete-state Markov processes, the entropy production is deduced and the network theory is developed, allowing us to obtain the affinities on the basis of the Hill–Schnakenberg cycle decomposition. Continuous-state Markov processes are described by their master equation, as well as stochastic differential equations. The spectral theory is also considered in the weak-noise limit. Furthermore, Langevin stochastic processes are presented in particular for Brownian motion and their deduction is carried out from the underlying microscopic dynamics.