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Theoretical Background

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Dissipation and Control in Microscopic Nonequilibrium Systems

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Abstract

This chapter provides a general background on the mathematical concepts used in the remainder of the thesis. We begin by discussing properties of random variables, probability distributions, and stochastic processes. We then use this framework to introduce three common methods used to model the dynamics of stochastic systems: the Master equation (for discrete-state systems), and the Fokker-Planck equation and the Langevin equation (continuous-space systems). Using these mathematical models for either trajectory- or probability distribution-level descriptions of stochastic systems, we introduce the framework of stochastic thermodynamics, and how it consistently defines physical quantities such as heat, work, and entropy for stochastic, strongly fluctuating systems. We also introduce the fluctuation theorems—a set of mathematical equalities central to the modern study of stochastic thermodynamics—and discuss their importance in modern nonequilibrium statistical mechanics, with a particular focus on the relationships between them. We further elaborate upon our discussion of entropy, introducing a number of related information-theoretical quantities that carry important physical interpretations later in the thesis. Finally, we discuss the role of control theory in nonequilibrium systems, and the historical utility of framing problems of efficient operation of molecular machines through the lens of control theory.

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Notes

  1. 1.

    For the interested reader, however, [4] provides a succinct introduction to the mathematical subject.

  2. 2.

    Or, in the language of probability theory, each individual event represents a set of zero measure.

  3. 3.

    See Gardiner [2] for a more in-depth discussion of these constraints, but simply put, the constraints relate to the continuity of the underlying stochastic process in the continuous-time limit.

  4. 4.

    This is analogous, in this case, to the Gillespie algorithm used to propagate the chemical master equation.

  5. 5.

    To get an underdamped equation of motion, we need to also include the time evolution of the distribution of velocities. The Kramers equation does just this for a 1D system, but we won’t discuss it in detail here. For the interested reader, however, an excellent overview is given in Hannes Risken’s book [12].

  6. 6.

    In fact, this is an argument made by Jacobs in [3] for the use of differential notation for SDEs in preference to writing them in terms of derivatives. The non-differentiability of sample paths implies that the derivatives themselves don’t exist, and thus it is more mathematically consistent to write the Langevin equation (or any SDE for that matter) in differential notation, such as in (2.30).

  7. 7.

    Specifically, the Reynolds number is defined mathematically as Re ≡ Ua∕ν where U is the speed of the object, a is a characteristic linear dimension, and ν is the kinematic viscosity of the fluid (for water, ν ≈ 10−6 m2/s). For a typical bacterium (such as, for instance, E. coli) the characteristic linear dimension is a ≈ 10−6 m and they travel at velocities of U ≈ 10−5 m/s and thus the Reynolds number is Re ≈ 10−5 [14].

  8. 8.

    The Kolmogorov-Sinai entropy is a definition of entropy, used often in dynamical systems theory, that is calculated from the Lyapunov exponents in chaotic systems [18].

  9. 9.

    This explanation of the entropy is clarified when the entropy is described in bits, by using base-2 logarithms. Here, the entropy is literally the average number of bits—yes or no questions—required to completely specify the state of a system.

  10. 10.

    Actually, this represents a further subfield known as stochastic optimal control, as it pertains to optimal control strategies in stochastic systems. However, in Chap. 6 we will refer to the control of systems through stochastic driving protocols as ‘stochastic control’, and thus, to avoid confusion, we will refer to stochastic optimal control in this context simply as optimal control.

  11. 11.

    Recent work on optimal bit erasure [34, 35] enforces a constraint on the final probability distribution rather than requiring that the control parameter reaches a particular final value.

  12. 12.

    David Chandler gives a particularly clear exposition on linear-response for step perturbations for Hamiltonian systems in [40], while Robert Zwanzig gives a clear overview of linear-response for time-dependent protocols in [41].

  13. 13.

    Put another way, the weakness of the perturbing field ϕ ensures that it only couples linearly to the conjugate force.

  14. 14.

    Originally, this result was derived for systems with Hamiltonian dynamics (as in, for instance, Ref. [40]), which led to skepticism about its broad applicability in microscopic systems (including a well-known critique from van Kampen in [42]). However, the linear-response predictions seemed to remain consistent with experimental observations in far more general situations than the Hamiltonian derivation would suggest. More recently, the analogous result has been derived for stochastic dynamics and nonequilibrium systems, however it is still an active area of research [43].

  15. 15.

    We will explore the limitations of this approximation in more depth in Chap. 6, with the particular motivation of how it can fail for stochastic control protocols.

  16. 16.

    A similar mathematical form of a friction coefficient was also derived, in a different context, by Berezhkovskii and Szabo in [49].

  17. 17.

    As it turns out, there are mutually incompatible assumptions made with regard to the continuous limit when deriving Crooks’ Fisher information metric or the generalized friction, but we will explore and elaborate on the details of these assumptions in Sect. 7.4.1.

  18. 18.

    To recover Crooks’ original measure of the thermodynamic length, we simply replace ζ ij(λ) in \(\mathcal {L}\) with the Fisher information matrix \(\mathcal {I}_{ij}(\boldsymbol {\lambda })\).

  19. 19.

    For periodic boundary conditions, the Zulkowski-DeWeese formula (2.67) cannot be used to calculate the generalized friction [63].

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    Steven J. Large

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Large, S.J. (2021). Theoretical Background. In: Dissipation and Control in Microscopic Nonequilibrium Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-85825-4_2

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