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Brownian Computation Is Thermodynamically Irreversible

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Abstract

Brownian computers are supposed to illustrate how logically reversible mathematical operations can be computed by physical processes that are thermodynamically reversible or nearly so. In fact, they are thermodynamically irreversible processes that are the analog of an uncontrolled expansion of a gas into a vacuum.

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Notes

  1. For an account of Einstein analysis, see Norton ([16], Sect. 3).

  2. While the process is not thermodynamically reversible, we recover the same thermodynamic entropy change for the environment by imagining another thermodynamically reversible process in which heat energy P. E trap is passed to the environment.

  3. The expression is simplified using S p (T)=E p (T)/T. This follows from considering the momentum degrees of freedom contribution to both entropy and energy during a thermodynamically reversible heating from T=0.

  4. It is a quite delicate matter to explain the cogency of the notion of a thermodynamically reversible process when proper realization of the process entails that nothing changes, so no process occurs. For my attempt see Norton [19].

  5. For isothermal, isobaric chemical reactions, the relevant generalized force is the chemical potential μ A=(∂G A /∂n A ) T,P , where G A =E+PV−TS is the Gibbs free energy of reagent A and n A the number of moles of A. In dilute solutions, \(\mu_{\mathrm{A}} = \mu_{\mathrm{A}_{0}} + RT \ln [A]\) for R the ideal gas constant, \(\mu_{A_{0}}\) the chemical potential at reference conditions and [A] the molar concentration. While each chemical reaction is reversible at the molecular level, the term RTln[A] contributes an entropic force, so that a chemical reaction will be thermodynamically irreversible if the concentrations of the reagents and products are not constantly adjusted to keep them at equilibrium levels.

  6. I believe the “nearly” refers to the small external force they add corresponding to the energy ramp of Sect. 3.5 above.

  7. Bennett ([6], p. 329) writes:

    When truly random data (e.g. a bit equally likely to be 0 or 1) is erased, the entropy increase of the surroundings is compensated by an entropy decrease of the data, so that the operation as a whole is thermodynamically reversible….When erasure is applied to such [nonrandom] data, the entropy increase of the environment is not compensated by an entropy decrease of the data, and the operation is thermodynamically irreversible.

    To interpret these remarks, one needs to know that Bennett tacitly assumes an inefficient erasure procedure that also creates kln2 of thermodynamic entropy that is passed to the environment.

  8. For other critiques of Landauer’s principle, see Maroney [14] and Hemmo and Shenker ([10], Chaps. 11–12). For historical perspective, see Bennett [5].

  9. See also Bennett ([2], pp. 905–906, 923) for similar remarks.

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  1. Department of History and Philosophy of Science, Center for Philosophy of Science, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    John D. Norton

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I thank Laszlo Kish for helpful discussion.

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Norton, J.D. Brownian Computation Is Thermodynamically Irreversible. Found Phys 43, 1384–1410 (2013). https://doi.org/10.1007/s10701-013-9753-1

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