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Physical Foundations of Landauer’s Principle

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 11106)

Abstract

We review the physical foundations of Landauer’s Principle, which relates the loss of information from a computational process to an increase in thermodynamic entropy. Despite the long history of the Principle, its fundamental rationale and proper interpretation remain frequently misunderstood. Contrary to some misinterpretations of the Principle, the mere transfer of entropy between computational and non-computational subsystems can occur in a thermodynamically reversible way without increasing total entropy. However, Landauer’s Principle is not about general entropy transfers; rather, it more specifically concerns the ejection of (all or part of) some correlated information from a controlled, digital form (e.g., a computed bit) to an uncontrolled, non-computational form, i.e., as part of a thermal environment. Any uncontrolled thermal system will, by definition, continually re-randomize the physical information in its thermal state, from our perspective as observers who cannot predict the exact dynamical evolution of the microstates of such environments. Thus, any correlations involving information that is ejected into and subsequently thermalized by the environment will be lost from our perspective, resulting directly in an irreversible increase in thermodynamic entropy. Avoiding the ejection and thermalization of correlated computational information motivates the reversible computing paradigm, although the requirements for computations to be thermodynamically reversible are less restrictive than frequently described, particularly in the case of stochastic computational operations. There are interesting possibilities for the design of computational processes that utilize stochastic, many-to-one computational operations while nevertheless avoiding net entropy increase that remain to be fully explored.

Keywords

  • Information theory
  • Statistical physics
  • Thermodynamics of computation
  • Landauer’s Principle
  • Reversible computing

This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories and by the Advanced Simulation and Computing program under the U.S. Department of Energy’s National Nuclear Security Administration (NNSA). Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for NNSA under contract DE-NA0003525. Approved for public release, SAND2018-7205 C. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

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Notes

  1. 1.

    Boltzmann’s constant \(k_\mathrm {B}\approx 1.38 \times 10^{-23}\ \mathrm {J}/\mathrm {K}\), in traditional units. This constant was actually introduced by Planck in [2]. We discuss this history further in Sect. 3.1.

  2. 2.

    The mathematical fact, not initially fully understood by Landauer, that reversible computational processes can indeed avoid information loss was rigorously demonstrated by Bennett [3], using methods anticipated by Lecerf [4].

  3. 3.

    In this equation, W counts the number of distinct microstates consistent with a given macroscopic description of a system.

  4. 4.

    Intuitively, the more different values \(v_i\) there are, the more unlikely or improbable each individual value would seem to be, proportionally—not knowing anything else about the situation.

  5. 5.

    The rule that probabilities must always sum to 1 can be derived by considering the implications, under our definitions, of breaking down all possible events (regardless of their probability) into a set of equally-likely micro-alternatives; only the probability distributions that sum to 1 turn out to be epistemologically self-consistent in that scenario, but we will not detail that argument here.

  6. 6.

    I gave a detailed example of this information capacity relation (Eq. 8) in [22].

  7. 7.

    Note that this information-theoretic concept of correlation differs from, and is more generally applicable than, a statistical correlation coefficient between scalar numeric variables. General discrete variables do not require any numerical interpretation.

  8. 8.

    Shannon’s formula (our Eq. 4) for H is usually credited to him, but Shannon himself credits Boltzmann, the true originator of this concept.

  9. 9.

    A Hilbert space is a (typically) many-dimensional vector space equipped with an inner product operator, defined over a field that is usually the complex numbers \(\mathbb {C}\).

  10. 10.

    I.e., if \(S(\, \varPhi (s)\, |\, C(s)\, ) = \hat{S}(\,\varPhi (s)\,|\,C(s)\,))\), or in other words, if \(K(\varPhi (s)) = K(C(s))\), so we have no more knowledge about the physical state than the computational state.

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Frank, M.P. (2018). Physical Foundations of Landauer’s Principle. In: Kari, J., Ulidowski, I. (eds) Reversible Computation. RC 2018. Lecture Notes in Computer Science(), vol 11106. Springer, Cham. https://doi.org/10.1007/978-3-319-99498-7_1

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