Microscopic Reversibility and Macroscopic Irreversibility: From the Viewpoint of Algorithmic Randomness

Hiura, Ken; Sasa, Shigehiko

doi:10.1007/s10955-019-02387-0

Cited by 4 publications

(8 citation statements)

References 45 publications

(59 reference statements)

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“…It is trivial to find configurations (x, y) where it is violated, but these become increasingly rare as N → ∞. I now state Theorem 3.5 in [81] due to Hiura and Sasa, which sharpens earlier results by Kac [92] in replacing a 'for P-almost every x' result by a 'for all P-random x' result that provides much more precise information on randomness. First, recall that if (x, y)…”

mentioning

confidence: 63%

“…The literature on this topic is enormous; I recommend [2][3][4]29]. My discussion is based on the pioneering work of Hiura and Sasa [81]. But before getting there, I would like to very briefly review Boltzmann's take on the general problem of irreversibility (which in my view is correct).…”

Section: Applications To Statistical Mechanicsmentioning

confidence: 99%

“…by noting that in an infinite system the recurrence time is infinite, whilst in a large system it is astronomically large (beyond the age of the universe). While its realization for the Boltzmann equation may still be remote (for mathematical rather than conceptual reasons or matters of principle), this scenario can be demonstrated in the Kac ring model [92] (see also [81,93,94]), which is a caricature of the Boltzmann equation. Namely:…”

Section: Applications To Statistical Mechanicsmentioning

confidence: 99%

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Typical = Random

Landsman

2023

Axioms

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This expository paper advocates an approach to physics in which “typicality” is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions state that some property Φ(x) holds for P-almost all x∈X, where P is a probability measure on some space X. Their more refined (and typically more recent) formulations show that Φ(x) holds for all P-random x∈X. The computational notion of P-randomness used here generalizes the one introduced by Martin-Löf in 1966 in a way now standard in algorithmic randomness. Examples come from probability theory, analysis, dynamical systems/ergodic theory, statistical mechanics, and quantum mechanics (especially hidden variable theories). An underlying philosophical theme, inherited from von Mises and Kolmogorov, is the interplay between probability and randomness, especially: which comes first?

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confidence: 63%

Section: Applications To Statistical Mechanicsmentioning

confidence: 99%

Section: Applications To Statistical Mechanicsmentioning

confidence: 99%

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Typical = Random

Landsman

2023

Axioms

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“…Our work contributes to the growing body of recent research on the relationship between algorithmic information theory and nonequilibrium thermodynamics [10,22,23,45,54], and it suggests possible directions of future research. For instance, it is interesting to ask whether other algorithmic fluctuation theorems may be derived within AIT, as well as resulting algorithmic analogues of thermodynamic uncertainty relations [55] and thermodynamic speed limits [56].…”

Section: Discussionmentioning

confidence: 85%

Generalized Zurek's bound on the cost of an individual classical or quantum computation

Kolchinsky¹

2023

Preprint

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We consider the minimal thermodynamic cost of an individual computation, where a single input x is transformed into a single output y. In prior work, Zurek proposed that this cost was given by K(x|y), the conditional Kolmogorov complexity of x given y (up to an additive constant which does not depend on x or y). However, this result was derived from an informal argument, applied only to deterministic computations, and had an arbitrary dependence on the choice of physical protocol (via the additive constant). Here we use stochastic thermodynamics to derive a generalized version of Zurek's bound from a rigorous Hamiltonian formulation. Our bound applies to all quantum and classical processes, whether noisy or deterministic, and it explicitly captures the dependence on the protocol. We show that K(x|y) is a fundamental cost of mapping x to y which must be paid using some combination of heat, noise, and protocol complexity, implying a tradeoff between these three resources. We also show that this bound is achievable. Our result is a kind of "algorithmic fluctuation theorem" which has implications for the relationship between the Second Law and the Physical Church-Turing thesis.

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“…We hope that the present paper provides an insight into studying thermodynamics from the viewpoints of the martingale structure and computability of strategies. As a study in the same spirit, see [36] showing the relevance of algorithmic randomness to thermodynamic irreversibility.…”

Section: Discussionmentioning

confidence: 99%

Gibbs Distribution from Sequentially Predictive Form of the Second Law

Hiura

2021

J Stat Phys

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We propose a prequential or sequentially predictive formulation of the work extraction where an external agent repeats the extraction of work from a heat engine by cyclic operations based on his predictive strategy. We show that if we impose the second law of thermodynamics in this situation, the empirical distribution of the initial microscopic states of the engine must converge to the Gibbs distribution of the initial Hamiltonian under some strategy, even though no probability distribution are assumed. We also propose a protocol where the agent can change only a small number of control parameters linearly coupled to the conjugate variables. We find that in the restricted situation the prequential form of the second law of thermodynamics implies the strong law of large numbers of the conjugate variables with respect to the control parameters. Finally, we provide a game-theoretic interpretation of our formulation and find that the prequential work extraction can be interpreted as a testing procedure for random number generator of the Gibbs distribution.

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Microscopic Reversibility and Macroscopic Irreversibility: From the Viewpoint of Algorithmic Randomness

Cited by 4 publications

References 45 publications

Typical = Random

Typical = Random

Generalized Zurek's bound on the cost of an individual classical or quantum computation

Gibbs Distribution from Sequentially Predictive Form of the Second Law

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