Computability at zero temperature

Burr, Michael A.; Wolf, Christian

doi:10.1088/1361-6544/ab9c71

Cited by 6 publications

(5 citation statements)

References 36 publications

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“…Additionally, computability at a point is a computable version of being continuous at a point. We refer the reader to section 3 and [15,27] for the precise definitions and details.…”

Section: Resultsmentioning

confidence: 99%

“…We prove that the space of potentials on a shift space X given by an oracle ψ with the finite alphabet A d is a computable metric space. For additional details, we refer the reader to [15]. We observe that the locally constant potentials with rational values (denoted by LC(X, Q)) are dense in C(X, R) with respect to the supremum norm.…”

Section: Computability Of Subshiftsmentioning

confidence: 99%

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Computability of topological pressure on compact shift spaces beyond finite type*

et al. 2022

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We investigate the computability (in the sense of computable analysis) of the topological pressure P top(ϕ) on compact shift spaces X for continuous potentials ϕ : X → R . This question has recently been studied for subshifts of finite type (SFTs) and their factors (sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalised gap shifts, and particular beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalised pressure function (X, ϕ) ↦ P top(X, ϕ| X ) is not computable for a large set of shift spaces X and potentials ϕ. In particular, the entropy map X ↦ h top(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

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“…Additionally, computability at a point is a computable version of being continuous at a point. We refer the reader to section 3 and [15,27] for the precise definitions and details.…”

Section: Resultsmentioning

confidence: 99%

Section: Computability Of Subshiftsmentioning

confidence: 99%

Computability of topological pressure on compact shift spaces beyond finite type*

et al. 2022

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“…It should be noted however that it has been recently established that for a large class of continuous potentials the Evolutionary entropy is computable in the sense of computable analysis [7], i.e., it can be computed by a Turing machine (a computer program for our purposes) at any pre-described accuracy.…”

Section: Evolutionary Entropy and Cycle Timesmentioning

confidence: 99%

Evolutionary entropy and the Second Law of Thermodynamics

Demetrius¹,

Wolf²

2020

Preprint

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The dynamics of molecular collisions in a macroscopic body are encoded by the parameter Thermodynamic entropy -a statistical measure of the number of molecular configurations that correspond to a given macrostate. Directionality in the flow of energy in macroscopic bodies is described by the Second Law of Thermodynamics: In isolated systems, that is systems closed to the input of energy and matter, thermodynamic entropy increases.The dynamics of the lower level interactions in populations of replicating organisms is encoded by the parameter Evolutionary entropy, a statistical measure which describes the number and diversity of metabolic cycles in a population of replicating organisms.Directionality in the transformation of energy in populations of organisms is described by the Fundamental Theorem of Evolution: In systems open to the input of energy and matter, Evolutionary entropy increases, when the energy source is scarce and diverse, and decreases when the energy source is abundant and singular.This article shows that when ρ → 0, and N → ∞, where ρ is the production rate of the external energy source, and N denote the number of replicating units, evolutionary entropy, an organized state of energy; and thermodynamic entropy, a randomized state of energy, coincide. Accordingly, the Fundamental Theorem of Evolution, is a generalization of the Second Law of Thermodynamics.

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“…We prove that the space of potentials on a shift space X given by an oracle ψ with the finite alphabet A d is a computable metric space. For additional details, we refer the reader to [13]. We observe that the locally constant potentials with rational values (denoted by LC(X, Q)) are dense in C(X, R) with respect to the supremum norm.…”

Section: Computability Of Subshiftsmentioning

confidence: 99%

“…Additionally, computability at a point is a computable version of being continuous at a point. We refer the reader to Section 3 and [13,25] for the precise definitions and details.…”

mentioning

confidence: 99%

Computability of topological pressure on compact shift spaces beyond finite type

Burr¹,

Das²,

Wolf³

et al. 2020

Preprint

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We investigate the computability (in the sense of computable analysis) of the topological pressure Ptop(φ) on compact shift spaces X for continuous potentials φ : X → R. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and Beta shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalized pressure function (X, φ) → Ptop(X, φ|X ) is not computable for a large set of shift spaces X and potentials φ. In particular, the entropy map X → htop(X) is computable at a shift space X if and only if X has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.

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Computability at zero temperature

Cited by 6 publications

References 36 publications

Computability of topological pressure on compact shift spaces beyond finite type*

Computability of topological pressure on compact shift spaces beyond finite type*

Evolutionary entropy and the Second Law of Thermodynamics

Computability of topological pressure on compact shift spaces beyond finite type

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