Entropy

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41 pages, 434 KiB

Open AccessArticle

The Generalized Stochastic Smoluchowski Equation

by Pierre-Henri Chavanis

Entropy 2019, 21(10), 1006; https://doi.org/10.3390/e21101006 - 15 Oct 2019

Cited by 32 | Viewed by 4325

Abstract

We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations [...] Read more.

We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

22 pages, 2685 KiB

Open AccessFeature PaperArticle

Entropy Production and Its Application to the Coupled Nonequilibrium Processes of ATP Synthesis

by Sunil Nath

Entropy 2019, 21(8), 746; https://doi.org/10.3390/e21080746 - 30 Jul 2019

Cited by 23 | Viewed by 3749

Abstract

Starting from the universal concept of entropy production, a large number of new results are obtained and a wealth of novel thermodynamic, kinetic, and molecular mechanistic insights are provided into the coupling of oxidation and ATP synthesis in the vital process of oxidative [...] Read more.

Starting from the universal concept of entropy production, a large number of new results are obtained and a wealth of novel thermodynamic, kinetic, and molecular mechanistic insights are provided into the coupling of oxidation and ATP synthesis in the vital process of oxidative phosphorylation (OX PHOS). The total dissipation,

Φ

, in OX PHOS with succinate as respiratory substrate is quantified from measurements, and the partitioning of

Φ

into the elementary components of ATP synthesis, leak, slip, and other losses is evaluated for the first time. The thermodynamic efficiency,

η

, of the coupled process is calculated from the data on

Φ

and shown to agree well with linear nonequilibrium thermodynamic calculations. Equations for the P/O ratio based on total oxygen consumed and extra oxygen consumed are derived from first principles and the source of basal (state 4) mitochondrial respiration is postulated from molecular mechanistic considerations based on Nath’s two-ion theory of energy coupling within the torsional mechanism of energy transduction and ATP synthesis. The degree of coupling,

q

, between oxidation and ATP synthesis is determined from the experimental data and the irreversible thermodynamics analysis. The optimality of biological free energy converters is explored in considerable detail based on (i) the standard biothermodynamic approach, and (ii) a new biothermokinetic approach developed in this work, and an effective solution that is shown to arise from consideration of the molecular aspects in Nath’s theory is formulated. New experimental data in state 4 with uncouplers and redox inhibitors of OX PHOS and on respiratory control in the physiological state 3 with ADP and uncouplers are presented. These experimental observations are shown to be incompatible with Mitchell’s chemiosmotic theory. A novel scheme of coupling based on Nath’s two-ion theory of energy coupling within the torsional mechanism is proposed and shown to explain the data and also pass the test of consistency with the thermodynamics, taking us beyond the chemiosmotic theory. It is concluded that, twenty years since its first proposal, Nath’s torsional mechanism of energy transduction and ATP synthesis is now well poised to catalyze the progress of experimental and theoretical research in this interdisciplinary field. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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33 pages, 644 KiB

Open AccessArticle

Time–Energy and Time–Entropy Uncertainty Relations in Nonequilibrium Quantum Thermodynamics under Steepest-Entropy-Ascent Nonlinear Master Equations

by Gian Paolo Beretta

Entropy 2019, 21(7), 679; https://doi.org/10.3390/e21070679 - 11 Jul 2019

Cited by 6 | Viewed by 3865

Abstract

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation

τ_{F} Δ_{H} \geq ℏ / 2

provides a general lower bound to the characteristic time [...] Read more.

In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation

τ_{F} Δ_{H} \geq ℏ / 2

provides a general lower bound to the characteristic time

τ_{F} = Δ_{F} / | d ⟨ F ⟩ / d t |

with which the mean value of a generic quantum observable F can change with respect to the width

Δ_{F}

of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty

Δ_{H}

(square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty

Δ_{S}

(square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation

{(2 τ_{F} Δ_{H} / ℏ)}^{2} + {(τ_{F} Δ_{S} / k_{B} τ)}^{2} \geq 1

where

τ

is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation

τ_{F} Δ_{S} \geq k_{B} τ

, meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty

Δ_{S}

. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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22 pages, 1968 KiB

Open AccessFeature PaperArticle

The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy

by Hector Zenil, Narsis A. Kiani and Jesper Tegnér

Entropy 2019, 21(6), 560; https://doi.org/10.3390/e21060560 - 3 Jun 2019

Cited by 6 | Viewed by 5636

Abstract

The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the [...] Read more.

The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the elements in the distribution. Because classical entropy-based Maxent collapses cases confounding all distinct degrees of randomness and pseudo-randomness, here we take into consideration the generative mechanism of the systems considered in the ensemble to separate objects that may comply with the principle under some restriction and whose entropy is maximal but may be generated recursively from those that are actually algorithmically random offering a refinement to classical Maxent. We take advantage of a causal algorithmic calculus to derive a thermodynamic-like result based on how difficult it is to reprogram a computer code. Using the distinction between computable and algorithmic randomness, we quantify the cost in information loss associated with reprogramming. To illustrate this, we apply the algorithmic refinement to Maxent on graphs and introduce a Maximal Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a generalisation over previous approaches. We discuss practical implications of evaluation of network randomness. Our analysis provides insight in that the reprogrammability asymmetry appears to originate from a non-monotonic relationship to algorithmic probability. Our analysis motivates further analysis of the origin and consequences of the aforementioned asymmetries, reprogrammability, and computation. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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13 pages, 522 KiB

Open AccessFeature PaperArticle

Where Was Past Low-Entropy?

by Carlo Rovelli

Entropy 2019, 21(5), 466; https://doi.org/10.3390/e21050466 - 4 May 2019

Cited by 10 | Viewed by 6188

Abstract

Where was the past low-entropy of the early universe located? Contrary to some popular answers, I argue that that the dominant source of low-entropy is the fact that a single degree of freedom, the scale factor, was not at equilibrium. I also discuss [...] Read more.

Where was the past low-entropy of the early universe located? Contrary to some popular answers, I argue that that the dominant source of low-entropy is the fact that a single degree of freedom, the scale factor, was not at equilibrium. I also discuss possible interpretations of the “improbability” of this early low-entropy. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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18 pages, 541 KiB

Open AccessArticle

Goal Identification Control Using an Information Entropy-Based Goal Uncertainty Metric

by Kai Xu and Quanjun Yin

Entropy 2019, 21(3), 299; https://doi.org/10.3390/e21030299 - 20 Mar 2019

Cited by 4 | Viewed by 3363

Abstract

Recent research has found situations where the identification of agent goals could be purposefully controlled, either by changing the underlying environment to make it easier, or exploiting it during agent planning to delay the opponent’s goal recognition. The paper tries to answer the [...] Read more.

Recent research has found situations where the identification of agent goals could be purposefully controlled, either by changing the underlying environment to make it easier, or exploiting it during agent planning to delay the opponent’s goal recognition. The paper tries to answer the following questions: what kinds of actions contain less information and more uncertainty about the agent’s real goal, and how to describe this uncertainty; what is the best way to control the process of goal identification. Our contribution is the introduction of a new measure we call relative goal uncertainty (rgu) with which we assess the goal-related information that each action contains. The rgu is a relative value associated with each action and represents the goal uncertainty quantified by information entropy after the action is taken compared to other executable ones in each state. After that, we show how goal vagueness could be controlled either for one side or for both confronting sides, and formulate this goal identification control problem as a mixed-integer programming problem. Empirical evaluation shows the effectiveness of the proposed solution in controlling goal identification process. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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12 pages, 2485 KiB

Open AccessArticle

From an Entropic Measure of Time to Laws of Motion

by Leonid M. Martyushev and Evgenii V. Shaiapin

Entropy 2019, 21(3), 222; https://doi.org/10.3390/e21030222 - 26 Feb 2019

Cited by 2 | Viewed by 3378

Abstract

A hypothesis proposed in the paper Entropy (Martyushev, L.M. Entropy 2017, 19, 345) on the deductive formulation of a physical theory based on explicitly- and universally-introduced basic concepts is further developed. An entropic measure of time with a number of properties [...] Read more.

A hypothesis proposed in the paper Entropy (Martyushev, L.M. Entropy 2017, 19, 345) on the deductive formulation of a physical theory based on explicitly- and universally-introduced basic concepts is further developed. An entropic measure of time with a number of properties leading to an analog of the Galileo–Einstein relativity principle is considered. Using this measure and a simple model, a kinematic law which relates time to the size and number of particles of a system is obtained. Corollaries of this law are examined. In particular, accelerated growth of the system size is obtained, whereas in systems with constant size, a decrease in the number of particles is observed. An interesting corollary is the emergence of repulsive and attractive forces inversely proportional to the square of the system size for relatively dense systems and constant for systems with sufficiently low density. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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Review

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20 pages, 2417 KiB

Open AccessReview

Maximum Entropy Production Theorem for Transitions between Enzyme Functional States and Its Applications

by Davor Juretić, Juraj Simunić and Željana Bonačić Lošić

Entropy 2019, 21(8), 743; https://doi.org/10.3390/e21080743 - 29 Jul 2019

Cited by 10 | Viewed by 3994

Abstract

Transitions between enzyme functional states are often connected to conformational changes involving electron or proton transport and directional movements of a group of atoms. These microscopic fluxes, resulting in entropy production, are driven by non-equilibrium concentrations of substrates and products. Maximal entropy production [...] Read more.

Transitions between enzyme functional states are often connected to conformational changes involving electron or proton transport and directional movements of a group of atoms. These microscopic fluxes, resulting in entropy production, are driven by non-equilibrium concentrations of substrates and products. Maximal entropy production exists for any chosen transition, but such a maximal transitional entropy production (MTEP) requirement does not ensure an increase of total entropy production, nor an increase in catalytic performance. We examine when total entropy production increases, together with an increase in the performance of an enzyme or bioenergetic system. The applications of the MTEP theorem for transitions between functional states are described for the triosephosphate isomerase, ATP synthase, for β-lactamases, and for the photochemical cycle of bacteriorhodopsin. The rate-limiting steps can be easily identified as those which are the most efficient in dissipating free-energy gradients and in performing catalysis. The last step in the catalytic cycle is usually associated with the highest free-energy dissipation involving proton nanocurents. This recovery rate-limiting step can be optimized for higher efficiency by using corresponding MTEP requirements. We conclude that biological evolution, leading to increased optimal catalytic efficiency, also accelerated the thermodynamic evolution, the synergistic relationship we named the evolution-coupling hypothesis. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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Other

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28 pages, 1457 KiB

Open AccessTutorial

An Introduction to the Non-Equilibrium Steady States of Maximum Entropy Spike Trains

by Rodrigo Cofré, Leonardo Videla and Fernando Rosas

Entropy 2019, 21(9), 884; https://doi.org/10.3390/e21090884 - 11 Sep 2019

Cited by 8 | Viewed by 4672

Abstract

Although most biological processes are characterized by a strong temporal asymmetry, several popular mathematical models neglect this issue. Maximum entropy methods provide a principled way of addressing time irreversibility, which leverages powerful results and ideas from the literature of non-equilibrium statistical mechanics. This [...] Read more.

Although most biological processes are characterized by a strong temporal asymmetry, several popular mathematical models neglect this issue. Maximum entropy methods provide a principled way of addressing time irreversibility, which leverages powerful results and ideas from the literature of non-equilibrium statistical mechanics. This tutorial provides a comprehensive overview of these issues, with a focus in the case of spike train statistics. We provide a detailed account of the mathematical foundations and work out examples to illustrate the key concepts and results from non-equilibrium statistical mechanics. Full article

(This article belongs to the Special Issue Entropy Production and Its Applications: From Cosmology to Biology)

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