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14 pages, 1627 KiB

Open AccessArticle

Species Richness Net Primary Productivity and the Water Balance Problem

by Allen G. Hunt, Muhammad Sahimi and Erica A. Newman

Entropy 2024, 26(8), 641; https://doi.org/10.3390/e26080641 - 28 Jul 2024

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Abstract

Species energy theory suggests that, because of limitations on reproduction efficiency, a minimum density of plant individuals per viable species exists and that this minimum correlates the total number of plant individuals N with the number of species S. The simplest assumption [...] Read more.

Species energy theory suggests that, because of limitations on reproduction efficiency, a minimum density of plant individuals per viable species exists and that this minimum correlates the total number of plant individuals N with the number of species S. The simplest assumption is that the mean energy input per individual plant is independent of the number of individuals, making N, and thus S as well, proportional to the total energy input into the system. The primary energy input to a plant-dominated ecosystem is estimated as its Net Primary Productivity (NPP). Thus, species energy theory draws a direct correspondence from NPP to S. Although investigations have verified a strong connection between S and NPP, strong influences of other factors, such as topography, ecological processes such as competition, and historical contingencies, are also at play. The lack of a simple model of NPP expressed in terms of the principal climate variables, precipitation P, and potential evapotranspiration, PET, introduces unnecessary uncertainty to the understanding of species richness across scales. Recent research combines percolation theory with the principle of ecological optimality to derive an expression for NPP(P, PET). Consistent with assuming S is proportional to NPP, we show here that the new expression for NPP(P, PET) predicts the number of plant species S in an ecosystem as a function of P and PET. As already demonstrated elsewhere, the results are consistent with some additional variation due to non-climatic inputs. We suggest that it may be easier to infer specific deviations from species energy predictions with increased accuracy and generality of the prediction of NPP(P, PET). Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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24 pages, 755 KiB

Open AccessArticle

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model

by Rubén Gómez González and Vicente Garzó

Entropy 2024, 26(3), 265; https://doi.org/10.3390/e26030265 - 15 Mar 2024

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Abstract

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real [...] Read more.

The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad’s moment method and with computer simulations performed in the Brownian limiting case (

m / m_{g} \to \infty

, where

m_{g}

and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress–strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio

m / m_{g}

increases. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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20 pages, 1465 KiB

Open AccessArticle

A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems

by Arash Edrisi, Hamza Patwa and Jose A. Morales Escalante

Entropy 2024, 26(3), 263; https://doi.org/10.3390/e26030263 - 14 Mar 2024

Cited by 1 | Viewed by 1401

Abstract

Kinetic theory provides modeling of open quantum systems subject to Markovian noise via the Wigner–Fokker–Planck equation, which is an alternate of the Lindblad master equation setting, having the advantage of great physical intuition as it is the quantum equivalent of the classical phase [...] Read more.

Kinetic theory provides modeling of open quantum systems subject to Markovian noise via the Wigner–Fokker–Planck equation, which is an alternate of the Lindblad master equation setting, having the advantage of great physical intuition as it is the quantum equivalent of the classical phase space description. We perform a numerical inspection of the Wehrl entropy for the benchmark problem of a harmonic potential, since the existence of a steady state and its analytical formula have been proven theoretically in this case. When there is friction in the noise terms, no theoretical results on the monotonicity of absolute entropy are available. We provide numerical results of the time evolution of the entropy in the case with friction using a stochastic (Euler–Maruyama-based Monte Carlo) numerical solver. For all the chosen initial conditions studied (all of them Gaussian states), up to the inherent numerical error of the method, one cannot disregard the possibility of monotonic behavior even in the case under study, where the noise includes friction terms. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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17 pages, 241 KiB

Open AccessArticle

It Ain’t Necessarily So: Ludwig Boltzmann’s Darwinian Notion of Entropy

by Steven Gimbel

Entropy 2024, 26(3), 238; https://doi.org/10.3390/e26030238 - 8 Mar 2024

Cited by 1 | Viewed by 1701

Abstract

Ludwig Boltzmann’s move in his seminal paper of 1877, introducing a statistical understanding of entropy, was a watershed moment in the history of physics. The work not only introduced quantization and provided a new understanding of entropy, it challenged the understanding of what [...] Read more.

Ludwig Boltzmann’s move in his seminal paper of 1877, introducing a statistical understanding of entropy, was a watershed moment in the history of physics. The work not only introduced quantization and provided a new understanding of entropy, it challenged the understanding of what a law of nature could be. Traditionally, nomological necessity, that is, specifying the way in which a system must develop, was considered an essential element of proposed physical laws. Yet, here was a new understanding of the Second Law of Thermodynamics that no longer possessed this property. While it was a new direction in physics, in other important scientific discourses of that time—specifically Huttonian geology and Darwinian evolution, similar approaches were taken in which a system’s development followed principles, but did so in a way that both provided a direction of time and allowed for non-deterministic, though rule-based, time evolution. Boltzmann referred to both of these theories, especially the work of Darwin, frequently. The possibility that Darwin influenced Boltzmann’s thought in physics can be seen as being supported by Boltzmann’s later writings. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

19 pages, 318 KiB

Open AccessArticle

Probability Turns Material: The Boltzmann Equation

by Lamberto Rondoni and Vincenzo Di Florio

Entropy 2024, 26(2), 171; https://doi.org/10.3390/e26020171 - 17 Feb 2024

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Abstract

We review, under a modern light, the conditions that render the Boltzmann equation applicable. These are conditions that permit probability to behave like mass, thereby possessing clear and concrete content, whereas generally, this is not the case. Because science and technology are increasingly [...] Read more.

We review, under a modern light, the conditions that render the Boltzmann equation applicable. These are conditions that permit probability to behave like mass, thereby possessing clear and concrete content, whereas generally, this is not the case. Because science and technology are increasingly interested in small systems that violate the conditions of the Boltzmann equation, probability appears to be the only mathematical tool suitable for treating them. Therefore, Boltzmann’s teachings remain relevant, and the present analysis provides a critical perspective useful for accurately interpreting the results of current applications of statistical mechanics. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

26 pages, 437 KiB

Open AccessArticle

Non-Equilibrium Wigner Function and Application to Model of Catalyzed Polymerization

by Ramon F. Alvarez-Estrada

Entropy 2024, 26(2), 104; https://doi.org/10.3390/e26020104 - 24 Jan 2024

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Abstract

The quantum Wigner function and non-equilibrium equation for a microscopic particle in one spatial dimension (

1 D

) subject to a potential and a heat bath at thermal equilibrium are considered by non-trivially extending a previous analysis. The non-equilibrium equation yields a [...] Read more.

The quantum Wigner function and non-equilibrium equation for a microscopic particle in one spatial dimension (

1 D

) subject to a potential and a heat bath at thermal equilibrium are considered by non-trivially extending a previous analysis. The non-equilibrium equation yields a general hierarchy for suitable non-equilibrium moments. A new non-trivial solution of the hierarchy combining the continued fractions and infinite series thereof is obtained and analyzed. In a short thermal wavelength regime (keeping quantum features adequate for chemical reactions), the hierarchy is approximated by a three-term one. For long times, in turn, the three-term hierarchy is replaced by a Smoluchovski equation. By extending that

1 D

analysis, a new model of the growth (polymerization) of a molecular chain (template or

t e

) by binding an individual unit (an atom) and activation by a catalyst is developed in three spatial dimensions (

3 D

). The atom,

t e

, and catalyst move randomly as solutions in a fluid at rest in thermal equilibrium. Classical statistical mechanics describe the

t e

and catalyst approximately. Atoms and bindings are treated quantum-mechanically. A mixed non-equilibrium quantum–classical Wigner–Liouville function and dynamical equations for the atom and for the

t e

and catalyst, respectively, are employed. By integrating over the degrees of freedom of

t e

and with the catalyst assumed to be near equilibrium, an approximate Smoluchowski equation is obtained for the unit. The mean first passage time (MFPT) for the atom to become bound to the

t e

, facilitated by the catalyst, is considered. The resulting MFPT is consistent with the Arrhenius formula for rate constants in chemical reactions. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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11 pages, 9899 KiB

Open AccessArticle

Evaluating the Adiabatic Invariants in Magnetized Plasmas Using a Classical Ehrenfest Theorem

by Abiam Tamburrini, Sergio Davis and Pablo S. Moya

Entropy 2023, 25(11), 1559; https://doi.org/10.3390/e25111559 - 18 Nov 2023

Viewed by 1361

Abstract

In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov’s. We introduce the Ehrenfest procedure as an [...] Read more.

In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov’s. We introduce the Ehrenfest procedure as an alternative tool that promises to address these challenges more efficiently. Based on the conjugate variable theorem and the well-known fluctuation-dissipation theorem, this procedure offers a less expensive way of deriving time evolution Equations for macroscopic properties in systems far from equilibrium. We investigate the application of the Ehrenfest procedure for the study of adiabatic invariants in magnetized plasmas. We consider charged particles trapped in a dipole magnetic field and apply the procedure to the study of adiabatic invariants in magnetized plasmas and derive Equations for the magnetic moment, longitudinal invariant, and magnetic flux. We validate our theoretical predictions using a test particle simulation, showing good agreement between theory and numerical results for these observables. Although we observed small differences due to time scales and simulation limitations, our research supports the utility of the Ehrenfest procedure for understanding and modeling the behavior of particles in magnetized plasmas. We conclude that this procedure provides a powerful tool for the study of dynamical systems and statistical mechanics out of equilibrium, and opens perspectives for applications in other systems with probabilistic continuity. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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6 pages, 1247 KiB

Open AccessOpinion

Walking with the Atoms in a Chemical Bond: A Perspective Using Quantum Phase Transition

by Sabre Kais

Entropy 2024, 26(3), 230; https://doi.org/10.3390/e26030230 - 3 Mar 2024

Viewed by 1587

Abstract

Phase transitions happen at critical values of the controlling parameters, such as the critical temperature in classical phase transitions, and system critical parameters in the quantum case. However, true criticality happens only at the thermodynamic limit, when the number of particles goes to [...] Read more.

Phase transitions happen at critical values of the controlling parameters, such as the critical temperature in classical phase transitions, and system critical parameters in the quantum case. However, true criticality happens only at the thermodynamic limit, when the number of particles goes to infinity with constant density. To perform the calculations for the critical parameters, a finite-size scaling approach was developed to extrapolate information from a finite system to the thermodynamic limit. With the advancement in the experimental and theoretical work in the field of ultra-cold systems, particularly trapping and controlling single atomic and molecular systems, one can ask: do finite systems exhibit quantum phase transition? To address this question, finite-size scaling for finite systems was developed to calculate the quantum critical parameters. The recent observation of a quantum phase transition in a single trapped ¹⁷¹ Yb⁺ ion indicates the possibility of quantum phase transitions in finite systems. This perspective focuses on examining chemical processes at ultra-cold temperatures, as quantum phase transitions—particularly the formation and dissociation of chemical bonds—are the basic processes for understanding the whole of chemistry. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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