Boltzmann Entropy Research Articles

Barrow’s nonlinear charged anti-de Sitter black hole and stability

AbstractAs we know, the horizon area of a black hole will increase when it absorbs matter. Based on Barrow’s concept of a fractal black hole horizon, it has been proposed (Phys Lett B 831:137181, 2022) that for a spherically fractal structure, the minimum increase in the horizon area is the area of the smallest bubble sphere. The corresponding black hole entropy is of a logarithmic form, which is similar to that of Boltzmann entropy under a certain condition. Based on this, we re-derive the entropy of Barrow’s Einstein–power-Yang–Mills (EPYM) anti-de Sitter (AdS) black hole, and calculate its temperature and heat capacity. There exists an interesting phenomenon wherein the ratio between Barrow’s temperature and the Hawking temperature of the EPYM AdS black hole is fully consistent with that of other Schwarzschild-like black holes. The Barrow and Hawking temperatures within a certain range of $$\Lambda $$ Λ are found to increase monotonically, and the corresponding heat capacities are all positive, which means that these black holes are thermodynamically stable. In addition, for Barrow’s EPYM AdS black hole, its heat capacity has a Schottky anomaly-like behavior, which may reflect the existence of a discrete energy level and microscopic degrees of freedom.

Entropy and Negative Specific Heat of Doped Graphene: Topological Phase Transitions and Nernst's Theorem Revisited.

This study explores the thermodynamic properties of doped graphene using an adapted electronic spectrum. We employed the one-electron tight-binding model to describe the hexagonal lattice structure. The dispersion relation for graphene is expressed in terms of the hopping energies using a compositional parameter that characterizes the different dopant atoms in the lattice. The focus of the investigation is on the impact of the compositions, specifically the presence of dopant atoms, on the energy spectrum, entropy, temperature, and specific heat of graphene. The numerical and analytical results reveal distinct thermodynamic behaviors influenced by the dopant composition, including topological transitions, inflection points in entropy, and specific heat divergences. In addition, the use of Boltzmann entropy and the revision of Nernst's theorem for doped graphene are introduced as novel aspects.

Monotonicity of the logarithmic energy for random matrices

It is well known that the semi-circle law, which is the limiting distribution in the Wigner theorem, is the minimizer of the logarithmic energy penalized by the second moment. A very similar fact holds for the Girko and Marchenko–Pastur theorems. In this work, we shed the light on an intriguing phenomenon suggesting that this functional is monotonic along the mean empirical spectral distribution in terms of the matrix dimension. This is reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann equation, the monotonicity of the free energy along ergodic Markov processes, and the Shannon monotonicity of entropy or free entropy along the classical or free central limit theorem. While we only verify this monotonicity phenomenon for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square Laguerre unitary ensemble, numerical simulations suggest that it is actually more universal. We obtain along the way explicit formulas of the logarithmic energy of the models which can be of independent interest.

Boltzmann's Mean Entropy Formula for Unbounded Spin Systems

Boltzmann's Mean Entropy Formula for Unbounded Spin Systems

A convergent finite-volume scheme for nonlocal cross-diffusion systems for multi-species populations

An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to “translate” the entropy calculations for the continuous equations to the finite-volume scheme, in particular to design discretizations of the mobilities, which guarantee a discrete chain rule even in the presence of nonlocal terms. Based on this idea, the existence of finite-volume solutions and the convergence of the scheme are proven. As a by-product, we deduce the existence of weak solutions to the continuous cross-diffusion system. Finally, we present some numerical experiments illustrating the behavior of the solutions to the nonlocal and associated local models.

Equivalence of information production and generalised entropies in complex processes.

Complex systems with strong correlations and fat-tailed distribution functions have been argued to be incompatible with the Boltzmann-Gibbs entropy framework and alternatives, so-called generalised entropies, were proposed and studied. Here we show, that this perceived incompatibility is actually a misconception. For a broad class of processes, Boltzmann entropy -the log multiplicity- remains the valid entropy concept. However, for non-i.i.d. processes, Boltzmann entropy is not of Shannon form, -k∑ipi log pi, but takes the shape of generalised entropies. We derive this result for all processes that can be asymptotically mapped to adjoint representations reversibly where processes are i.i.d. In these representations the information production is given by the Shannon entropy. Over the original sampling space this yields functionals identical to generalised entropies. The problem of constructing adequate context-sensitive entropy functionals therefore can be translated into the much simpler problem of finding adjoint representations. The method provides a comprehensive framework for a statistical physics of strongly correlated systems and complex processes.

Energy Structure Theory: A General Unified Thermodynamics Theory

This study, while reviewing some of the established unified equations and fundamentals of the energy structure and providing a detailed interpretation of their physical concepts, expands the relevant equations for new topics and applications, and in fact, establishes novel results and equations from the energy structure analysis. In fact, this paper establishes an energy components-based-general model inspired by the first and second laws of thermodynamics as well as using a new division to the total energy of the system. The established model is completed by extracting the physical direction for the feasible processes based on the energy components of the system. As two of the most important achievements of the energy components approach, using a new quasi-statistical approach as well as a novel energy conservation principle, an entropy equation is gained that has a common basis as the Boltzmann entropy equation as well as a general solution to the different formulations of the second law of thermodynamics is established. The established equations are gained without any limiting assumptions, and are governed to any physical system. Several basic examples have been studied, and matching the obtained results with expected ones is shown.

Exploring Relationships between Boltzmann Entropy of Images and Building Classification Accuracy in Land Cover Mapping.

Remote sensing images are important data sources for land cover mapping. As one of the most important artificial features in remote sensing images, buildings play a critical role in many applications, such as population estimation and urban planning. Classifying buildings quickly and accurately ensures the reliability of the above applications. It is known that the classification accuracy of buildings (usually indicated by a comprehensive index called F1) is greatly affected by image quality. However, how image quality affects building classification accuracy is still unclear. In this study, Boltzmann entropy (an index considering both compositional and configurational information, simply called BE) is employed to describe image quality, and the potential relationships between BE and F1 are explored based on images from two open-source building datasets (i.e., the WHU and Inria datasets) in three cities (i.e., Christchurch, Chicago and Austin). Experimental results show that (1) F1 fluctuates greatly in images where building proportions are small (especially in images with building proportions smaller than 1%) and (2) BE has a negative relationship with F1 (i.e., when BE becomes larger, F1 tends to become smaller). The negative relationships are confirmed using Spearman correlation coefficients (SCCs) and various confidence intervals via bootstrapping (i.e., a nonparametric statistical method). Such discoveries are helpful in deepening our understanding of how image quality affects building classification accuracy.

Land System Simulation of Ruoergai Plateau by Integrating MaxEnt and Boltzmann Entropy into CLUMondo

In the context of global change, land cover change is significantly influenced by human activities. However, there is limited knowledge about the potential economic and ecological benefits that land cover change on the Ruoergai Plateau will bring by 2035, considering the existing development plans. In our study, the CLUMondo model was improved by integrating the MaxEnt model and Boltzmann entropy and used to predict the structure and intensity of land change in China’s Ruoergai Plateau. The results show that the model integrated with MaxEnt and Boltzmann entropy is the most accurate in four contrasting experiments that have a Kappa of 0.773. The predicted results show that with the increase in the demand for ecological benefits, the total area of the water area shows a clear increasing trend. With 0.25% GDP growth, the water area is about 178 km2. With 2.5% GEP growth, the water area is about 202 km2. The latter is 24 km2 more than the former, an increase of about 13.6%. With the increase in the demand for economic benefits, the total area of construction land shows a clear increasing trend. Grassland, forest, and cropland are partly converted into construction land, because of the higher economic benefits of construction land. At the same time, the density of construction land will increase. With 12.6% GDP growth, the high-density construction area is about 399 km2. With 126.1% GEP growth, the water area is about 761 km2. High-density construction land increased by 90.7% (about 362 km2). In the low elevation area near the mountains of Ruoergai County, a new concentration of construction land will appear. The simulation results are of great significance for guiding ecological protection and urban construction in Ruoergai.

Entanglement entropy, single-particle occupation probabilities, and short-range correlations

For quantum many-body systems with short-range correlations (SRCs), the intimate relationship between their magnitude, the behavior of the single-particle occupation probabilities at momenta larger than the Fermi momentum, and the entanglement entropy is a new qualitative aspect not studied and exploited yet. A large body of recent condensed matter studies indicates that the time evolution of the entanglement entropy describes the nonequilibrium dynamics of isolated and strongly interacting many-body systems, in a manner similar to the Boltzmann entropy, which is strictly defined for dilute and weakly interacting many-body systems. Both theoretical and experimental studies in nuclei and cold atomic gases have shown that the fermion momentum distribution has a generic behavior $n(k)=C/{k}^{4}$ at momenta larger than the Fermi momentum, due to the presence of SRCs, with approximately 20% of the particles having momenta larger than the Fermi momentum. The presence of the long momentum tails in the presence of SRCs changes the textbook relation between the single-particle kinetic energy and occupation probabilities, ${n}_{\text{mf}}(k)=1/{1+exp\ensuremath{\beta}[\ensuremath{\epsilon}(k)\ensuremath{-}\ensuremath{\mu}]}$ for momenta very different form the Fermi momentum, particularly for dynamics processes. SRCs induced high-momentum tails of the single-particle occupation probabilities increase the entanglement entropy of fermionic systems, which in its turn affects the dynamics of many nuclear reactions, such as heavy-ion collisions and nuclear fission.

The Entropy Density Behavior across a Plane Shock Wave.

Entropy density behavior poses many problems when we study non-equilibrium situations. In particular, the local equilibrium hypothesis (LEH) has played a very important role and is taken for granted in non-equilibrium problems, no matter how extreme they are. In this paper we would like to calculate the Boltzmann entropy balance equation for a plane shock wave and show its performance for Grad's 13-moment approximation and the Navier-Stokes-Fourier equations. In fact, we calculate the correction for the LEH in Grad's case and discuss its properties.

Work Extraction from Unknown Quantum Sources.

Energy extraction is a central task in thermodynamics. In quantum physics, ergotropy measures the amount of work extractable under cyclic Hamiltonian control. As its full extraction requires perfect knowledge of the initial state, however, it does not characterize the work value of unknown or untrusted quantum sources. Fully characterizing such sources would require quantum tomography, which is prohibitively costly in experiments due to the exponential growth of required measurements and operational limitations. Here, we therefore derive a new notion of ergotropy applicable when nothing is known about the quantum states produced by the source, apart from what can be learned by performing only a single type of coarse-grained measurement. We find that in this case the extracted work is defined by the Boltzmann and observational entropy in cases where the measurement outcomes are, or are not, used in the work extraction, respectively. This notion of ergotropy represents a realistic measure of extractable work, which can be used as the relevant figure of merit to characterize a quantum battery.

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

More on the demons of thermodynamics

Robertson replies: Daniel Sheehan, Garret Moddel, and James Lee draw attention to an interesting book, Challenges to the Second Law of Thermodynamics: Theory and Experiment, which provides a survey of recent work that throws doubt on the inviolability of the second law. Do those avenues resurrect Maxwell’s demon? Do they conjure new ones?Of course, no scientific law is immutably beyond reproach, no matter how many famous physicists have sworn by it. But in considering the avenues suggested in the book, it can be helpful to scrutinize what we mean by the “second law.”11. For more on this theme, see J. Uffink, in Time and History: Proceedings of the 28 International Ludwig Wittgenstein Symposium; Kirchberg am Wechsel, Austria 2005, F. Stadler, M. Stöltzner, eds., De Gruyter (2006), p. 275. If the second law is that entropy—of which there are many forms—cannot decrease, then it can surely be violated; the Boltzmann entropy decreases in macroscopically indeterministic processes. But if the second law is taken to be that no engine is more efficient than a Carnot engine, then at least the previous example is not necessarily a violation.As David Wallace said in a talk at the University of Cambridge in November 2015, the distinction is between whether we can find ingenious and cunning devices or whether we can solve the energy crisis! The faith in the implausibility of the latter—that we will not find a perpetual motion machine of the second kind—is what those aggrandizing thermodynamics attest to. I meant to have captured that distinction between different kinds of “violations” of the second law in my discussion of “deft illusionists” versus “true magicians.” But that may have sounded dismissive; to be clear, important insights are revealed by studying the cases that Sheehan, Moddel, and Lee highlight, especially with respect to how the macroscopic domain may differ from the mesoscopic and microscopic ones. And if there were a true magician, then that would be welcome news in the current energy crisis.Similarly, whether the spin-echo experiment counts as a violation depends on precisely what we mean by the second law—no one is expecting to create a greater-than-Carnot efficiency engine out of that scenario. The point is merely that a system “retracing its steps” may have seemed nigh on impossible to Josef Loschmidt, but the spin-echo case provides a nice illustration of its feasibility. I am sympathetic to Jean Jeener’s view that the spin echo is not a case of increasing and then decreasing entropy, but then again, if we just look at the unitary dynamics, then entropy is neither increasing nor decreasing.One feature absent from my original article—as rightly emphasized by Harvey Leff and Andrew Rex—is the connection between thermal physics and information forged by Maxwell’s demon. That is often the lynchpin or starting place for those interested in quantum thermodynamics (referenced as “quantum steampunk” in my article) and, as Leff and Rex emphasize, stochastic thermodynamics. That brings up a question though: If information is central to thermodynamics, does that raise the specter of anthropocentrism?ReferenceSection:ChooseTop of pageReference <<1. For more on this theme, see J. Uffink, in Time and History: Proceedings of the 28 International Ludwig Wittgenstein Symposium; Kirchberg am Wechsel, Austria 2005, F. Stadler, M. Stöltzner, eds., De Gruyter (2006), p. 275. Google Scholar© 2023 American Institute of Physics.

Euler fluid in two dimensions: Statistical approach.

We use Kirchhoff's vortex formulation of 2D Euler fluid equationsto explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equationsof motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.

A Quasi-Statistical Approach to the Boltzmann Entropy Equation Based on a Novel Energy Conservation Principle

Boltzmann entropy equation is gained according to the statistical mechanics directly and general dependence between entropy and probability is obtained. Based on the second law of thermodynamics with a glance at the Boltzmann entropy equation, it can be deduced that physical processes are done in a direction that the probability of the system and total entropy increase. In fact, the possible process performing states and their entropy variations will be determined at a specific energy level. In this paper, an entropy equation is gained by using a new quasi-statistical approach to the physical processes as well as a novel energy conservation principle. The variation of the "energy structure equation”, as an equation to formulate the performed process using activated energy components of the system and their dependence, is studied in different possible paths by using the energy conservation principle directly. Despite the classical mechanics that all particles are studied, in the novel approach, "particular processes" as all processes that have the same active independent energy components are studied at "various conditions"; in other words, all conditions that same energy amount is applied to the system. One of the advantages of this novel approach is that the volume of the needed calculations will be decreased mainly in comparison with the Boltzmann entropy equation. Dependence of the entropy and rate of the energy components is gained from the novel energy conservation principle. The gained relation, expressed by energy components of the system, is considered with no constraints on the structure of the system but has a common basis with the Boltzmann entropy equation. In fact, by using a novel macroscopic-statistical approach, the entropy variation of a physical system is studied.

Observational-entropic study of Anderson localization

The notion of the thermodynamic entropy in the context of quantum mechanics is a controversial topic. Although there were proposals to refer von Neumann entropy as the thermodynamic entropy, it has its own limitations. The observational entropy has been developed as a generalization of Boltzmann entropy, and it is presently one of the most promising candidates to provide a clear and well-defined understanding of the thermodynamic entropy in quantum mechanics. In this paper, we study the behavior of the observational entropy in the context of localization-delocalization transition for one-dimensional Aubrey-Andr\'e (AA) model. We find that for the typical midspectrum states, in the delocalized phase the observation entropy grows rapidly with coarse-grain size and saturates to the maximal value, whereas in the localized phase the growth is logarithmic. Moreover, for a given coarse-graining, it increases logarithmically with system size in the delocalized phase, and obeys area law in the localized phase. We also find the increase in the observational entropy followed by the quantum quench is logarithmic in time in the delocalized phase as well as at the transition point, whereas in the localized phase it oscillates. Finally, we also venture the self-dual property of the AA model using momentum space coarse graining.

On Gage-Hamilton's entropy formula and Harnack inequality for the curve shortening flow

By the first two derivatives of the Boltzmann entropy of the curvature, which was first studied by Gage and Hamilton for the curve shortening flow in the plane, we define a monotonicity formula which is strictly increasing unless on a shrinking circle. By calculating pointwisely we give an alternate proof of Gage-Hamilton's Harnack inequality.

Globally time-reversible fluid simulations with smoothed particle hydrodynamics

This paper describes an energy-preserving and globally time-reversible code for weakly compressible smoothed particle hydrodynamics (SPH). We do not add any additional dynamics to the Monaghan's original SPH scheme at the level of ordinary differential equation, but we show how to discretize the equations by using a corrected expression for density and by invoking a symplectic integrator. Moreover, to achieve the global-in-time reversibility, we have to correct the initial state, implement a conservative fluid-wall interaction, and use the fixed-point arithmetic. Although the numerical scheme is reversible globally in time (solvable backwards in time while recovering the initial conditions), we observe thermalization of the particle velocities and growth of the Boltzmann entropy. In other words, when we do not see all the possible details, as in the Boltzmann entropy, which depends only on the one-particle distribution function, we observe the emergence of the second law of thermodynamics (irreversible behavior) from purely reversible dynamics.

Kinetic theory of one-dimensional inhomogeneous long-range interacting N-body systems at order 1/N^{2} without collective effects.

Long-range interacting systems irreversibly relax as a result of their finite number of particles, N. At order 1/N, this process is described by the inhomogeneous Balescu-Lenard equation. Yet, this equationexactly vanishes in one-dimensional inhomogeneous systems with a monotonic frequency profile and sustaining only 1:1 resonances. In the limit where collective effects can be neglected, we derive a closed and explicit 1/N^{2} collision operator for such systems. We detail its properties, highlighting in particular how it satisfies an H theorem for Boltzmann entropy. We also compare its predictions with direct N-body simulations. Finally, we exhibit a generic class of long-range interaction potentials for which this 1/N^{2} collision operator exactly vanishes.