Boltzmann Entropy Research Articles

Toverline{T} -deformation of q-Yang-Mills theory

We derive the Toverline{T} -perturbed version of two-dimensional q-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the Toverline{T} -deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large N factorization into chiral and anti-chiral sectors. For the U(N) gauge theory on the sphere, we show that the large N phase transition persists, and that it is of third order and induced by instantons. The effect of the Toverline{T} -deformation is to decrease the critical value of the ’t Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for (q, t)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large N limit of Yang-Mills theory, showing that the Toverline{T} -deformation decreases the contribution of the Boltzmann entropy.

A semi-classical estimate for the q-parameter and decay time with Tsallis entropy of black holes in quantum geometry

In this letter, using the non-extensive entropy of Tsallis, we study some properties of the Schwarzschild black holes (BHs), based on the loop quantum gravity (LQG), some novel characteristics and results of the Schwarzschild BH can be obtained in Mejrhit and Ennadifi (Phys Lett B 794:45–49, 2019). Here we find that these findings are strikingly identical to ones obtained by Hawking and Page in anti-de Sitter space within the original of the Boltzmann entropy formula. By using the semi-classical estimate analysis on the energy at this minimum M_{min}, an approximate relationship between the q and gamma parameters of BHs can be found, (qapprox frac{sqrt{3}gamma }{pi ln 2}+1), which is remarkable approaching to q-parameters of cosmic ray spectra and quarks coalescing to hadrons in high energy.

On the relationship between: L Boltzmann entropy, M Planck distribution, C Shannon information and the principle of particles indistinguishability

The relationship between the fundamental formulas of statistical physics and information theory is considered. It is shown that the M Plank distribution for blackbody radiation and the quantum analog of C Shannon’s capacity for a frequency band-limited classical-quantum channel have a common basis – the principle of particles indistinguishability.

Classical dynamical coarse-grained entropy and comparison with the quantum version.

We develop the framework of classical observational entropy, which is a mathematically rigorous and precise framework for nonequilibrium thermodynamics, explicitly defined in terms of a set of observables. Observational entropy can be seen as a generalization of Boltzmann entropy to systems with indeterminate initial conditions, and it describes the knowledge achievable about the system by a macroscopic observer with limited measurement capabilities; it becomes Gibbs entropy in the limit of perfectly fine-grained measurements. This quantity, while previously mentioned in the literature, has been investigated in detail only in the quantum case. We describe this framework reasonably pedagogically, then show that in this framework, certain choices of coarse-graining lead to an entropy that is well-defined out of equilibrium, additive on independent systems, and that grows toward thermodynamic entropy as the system reaches equilibrium, even for systems that are genuinely isolated. Choosing certain macroscopic regions, this dynamical thermodynamic entropy measures how close these regions are to thermal equilibrium. We also show that in the given formalism, the correspondence between classical entropy (defined on classical phase space) and quantum entropy (defined on Hilbert space) becomes surprisingly direct and transparent, while manifesting differences stemming from noncommutativity of coarse-grainings and from nonexistence of a direct classical analog of quantum energy eigenstates.

Belg: A Tool for Calculating Boltzmann Entropy of Landscape Gradients.

Entropy is a fundamental concept in thermodynamics that is important in many fields, including image processing, neurobiology, urban planning, and sustainability. As of recently, the application of Boltzmann entropy for landscape patterns was mostly limited to the conceptual discussion. However, in the last several years, a number of methods for calculating Boltzmann entropy for landscape mosaics and gradients were proposed. We developed an R package belg as an open source tool for calculating Boltzmann entropy of landscape gradients. The package contains functions to calculate relative and absolute Boltzmann entropy using the hierarchy-based and the aggregation-based methods. It also supports input raster with missing (NA) values, allowing for calculations on real data. In this study, we explain ideas behind implemented methods, describe the core functionality of the software, and present three examples of its use. The examples show the basic functions in this package, how to adjust Boltzmann entropy values for data with missing values, and how to use the belg package in larger workflows. We expect that the belg package will be a useful tool in the discussion of using entropy for a description of landscape patterns and facilitate a thermodynamic understanding of landscape dynamics.

INFORMATION THEORY OF CARTOGRAPHY: A FRAMEWORK FOR ENTROPY-BASED CARTOGRAPHIC COMMUNICATION THEORY

Abstract. Map is an effective communication means. It carries and transmits spatial information about spatial objects and phenomena, from map makers to map users. Therefore, cartography can be regarded as a communication system. Efforts has been made on the application of Shannon Information theory developed in digital communication to cartography to establish an information theory of cartography, or simply cartographic information theory (or map information theory). There was a boom during the period from later 1960s to early 1980s. Since later 1980s, researcher have almost given up the dream of establishing the information theory of cartography because they met a bottleneck problem. That is, Shannon entropy is only able to characterize the statistical information of map symbols but not capable of characterizing the spatial configuration (patterns) of map symbols. Fortunately, break-through has been made, i.e. the building of entropy models for metric and thematic information as well as a feasible computational model for Boltzmann entropy. This paper will review the evolutional processes, examine the bottleneck problems and the solutions, and finally propose a framework for the information theory of cartography. It is expected that such a theory will become the most fundamental theory of cartography in the big data era.

Entropy formula of N-body system

We prove a proposition that the entropy of the system composed of finite N molecules of ideal gas is the q-entropy or Havrda–Charvát–Tsallis entropy, which is also known as Tsallis entropy, with the entropic index q=frac{D(N-1)-4}{D(N-1)-2} in D-dimensional space. The indispensable infinity assumption used by Boltzmann and others in their derivation of entropy formulae is not involved in our derivation, therefore our derived formula is exact. The analogy of the N-body system brings us to obtain the entropic index of a combined system q_C formed from subsystems having different entropic indexes q_A and q_B as frac{1}{1-q_C}=frac{1}{1-q_A}+frac{1}{1-q_B}+frac{D+2}{2}. It is possible to use the number N for the physical measure of deviation from Boltzmann entropy.

Generalized entropies and corresponding holographic dark energy models

Using Tsallis statistics and its relation with Boltzmann entropy, the Tsallis entropy content of black holes is achieved, a result in full agreement with a recent study (Mejrhit and Ennadifi in Phys Lett B 794:24, 2019). In addition, employing Kaniadakis statistics and its relation with that of Tsallis, the Kaniadakis entropy of black holes is obtained. The Sharma-Mittal and Rényi entropy contents of black holes are also addressed by employing their relations with Tsallis entropy. Thereinafter, relying on the holographic dark energy hypothesis and the obtained entropies, two new holographic dark energy models are introduced and their implications on the dynamics of a flat FRW universe are studied when there is also a pressureless fluid in background. In our setup, the apparent horizon is considered as the IR cutoff, and there is not any mutual interaction between the cosmic fluids. The results indicate that the obtained cosmological models have (i) notable powers to describe the cosmic evolution from the matter-dominated era to the current accelerating universe, and (ii) suitable predictions for the universe age.

Simulations of the Water Food Energy Nexus for policy driven intervention

Water-Food-Energy (WFE) resources exert mutual influences upon each other and thus cannot be managed separately. Information on household WFE expenditures addresses knowledge that distinguishes between geospatial districts' social welfare. Social welfare and investment in districts' WFE resources are interconnected. District (node) product of WFE normalized expenditures (Volume) is considered as a representative WFE Nexus holistic quantity. This Volume is assumed to be a function of residents' knowledge of welfare level across districts. We prove that the Volume rate conforms to Boltzmann entropy, and this is the premise of our hypothesis for directed information from high to low welfare between network nodes. Welfare mass (WM) represents the district's Volume combined with its income and population density. This WM is used as input into a model balancing between all domain nodes that allows policymakers to simulate the effects of potential quantifiable policy decisions targeted to individual districts at a domain level while also considering influences between districts. Based on existing historic data, the established tool exemplifies its potential by providing outcomes for Israel districts showing the influence of imposing different temporal allocation/deallocation actions as managerial regulations to prescribed districts. It is found that districts with a high WM do not suffer when a defund is applied, but districts that have a low WM gain from subsidies.

Generalization of the Dynamical Lack-of-Fit Reduction from GENERIC to GENERIC

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution. The dissipation potential is obtained by solving a Hamilton–Jacobi equation. In summary, the lack-of-fit reduction can start with GENERIC and obtain GENERIC for the reduced state variables.

Typical and extreme entropies of long-lived isolated quantum systems

In this paper, we investigate and compare two well-developed definitions of entropy relevant for describing the dynamics of isolated quantum systems: bipartite entanglement entropy and observational entropy. In a model system of interacting particles in a one-dimensional lattice, we numerically solve for the full quantum behavior of the system. We characterize the fluctuations, and find the maximal, minimal, and typical entropy of each type that the system can eventually attain through its evolution. While both entropies are low for some "special" configurations and high for more "generic" ones, there are several fundamental differences in their behavior. Observational entropy behaves in accord with classical Boltzmann entropy (e.g. equilibrium is a condition of near-maximal entropy and uniformly distributed particles, and minimal entropy is a very compact configuration). Entanglement entropy is rather different: minimal entropy "empties out" one partition while maximal entropy apportions the particles between the partitions, and neither is typical. Beyond these qualitative results, we characterize both entropies and their fluctuations in some detail as they depend on temperature, particle number, and box size.

A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils

In this paper we propose a robust and efficient primal-dual interior-point method for a nonlinear ill-conditioned problem with associated errors which are arising in the unfolding procedure for neutron energy spectrum from multiple activation foils. Based on the maximum entropy principle and Boltzmann's entropy formula, the discrete form of the unfolding problem is equivalent to computing the analytic center of the polyhedral set \begin{document}$ P = \{x \in R^n \mid Ax = b, x \ge 0\} $\end{document} , where the matrix \begin{document}$ A \in R^{m\times n} $\end{document} is ill-conditioned, and both \begin{document}$ A $\end{document} and \begin{document}$ b $\end{document} are inaccurate. By some derivations, we find a new regularization method to reformulate the problem into a well-conditioned problem which can also reduce the impact of errors in \begin{document}$ A $\end{document} and \begin{document}$ b $\end{document} . Then based on the primal-dual interior-point methods for linear programming, we propose a hybrid algorithm for this ill-conditioned problem with errors. Numerical results on a set of ill-conditioned problems for academic purposes and two practical data sets for unfolding the neutron energy spectrum are presented to demonstrate the effectiveness and robustness of the proposed method.

Entanglement entropy and the large N expansion of two-dimensional Yang-Mills theory

Two-dimensional Yang-Mills theory is a useful model of an exactly solvable gauge theory with a string theory dual at large N. We calculate entanglement entropy in the 1/N expansion by mapping the theory to a system of N fermions interacting via a repulsive entropic force. The entropy is a sum of two terms: the “Boltzmann entropy,” log dim(R) per point of the entangling surface, which counts the number of distinct microstates, and the “Shannon entropy,” − Σ pR log pR, which captures fluctuations of the macroscopic state. We find that the entropy scales as N2 in the large N limit, and that at this order only the Boltzmann entropy contributes. We further show that the Shannon entropy scales linearly with N, and confirm this behaviour with numerical simulations. While the term of order N is surprising from the point of view of the string dual — in which only even powers of N appear in the partition function — we trace it to a breakdown of large N counting caused by the replica trick. This mechanism could lead to corrections to holographic entanglement entropy larger than expected from semiclassical field theory.

Calculating the Wasserstein Metric-Based Boltzmann Entropy of a Landscape Mosaic.

Shannon entropy is currently the most popular method for quantifying the disorder or information of a spatial data set such as a landscape pattern and a cartographic map. However, its drawback when applied to spatial data is also well documented; it is incapable of capturing configurational disorder. In addition, it has been recently criticized to be thermodynamically irrelevant. Therefore, Boltzmann entropy was revisited, and methods have been developed for its calculation with landscape patterns. The latest method was developed based on the Wasserstein metric. This method incorporates spatial repetitiveness, leading to a Wasserstein metric-based Boltzmann entropy that is capable of capturing the configurational disorder of a landscape mosaic. However, the numerical work required to calculate this entropy is beyond what can be practically achieved through hand calculation. This study developed a new software tool for conveniently calculating the Wasserstein metric-based Boltzmann entropy. The tool provides a user-friendly human–computer interface and many functions. These functions include multi-format data file import function, calculation function, and data clear or copy function. This study outlines several essential technical implementations of the tool and reports the evaluation of the software tool and a case study. Experimental results demonstrate that the software tool is both efficient and convenient.

Boltzmann entropy for quantum field systems

A way to construct Boltzmann entropy, i.e., the entropy as a function of a microscopic pure state, for quantum field systems is proposed. Operators that shift the field in wavevector space are used in the construction. By employing an assumption, it is shown that, for almost all states in the ensemble of pure states corresponding to a thermodynamic state, the value of the proposed Boltzmann entropy coincides with that of the thermodynamic entropy for the thermodynamic state. For general self-interacting fields, the Boltzmann entropy evolves with time under Hamiltonian dynamics, so that it is capable of characterizing the thermalization of isolated quantum field systems.

Entanglement, partial set of measurements, and diagonality of the density matrix in the parton model

We define the "entropy of ignorance" which quantifies the entropy associated with ability to perform only a partial set of measurement on a quantum system. For a parton model the entropy of ignorance is equal to a Boltzmann entropy of a classical system of partons. We analyze a calculable model used for describing low x gluons in Color Glass Condensate approach, which has similarities with the parton model of QCD. In this model we calculate the entropy of ignorance in the particle number basis as well as the entanglement entropy of the observable degrees of freedom. We find that the two are similar at high momenta, but differ by a factor of order unity at low momenta. This holds for the Renyi as well as von Neumann entropies. We conclude that the entanglement does not seem to play an important role in the context of the parton model.

A Head/Tail Breaks-Based Method for Efficiently Estimating the Absolute Boltzmann Entropy of Numerical Raster Data

Shannon entropy is the most popular method for quantifying information in a system. However, Shannon entropy is considered incapable of quantifying spatial data, such as raster data, hence it has not been applied to such datasets. Recently, a method for calculating the Boltzmann entropy of numerical raster data was proposed, but it is not efficient as it involves a series of numerical processes. We aimed to improve the computational efficiency of this method by borrowing the idea of head and tail breaks. This paper relaxed the condition of head and tail breaks and classified data with a heavy-tailed distribution. The average of the data values in a given class was regarded as its representative value, and this was substituted into a linear function to obtain the full expression of the relationship between classification level and Boltzmann entropy. The function was used to estimate the absolute Boltzmann entropy of the data. Our experimental results show that the proposed method is both practical and efficient; computation time was reduced to about 1% of the original method when dealing with eight 600 × 600 pixel digital elevation models.

Landscape Sustainability Evaluation of Ecologically Fragile Areas Based on Boltzmann Entropy

From the perspective of landscape, it is important to evaluate the landscape sustainability of ecologically fragile areas and explore temporal and spatial evolution laws to promote their sustainable development. Presently, most studies on the analysis of landscape Boltzmann entropy (also called configurational entropy) are based on a single landscape, and most of these studies are theoretical discussions. However, there are few case studies on landscape ecology. The main objectives of this paper are to explore a quantitative relationship between Boltzmann entropy and landscape sustainability, to propose a method for evaluating landscape sustainability based on Boltzmann entropy, and to evaluate the sustainability of diverse landscapes in Mizhi County, Shaanxi Province, China. This article uses digital elevation model (DEM) data with a spatial resolution of 30 m in Mizhi County. The remote sensing data on Mizhi County from 2000 were obtained by the Landsat Enhanced Thematic Mapper (ETM) + sensor, and the high-resolution image of Mizhi County from 2015 was obtained by the Gaofen-1 satellite. In this article, the subbasins are taken as the evaluation unit, and the Boltzmann entropy of Mizhi County is calculated based on the experts’ scoring of landscape sustainability in the study area. Through the analysis of landscape sustainability results from 216 subbasins in Mizhi County in 2000 and 2015, the following conclusions are drawn: (1) the evaluation matrix proposed in this paper is effective, and the Boltzmann entropy obtained by this method can directly reflect the level of landscape sustainability; (2) during the research period, the landscape sustainability of Mizhi County showed a good trend overall, especially the three townships of Taozhen, Shadian, and Shigou, which were significantly improved, and these findings were consistent with the field investigation; (3) on the spatial level, the landscape sustainability of mid-eastern Mizhi County is relatively poor compared to that in other regions, but the sustainability is also slowly increasing.

Structuredness as a Measure of the Complexity of the Structure and the Role of Post-Dissipative Structures and Ratchet Processes in Evolution

As shown earlier, the algorithmic complexity, like Shannon information and Boltzmann entropy, tends to increase in accordance with the general law of complification. However, the algorithmic complexity of most material systems does not reach its maximum, i.e. chaotic state, due to the various laws of nature that create certain structures. The complexity of such structures is very different from the algorithmic complexity, and we intuitively feel that its maximal value should be somewhere between order and chaos. I propose a formula for calculation such structural complexity, which can be called - structuredness. The structuredness of any material system is determined by structures of three main types: stable, dissipative, and post-dissipative. The latter are defined as stable structures created by dissipative ones, directly or indirectly. Post-dissipative structures, as well as stable, can exist for an unlimited time, but at the micro level only, without energy influx. The appearance of such structures leads to the “ratchet” process, which determines the structure genesis in non-living and, especially, in living systems. This process allows systems with post-dissipative structures to develop in the direction of maximum structuring due to the gradual accumulation of these structures, even when such structuring contradicts the general law of complification.

Optimization of guide star catalog for daytime star sensors

Daytime star sensor provides accuracy navigation information to air vehicles near the ground in the daytime by observing stars. It has been an important development of modern star sensors. In order to achieve a high signal-to-noise ratio, daytime star sensors work in the infrared band to avoid interferences from sky background. Daytime star sensors output accurate attitudes by identifying the observed stars in the field of view (FOV) according to the loaded guide star catalog. Guide stars are usually required to be distributed uniformly on the celestial sphere to improve the performance of star pattern identification. The parameters including limiting magnitude and FOV are determined by processing the 2MASS star catalog as the original star data and performing star distribution statistics. After constellation features are discussed, the idea of distributing stars in the local FOV to constellations is put forward by using the star pair angular separations. An optimization algorithm to build the guide star catalog for daytime stars is proposed to achieve evenly distributed guide stars. The guide star catalog is established and analyzed, proving that the proposed algorithm has simple calculation and easy realization. The Boltzmann entropy of obtained guide star catalog drops two orders of magnitude. Guide stars are distributed more uniformly.