信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系--兼论玻尔兹曼熵和克劳修斯熵是否等价

论述了信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系;由不涉及具体系统的方法从玻尔兹曼关系、信息熵推导出了克劳修斯熵的表达式;指出玻尔兹曼熵与克劳修斯熵不是等价关系,而是玻尔兹曼熵包含克劳修斯熵,信息熵又包含玻尔兹曼熵。     

Upper entropy axioms and lower entropy axioms

The paper suggests the concepts of an upper entropy and a lower entropy. We propose a new axiomatic definition, namely, upper entropy axioms, inspired by axioms of metric spaces, and also formulate lower entropy axioms. We also develop weak upper entropy axioms and weak lower entropy axioms. Their conditions are weaker than those of Shannon–Khinchin axioms and Tsallis axioms, while these conditions are stronger than those of the axiomatics based on the first three Shannon–Khinchin axioms. The subadditivity and strong subadditivity of entropy are obtained in the new axiomatics. Tsallis statistics is a special case of satisfying our axioms. Moreover, different forms of information measures, such as Shannon entropy, Daroczy entropy, Tsallis entropy and other entropies, can be unified under the same axiomatics.     

Physical information entropy and probability Shannon entropy

In a previous work by one of us (R. Urigu) concerning open quantum systems it was remarked that in processes of the type , when evaluating the information entropy of the environment as the Shannon entropy of the outcome probabilities in the channels , the total information entropy may decrease. We remark here that this decrease is easily excluded by requiring a condition of quantum modelizability of the environment even with respect to Shannon entropy (“cybernetic interpretability” of the environment). Further conditions on the quantum model of the environment are defined (“maximal observability” and “Boolean interpretability”), which are proved to be equivalent, and it turns out that, once satisfied in one model, they also are in any model with pure initial state; furthermore, these conditions turn out to be equivalent to the condition that the process consists of pure operations of the first kind. The relevance to the concept of macroscopicity and to the “von Neumann chain” is discussed.     

Sharp continuity bounds for entropy and conditional entropy

<正>The von Neumann entropy captures many operational quantities in the quantum information theory such as quantum capacity of the communication channel.Von Neumann entropy is continuous and is represented by Fannes inequality,which was originally given in ref.[1].Quantum correlations such as entanglement and quantum discord etc.,are important resources in quantum information processing.In the last year enormous progress on the generation,concentration,detection and quantification of entanglement has been achieved     

Non-equilibrium entropy

The correction, which is due to interactions between particles, to the non-equilibrium entropy of an ideal Bose-gas is calculated.     

漫谈熵

熵是物理中的一个既重要又微妙的概念。文章从物理学引入熵谈起,依次讨论熵与热力学第二定律、熵的统计力学定义、熵增与基础物理理论的矛盾,以及时间箭头与玻尔兹曼大脑,最后介绍著名的黑洞熵。     

Entanglement entropy converges to classical entropy around periodic orbits

We consider oscillators evolving subject to a periodic driving force that dynamically entangles them, and argue that this gives the linearized evolution around periodic orbits in a general chaotic Hamiltonian dynamical system. We show that the entanglement entropy, after tracing over half of the oscillators, generically asymptotes to linear growth at a rate given by the sum of the positive Lyapunov exponents of the system. These exponents give a classical entropy growth rate, in the sense of Kolmogorov, Sinai and Pesin. We also calculate the dependence of this entropy on linear mixtures of the oscillator Hilbert-space factors, to investigate the dependence of the entanglement entropy on the choice of coarse graining. We find that for almost all choices the asymptotic growth rate is the same.     

Topological entanglement entropy

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator rho for the degrees of freedom in the interior. The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho) = alphaL - gamma + ..., where the ellipsis represents terms that vanish in the limit L --> infinity. We show that - gamma is a universal constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for gamma in terms of properties of the superselection sectors of the medium.     

Entropy, entropy flux and entropy rate of granular materials

The aim of this work is to analyze the entropy, entropy flux and entropy rate of granular materials within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the entropy density rate. From the knowledge of a non-equilibrium distribution function-valid for processes closed to equilibrium-it is obtained that the entropy density rate is proportional to the internal energy density rate divided by the temperature, while the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular material is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the entropy density rate and entropy flux are valid in general for processes close to equilibrium of granular materials, where linearized constitutive equations hold.     

Local quantum entropy

It is shown that it is possible to define true local entropy in velocity space, in an approximate version of the two-fluid formulation of quantum theory introduced by the present author in earlier papers. Using this definition, it is then shown that it is possible to define finite forms for total entropy at all points in configuration space. This important step is achieved by the introduction of a responding velocity space. The use of a basis system which responds to occupation number density, makes possible a clear separation of the statistics and the dynamics of the underlying quantum process, and also makes possible the unambiguous use of certain divergent and oscillatory integrals.     

Quantum games entropy

We propose the study of quantum games from the point of view of quantum information theory and statistical mechanics. Every game can be described by a density operator, the von Neumann entropy and the quantum replicator dynamics. There exists a strong relationship between game theories, information theories and statistical physics. The density operator and entropy are the bonds between these theories. The analysis we propose is based on the properties of entropy, the amount of information that a player can obtain about his opponent and a maximum or minimum entropy criterion. The natural trend of a physical system is to its maximum entropy state. The minimum entropy state is a characteristic of a manipulated system, i.e., externally controlled or imposed. There exist tacit rules inside a system that do not need to be specified or clarified and search the system equilibrium based on the collective welfare principle. The other rules are imposed over the system when one or many of its members violate this principle and maximize its individual welfare at the expense of the group.     

On entropy production

We investigate the case of a dynamical system when irreversible time evolution is generated by a nonHermitian superoperator on the states of the system. We introduce a generalized scalar product which can be used to construct a monotonically changing functional of the state, a generalized entropy. This will depend on the level of system dynamics described by the evolution equation. In this paper we consider the special case when the irreversibility derives from imbedding the system of interest into a thermal reservoir. The ensuing time evolution is shown to be compatible both with equilibrium thermodynamics and the entropy production near the final steady state. In particular, Prigogine’s principle of minimum entropy production is discussed. Also the limit of zero temperature is considered. We present comments on earlier treatments.     

Symbolic transfer entropy

We propose to estimate transfer entropy using a technique of symbolization. We demonstrate numerically that symbolic transfer entropy is a robust and computationally fast method to quantify the dominating direction of information flow between time series from structurally identical and nonidentical coupled systems. Analyzing multiday, multichannel electroencephalographic recordings from 15 epilepsy patients our approach allowed us to reliably identify the hemisphere containing the epileptic focus without observing actual seizure activity.     

Topological permutation entropy

Permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take, just as Shannon entropy quantifies the diversity of the values themselves. When the observable or state variable has a natural order relation, making permutation entropy possible to compute, then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source. Herein, extending a previous result on metric entropy rate, we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics. This result is not limited to one-dimensional maps.     

Black hole entropy quantization

Ever since the pioneering works of Bekenstein and Hawking, black hole entropy has been known to have a quantum origin. Furthermore, it has long been argued by Bekenstein that entropy should be quantized in discrete (equidistant) steps given its identification with horizon area in (semi-)classical general relativity and the properties of area as an adiabatic invariant. This lead to the suggestion that the black hole area should also be quantized in equidistant steps to account for the discrete black hole entropy. Here we shall show that loop quantum gravity, in which area is not quantized in equidistant steps, can nevertheless be consistent with Bekenstein's equidistant entropy proposal in a subtle way. For that we perform a detailed analysis of the number of microstates compatible with a given area and show consistency with the Bekenstein framework when an oscillatory behavior in the entropy-area relation is properly interpreted.