Dual-phase-lag transfer equations and entropy behavior in relaxational hydrodynamics

Extended thermodynamics of irreversible processes is developed; based on two postulates by which additional variables of the entropy density are dissipative fluxes and material time derivatives of the ordinary thermodynamic variables. Within these theories a more general approximation of entropy production is obtained. As a consequence of the proposed formalism, the constitutive dual-phase-lag equations, as well as equations of the conventional version of extended irreversible thermodynamics are obtained. The behavior of the entropy during oscillatory approach to equilibrium is considered. The proposed theory leads to a strictly monotonic dependency of the entropy on time.     

Molecular extended thermodynamics of rarefied polyatomic gases and wave velocities for increasing number of moments

Molecular extended thermodynamics of rarefied polyatomic gases is characterized by two hierarchies of equations for moments of a suitable distribution function in which the internal degrees of freedom of a molecule is taken into account. On the basis of physical relevance the truncation orders of the two hierarchies are proven to be not independent on each other, and the closure procedures based on the maximum entropy principle (MEP) and on the entropy principle (EP) are proven to be equivalent.     

New Exact Solutions for Konopelchenko-Dubrovsky Equation Using an Extended Riccati Equation Rational Expansion Method

Taking the Konopelchenko-Dubrovsky system as a simple example, some families of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by using the extended Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.     

Onsager-Casimir symmetry properties of the Burnett equations

We demonstrate the approximate nature of the Onsager-Casimir relations for the example of the linearized Burnett equations for a dilute gas. For any discussion of Onsager relations the choice of a correct set of thermodynamic forces and fluxes is of course crucial. By retracing the Chapman-Enskog procedure, we show that the usual expressions for the thermodynamic forces require modifications at the Burnett level. However, inclusion of these terms does not remedy the violation of Onsager symmetry first noticed by McLennan. A modified version of the Onsager symmetry that involves thermodynamic forces derived from an entropy Lagrangian rather than from the entropy itself does remain valid on the Burnett level. Throughout, we allow for the presence of an external potential; the Burnett equations including potential terms are derived in an appendix for a set of variables particularly suited for our discussion. We stress that in discussing Onsager relations one should use the full thermodynamic fluxes rather than their dissipative parts only, in spite of the fact that only the latter contribute to the entropy production.     

Analytical Investigation of Soliton Solutions to Three Quantum Zakharov-Kuznetsov Equations

In the present paper, the two-dimensional quantum Zakharov-Kuznetsov (QZK) equation, three-dimensional quantum Zakharov-Kuznetsov equation and the three-dimensional modified quantum Zakharov-Kuznetsov equation are analytically investigated for exact solutions using the modified extended tanh-expansion based method. A variety of new and important soliton solutions are obtained including the dark soliton solution, singular soliton solution, combined dark-singular soliton solution and many other trigonometric function solutions. The used method is implemented on the Mathematica software for the computations as well as the graphical illustrations.     

Dynamic statistical information theory

In recent years we extended Shannon static statistical information theory to dynamic processes and established a Shannon dynamic statistical information theory, whose core is the evolution law of dynamic entropy and dynamic information. We also proposed a corresponding Boltzmman dynamic statistical information theory. Based on the fact that the state variable evolution equation of respective dynamic systems, i.e. Fokker-Planck equation and Liouville diffusion equation can be regarded as their information symbol evolution equation, we derived the nonlinear evolution equations of Shannon dynamic entropy density and dynamic information density and the nonlinear evolution equations of Boltzmann dynamic entropy density and dynamic information density, that describe respectively the evolution law of dynamic entropy and dynamic information. The evolution equations of these two kinds of dynamic entropies and dynamic informations show in unison that the time rate of change of dynamic entropy densities is caused by their drift, diffusion and production in state variable space inside the systems and coordinate space in the transmission processes; and that the time rate of change of dynamic information densities originates from their drift, diffusion and dissipation in state variable space inside the systems and coordinate space in the transmission processes. Entropy and information have been combined with the state and its law of motion of the systems. Furthermore we presented the formulas of two kinds of entropy production rates and information dissipation rates, the expressions of two kinds of drift information flows and diffusion information flows. We proved that two kinds of information dissipation rates (or the decrease rates of the total information) were equal to their corresponding entropy production rates (or the increase rates of the total entropy) in the same dynamic system. We obtained the formulas of two kinds of dynamic mutual informations and dynamic channel capacities reflecting the dynamic dissipation characteristics in the transmission processes, which change into their maximum—the present static mutual information and static channel capacity under the limit case where the proportion of channel length to information transmission rate approaches to zero. All these unified and rigorous theoretical formulas and results are derived from the evolution equations of dynamic information and dynamic entropy without adding any extra assumption. In this review, we give an overview on the above main ideas, methods and results, and discuss the similarity and difference between two kinds of dynamic statistical information theories.     

Solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions

The known solution to the spatially homogeneous nonlinear Boltzmann equation for Maxwell models in a series of Laguerre polynomials is extended to include nonisotropic initial conditions. Existence proofs for a class of solutions are supplied. The equations for the generalized (nonisotropic Laguerre) moments are derived in explicit form for two- and three-dimensional models. Further it is shown that the ordinary moments satisfy the same set of equations as the (Hermite) polynomial moments.     

Entropy as a measure of diffusion

The time variation of entropy, as an alternative to the variance, is proposed as a measure of the diffusion rate. It is shown that for linear and time-translationally invariant systems having a large-time limit for the density, at large times the entropy tends exponentially to a constant. For systems with no stationary density, at large times the entropy is logarithmic with a coefficient specifying the speed of the diffusion. As an example, the large-time behaviors of the entropy and the variance are compared for various types of fractional-derivative diffusions.     

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.     

论动态信息理论

本文综述了作者的研究成果.近十年,作者将现有静态统计信息理论拓展至动态过程,建立了以表述动态信息演化规律的动态信息演化方程为核心的动态统计信息理论.基于服从随机性规律的动力学系统(如随机动力学系统和非平衡态统计物理系统)与遵守确定性规律的动力学系统(如电动力学系统)的态变量概率密度演化方程都可看成是其信息符号演化方程,推导出了动态信息(熵)演化方程.它们表明:对于服从随机性规律的动力学系统,动态信息密度随时间的变化率是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和耗损三者引起的,而动态信息熵密度随时间的变化率则是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和产生三者引起的.对于遵守确定性规律的动力学系统,动态信息(熵)演化方程与前者的相比,除动态信息(熵)密度在系统内部的态变量空间仅有漂移外,其余皆相同.信息和熵已与系统的状态和变化规律结合在一起,信息扩散和信息耗损同时存在.当空间噪声可略去时,将会出现信息波.若仅研究系统内部的信息变化,动态信息演化方程就约化为与表述上述动力学系统变化规律的动力学方程相对应的信息方程,它既可看成是表述动力学系统动态信息的演化规律,亦可看成是动力学系统的变化规律都可由信息方程表述.进而给出了漂移和扩散信息流公式、信息耗散率公式和信息熵产生率公式及动力学系统退化和进化的统一信息表述公式.得到了反映信息在传递过程中耗散特性的动态互信息公式和动态信道容量公式,它们在信道长度和信号传递速度之比趋于零的极限情况下变为现有的静态互信息公式和静态信道容量公式.所有这些新的理论公式和结果都是从动态信息演化方程统一推导出的.     

On the diffusive nature of entropy flow in infinite systems: Remarks to a paper by Guo-Papanicolau-Varadhan

The hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy (free energy) arguments. The methods of [13] are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise. Following [14, 15] and [3–5] we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended. The flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems [4].This work was supported in part by the Hungarian National Foundation for Scientific Research Grant 1815, NSF Grant DMR 86-12369, and by the Institut des Hautes Etudes Scientifiques     

Symmetry groups of mathematical physics

Recent work on Lie’s method of extended groups to obtain symmetry groups and invariants of differential equations of mathematical physics is surveyed. As an essentially new contribution one-parameter Lie groups admitted by three-dimensional harmonic oscillator, three-dimensional wave equation, Klein-Gordon equation, two-component Weyl’s equation for neutrino and four-component Dirac equation for Fermions are obtained.     

Mathematical inequalities for some divergences

Some natural phenomena are deviating from standard statistical behavior and their study has increased interest in obtaining new definitions of information measures. But the steps for deriving the best definition of the entropy of a given dynamical system remain unknown. In this paper, we introduce some parametric extended divergences combining Jeffreys divergence and Tsallis entropy defined by generalized logarithmic functions, which lead to new inequalities. In addition, we give lower bounds for one-parameter extended Fermi–Dirac and Bose–Einstein divergences. Finally, we establish some inequalities for the Tsallis entropy, the Tsallis relative entropy and some divergences by the use of the Young’s inequality.     

一种基于新型斜率限制器的理想磁流体方程的高分辨率熵相容格式

针对磁流体动力学方程, 通过分析数据重建所需的条件, 构造一种基于MUSCL(Monotone Upstream-Centred Scheme for Conservation Laws)型重建方法的斜率限制器, 获得了一种求解理想磁流体动力学方程的高分辨率熵相容格式。该格式在解的光滑区域具有高精度; 在解的间断区域可以合理地控制耗散, 可有效避免非物理现象的产生。采用熵稳定格式、熵相容格式和新的高分辨率熵相容格式对一维、二维理想磁流体动力学方程进行数值模拟。结果表明: 新格式能准确地捕捉解的结构, 且具有无振荡、高分辨、鲁棒等特性。     

Mapping deformation method and its application to nonlinear equations

An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein－Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.     

Maximum entropy principle for rarefied polyatomic gases

The aim of this paper is to show that the procedure of maximum entropy principle for the closure of the moments equations for rarefied monatomic gases can be extended also to polyatomic gases. The main difference with respect to the usual procedure is the existence of two hierarchies of macroscopic equations for moments of suitable distribution function, in which the internal energy of a molecule is taken into account. The field equations for 14 moments of the distribution function, which include dynamic pressure, are derived. The entropy and the entropy flux are shown to be a generalization of the ones for classical Grad’s distribution. The results are in perfect agreement with the recent macroscopic approach of extended thermodynamics for real gases.     

Relating Entropies of Quantum Channels

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi–Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.     

Extended Fan's Algebraic Method and Its Application to KdV and Variant Boussinesq Equations

An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinear partial differential equations. The key idea of this method is to introduce an auxiliary ordinary differential equation which is regarded as an extended elliptic equation and whose degree r is expanded to the case of r>4. The efficiency of the method is demonstrated by the KdV equation and the variant Boussinesq equations. The results indicate that the method not only offers all solutions obtained by using Fu's and Fan's methods, but also some new solutions.     

Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation Using an Extended sinh-Gordon Equation Expansion Method

The sinh-Gordon equation expansion method is further extended by generalizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the (2+1)-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.