Soliton solutions,travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.     

On the Non-linear Generalization of the Gyarmati Principle and Theorem

The validity of the Gyarmati theorem has been extended to non-linear types of constitutive equations governing dissipative processes which have recently been suggested. It has been proved that by extending the validity of the theorem, the validity of Gyarmati's variational principle of thermodynamics is guaranted for non-linear theories adequate to constitutive equations in question as well. The theory was applied for stationary temperature distribution in a rigid bar.     

Semiclassical calculation of decay rates

Several relevant aspects of quantum-field processes can be well described by semiclassical methods. In particular, the knowledge of non-trivial classical solutions of the field equations, and the thermal and quantum fluctuations around them, provide non-perturbative information about the theory. In this work, we discuss the calculation of the one-loop effective action from the semiclasssical viewpoint. We intend to use this formalism to obtain an accurate expression for the decay rate of non-static metastable states.     

Determination of length scale parameters of strain gradient continuum theory for crystalline solids using a computational quantum mechanical model based on density functional theory

Although the classical continuum theory is advantageous in finding solutions to a wide range of engineering problems, it cannot describe some phenomena such as dispersion of acoustic waves, effects of surfaces and interfaces on the mechanical behavior of small-scale structures, and microstructure contribution in special materials. Owing to this fact, several enhanced continuum theories have evolved in the literature. However, the difficulty in determination of the length scale parameters that appear in the governing equations of such theories hampers their widespread use in practice. To date, except for a very limited number of materials, there is no known experimental procedure for the identification of these parameters. In this research, the internal length scales for an augmented continuum theory, i.e., Mindlin's strain gradient theory, have been theoretically determined for some crystalline materials with cubic structure that are of engineering interest, using ab initio DFT. According to the values obtained for these parameters, it can be perceived that the strain gradient theory is a valuable tool for capturing the size effects at even the smallest scales comparable to the dimensions of a unit cell of a crystal lattice.     

On Treder-Type Tetrad Theories of Gravitation II. Treder's Tetrad Theories in Linear Approximation

We consider some properties of TREDER'S tetrad theories, derived in I, using the field equations proposed by KASPER and LIEBSCHER . The linearized theory is considered, because the field energy becomes positive, if the energy of the weak field is a positive one. Using the dynamical equations, the field equations lead for the symmetric part of the field to the gauge invariant field equations in Hilbert gauge and to corresponding equations for the antisymmetric part. This means that in this approximation the dynamical equations replace the gauge invariance and the tetrad field corresponds to a mixture of tensor and scalar gravitons. We discuss possible experiments for showing the existence of scalar gravitons and limiting the free parameter of the theory.     

Classical versus quantum gravity

Is Einstein's metric theory of gravitation to be quantized to yield a complete and logically consistent picture of the geometry of the real world in the presence of quantized material sources? To answer this question, we give arguments that there is a consistent way to extend general relativity to small distances by incorporating further geometric quantities at the level of the connection into the theory and introducing corresponding field equations for their determination, allowing thereby the metric and the Levi-Civita connection to remain classical quantities. The dualism between matter and geometry is extended to quantized fields with the help of a Hibert bundle ? raised over a Riemann-Cartan spacetime. Quantized subnuclear matter fields (generalized quantum mechanical wave functions) are sections on ? which determine generalized bilinear currents acting as sourc currents for the bundle geometry at small distances. The established dualism between matter and the underlying bundle geometry contains general relativity as a classical part.     

The Mathematical Role of Time and Space-Time in Classical Physics

We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and spacetime in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that spacetime is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity.     

Novel Rotating Hairy Black Hole in (2+1)-Dimensions and Shear Viscosity to Entropy Ratio

The novel rotating hairy black hole metric in (2 + 1) dimensions, which is an exact solution to the field equations of the Einstein-scalar AdS theory with a non-minimal coupling, considered in this paper and some hydrodynamics quantities such as diffusion constant and shear viscosity investigated. By using thermodynamics quantities such as temperature and entropy we can use diffusion constant to obtain shear viscosity and then calculate shear viscosity to entropy ratio.     

Hierarchy of equations for the surface tension of solids

Of the four main equations in thermodynamics for the surface tension of condensed matter, i.e. the generalized and classical Lippmann equations and the Shuttleworth and Gokhshtein equations, only the classical Lippmann and Gokhshtein equations have been confirmed experimentally. The generalized Lippmann (Couchman–Davidson) equation is considered to be more universal, since three other equations could be derived from it. Although this fact has been widely accepted, it was recently reevaluated in two opposite ways. In the first approach, the experimental verification of the Gokhshtein equation should support the correctness of the generalized Lippmann and Shuttleworth equations. In the second approach, the incompatibility of the Shuttleworth equation with Hermann's mathematical structure of thermodynamics throws doubts upon all its corollaries, including the generalized Lippmann and Gokhshtein equations. However, both of these approaches are here shown to be erroneous, since the Gokhshtein equation cannot be correctly derived from any of the above-mentioned equations, and the opposite is also true: neither the generalized Lippmann nor Shuttleworth equations could be derived from the Gokhshtein equation.     

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.     

Prequantum Classical Statistical Field Theory: Complex Representation, Hamilton-Schrödinger Equation, and Interpretation of Stationary States

We develop a prequantum classical statistical model in that the role of hidden variables is played by classical (vector) fields. We call this model Prequantum Classical Statistical Field Theory (PCSFT). The correspondence between classical and quantum quantities is asymptotic, so we call our approach asymptotic dequantization. We construct the complex representation of PCSFT. In particular, the conventional Schrödinger equation is obtained as the complex representation of the system of Hamilton equations on the infinite-dimensional phase space. In this note we pay the main attention to interpretation of so called pure quantum states (wave functions) in PCSFT, especially stationary states. We show, see Theorem 2, that pure states of QM can be considered as labels for Gaussian measures concentrated on one dimensional complex subspaces of phase space that are invariant with respect to the Schrödinger dynamics. “A quantum system in a stationary state ψ” in PCSFT is nothing else than a Gaussian ensemble of classical fields (fluctuations of the vacuum field of a very small magnitude) which is not changed in the process of Schrödinger's evolution. We interpret in this way the problem of stability of hydrogen atom. One of unexpected consequences of PCSFT is the infinite dimension of physical space on the prequantum scale.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Effect of arbitrary matter-geometry coupling on thermodynamics in f(R) theories of gravity

In this paper, the thermodynamics of the Friedmann–Lemaître–Robertson–Walker universe have been explored in f(R) theories of gravity with arbitrary matter-geometry coupling. The equivalence between the modified Friedmann equations with any spatial curvature and the first law of thermodynamics is confirmed, where the assumption of the entropy plays a crucial role. Then laws of thermodynamics in our considering case are obtained. They can reduce to the ones given in Einstein’s general theory of relativity under certain conditions. Moreover, a particular model is investigated through the obtained generalized second law of thermodynamics with observational results of cosmographic parameters.     

Equilibrium theories of simple liquids

In this article, the theory of equilibrium properties of simple classical fluids is reviewed. The various relationships between the pair potential φ(r) and the pair correlation function g(r) are explored, from the usual integral equations and the perturbation theories to the generalized random phase approximation proposed recently. Particular attention is devoted to the extraction of the intermolecular forces from a given experimental data on the structure and thermodynamics of fluids. In particular, the propagation of errors in these calculations arising due to uncertainties in the input data is discussed. Finally, the recent use of BGY integral equation and the vacancy-cell model in the study of solid-liquid transition and melting is discussed.     

Relativistic Rational Extended Thermodynamics of Polyatomic Gases with a New Hierarchy of Moments

A relativistic version of the rational extended thermodynamics of polyatomic gases based on a new hierarchy of moments that takes into account the total energy composed by the rest energy and the energy of the molecular internal mode is proposed. The moment equations associated with the Boltzmann–Chernikov equation are derived, and the system for the first 15 equations is closed by the procedure of the maximum entropy principle and by using an appropriate BGK model for the collisional term. The entropy principle with a convex entropy density is proved in a neighborhood of equilibrium state, and, as a consequence, the system is symmetric hyperbolic and the Cauchy problem is well-posed. The ultra-relativistic and classical limits are also studied. The theories with 14 and 6 moments are deduced as principal subsystems. Particularly interesting is the subsystem with 6 fields in which the dissipation is only due to the dynamical pressure. This simplified model can be very useful when bulk viscosity is dominant and might be important in cosmological problems. Using the Maxwellian iteration, we obtain the parabolic limit, and the heat conductivity, shear viscosity, and bulk viscosity are deduced and plotted.     

Fluids with highly directional attractive forces. IV. Equilibrium polymerization

We investigate approximation methods for systems of molecules interacting by core repulsion and highly directional attraction due to several attraction sites. The force model chosen imitates a chemical bond by providing for bond saturation when binding occurs. The dense fluid is an equilibrium mixture ofs-mers with mutual repulsion. We use a previously derived reformulation of statistical thermodynamics, in which the particle species are monomeric units with a specified set of attraction sites bonded. Thermodynamic perturbation theory (TPT) and integral equations of two types are derived. The use of TPT is illustrated by explicit calculation for a molecular model with two attraction sites, capable of forming chain and ring polymers. Successes and defects of TPT are discussed. The integral equations for pair correlations between particles of specified bonding include calculation of self-consistent densities of species. Methods of calculating thermodynamic properties from the solutions of integral equations are given.Supported by the NSF under grant No. CHE-82-11236.     

Remarks on the Idea of Spontaneous Break-Down of Symmetry,Cosmic Variability of Eddington's Number,Mach's Idea Concering the Origin of Inertia,and Field Equations Describing Global and Local Gravitational Fields

Subject is considered on the level of classical field theory. We start from some aspects of the theory of ferromagnets. Their counterpart in classical field theory is pointed out using the over simplified model of a selfinteracting scalar field. The ground state (“vacuum expection value”) of the scalar field is interpreted as cosmic background field, which can be considered as constant for local physical phenomena. In practice, however, it is a function of the age of universe. Which kind of function it could be is suggested by a discussion of the cosmic variability of Eddington's number γ = 10⁴⁰, which refers to Dirac's consideration of this problem. But contrary to Dirac's assumption that atomic quantities are constant, we suppose that the inertial mass of elementary particles is a function of the age of universe. The cosmic gravitational field is described by other equations than the gravitational field created by local matter distributions. The field equations for the local gravitational field we start from reduce to Einstein's equations, if we neglect the possible influence of the universe on local phenomena. In case that the cosmic matter is homogeneously and isotropically distributed, the field equations for the cosmic gravitational field permit only such a time dependent solution the three-spaces of which are linearly expanding and spherically closed. The different field equations for cosmic and local gravitational fields are considered approximations of more fundamental field equations which approximately split into two sets of equations, if it is possible to contrast local physical systems with the universe. The described cosmological model taken as a basis, the inertial mass of elementary particles becomes a function of the matter density creating the cosmic gravitational field. This could be considered as, at least, partly realisation of Mach's idea concerning the origin of inertia. Starting from the interpretation of the ground state (vacuum expection value) as a function of a certain cosmic background field, more realistic gage field models could give the following picture of cosmic development: In the far past there was a state of the universe characterized by enormous contraction of matter. In this stage of development, it was impossible to contrast particles with the universe. Matter expands and it becomes possible to contrast certain physical systems with the universe. But the ground state is such a symmetric one that only fields with vanishing rest mass can be contrasted with the universe (ferromagnet above Curie temperature). With further expansion of the universe the ground state will lose certain symmetry properties. By this it becomes possible that you get the impression there are particles with nonvanishing rest mass (ferromagnet below Curie temperature). Finally, the influence of the universe on local physical systems goes to zero with further expansion. Especially, this means the inertial mass of elementary particles goes to zero, too (Curie temperature of ferromagnetic material goes to zero with cosmic expansion).     

玻璃转变研究进展

玻璃转变是凝聚态物理基础理论中的一个重要问题和难题，是涉及动力学和热力学的众多前沿问题．玻璃转变的理论一直在不断的发展和更新．从20世纪50年代出现的自由体积理论到现在还在不断完善的模态耦合理论及其他众多理论，都只能解决玻璃转变中的某些问题．一个完整的玻璃转变理论仍需要人们作艰苦的努力．为了澄清混淆不清的玻璃转变概念，文章就玻璃转变的概念、研究内容和有关理论的发展进行简述．在分析了几个占主导地位的玻璃转变理论后，阐述了玻璃转变中需要进一步深入研究的问题．     

Description of far-from-equilibrium processes by mean-field lattice gas models

Mean-field kinetic equations are a valuable tool to study the atomic dynamics and spin dynamics of simple lattice gas and Ising models. They can be derived from the microscopic master equation of the system and contain analytical expressions for kinetic coefficients and thermodynamic quantities which are usually introduced phenomenologically. We review several methods to obtain such equations, and discuss applications to the dynamics of order–disorder transitions, spinodal decomposition, and dendritic growth in the isothermal or chemical model. In the case of dendritic growth we show that the mean-field kinetic equations are equivalent to standard continuum equations for this problem and derive expressions for macroscopic quantities, e.g. the surface tension and kinetic coefficients, as functions of the microscopic order parameters. In spinodal decomposition, we focus our attention on the vacancy mechanism, which is a more faithful picture of diffusion in solids than the more widely examined exchange mechanism. We study the interfaces between an unstable mixture and a stable ‘vapour’ phase, and analyse surface modes that lead to specific surface patterns. For order–disorder transitions, studied in the framework of a repulsive two-sublattice model, we derive sets of coupled equations for the mean concentration (a conserved quantity) and for the occupational difference between the two sublattices emerging from the symmetry breaking due to ordering (non-conserved order parameter). These equations are applied to transport in the presence of ordered domains. Finally, we discuss the possibilities of improving the simple mean-field approximation by density functional theories and various forms of the dynamic pair approximation, including the path-probability method.