On the multicomponent surface phase balance equations

The system of the field equations is formulated at the phenomenologically modelledn-component surface phase (e.g. interface, solidification front, shock wave, membrane etc.). The surface phase balance equations then constitute a closed set of the general boundary conditions of the appropriate balance equations of volume phases (as, for example, the heat conduction or diffusion equations etc.).     

A Modified Homogeneous Balance Method and Its Applications

A modified homogeneous balance method is proposed by improving some key steps in the homogeneous balance method. Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneous balance method. Generalized Boussinesq equation, KP equation, and mKdV equation are chosen as examples to llustrate our method. This approach is also applicable to a large variety of nonlinear evolution equations.     

Microscopic derivation of hydrodynamic balance equations in the presence of an electromagnetic field

Using field-theoretic methods we derive balance equations for a charged fluid in an external electromagnetic field the effects of which are included by minimal coupling. An infinite hierarchy of balance equations for tensor operators is derived. A macroscopic velocity field is introduced by a unitary transformation on the field operators. Suitable statistical averages in the local equilibrium approximation yield macroscopic balance equations. The significance of new terms is discussed.     

Application of an extended homogeneous balance method to new exact solutions of nonlinear evolution equations

The multiple soliton solutions of the approximate equations for long water waves and soliton-like solutions for the dispersive long-wave equations in 2+1 dimensions are constructed by using an extended homogeneous balance method. Solitary wave solutions are shown to be a special case of the present results. This method is simple and has a wide-ranging practicability, and can solve a lot of nonlinear partial differential equations.     

Energy-momentum description of electrostatic waves in magnetoactive plasmas

The physical mechanism of the energy-momentum transfer governing the propagation of electrostatic waves in collisionsless plasmas is presented. Plasma is supposed to be immersed in external uniform crossed magnetic and electric field. The equilibrium plasma, determined by a stationary distribution of charged particles, is assummed to be generally anisotropic and weakly nonuniform. The changes in macroscopic quantities (in the kinetic energy of perpendicular and parallel motion, etc.) due to the self-consistent wave-particle interaction are derived. It is shown, that the corresponding dispersion equation is identical with the energy-momentum balance equations expressed in the wave frame. A new expression of the energy of waves (plasmons) is given, which ensures the energy-momentum balance equations to be mutually independent equations. This differs from the usual expression of the wave energy leading to energy-momentum balance equations which are not mutually independent.     

Extension of Hydrodynamic Balance Equations for Simple Fluids

A systematic approach to derivation of hydrodynamic-like balance equations for systems with a smooth continuous potential as well as hard sphere repulsion is represented. Cases of many-particle local and two-particle nonlocal hydrodynamic densities are considered. The results are applied for construction of balance equations for fluxes of momentum and energy, which form the first extension of conventional hydrodynamics. Explicit balance equations for the stress tensor and the heat flux in the local frame of reference are obtained.     

Model of discontinuity surface in continuum physics

In this paper we adopt the division of the discontinuity surfaces into autonomous, nonautonomous and surfaces of jump. A uniform, general integral form of balance equations (conservation laws) is derived for all three types and its localisation to the point on the discontinuity surface is carried out. In the case of the surfaces of jump the local balance equation takes the form of the Kotchine condition. Local form of the balance equations is specified for the mass, momentum, energy (total, kinetic and internal) and entropy. The equation which expresses the hypothesis of local equilibrium for a discontinuity surface is derived. This equation reflects also the phase transitions that take place in equilibrium. The relations of the derived results to other theories are discussed.     

Evolution Equations for Fermion Systems in the Continual Representation

The quantum balance equations are derived for the number of particles, the momentum, the energy, and the magnetic moment density. These equations in the classical limit 0 transform into the well-known balance equations. The equation for the magnetic moment is a generalization of the Bloch equation. It is also shown that the spin-spin interaction Hamiltonian, conventionally used in the quantum theory of a many-particle system, yields incorrect equations for the magnetic field in a medium and that this defect can be eliminated. An important role of the Bohm quantum potential is demonstrated for a system of identical bosons and fermions.     

Some properties of Fokker-Planck equations with memory under detailed balance

Some properties of the Fokker-Planck equations with memory under the condition that detailed balance holds are discussed. The appropriate conditions which the drift and diffusion coefficients for a system in detailed balance should satisfy are obtained. The analysis is then restricted to a special class of Fokker-Planck equations namely the one studied previously by Zwanzig. Analog of Onsager's reciprocity relation is obtained and the eigenfunction expansion of the conditional probability is given. Finally the linear response theory of such systems is also briefly discussed.     

Fluctuations and stability of stationary non-equilibrium systems in detailed balance

A generalized thermodynamic potential for Markoffian systems with detailed balance and far from thermal equilibrium has been derived in a previous paper. It was shown that the principle of detailed balance is equivalent to a set of conditions fulfilled by this potential (“potential conditions”). The properties of this potential allow us to extend the validity of a number of thermodynamic concepts well known for systems in or near thermal equilibrium to stationary states far from thermal equilibrium. The concept of symmetry breaking phase transitions for these systems is introduced in analogy to thermal equilibrium systems by considering the dependence of the stationary probability density of the system on a set of externally controlled parameters {λ}. A functional of the time dependent probability density of the system is defined in close analogy to the Gibb's definition of entropy. This functional has the properties of a Ljapunov functional of the governing Fokker-Planck equation showing the stability of the stationary probability density. The Langevin equations connected with the Fokker-Planck equation are considered. It is shown that, by means of the potential conditions, generalized “thermodynamic” fluxes and forces may be defined in such a way that the smoothly varying part of the Langevin equations (kinetic equations) constitutes a linear relation between fluxes and forces. The matrix of coefficients is given by the diffusion matrix of the Fokker-Planck equation. The symmetry relations which hold for this matrix due to the potential conditions then lead to the Onsager-Casimir symmetry relations extended to systems with detailed balance near stationary states far from thermal equilibrium. Finally it is shown that under certain additional assumptions the generalized thermodynamic potential may be used as a Ljapunov function of the kinetic equations.     

Why solve the Hamiltonian constraint in numerical relativity?

The indefinite sign of the Hamiltonian constraint means that solutions to Einstein's equations must achieve a delicate balance – often among numerically large terms that nearly cancel. If numerical errors cause a violation of the Hamiltonian constraint, the failure of the delicate balance could lead to qualitatively wrong behavior rather than just decreased accuracy. This issue is different from instabilities caused by constraint-violating modes. Examples of stable numerical simulations of collapsing cosmological spacetimes exhibiting local mixmaster dynamics with and without Hamiltonian constraint enforcement are presented.     

Kinetic modeling of a slow flow CW CO2 laser

A kinetic model for analysis of the slow-flow CW-discharge CO₂ laser with diffusion cooling has been developed in which the gas temperature is obtained from energy balance equations. The method is based on the numerical solution of a set of nonlinear differential equations for vibrational kinetics. The numerical predictions from the model are compared with some experimental results and a good agreement is obtained.     

The residue harmonic balance for fractional order van der Pol like oscillators

When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time.     

Solution of Fokker Planck equations with and without manifest detailed balance

Previous work on Fokker Planck equations with manifest detailed balance is generalized to include also the case without manifest detailed balance. The two cases are unified by exhibiting a general time reversal transformation with respect to which any Fokker Planck equation satisfies detailed balance, provided its steady state distribution exists. We also introduce a new method for solving some Fokker Planck equations with nonvanishing steady state drift by analytic continuation of the solution of a hermitian eigenvalue problem.     

Solution of Fokker Planck equations with and without manifest detailed balance

Previous work on Fokker Planck equations with manifest detailed balance is generalized to include also the case without manifest detailed balance. The two cases are unified by exhibiting a general time reversal transformation with respect to which any Fokker Planck equation satisfies detailed balance, provided its steady state distribution exists. We also introduce a new method for solving some Fokker Planck equations with nonvanishing steady state drift by analytic continuation of the solution of a hermitian eigenvalue problem.     

Convergence to Equilibrium for the Discrete Coagulation-Fragmentation Equations with Detailed Balance

Under the condition of detailed balance and some additional restrictions on the size of the coefficients, we identify the equilibrium distribution to which solutions of the discrete coagulation-fragmentation system of equations converge for large times, thus showing that there is a critical mass which marks a change in the behavior of the solutions. This was previously known only for particular cases as the generalized Becker–Döring equations. Our proof is based on an inequality between the entropy and the entropy production which also gives some information on the rate of convergence to equilibrium for solutions under the critical mass.     

Dynamics of a Fermi system with resonant dissipation and dynamical detailed balance

The dissipative dynamics of a system of Fermions is described in the framework of a resonance model—the quantum master equation describes two-body correlations of the system with the environment particles. This equation, with microscopic coefficients depending on the exactly known two-body potential between the system and the environment particles, is discussed in comparison with other master equations, obtained on axiomatic grounds, or derived from a coupling with an environment of harmonic oscillators without altering the quantum conditions. The asymptotic solution is in accordance with the detailed balance principle, and with other generally accepted conditions satisfied during the whole time-evolution: Pauli master equations for the diagonal elements of the density matrix, and damped Bloch–Feynman equations for the non-diagonal ones, that we call dynamical detailed balance. For a harmonic oscillator coupled with the electromagnetic field through dipole interaction, a master equation with transition operators between successive levels is obtained. As an application, the decay width of a quantum logic gate is calculated.     

Analysis of Equilibrium States of Markov Solutions to the 3D Navier-Stokes Equations Driven by Additive Noise

We prove that every Markov solution to the three dimensional Navier-Stokes equations with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.     

Relativistic kinetics of phonon gas in superfluids

The relativistic kinetic theory of the phonon gas in superfluids is developed. The technique of the derivation of macroscopic balance equations from microscopic equations of motion for individual particles is applied to an ensemble of quasi-particles. The necessary expressions are constructed in terms of a Hamilton function of a (quasi-)particle. A phonon contribution into superfluid dynamic parameters is obtained from energy-momentum balance equations for the phonon gas together with the conservation law for superfluids as a whole. Relations between dynamic flows being in agreement with results of relativistic hydrodynamic consideration are found. Based on the kinetic approach a problem of relativistic variation of the speed of sound under phonon influence at low temperature is solved.     

The Solitary Waves for Cubic-Quintic Nonlinear Schrödinger Equation with Variable Coefficients

The cubic-quintic nonlinear Schrödinger equation (CQNLS) plays important parts in the optical fiber and the nuclear hydrodynamics. By using the homogeneous balance principle, the bell type, kink type, algebraic solitary waves, and trigonometric traveling waves for the cubic-quintic nonlinear Schrödinger equation with variable coefficients (vCQNLS) are derived with the aid of a set of subsidiary high-order ordinary differential equations (sub-equations for short). The method used in this paper might help one to derive the exact solutions for the other high-order nonlinear evolution equations, and shows the new application of the homogeneous balance principle.