Nonlinear waves in the nonequilibrium systems of the oscillatory type. Part II

The paper is the continuation of the preceding one (Part I). In the part II we consider the inhomogeneous quasistationary wave patterns which result from collisions of the travelling waves with close wave-vectors, and which may be regarded as narrow stationary spectral packages consisting of the quasimonochromatic travelling waves. Then we consider the long-wave oscillatory system. It is well known that in the absence of dispersion this system possesses the potential which makes it possible to compare the absolute stability of different quasistationary states. We make an attempt to construct an analogous quantity for the dispersive system. We argue that this quantity should be of the type of the potential production (an analogue of the entropy production in the nonequilibrium thermodynamics) rather than a generalization of the potential itself. Though we have no rigorous proof that our quantity indeed determines the absolute stability, we demonstrate that this hypothesis is verisimilar.     

Nonlinear standing waves,resonance phenomena,and frequency characteristics of distributed systems

This review is dedicated to resonator oscillations under conditions of a strongly expressed nonlinearity under which steep shock fronts emerge in the wave profiles. Models and approximated methods for their analysis for quadratic and cubic nonlinear media are examined, as well as for nonlinearity when taking into account the mobility of boundaries. The forms of the profiles are calculated both for a steady-state oscillation regime and during the establishment of the profiles. Dissipative losses and selective losses at specially chosen frequencies are considered. An analysis of nonlinear Q-factor is given. The possibility of increasing the acoustic energy accumulated in the cavity of the resonator is discussed. Special attention is given to various physical phenomena that are exhibited only in nonlinear acoustic fields.     

Nonlinear and nonequilibrium dynamics in geomaterials

The transition from linear to nonlinear dynamical elasticity in rocks is of considerable interest in seismic wave propagation as well as in understanding the basic dynamical processes in consolidated granular materials. We have carried out a careful experimental investigation of this transition for Berea and Fontainebleau sandstones. Below a well-characterized strain, the materials behave linearly, transitioning beyond that point to a nonlinear behavior which can be accurately captured by a simple macroscopic dynamical model. At even higher strains, effects due to a driven nonequilibrium state, and relaxation from it, complicate the characterization of the nonlinear behavior.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

Weak shock waves in negative-dispersion nonequilibrium media

The “frozen” and equilibrium shock adiabats for a gas with sustained steady-state nonequilibrium are constructed accurate to the second order of smallness. With these adiabats, the pattern of and stability conditions for weak shock waves in negative-dispersion nonequilibrium media, where the speed of low-frequency (equilibrium) sound exceeds that of high-frequency (frozen) sound, are considered. On the basis of a model nonlinear equation describing the evolution of gasdynamic perturbations in low-dispersion media, the nonstationary evolution of shock waves at a negative low-frequency nonlinearity coefficient is analyzed. This situation corresponds to a low-frequency adiabat convex upwards. It is shown that a step autowave may arise in this case whose amplitude is entirely specified by the nonequilibrium parameters of the medium and correlates with the point where the low-frequency and high-frequency adiabats intersect. In addition, it is found that the initial unsteady shock wave may split into two steady ones: a step autowave followed by a steady smooth-front expansion shock.     

Sound waves in a nonequilibrium molecular gas

We consider relaxation processes of dynamics of a nonequilibrium vibrationally excited gas (T_v>T₀). It is shown that the effective heat capacities can become negative in such a medium. This leads to anomalous effects during propagation of long-wave sound: there exist regions (in the parameter T_v/T₀) of sharp enhancement and reduction of its speed, as well as regions where low-frequency sound can generally not propagate. Instability conditions are investigated for various temperature dependences of vibrational relaxation times. It is shown that in a certain region of T_v/T₀ values one has amplification of sound waves, related to the formation of second viscosity in these media. In this case a change in the amount of medium nonequilibrium can vary the sound frequency corresponding to maximum amplification.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 53–58, July, 1986.     

Sensitivity of nonequilibrium systems

Simple mechanisms through which nonequilibrium structures can be influenced by external fields are discussed. It is shown that a very weak gravitational or electric field can have a large influence on selection or creation of structures. In the absence of cooperativity, the influence of a weak field, to the leading order, is characterized by the ratio (E_int/kT), where E_int is the energy of interaction; however, when there is far-from-equilibrium cooperativity, it is shown that the influence of the field is characterized by $\text{(E}{}_{\mathrm{int}}\text{kT)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$.     

Nonlinear longitudinal waves in the presence of the interaction between the strain and temperature fields and the field of nonequilibrium atomic defects

The propagation of nonlinear longitudinal waves in a plate is studied by taking into account the interaction of the longitudinal displacement component with the temperature field and the field of concentration of nonequilibrium atomic point defects. A nonlinear evolution equation is derived for describing the self-consistent thermoelastic longitudinal strain fields. It is shown that the thermoelastic effect on the strain waves manifests itself in the appearance of dissipative terms, which describe the heat transfer and the thermoelastic interaction caused by the strain-induced heat release due to the recombination of nonequilibrium atomic defects. The soliton solutions to the evolution equation are investigated, and the characteristic features of their damping are considered with allowance for the low-frequency and high-frequency losses.     

Theory of irreversible processes in nonequilibrium quantum systems. I. General formalism

The theory of Van Hove for nonequilibrium quantum statistical mechanics is extensively reformulated in terms of a superspace (a kind of operator space). This reformulation enables us to introduce a diagrammatic method which makes it convenient to deal with practical problems in physical systems. In our formalism, quantum statistical effects are considered on the basis of a systematic rule for the contraction technique. A complicated statistical effect in boson or fermion systems can be treated by starting with a simple unsymmetrized formalism in the Boltzmann statistics.     

Bose-Einstein condensation in nonequilibrium systems

The properties of quantum statistically degenerate systems of Bosons which are created by an external pump field and decay within a finite lifetime are investigated by means of a Green's function treatment. These investigations help to understand the physical properties of such condensed Bose-systems as excitons, excitonic molecules and spin aligned hydrogen atoms. As an example, recent experiments by Hulin et al. on degenerate excitons in Cu₂O are analyzed and a condensate fraction of about 5% is obtained.     

Time-symmetric fluctuations in nonequilibrium systems

For nonequilibrium steady states, we identify observables whose fluctuations satisfy a general symmetry and for which a new reciprocity relation can be shown. Unlike the situation in recently discussed fluctuation theorems, these observables are time-reversal symmetric. That is essential for exploiting the fluctuation symmetry beyond linear response theory. In addition to time reversal, a crucial role is played by the reversal of the driving fields that further resolves the space-time action. In particular, the time-symmetric part in the space-time action determines the second order effects of the nonequilibrium driving.     

Fluctuation,relaxation, and extensivity of macrovariables in nonequilibrium systems

The extensive property of a macrovariable is proved for a quantal system whose Hamiltonian depends on time and for a stochastic system whose temporal evolution operator depends on time. These generalized situations are concerned with bulk-contact open systems. The extensive property, fluctuation, and nonlinear relaxation are investigated explicitly by calculating rigorously generating functions in exactly soluble models such as the linear stochastic model and linearXY model. The relation between the nonlinear critical slowing down and linear critical slowing down is also discussed.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.     

Dust acoustic waves in a nonequilibrium dusty plasma

With the use of the method of moments applicable for any values of the parameter of the nonideality of a dusty plasma and the hydrodynamic approach applicable only for small nonideality parameters, the theory of waves and oscillations of a complex plasma has been generalized to the case of a two-exponential interaction potential. It has been shown that the hydrodynamic approach and method of moments give the same dispersion relation for small nonideality parameters. It has been demonstrated that the velocity of dust acoustic waves in the long- and short-wavelength regions is determined by the small and large screening constants, respectively. It has been shown that the velocity of dust acoustic waves in nonequilibrium plasma is much higher than that obtained in the Debye screening theory for equilibrium plasma. In the hydrodynamic approach, the importance of the inclusion of the self-consistent mutual effect of the dust, electron, and ion components, and sinks of electrons and ions on dust particles, which lead to a noticeable change in the parameters of the interaction potential of dust particles, has been demonstrated.