Nonlinear waves in nonequilibrium systems of the oscillatory type,part I

Nonlinear waves in mathematical models of nonequilibrium spatially uniform media with the oscillatory instability of the trivial state are considered. The models are based on the generalized Ginsburg-Landau equations. For the long-wave system, i.e. that described by two-component reaction-diffusion equations, we obtain the full stability conditions for monochromatic plane travelling waves. The basic part of the paper is devoted to the short-wave system which can be described by reaction-diffusion equations with not less than three components or by a two-component system with residual nonlocality. We construct the Ginsburg-Landau equation for this system, and we find its general quasistationary one-dimensional solution which is a travelling wave modulated by a travelling envelope wave. The stability of this solution is investigated with the especial emphasis on different important particular cases. The obtained results are compared with experimental observations of different waves on fronts of detonation and non-gaseous combustion (which also are characterized by the oscillatory short-wave instability of the trivial state), and the qualitative agreement between theoretical and experimental results is demonstrated.     

FT NMR of nonequilibrium states of complex spin systems. II. Weakly coupled systems

Since the Fourier transform NMR spectrum of a nonequilibrium state may be different from the slow-passage spectrum, a theory is developed which describes the phenomenon in any weakly coupled system, including those with magnetic equivalence. Liouville space techniques are used to derive simple expressions for the line intensities as a function of flip angle, in terms of Legendre polynomials. The theory uses F² representations directly and is applicable to any weakly coupled system. The results are tabulated for A_mX_n systems, m,n = 1, 2, 3, and 6. The theory is still applicable when. the weak coupling approximation starts to break down and line intensities are perturbed. It is shown that if the nonequilibrium intensities are corrected by a simple factor, as in the equilibrium case, these systems may be treated as if they were weakly coupled. In practice, the theory may be used to extrapolate to the slow-passage, zero flip angle spectrum. Furthermore, different parts of a degenerate line may be separated by means of the flip angle dependence. As an illustration, a nonequilibrium state was created among the protons in ethyl acetate. The experimental flip angle dependence of its spectra is well represented by the theory.     

Stochastic theory of nonequilibrium steady states. Part II: Applications in chemical biophysics

The mathematical theory of nonequilibrium steady state (NESS) has a natural application in open biochemical systems which have sustained source(s) and sink(s) in terms of a difference in their chemical potentials. After a brief introduction in Section 1, in Part II of this review, we present the widely studied biochemical enzyme kinetics, the workhorse of biochemical dynamic modeling, in terms of the theory of NESS (Section 2.1). We then show that several phenomena in enzyme kinetics, including a newly discovered activation-inhibition switching (Section 2.2) and the well-known non-Michaelis-Menten-cooperativity (Section 2.3) and kinetic proofreading (Section 2.4), are all consequences of the NESS of driven biochemical systems with associated cycle fluxes. Section 3 is focused on nonlinear and nonequilibrium systems of biochemical reactions. We use the phosphorylation-dephosphorylation cycle (PdPC), one of the most important biochemical signaling networks, as an example (Section 3.1). It starts with a brief introduction of the Delbrück-Gillespie process approach to mesoscopic biochemical kinetics (Sections 3.2 and 3.3). We shall discuss the zeroth-order ultrasensitivity of PdPC in terms of a new concept — the temporal cooperativity (Sections 3.4 and 3.5), as well as PdPC with feedback which leads to biochemical nonlinear bistability (Section 3.6). Also, both are nonequilibrium phenomena. PdPC with a nonlinear feedback is kinetically isomorphic to a self-regulating gene expression network, hence the theory of NESS discussed here could have wide applications to many other biochemical systems.     

Theory of irreversible processes in nonequilibrium quantum systems. II. Cluster expansions with contraction for homogenous systems

On the basis of a method developed in a previous paper, a systematic rule for obtaining a symmetrized collision superoperator of the Van Hove generalized master equation including an arbitrary number of particles is given. In the formalism, the quantum statistical effect is taken into account through the use of contractions (internal and external contractions) on the basis of the cluster expansion. As an application of this general rule, a symmetrized collision superoperator including the effect of three-particle collisions is obtained.     

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.     

Convective excitation of quasistatic waves in an inhomogeneous anisothermic plasma II. Nonlinear estimates

Nonlinear effects stabilizing the convective instabilities excited in an anisothermic plasma (T _etT _i) at the plasma boundaryaV_s/ _Bi) are discussed. Waves having in the linear theory (Part I) the highest growth rates ( _Bi) saturate at first. Being excited by a small part of slow plasma electrons ( _zTe) only, they saturate at a relatively low level. Further, surface waves with lower frequencies and higher phase velocities ( _ph/k_z) become dominant and a broadening of the plasma boundary occurs. For their saturation nonlinear interaction is more important than the quasilinear effects. During the time interval of several _Bi ^–1 the longest surface waves withk _yBi/V_s, _Bik_yV_s and _{ph Te} saturate at the absolutely highest level. The plasma boundary broadens in the meanwhile up toaV _s/_Bi. The wave energy is comparable to the whole energy connected with the longitudinal motion of the initially thermal electrons inside this boundary layer. The wave amplitude is large enough to trap the initially cold ions belonging to this layer and heat them up to energies comparable to those of the electron component. The heating process occurs again within several _Bi ^–1 and the Larmor radius of the ions is then comparable toV _s/_Bi. Further evolution of the system is governed by the unstable local perturbations.He leaves of absence from thePhysical and Technical Institute, Kharkov, USSR.     

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.     

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.     

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 wave processes in a plasma with stationary random irregularities. Part II

We analyze the influence of large-scale random irregularities of a medium on nonlinear wave interactions. First, we study three-wave interactions in an equilibrium medium, and then we consider the problems of stabilizing the explosive instability in the presence of random irregularities. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 50, No. 1, pp. 54–71, January 2007     

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.