We present the relationship between nonlinear-relaxation-time (NLRT) and quasideterministic approaches to characterize the decay of an unstable state. The universal character of the NLRT is established. The theoretical results are applied to study the dynamical relaxation of the Landau model in one and n variables and also a laser model.

We study the effect of quenched impurities in a driven diffusive lattice gas by means of numerical Monte Carlo simulations. The particles are driven by an external field. As the temperature is varied the system is found to undergo a second-order kinetic phase transition to an ordered strip-geometry state. In the ferromagnetic case an anisotropic finite-size-scaling analysis shows that the critical temperature decreases linearly with increasing impurity concentration. We determine the power spectrum of the current correlations at the determined critical temperatures and we also determine the conductivity as a function of the impurity concentration. In the antiferromagnetic case it follows from simulations and a finite-size-scaling analysis that the critical temperature is a decreasing function of impurity concentration and that the model is described by the Ising critical exponents.

We examine the model of self-organized criticality investigated by Hwa and Kardar [Phys. Rev. Lett. 62, 1813 (1989)] based on a continuum, nonlinear driven diffusion equation with stochastic noise. We point out some problems in the heuristic arguments used in relating the temporal response functions of the model to the exponents calculated exactly by dynamic-renormalization-group analysis. We propose an alternative, direct calculation of the response function and its power spectrum for the total energy dissipated in the system by a mode-coupling theory, making use of the renormalization-group recursion relations for the renormalized parameters of the theory. We find a significant difference from the previous results. Simulation results of the ‘‘running-sandpile’’ model agree with our mode-coupling-theory predictions.

The convergence properties of cycle-expanded periodic-orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three-disk billiard. We present evidence that the semiclassical and perhaps the classical Selberg zeta functions have poles. Applying a Padé approximation on the expansions of the full Euler products, as well as on the individual dynamical zeta functions in the products, we calculate the leading poles and the zeros of the improved expansions with the first few poles removed. The removal of poles tends to change the simple linear exponential convergence of the Selberg zeta functions to an exp{-${\mathit{n}}^2$} decay in the semiclassical case. The classical Selberg zeta function decays like exp{-${\mathit{n}}^{3/2}$}. The leading poles of the jth dynamical zeta function are found to equal the leading zeros of the (j+1)th one: However, in contrast to the zeros, which are all simple, the poles seem without exception to be double. The poles are therefore in general not completely canceled by zeros in the way suggested by Artuso, Aurell, and Cvitanović [Nonlinearity 3, 325 (1990)]. The only complete cancellations of this kind occur in the classical Selberg zeta function between the poles (double) of the first and the zeros (squared) of the second dynamical zeta function. Furthermore, we find strong indications that poles are responsible for the presence of spurious zeros in periodic-orbit quantized spectra and that these spectra can be greatly improved by removing the leading poles, e.g., by using the Padé technique.

An approach to compute the mean-first-passage time (MFPT) in bistable systems driven by colored noise is presented. The approach is valid in the limit of large but finite noise correlation time and finite noise strength, a case for which no satisfactory theory exists at present. Our approach is a modification of the fluctuating-potential theory proposed in Phys. Rev. Lett. 61, 7 (1988). Excellent agreement is found between our results for MFPT and that of the simulation results of Mannella, Palleschi, and Grigolini [Phys. Rev. A 42, 5946 (1990)]. Interesting similarity between stochastic resonance and the colored-noise problem is brought out.

The scaling behavior of the interface width for the generalized Kardar-Parisi-Zhang (KPZ) equation ∂h/∂t=ν${\mathrm{\nabla }}^2$h+λ‖∇h${\mathrm{\Vert }}^{\mathrm{\mu }}$+η(x,t) is studied in two dimensions as a function of μ with and without the additive noise term η. In the case of additive noise, the scaling of the surface width is found for all μ to be the same as for the ordinary KPZ equation (μ=2) in contrast to a previous conjecture. This appears to be due to the combined action of the nonlinearity ‖∇h${\mathrm{\Vert }}^{\mathrm{\mu }}$ and the noise under renormalization, which together induce a ‖∇h${\mathrm{\Vert }}^2$ term. For the deterministic case corresponding to the smoothing of an initially rough interface with roughness exponent α, good agreement with the scaling relation z=min[2,α(1-μ)+μ] is obtained for μ≥1. However, for μ<1, an instability is observed, which leads to a fluctuating grooved surface. For an asymptotically large system, the roughness exponent is 1 and the growth exponent is approximately equal to 1/2. The evolution of two slightly different surfaces is studied in order to determine a Lyapunov exponent characterizing this instability.

There has been much interest recently in simple lattice-gas models of catalytic reactions. Using Monte Carlo simulations, we investigate interfacial roughening in a model of a Langmuir-Hinshelwood reaction which includes an effective repulsive interaction between the two reacting species. We measure the width of the reactive zone between the two reactants and the roughness of this interface. It is found that, when the interspecies repulsive interaction is nonzero, the interface roughness obeys the scaling law w∼${\mathit{t}}^{\mathrm{\beta }}$, with β=1/4. When there is no interspecies interaction, the interface roughness scales as ${\mathit{t}}^{1/2}$. The reactive-zone width also scales as ${\mathit{t}}^{1/2}$ when there is no interspecies repulsion. With interspecies repulsion, however, we find that the reactive-zone width tends to a finite value at long times. We argue that in this case the interface should be described by the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889 (1986)] with ν≠0 and λ=0.

We have investigated the random walk of particles in the frame of a conventional master equation for directed random walks. The transfer rates are supposed to be random variables and we incorporate the possibility of correlations. We assume that the chain consists of successive segments of random lengths. Within a given segment, the transfer rates are equal to a single random variable. The transfer rates belonging to two different segments are supposed to be independent and distributed according to the same probability law. We have calculated the time-asymptotic behavior of the mean coordinate of the particle. The resulting character of the motion emerges from the interplay between two basic features: the probability of having a small value of the transfer rate and the probability of having long segments. If the first moment of the segment-length distribution diverges, the asymptotic regime undergoes radical changes as compared to the noncorrelated model.

We studied the evolutionary dynamics of a population undergoing asexual reproduction in a flat-fitness landscape. The quantity of interest is the distribution of the overlap function q which is a measure of the similarity in the genome structure between two individuals. We obtain analytical expressions for 〈q〉, 〈${\mathit{q}}^2$〉, and p(q) in a model with the following features: continuous time, fluctuating population divided into many compartments, and a finite number of genes per genome. A few special cases of interest are also discussed.

In contrast to earlier nonlinear-dynamics investigations concerning the consequences of coupling limit-cycle oscillators, we propose the conceptionally simple extension of studying the interaction dynamics of chaotic subsystems. We illustrate this by simulating a ‘‘toy system,’’ the dynamics of a linear chain of damped-driven pendulums where the state of the isolated individual pendulum is chaotic. The harmonic coupling between these chaotic oscillators results in a very complex and rich spatiotemporal dynamics as a function of coupling strength and system size. This suggests that the extension to realistic representations of physical systems may provide a fruitful paradigm for studying dynamical disorder in the real world.

The response of a bistable system to a low-frequency driving force is demonstrated to display a giant nonlinearity: a sinusoidal force can induce a nearly rectangular signal in the system even when its amplitude is small. The nonlinearity comes about through the force modulating the effective activation energies of fluctuational transitions between the stable states, which in turn can give rise to strong modulation of the transition probabilities and the populations of the states, for low noise intensities. Theoretical results are shown to be in good agreement with the results of analog and digital simulations.

We consider mean-first-passage times and transition rates in bistable systems driven by white shot noise. We obtain closed analytical expressions, asymptotic approximations, and numerical simulations in two cases of interest: (i) jumps sizes exponentially distributed and (ii) jumps of the same size.

Previous studies of the universality hypothesis for the classification of the critical behavior of statistical systems, based on the construction of gauge models as equivalent to random-surface theories, were strongly restricted to the n=0 (spin components) limit. This difficulty is overcome here, thus rendering possible a precise description of the correspondence between the generating functions of ensembles of random surfaces and the (more simple) partition functions of the related Hamiltonian gauge models for a wide spectrum of physically relevant processes.

The classical and quantum dynamics of a periodically driven anharmonic oscillator is studied. For a monochromatic field it is demonstrated that quantum mechanics imposes only weak limitations on the classical chaotic spreading, which are destroyed by introducing weak random noise in the field amplitude or when the time-reversal symmetry of the system is broken.

The motion of energy levels in quantum systems that show a chaotic classical limit is statistically analyzed. A quantitative comparison is made between the tails of the curvature distribution and numerical results obtained for various physical models. Approximate analytic expressions for the full curvature distribution are derived from the statistical mechanics of a fictitious gas in a refined formulation that recovers the random-matrix theory for an arbitrary number of levels. They provide a better description of numerical data than just the tail-limiting expressions available previously. Good agreement with numerical data for various physical systems as well as for a model random dynamics is obtained with the ad hoc introduced, very simple analytic expressions containing no free parameter. The nonuniversal behavior of small curvatures is discussed. The data obtained for the magnetized hydrogen atom support the previous interpretation of this phenomenon as due to the ‘‘scarred’’ wave functions. A large number of analyzed data allows us to show that the details of the curvature distribution provide a qualitative measure of the degree of scarring in different systems.

Statistical properties of levels of quantum systems chaotic in the classical limit are studied using the distribution of avoided crossings, i.e., of the sizes of local minima of adjacent-level spacings. The results obtained previously for the two-level random-matrix theory are compared with the prediction of the statistical-mechanics description of the equivalent fictitious-particle system. The distributions derived are compared with numerical results obtained for several physical systems. The origin of the discrepancies (in former numerical calculations) of small-avoided-crossing behavior is found. The ratio of the average crossing to the average spacing is shown to have a nonuniversal behavior and seems to provide information on the degree of scarring in the system studied.

Both inelastic-neutron-scattering experiments and computer simulations have established that liquid water can support a longitudinal mode propagating with a velocity much larger than the ordinary speed of sound. The physics behind this peculiar behavior is studied by a comprehensive microscopic approach. The effect is found to be a rather extreme case of a well-known phenomenon occurring even in monatomic liquids. To make the analogies and the differences more transparent, simulation data in water and in a simple alkali metal (liquid Cs near melting) are also reported. Finally, we discuss the ultimate microscopic origin of fast-sound effect in liquid water.

The potential contribution to the viscosity in liquids is calculated from the pair-correlation function, which may be obtained by solving the extended Kirkwood-Smoluchowski (KS) equation. A secondary boundary condition near the hard core (r=$1^{+}$) for the excess pair probability current density in the relative pair space is derived rigorously and applied to the extended KS equation. The intermolecular potential consists of hard core plus arbitrary soft tail. The viscosity coefficients calculated in this work prove to be essentially functions of the square root of the shear rate rather than functions of the shear rate itself. We give the explicit representation for the viscosity coefficients in the case of hard spheres. The shear thinning of the shear viscosity is recovered. The viscosity of the normal pressure difference 1/2(${\mathit{p}}_{\mathit{x}\mathit{x}}$-${\mathit{p}}_{\mathit{y}\mathit{y}}$) is found to be positive. The normal pressure difference of the second kind 1/2[${\mathit{p}}_{\mathit{z}\mathit{z}}$-1/2(${\mathit{p}}_{\mathit{x}\mathit{x}}$+${\mathit{p}}_{\mathit{y}\mathit{y}}$)] vanishes in the first-order perturbation calculation with respect to the deformation.

A detailed mode-coupling analysis of the atomic self-motion in liquid lead at 623 K is presented and discussed. Time-correlation functions calculated in a large-scale molecular-dynamics simulation were used to derive the relevant memory functions, which were compared with those calculated within the framework of mode-coupling theory. The mode-coupling formalism for the memory function of the velocity autocorrelation function has been extended to wave-vector-dependent memory functions. The detailed analysis of the two sets of memory functions in terms of constituent coupling modes leads to the conclusion that some assumptions incorporated in the present formulation of mode-coupling theory are of limited validity. In particular, the phenomenological form for the memory function proposed by Levesque and Verlet [Phys. Rev. A 14, 408 (1970)] for the Lennard-Jones liquid is found to be inadequate in the case of liquid lead.

A proof of the validity of the Chialvo-Debenedetti conjecture [Phys. Rev. A 43, 4289 (1991)], the crucial element to achieve an equivalence between the McQuarrie [Statistical Mechanics (Harper & Row, New York, 1976)] and Helfand [Phys. Rev. 119, 1 (1960)] shear-viscosity equations, is presented here. Some theoretical consequences of that validity are also discussed, such as the unification of most shear-viscosity expressions into one given by Andrews for first-order transport coefficients [J. Chem. Phys. 47, 3161 (1967)]. The system-size dependence of the McQuarrie shear-viscosity values is analyzed and an extrapolation method is proposed and tested to determine the asymptotic values.