We study the range of multifractality for the probability density P(r,t) of random walks on linear random fractals, for a given distance r and time t. Analytical study of the moments 〈${\mathit{P}}^{\mathit{q}}$(r,t)〉 shows that multifractality exists only when 1<${\mathit{qr}}_{\mathit{w}}^{\mathit{d}}$/t and qr/t<1, with ${\mathit{d}}_{\mathit{w}}$=2${\mathit{d}}_{\mathit{f}}$, where ${\mathit{d}}_{\mathit{f}}$ is the fractal dimension of the linear fractal. The results can be extended to more general random fractals and are consistent with recent numerical data for the form of 〈P(r,t)〉.

We have developed in this paper a nonperturbative cluster-expansion strategy for generating the stochastically averaged time-evolution operator for quantum systems driven by Ornstein-Uhlenbeck (OU) colored noise. The method induces a boson mapping of the (real or complex) stochastic variable f, and interprets the stochastic average of a pair of variables f at two different times as the expectation value of the time-ordered product of the associated bosons with respect to the boson vacuum ‖$0_{\mathit{B}}$〉. The stochastic evolution is thus mapped onto a deterministic evolution in an expanded Fock space. The evolution of the system from the groups of state of interest is monitored using our recently developed nonperturbative time-dependent multireference coupled-cluster (TDMRCC) method. In this, the evolution operator U is written in a factorized form ${\mathit{U}}_{\mathrm{ex}}$${\mathit{U}}_{\mathit{M}}$, where ${\mathit{U}}_{\mathit{M}}$ evolves in the space of the starting functions (model space) and ${\mathit{U}}_{\mathrm{ex}}$ brings in the virtual functions. ${\mathit{U}}_{\mathrm{ex}}$ and ${\mathit{U}}_{\mathit{M}}$ are both written as normal ordered exponentials involving cluster operators. In the present context, the TDMRCC method translates into one for generating the evolution operator ${\mathit{U}}_{\mathit{B}}$ for the Hamiltonian containing the additional boson variables, and the stochastic averaging of ${\mathit{U}}_{\mathit{B}}$ is realized as the expectation value 〈$0_{\mathit{B}}$‖${\mathit{U}}_{\mathit{B}}$‖$0_{\mathit{B}}$〉.

We call the TDMRCC method involving the expanded Fock space a cluster cumulant method. We have analyzed the relation of the cluster cumulant approach with the Fox-Kubo operator cumulant expansion and the method of marginal averages involving the Fokker-Planck operator of the OU process. It has been shown that an order by order expansion of our equations in the power of stochastic coupling generates the perturbative cumulant results of Fox and Kubo. It is also demonstrated that the traditional Fokker-Planck method of marginal averages uses a Kubo-Schrödinger operator which is related to our boson mapped Hamiltonian by a transformation which converts the Fokker-Planck operator Γ to a manifestly Hermitian form. However, the method of marginal averages involves a linear expansion of ${\mathit{U}}_{\mathit{B}}$ in terms of eigenfunctions of Γ, while the cluster cumulant method uses a cluster expansion, which is nonlinear in the expansion parameters. The former is thus analogous to a configuration-interaction expansion, while the latter is analogous to a coupled-cluster strategy. The method is illustrated by applying it to a simple yet nontrivial problem, viz., a harmonic oscillator linearly perturbed by a (real or complex) stochastic variable. Survivalities of both the ground and the first excited states have been considered for a wide range of coupling strengths and noise correlation times. Our method provides exact results in this case; these have been used as benchmarks for assessing performance of the second- and fourth-order cumulant results.

We demonstrate that two identical chaotic systems can be made to synchronize by applying small, judiciously chosen, temporal-parameter perturbations to one of them. This idea is illustrated with a numerical example. Other issues related to synchronization are also discussed.

The relaxation from an intrinsically unstable state to the metastable state of a bistable system, which is driven by colored noise, is investigated in the weak-noise strength D and general correlation time τ. The influence of τ on the relaxation of the system is discussed.

Two models with nonconserving dynamics and slow continuous deterministic driving, a stick-slip model (SSM) of earthquake dynamics and a toy forest-fire model (FFM), have recently been argued to show numerical evidence of self-organized criticality (generic, scale-invariant steady states). To determine whether the observed criticality is indeed generic, we study these models as a function of a parameter γ which was implicitly tuned to a special value, γ=1, in their original definitions. In both cases, the maximum Lyapunov exponent vanishes at γ=1. We find that the FFM does not exhibit self-organized criticality for any γ, including γ=1; nor does the SSM with periodic boundary conditions. Both models show evidence of macroscopic periodic oscillations in time for some range of γ values. We suggest that such oscillations may provide a mechanism for the generation of scale-invariant structure in nonconserving systems, and, in particular, that they underlie the criticality previously observed in the SSM with open boundary conditions.

The velocity autocorrelation function (VACF) of a disk moving in a two-dimensional viscous fluid is calculated under the condition that the momentum leaks out of the fluid with a relaxation time τ. In the absence of any memory effects, VACF decays exponentially. Even in the presence of memory effects, VACF essentially decays exponentially being accompanied by the correction of algebraic decay. This correction depends on the coupling strength between the sheet and the outer fluid, i.e., ∼${\mathit{e}}^{\mathrm{-}\mathit{t}/\mathrm{\tau }}$${\mathit{t}}^{\mathrm{-}3/2}$ and ∼${\mathit{e}}^{\mathrm{-}\mathit{t}/\mathrm{\tau }}$${\mathit{t}}^{\mathrm{-}1}$ for strong- and weak-coupling limits, respectively. The correction of ${\mathit{t}}^{\mathrm{-}1}$ reflects the two-dimensional character of the fluid sheet.

We study the dynamics of a two-dimensional system with complex conserved order parameter, following a deep quench. In accordance with a recent theoretical prediction, we observe an effective value of the dynamical exponent at late times of φ=0.245±0.010. While interesting dynamical effects were noticed in the study of the equal-time correlation functions, no evidence of multiscaling behavior was found. A correspondence between the quenched dynamics of this system in two dimensions and the dynamics of roughening of crystal-vapor interfaces is pointed out. We also study the dynamics in one dimension and find that φ≊0.18.

We have derived an exact expression for the determinant of the toppling matrix of the Abelian sandpile model in any finite dimension with open boundary conditions. The result can also apply to the semiopen boundary cases in certain limiting cases. An analytic result can also be obtained in the thermodynamic limit where the grid size tends to infinite: namely, the total number of system configurations in the self-organized critical state follows a power law as the grid size increases. Therefore, under a uniform but random particle addition, 1/${\mathit{f}}^2$ instead of 1/f scaling is observed. The relation with the site percolation problem is also discussed.

A very simple method to reduce noise in experimental data with nonlinear time evolution is presented. Locally constant fits are used to obtain a less noisy trajectory consistent with the dynamics as well as with the measured data. Neighborhoods are defined by coordinates both from the past and from the future. The method is applied to the Hénon map and to a discretized form of the Mackey-Glass equation.

We examine the eigenvector component statistics for a quantum system whose classical analog exhibits chaos. Combining the unfolded components from all eigenvectors to achieve high statistics, we are able to compare random-matrix theory and numerical experiment in detail by fitting the data to a ${\mathrm{\chi }}^2$ distribution with ν degrees of freedom. Although there are statistically significant deviations between the fit and data, component fluctuations are shown to be well modeled by Gaussian-orthogonal-ensemble fluctuations. By examining the component statistics of the scarred eigenvectors alone, we find that they are also in qualitative agreement with random-matrix theory, giving evidence that scarring by periodic orbits and statistics of fluctuations are compatible features of the eigenstates.

This work treats many-body aspects in an idealized class of reversible binding problems involving a static binding site with many diffusing point particles. In the noncompetitive limit, where no restriction exists on the number of simultaneously bound particles, the problem reduces to reversible aggregation. In the competitive limit, where only one particle may be simultaneously bound, it becomes a model for a pseudounimolecular reaction. The general formalism for both binding limits involves the exact microscopic hierarchy of diffusion equations for the N-body density functions. In the noncompetitive limit of independent particles, the hierarchy admits an analytical solution which may be viewed as a generalization of the Smoluchowski aggregation theory to the (idealized) reversible case. In the competitive limit, the hierarchy enables straightforward derivations of useful identities, determination of the ultimate equilibrium solution, and justification for approximations. In particular, the utility of a density-expansion, short-time approximation is investigated. The approximation relies on the ability to solve the hierarchy numerically for a small number of particles. This direct-propagation algorithm is described in the numerical section.

We present an exact and Monte Carlo renormalization-group (MCRG) study of the self-avoiding walks (SAW’s) on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension ${\mathit{d}}_{\mathit{f}}$ is equal to 2 for all values of the fractal enumerator b (an odd integer, 3≤b<∞). On the other hand, we demonstrate, via precise calculations, that the corresponding spectral dimension ${\mathit{d}}_{\mathit{s}}$ monotonically increases from 1.393 85 to 2 when b varies from 3 to ∞. For the PF fractals, we calculate exactly (for 3≤b≤9) and through the MCRG approach (for b≤121) the SAW critical exponents ν (associated with the mean-squared end-to-end distance) and γ (associated with the total number of distinct SAW’s). The MCRG results for 3≤b≤9 deviate from exact results at most 0.04% in the case of ν and 0.06% in the case of γ. Our results show clearly that ν monotonically decreases with b and crosses the Euclidean value ν=3/4 between b=27 and b=29. This is in contrast with all available Flory-type theories, as they predict that ν should be, in the case under study, strictly less than 3/4. In addition, our results show that the critical exponent γ, being always larger than the Euclidean value 43/32, monotonically increases with b. We discuss, in a framework of the finite-size scaling approach, behavior of ν and γ in the fractal-to-Euclidean crossover region that occurs when b→∞. Finally, we discuss a possible relevance of our results to the problem of SAW’s on the two-dimensional percolation clusters.

The behavior of the ensemble-averaged spectral density 〈G(z)〉 for a given eigenvalue problem (z${\mathrm{\delta }}_{\mathit{i}\mathit{j}}$-${\mathrm{\Phi }}_{\mathit{i}\mathit{j}}$)${\mathit{y}}_{\mathit{j}}$=0 with the random matrix Φ, distributed by the probability P(Φ) will be examined. By using the replica method, the change of 〈G(z)〉 for an infinitesimal neighboring distribution is calculated and, as a result, it was found that for systems with a higher disorder of the random matrix Φ the averaged spectral density increases for very small and very large values z, respectively.

We study the relation between the energy-level statistics of a quantal system and the integrability of its classical counterpart, derived using coherent states. Energy levels of any given distribution are reproduced using models based either on the harmonic oscillator or on an su(2) algebraic model. The energy-level fluctuations seen in the quantum models are greatly suppressed in their classical counterparts. This is opposite to what is usually seen, namely, the quantum system suppresses sensitivity on the initial conditions of its classical chaotic counterpart.

Dissipative corrections to the logarithm of the probability of escape from a metastable state of an underdamped Markov system are considered. The approach proposed is based on the Poincaré cross-section method applied to the optimal path along which a system moves, with an overwhelming probability, in the course of a fluctuation resulting in escape. The corrections depend crucially on the structure of the trajectories of motion of the system near the saddle point in the absence of dissipation and fluctuations. For two important patterns of the trajectories the corrections are ∼${\mathrm{\eta }}^2$ lnη and ∼η, where η is a dissipation parameter. Numerical analysis of the escape probabilities for a nonlinear oscillator bistable in a nearly resonant field is fulfilled and the results are shown to be in good agreement with the analytical predictions. A new feature of the pattern of the optimal paths in systems without detailed balance, the onset of caustics, is revealed.

We present results of an experimental study on the reaction kinetics of a one-dimensional diffusion-reaction system, on a picosecond to millisecond time scale. Tetramethylammonium manganese trichloride (TMMC) is a perfect model system to study this problem. Time-resolved luminescence of TMMC has been measured over nine decades in time. The nonexponential shape of the luminescence decay curves depends strongly on exciting laser power. This is shown to result from a fusion reaction (A+A→A) between photogenerated excitons, which is very well described by the diffusion-limited single-species fusion model for initial exciton densities <2×${10}^{\mathrm{-}3}$ (fraction of the number of sites). At higher initial exciton densities the diffusion process, and thus the reaction rate, is significantly influenced by the heat produced in the fusion reaction. This is supported by Monte Carlo simulations.

Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.

The existence of accurately measured oxygen pair-correlation functions for liquid water suggests a natural form of linear perturbation theory for that substance. The unperturbed potential involves isotropically interacting molecules, and reproduces the measured oxygen-pair short-range order. Monte Carlo simulations in this theoretical context have been carried out for the unperturbed system and for the fully coupled system in the liquid state at 25 °C. Inherent structures (potential-energy minima) have been generated and examined in both of these cases; they demonstrate that the directionality intrinsic to hydrogen bonding plays an important role in structurally stabilizing short-range order at the pair and higher-order correlation levels.

Functional differentiation of systematic expansions for the entropy, in the grand ensemble [B. B. Laird and A. D. J. Haymet, Phys. Rev. A 45, 5680 (1992)], leads directly to consistent integral equations for classical systems interacting via two-body, three-body, and even higher-order forces. This method is both a concise method for organizing existing published results and for deriving previously unpublished, higher-order integral equations. The equations are automatically consistent in the sense that all thermodynamic quantities may be derived from a minimum on an approximate free-energy surface, without the need to introduce weighting functions or numerically determined crossover functions. A number of existing approximate theories are recovered by making additional approximations to the equations. For example, the Kirkwood superposition approximation is shown to arise from a particular approximation to the entropy. The lowest-order theory is then used to obtain integral-equation predictions for the well-known Stillinger-Weber model for silicon, with encouraging results. Further connections are made with increasingly popular density-functional methods in classical statistical mechanics.

An alternative development of the recursive-renormalization-group (RNG) theory for the subgrid modeling of turbulence is presented which is now independent of the order in which the subgrid averaging is performed. The relevant approximations, perturbation ordering, and the averaging process are explicitly considered. In particular, it is shown that, of the higher-order nonlinearities introduced into the RNG Navier-Stokes equation, only the third-order nonlinearity appears at the desired level of the perturbation expansion. Moreover, these triple-velocity product terms appear at the same order as that of the eddy viscosity which is generated by the RNG subgrid-elimination procedure. These third-order nonlinearities also play a major role in the energy-balance equation with the corresponding energy-transfer process resulting in an analytic eddy-viscosity formulation which is in agreement with that from closure theories and the results of direct numerical simulations (DNS). This is also confirmed further here by a direct analysis of both large-eddy-simulation and DNS databases for the fluid velocity. Moreover, it is shown that these RNG-induced triple nonlinearities give rise to a backscatter in the energy from small scales to large spatial scales, in agreement with recent closure theories and numerical simulations.