Phase Diagram of a Two-Species Lattice Model with a Linear Instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy ( Phys . Rev. Lett ., 79 , 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.     

Dynamics of a disordered, driven zero-range process in one dimension

We study a disordered, driven zero range process which models a closed system of attractive particles that hop with site-dependent rates and whose steady state shows a condensation transition with increasing density. We characterize the dynamical properties of the mass fluctuations in the steady state in one dimension both analytically and numerically and show that there is a dynamic phase transition in the density-disorder plane. We also determine the form of the scaling function which describes the growth of the condensate as a function of time, starting from a uniform density distribution.     

Energy fluctuations in vibrated and driven granular gases

We investigate the behavior of energy fluctuations in several models of granular gases maintained in a non-equilibrium steady state. In the case of a gas heated from a boundary, the inhomogeneities of the system play a predominant role. Interpreting the total kinetic energy as a sum of independent but not identically distributed random variables, it is possible to compute the probability density function (pdf) of the total energy. Neglecting correlations and using the analytical expression for the inhomogeneous temperature profile obtained from the granular hydrodynamic equations, we recover results that have previously been observed numerically and that had been attributed to the presence of correlations. In order to separate the effects of spatial inhomogeneities from those ascribable to velocity correlations, we have also considered two models of homogeneously thermostated gases: in this framework it is possible to reveal the presence of non-trivial effects due to velocity correlations between particles. Such correlations stem from the inelasticity of collisions. Moreover, the observation that the pdf of the total energy tends to a Gaussian in the large system limit suggests that they are also due to the finite size of the system.     

Fluctuation Relation beyond Linear Response Theory

The Fluctuation Relation (FR) is an asymptotic result onthe distribution of certain observables averaged over timeintervals τ as τ → ∞ and it is a generalization of thefluctuation–dissipation theorem to far from equilibrium systemsin a steady state, which reduces to the usual Green–Kubo (GK)relation in the limit of small external non-conservative forces.FR is a theorem for smooth uniformly hyperbolic systems, and it isassumed to be true in all dissipative ‘chaotic enough’ systemsin a steady state. In this paper, we develop a theory of finitetime corrections to FR, needed to compare the asymptoticprediction of FR with numerical observations, which necessarilyinvolve fluctuations of observables averaged over finite timeintervals τ. We perform a numerical test of FR in two cases inwhich non-Gaussian fluctuations are observable, while GK does notapply and we get a non-trivial verification of FR that is independent of and different from linear response theory.Our results are compatible with the theory of finite timecorrections to FR, while FR would be observably violated,well within the precision of our experiments, if such correctionswere neglected.     

Taming the cosmological constant in 2D causal quantum gravity with topology change

As shown in previous work, there is a well-defined nonperturbative gravitational path integral including an explicit sum over topologies in the setting of causal dynamical triangulations in two dimensions. In this paper we derive a complete analytical solution of the quantum continuum dynamics of this model, obtained uniquely by means of a double-scaling limit. We show that the presence of infinitesimal wormholes leads to a decrease in the effective cosmological constant, reminiscent of the suppression mechanism considered by Coleman and others in the four-dimensional Euclidean path integral. Remarkably, in the continuum limit we obtain a finite spacetime density of microscopic wormholes without assuming fundamental discreteness. This shows that one can in principle make sense of a gravitational path integral which includes a sum over topologies, provided suitable causality restrictions are imposed on the path integral histories.     

Fokker-Planck equation approach to fluctuations about nonequilibrium steady states

We examine the properties of steady states in systems which interact at the boundary with a nonequilibrium environment. The examination is based on a nonlinear Fokker-Planck equation, the structure of which is determined by the fact that it also governs the time evolution of the equilibrium fluctuations of the system. The nonlinearities in the Fokker-Planck equation may have two origins: thermodynamic nonlinearities which arise if the thermodynamic potential is not a bilinear function of the state variables, and nonlinear mode coupling which arises if the transport coefficients depend on the state. While these nonlinearities have only a small effect on the equilibrium fluctuations of a system away from critical points, they are shown to be important for the determination of fluctuations about nonequilibrium steady states. In particular the state dependence of the transport coefficients may lead to deviations from local equilibrium and to a breakdown of detail balance. An explicit formula for the time correlations of fluctuations about the nonequilibrium steady state is obtained. The formula leads to long-range correlations in fluids in the presence of a temperature gradient. The result is compared with earlier approaches to the same problem. Finally, we study the linear response to external forces and obtain a generalization of the fluctuation-dissipation formula relating the response functions with the nonequilibrium correlation functions.     

Singular scaling functions in clustering phenomena

We study clustering in a stochastic system of particles sliding down a fluctuating surface in one and two dimensions. In steady state, the density-density correlation function is a scaling function of separation and system size. This scaling function is singular for small argument — it exhibits a cusp singularity for particles with mutual exclusion, and a divergence for noninteracting particles. The steady state is characterized by giant fluctuations which do not damp down in the thermodynamic limit. The autocorrelation function is a singular scaling function of time and system size. The scaling properties are surprisingly similar to those for particles moving in a quenched disordered environment that results if the surface is frozen.     

Electronic superstructures in doped superlattices

Using an approach based on the density functional, we show that the exchange-correlation contribution to the system energy can be bigger than the sum of the kinetic energy and the Hartree contribution due to redistribution of carriers over the quantum wells in doped composite superlattices at low temperatures and moderate impurity densities. As a result, the ground state of the system can correspond to an inhomogeneous electron distribution over the quantum wells. Conditions when the homogeneous state is stable against small and finite density fluctuations are determined, and a phase diagram is plotted. A nonlinear theory of the inhomogeneous state is considered. Zh. éksp. Teor. Fiz. 114, 1089–1100 (September 1998)     

Translation by Ribosomes with mRNA Degradation: Exclusion Processes on Aging Tracks

We investigate the role of degradation of mRNA on protein synthesis using the totally asymmetric simple exclusion process (TASEP) as the underlying model for ribosome dynamics. mRNA degradation has a strong effect on the lifetime distribution of the mRNA, which in turn affects polysome statistics such as the number of ribosomes present on an mRNA strand of a given size. An average over mRNA of all ages is equivalent to an average over possible configurations of the corresponding TASEP—both before steady state and in steady state. To evaluate the relevant quantities for the translation problem, we first study the approach towards steady state of the TASEP, starting with an empty lattice representing an unloaded mRNA. When approaching the high density phase, the system shows two distinct phases with the entry and exit boundaries taking control of the density at their respective ends in the second phase. The approach towards the maximal current phase exhibits the surprising property that the ribosome entry flux can exceed the maximum possible steady state value. In all phases, the averaging over the mRNA age distribution shows a decrease in the average ribosome density profile as a function of distance from the entry boundary. For entry/exit parameters corresponding to the high density phase of TASEP, the average ribosome density profile also has a maximum near the exit end.     

A Diffusive System Driven by a Battery or by a Smoothly Varying Field

We consider the steady state of a one dimensional diffusive system, such as the symmetric simple exclusion process (SSEP) on a ring, driven by a battery at the origin or by a smoothly varying field along the ring. The battery appears as the limiting case of a smoothly varying field, when the field becomes a delta function at the origin. We find that in the scaling limit the long range pair correlation functions of the system driven by a battery are very different from the ones known in the steady state of the SSEP maintained out of equilibrium by contact with two reservoirs, even when the steady state density profiles are identical in both models.     

Prospects for studying chemisorption kinetics using current fluctuations

The rates of surface catalytic reactions are, in principle, reflected in the frequency spectrum of adsorbate density fluctuations. These fluctuations change the incremental current (i.e., increment over the average current corresponding to the average surface coverage) noise due to electron transfer between the adsorbate and the substrate and to the scattering of the substrate electrons from the adsorbates. In an adiabatic limit, the incremental current noise is proportional to the conductivity modulation induced by the adsorbate density fluctuations and to the square of the applied voltage. Thus, the noise in the substrate current provides information about the surface reaction kinetics. This reaction may proceed either in equilibrium or in nonequilibrium. The adsorbate density spectrum has been calculated both in and out of equilibrium for two specific reactions. No major difference is found between the in and out of equilibrium cases. When these involve two-step processes, the adsorbate density correlation function is the sum of two decaying exponentials and the spectrum is the sum of two Lorentzians.     

Dynamic density and spin responses of a superfluid Fermi gas in the BCS–BEC crossover: Path integral formulation and pair fluctuation theory

We present a standard field theoretical derivation of the dynamic density and spin linear response functions of a dilute superfluid Fermi gas in the BCS–BEC crossover in both three and two dimensions. The derivation of the response functions is based on the elegant functional path integral approach which allows us to calculate the density–density and spin–spin correlation functions by introducing the external sources for the density and the spin density. Since the generating functional cannot be evaluated exactly, we consider two gapless approximations which ensure a gapless collective mode (Goldstone mode) in the superfluid state: the BCS–Leggett mean-field theory and the Gaussian-pair-fluctuation (GPF) theory. In the mean-field theory, our results of the response functions agree with the known results from the random phase approximation. We further consider the pair fluctuation effects and establish a theoretical framework for the dynamic responses within the GPF theory. We show that the GPF response theory naturally recovers three kinds of famous diagrammatic contributions: the Self-Energy contribution, the Aslamazov–Lakin contribution, and the Maki–Thompson contribution. We also show that unlike the equilibrium state, in evaluating the response functions, the linear (first-order) terms in the external sources as well as the induced order parameter perturbations should be treated carefully. In the superfluid state, there is an additional order parameter contribution which ensures that in the static and long wavelength limit, the density response function recovers the result of the compressibility (compressibility sum rule). We expect that the f$f$-sum rule is manifested by the full number equation which includes the contribution from the Gaussian pair fluctuations. The dynamic density and spin response functions in the normal phase (above the superfluid critical temperature) are also derived within the Nozières–Schmitt–Rink (NSR) theory.     

Stochastic resonance and scale invariance in nonequilibrium metastable states

Using computer simulations, we study metastability in a two-dimensional Ising ferromagnet relaxing toward a nonequilibrium steady state. The interplay between thermal and nonequilibrium fluctuations induces resonant and scale-invariant phenomena not observed in equilibrium. In particular, we measure noise-enhanced stability of the metastable state in a nonequilibrium environment. The limit of metastability, or pseudospinodal separating the metastable regime from the unstable one, exhibits reentrant behavior as a function of temperature for strong nonequilibrium conditions. Furthermore, when subject to both open boundaries and nonequilibrium fluctuations, the metastable system decays via well-defined avalanches. These exhibit power-law size and lifetime distributions, resembling the scale-free avalanche dynamics observed in real magnets and other complex systems. We expect some of these results to be verifiable in actual (impure) specimens.     

Diffusive Fluctuations for One-Dimensional Totally Asymmetric Interacting Random Dynamics

We study central limit theorems for a totally asymmetric, one-dimensional interacting random system. The models we work with are the Aldous–Diaconis–Hammersley process and the related stick model. The A-D-H process represents a particle configuration on the line, or a 1-dimensional interface on the plane which moves in one fixed direction through random local jumps. The stick model is the process of local slopes of the A-D-H process, and has a conserved quantity. The results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling. We look at diffusive fluctuations, by which we mean fluctuations on the scale of the classical central limit theorem. In the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations. Of particular interest is the effect of macroscopic shocks, which play a dominant role because dynamical noise is suppressed on the scale we are working. Received: 4 October 2001 / Accepted: 12 March 2002     

Nonequilibrium Thermodynamics of the First and Second Kind: Averages and Fluctuations

We elaborate and compare two approaches to nonequilibrium thermodynamics, the two-generator bracket formulation of time-evolution equations for averages and the macroscopic fluctuation theory, for a purely dissipative isothermal driven diffusive system under steady state conditions. The fluctuation dissipation relations of both approaches play an important role for a detailed comparison. The nonequilibrium Helmholtz free energies introduced in these two approaches differ as a result of boundary conditions. A Fokker-Planck equation derived by projection operator techniques properly reproduces long range fluctuations in nonequilibrium steady states and offers the most promising possibility to describe the physically relevant fluctuations around macroscopic averages for time-dependent nonequilibrium systems.     

Vacuum effects of non-nucleonic baryons in nuclear matter

The role of vacuum properties of non-nucleonic baryons in hadronic field theories is discussed. We use an extension of Walecka's σω model treated at the mean-field level. Due to energy considerations the non-nucleonic baryons yield only vacuum effects near normal nuclear matter density. We study the effect of Roper resonance vacuum fluctuations on the nuclear matter equation of state and Coulomb sum. In addition we perform a schematic calculation of the effect of Δ-isobar vacuum fluctuations on the nuclear matter equation of state. Although the Roper resonance produces only a marginal change in the Coulomb sum, the Roper and delta may produce a significant softening of the equation of state. Thus these normally neglected degrees of freedom might serve an important role.     

Power spectrum crossover in sediments of a paleolake disturbed by volcanism

We study density fluctuations from sediments of a paleolake in central Mexico that was subjected to volcanic perturbations by means of computed tomography (CT) measurements on blocks chiselled out of mines at the lake's bed. The mine walls show laminations corresponding to the alternation of low density diatom sediments and high density volcanic ash depositions. We have previously shown that there is a range of scales where these fluctuations present a self-similar behavior [1]. Here we relate density correlation calculations to the power spectrum of the fluctuations. We show that a scaling region in the power spectrum coincides with the scaling region in the correlations produced by relaxation from intense volcanic perturbations to steady state fluctuations. There appears to be a kink-like crossover in the power spectrum from mid range scaling to a shorter range scale invariance. This, together with the density probability distribution of the fluctuations, draws attention to the dominant role of rare events. We believe that our analysis may be useful for the understanding of other phenomena with similar power spectrum properties, in which a scale invariance in the unperturbed system is altered by external perturbations that induce an additional scaling behavior.     

Nonequilibrium phase transition in the sedimentation of reproducing particles

We study numerically and analytically the dynamics of a sedimenting suspension of active, reproducing particles, such as growing bacteria in a gravitational field. In steady state we find a nonequilibrium phase transition between a "sedimentation" regime, analogous to the sedimentation equilibrium of passive colloids, and a "uniform" regime, in which the particle density is constant in all but the top and bottom of the sample. We discuss the importance of fluctuations in particle density in locating the phase-transition point, and report the kinetics of sedimentation at early times.     

Quadratic-nonlinear Landau-Zener transition for association of an atomic Bose-Einstein condensate with inter-particle elastic interactions included

We study the strong coupling limit of a quadratic-nonlinear Landau-Zener problem for coherent photo- and magneto-association of cold atoms taking into account the atom-atom, atom-molecule, and molecule-molecule elastic scattering. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct a highly accurate and simple analytic approximation describing the time dynamics of the coupled atom-molecule system. We show that the approximation describing time evolution of the molecular state probability can be written as a sum of two distinct terms; the first one, being a solution to a limit first-order nonlinear equation, effectively describes the process of the molecule formation while the second one, being a scaled solution to the linear Landau-Zener problem (but now with negative effective Landau-Zener parameter as long as the strong coupling regime is considered), corresponds to the remaining oscillations which come up when the process of molecule formation is over.     

Lyapunov Instability for a Hard-Disk Fluid in Equilibrium and Nonequilibrium Thermostated by Deterministic Scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.