We numerically simulate thermal dynamics in an energy landscape with a hierarchical organization ... more We numerically simulate thermal dynamics in an energy landscape with a hierarchical organization of metastable states. In the model, record sized energy fluctuations are needed to trigger changes from one metastable state to another. The initial configuration is chosen to have a large energy excess, relative to the thermal equilibrium value at the running temperature. We show that the energy surplus trapped in the initial state is dissipated in a series of intermittent bursts, whose rate decreases as the inverse of the age of the system. In addition, one observes energy fluctuations with a zero centered Gaussian distribution. These describe the pseudo equilibrium dynamics within a single metastable state and do not contribute to the energy dissipation. The derivative of the thermal energy with respect to the logarithm of time is asymptotically constant, and comprises a temperature independent part, and a part with an Arrhenius temperature dependence. These findings closely mirror th...
Cellular Automata and Complex Dynamics of Driven Elastic Media
Several systems of importance in condensed matter physics can be modelled as an elastic medium in... more Several systems of importance in condensed matter physics can be modelled as an elastic medium in a disordered environment and driven by an external force. In the simplest cases, the equation of motion involves competition between a local non-linear potential (fluctuating in space) and elastic coupling, as well as relaxational (inertialess) dynamics. Despite a simple mathematical description, the interactions between many degrees of freedom lead to the emergence of time and length scales much longer than those set by the microscopic dynamics. Extensive computations have improved the understanding of the behavior of such models, but full solutions of the equations of motion for very large systems are time-consuming and may obscure important physical principles in a massive volume of output. The development of cellular automata models has been crucial, both in conceptual simplification and in allowing the collection of data on many replicas of very large systems. We will discuss how the marriage of cellular automata models and parallel computation on a MasPar MP-1216 computer has helped to elucidate the dynamical properties of these many-degree-of-freedom systems.
Record Dynamics (RD) deals with complex systems evolving through a sequence of metastable stages.... more Record Dynamics (RD) deals with complex systems evolving through a sequence of metastable stages. These are macroscopically distinguishable and appear stationary, except for the sudden and rapid changes, called quakes, which induce the transitions from one stage to the next. This phenomenology is well known in physics as “physical aging”, but from the vantage point of RD, the evolution of a class of systems of physical, biological, and cultural origin is rooted in a hierarchically structured configuration space and can, therefore, be analyzed by similar statistical tools. This colloquium paper strives to present in a coherent fashion methods and ideas that have gradually evolved over time. To this end, it first describes the differences and similarities between RD and two widespread paradigms of complex dynamics, Self-Organized Criticality and Continuous Time Random Walks. It then outlines the Poissonian nature of records events in white noise time-series, and connects it to the sta...
Mesoscopic aging systems are characterized by large intermittent noise fluctuations. In a record ... more Mesoscopic aging systems are characterized by large intermittent noise fluctuations. In a record dynamics scenario [P. Sibani and J. Dall, Europhys. Lett. 64, 2003] these events, quakes, are treated as a Poisson process with average α ln(1 + t/tw), where t is the observation time, tw is the age and α is a parameter. Assuming for simplicity that quakes constitute the only source of de-correlation, we present a model for the probability density function (PDF) of the configuration autocorrelation function. Beside α, the model has the average quake size 1/q as a parameter. The model autocorrelation PDF has a Gumbel-like shape, which approaches a Gaussian for large t/tw and becomes sharply peaked in the thermodynamic limit. Its average and variance, which are given analytically, depend on t/tw as a power-law and a power-law with a logarithmic correction, respectively. Most predictions are in good agreement with data from the literature and with the simulations of the Edwards-Anderson spin glass carried out as a test. Introduction.-After a rapid quench of an external parameter, e.g. the temperature, many complex materials age, i.e. their properties slowly change with the waiting time, t w , elapsed from the quench. Ever since the initial observations in polymers [1], evidence has accumulated that spin-glasses [2], type II superconductors [3], glasses [4], and soft condensed matter [5], among others, age in similar ways, e.g. : For observation times t ≪ t w physical averages are nearly constant, and autocorrelations and their conjugate linear response functions are connected by an equilibrium-like fluctuation-dissipation theorem (FDT). Conversely, for t ≫ t w they visibly drift and the FDT is violated. As was recently discovered, the drift happens in an intermittent fashion [6, 7], i.e. through rare, large, and spatially heterogeneous rearrangements , which appear as non-Gaussian tails in the probability density function (PDF) of configurational probes such as colloidal particle displacement [8, 9] and correlation [10] or voltage noise fluctuations in glasses [11]. As aging phenomena are similar for a broad class of interactions, we seek a mesoscopic description, and assume that intermittent events, for short quakes, are the main source of de-correlation in non-equilibrium aging. In the framework of record dynamics [12, 13], quakes are irreversible and are triggered by (energy) fluctuations of record magnitude. We show how this leads to a description of the configurational autocorrelation function, more specifically, the dependence of the shape of its PDF on t, t w , the temperature T and the system size N , which resembles observations for colloidal gels [10] spin-glasses and kinetically constrained models [15, 16]. The model PDF is closely approximated by the Gumbel distributions widely
We consider the concept of mutual information in ecological networks, and use this idea to analys... more We consider the concept of mutual information in ecological networks, and use this idea to analyse the Tangled Nature model of co-evolution. We show that this measure of correlation has two distinct behaviours depending on how we define the network in question: if we consider only the network of viable species this measure increases, whereas for the whole system it decreases. It is suggested that these are complimentary behaviours that show how ecosystems can become both more stable and better adapted.
We present extensive numerical investigations of the structural relaxation dynamics of a realisti... more We present extensive numerical investigations of the structural relaxation dynamics of a realistic model of the amorphous high-temperature ceramic a-Si 3 B 3 N 7 , probing the mean square displacement (MSD) of the atoms, the bond survival probability (BSP), the average energy, the specific heat, and the two-point energy average. Combining the information from these different sources, we identify a transition temperature T c ≈ 2000 K below which the system is no longer ergodic and physical quantities observed over a time t obs show a systematic parametric dependence on the waiting time t w , or age, elapsed after the quench. The aging dynamics 'stiffens' as the system becomes older, which is similar to the behavior of highly idealized models such as Ising spin-glasses and Lennard-Jones glasses.
We study numerically the dynamical equilibrium behavior for the uniformly driven, elastic model o... more We study numerically the dynamical equilibrium behavior for the uniformly driven, elastic model of Fukuyama, Lee, and Rice for pinned charge-density waves, in the critical region close to the threshold field for sliding, in two and three dimensions. We obtain a critical exponent for the mean velocity in good agreement with recent experiments, and scaling for the velocity correlation function, from which we extract a diverging correlation length. The correlation-length exponent v is found to be less than 2/d (d is the dimension), suggesting unusual critical behavior for this model.
Using a non-thermal local search, called Extremal Optimization (EO), in conjunction with a recent... more Using a non-thermal local search, called Extremal Optimization (EO), in conjunction with a recently developed scheme for classifying the valley structure of complex systems, we analyze a shortrange spin glass. In comparison with earlier studies using a thermal algorithm with detailed balance, we determine which features of the landscape are algorithm dependent and which are inherently geometrical. Apparently a characteristic for any local search in complex energy landscapes, the time series of successive energy records found by EO is also characterized approximately by a Poisson statistic with logarithmic time arguments. Differences in the results provide additional insights into the performance of EO. In contrast with a thermal search, the extremal search visits dramatically higher energies while returning to more widely separated low-energy configurations. Two important properties of the energy landscape are independent of either algorithm: first, to find lower energy records, progressively higher energy barriers need to be overcome. Second, the Hamming distance between two consecutive low-energy records is linearly related to the height of the intervening barrier.
Journal of Statistical Mechanics: Theory and Experiment, 2004
The memory and rejuvenation properties of intermittent heat transport are explored theoretically ... more The memory and rejuvenation properties of intermittent heat transport are explored theoretically and by numerical simulation for Ising spin glasses with short-ranged interactions. The theoretical part develops a picture of non-equilibrium glassy dynamics recently introduced by the authors, which links irreversible 'intermittent' events, or 'quakes' to thermal fluctuations of record magnitude. The pivotal idea is that the largest energy barrier b(tw, T) surmounted prior to tw at temperature T determines the rate rq ∝ 1/tw of the intermittent events near tw. The idea implies that the same rate after a negative temperature shift should be given by rq ∝ 1/t ef f w. The 'effective age' t ef f w ≥ tw has an algebraic dependence on tw, whose exponent contains the temperatures before and after the shift. This analytical expression is confirmed by numerical simulations. Marginal stability suggests an asymmetry between cooling and heating, i.e. a positive temperature shift T → T ′ should erase the memory b(tw, T). This is confirmed by the simulations, which show a rate rq ∝ 1/tw, controlled by the barrier b(tw, T ′) ≥ b(tw, T). Additional 'rejuvenation' effects are also identified in the intermittency data for shifts of both signs.
Aging in spin glasses is analyzed via the Probability Density Function (PDF) of the heat transfer... more Aging in spin glasses is analyzed via the Probability Density Function (PDF) of the heat transfer over a short time δt between system and heat bath. The PDF contains a Gaussian part, describing reversible fluctuations, and an exponential 'intermittent' tail. We find that the relative weight of these two parts depends, for fixed δt on the ratio of the total sampling time t to the age tw. Fixing this ratio, the intensity of the intermittent events is proportional to δt/tw and independent of the temperature. The Gaussian part has a variance with the same temperature dependence as the variance of the equilibrium energy in a system with an exponential density of states. All these observations are explained assuming that for any tw, intermittent events are triggered by local energy fluctuations exceeding those previously occurred.
A simple geometrical characterization of configuration space neighborhoods of local energy minima... more A simple geometrical characterization of configuration space neighborhoods of local energy minima in spin glass landscapes is found by exhaustive search. Combined with previous Monte Carlo investigations of thermal domain growth, it allows a discussion of the connection between real and configuration space descriptions of low temperature relaxational dynamics. We argue that the part of state-space corresponding to a single growing domain is adequately modeled by a hierarchically organized set of states and that thermal (meta)stability in spin glasses is related to the nearly exponential local density of states present within each trap. 75.10Nr, 02.50Ga, 05.40.+j, 02.70.-c Typeset using REVT E X
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