A solvable non-conservative model of Self-Organized Criticality

We present the first solvable non-conservative sandpile-like critical model of Self-Organized Criticality (SOC), and thereby substantiate the suggestion by Vespignani and Zapperi [A. Vespignani and S. Zapperi, Phys. Rev. E 57, 6345 (1998)] that a lack of conservation in the microscopic dynamics of an SOC-model can be compensated by introducing an external drive and thereby re-establishing criticality. The model shown is critical for all values of the conservation parameter. The analytical derivation follows the lines of Broeker and Grassberger [H.-M. Broeker and P. Grassberger, Phys. Rev. E 56, 3944 (1997)] and is supported by numerical simulation. In the limit of vanishing conservation the Random Neighbor Forest Fire Model (R-FFM) is recovered.Comment: 4 pages in RevTeX format (2 Figures) submitted to PR

Connecting the Micro-dynamics to the Emergent Macro-variables: Self-Organized Criticality and Absorbing Phase Transitions in the Deterministic Lattice Gas

We reinvestigate the Deterministic Lattice Gas introduced as a paradigmatic model of the 1/f spectra (Phys. Rev. Lett. V26, 3103 (1990)) arising according to the Self-Organized Criticality scenario. We demonstrate that the density fluctuations exhibit an unexpected dependence on systems size and relate the finding to effective Langevin equations. The low density behavior is controlled by the critical properties of the gas at the absorbing state phase transition. We also show that the Deterministic Lattice Gas is in the Manna universality class of absorbing state phase transitions. This is in contrast to expectations in the literature which suggested that the entirely deterministic nature of the dynamics would put the model in a different universality class. To our knowledge this is the first fully deterministic member of the Manna universality class.Comment: 8 pages, 12 figures. Changes in the new version: Reference list has been correcte

Accuracy of the cluster-approximation method in a nonequilibrium model

We examine a model in which a nonequilibrium phase transition from an active to an extinct state is observed. The order of this phase transition has been shown to be either continuous or first-order, depending on the parameter values and the dimension of the system. Using increasingly large clusters, we use the cluster approximation method to obtain estimates for the critical points in 1+1 dimensions. For the continuous phase transitions only, extrapolations of these approximations show excellent agreement with simulation results. Further, the approximations suggest that, consistent with simulation results, in 1+1 dimensions no first-order phase transitions are observed.Comment: 8 pages, 3 figures and 1 tabl

Vortices Clustering: The Origin of the Second Peak in the Magnetisation Loops of High Temperature Superconductors

We study vortex clustering in type II Superconductors. We demonstrate that the ``second peak'' observed in magnetisation loops may be a dynamical effect associated with a density driven instability of the vortex system. At the microscopic level the instability shows up as the clustering of individual vortices at (rare) preferential regions of the pinning potential. In the limit of quasi-static ramping the instability is related to a phase transition in the equilibrium vortex system.Comment: 11 pages + 3 figure

Off equilibrium magnetic properties in a model for vortices in superconductors

We study the properties of a simple lattice model of repulsive particles diffusing in a pinning landscape. The behaviour of the model is very similar to the observed physics of vortices in superconductors. We compare and discuss the equilibrium phase diagram, creep dynamics, the Bean state profiles, hysteresis of magnetisation loops (including the second peak feature), and, in particular, ``off equilibrium'' relaxations. The model is analytically tractable in replica mean field theory and numerically via Monte Carlo simulations. It offers a comprehensive schematic framework of the observed phenomenology

Memory effects in response functions of driven vortex matter

Vortex flow in driven type II superconductors shows strong memory and history dependent effects. Here, we study a schematic microscopic model of driven vortices to propose a scenario for a broad set of these kind of phenomena ranging from ``rejuvenation'' and ``stiffening'' of the system response, to ``memory'' and ``irreversibility'' in I-V characteristics

Tangent map intermittency as an approximate analysis of intermittency in a high dimensional fully stochastic dynamical system: The Tangled Nature model

It is well known that low-dimensional nonlinear deterministic maps close to a tangent bifurcation exhibit intermittency and this circumstance has been exploited, e.g. by Procaccia and Schuster [Phys. Rev. A 28, 1210 (1983)], to develop a general theory of 1/f spectra. This suggests it is interesting to study the extent to which the behavior of a high-dimensional stochastic system can be described by such tangent maps. The Tangled Nature (TaNa) Model of evolutionary ecology is an ideal candidate for such a study, a significant model as it is capable of reproducing a broad range of the phenomenology of macroevolution and ecosystems. The TaNa model exhibits strong intermittency reminiscent of Punctuated Equilibrium and, like the fossil record of mass extinction, the intermittency in the model is found to be non-stationary, a feature typical of many complex systems. We derive a mean-field version for the evolution of the likelihood function controlling the reproduction of species and find a local map close to tangency. This mean-field map, by our own local approximation, is able to describe qualitatively only one episode of the intermittent dynamics of the full TaNa model. To complement this result we construct a complete nonlinear dynamical system model consisting of successive tangent bifurcations that generates time evolution patterns resembling those of the full TaNa model in macroscopic scales. The switch from one tangent bifurcation to the next in the sequences produced in this model is stochastic in nature, based on criteria obtained from the local mean-field approximation, and capable of imitating the changing set of types of species and total population in the TaNa model. The model combines full deterministic dynamics with instantaneous parameter random jumps at stochastically drawn times. In spite of the limitations of our approach, that entails a drastic collapse of degrees of freedom, the description of a high-dimensional model system in terms of a low-dimensional one appears to be illuminating

Elastic Chain in a Random Potential: Simulation of the Displacement Function <(u(x)−u(0))2><(u(x)-u(0))^2> and Relaxation

We simulate the low temperature behaviour of an elastic chain in a random potential where the displacements $u(x)$ are confined to the {\it longitudinal} direction ($u(x)$ parallel to $x$) as in a one dimensional charge density wave--type problem. We calculate the displacement correlation function $g(x)=< (u(x)-u(0))^2>$ and the size dependent average square displacement $W(L)=$. We find that $g(x)\sim x^{2\eta}$ with $\eta\simeq3/4$ at short distances and $\eta\simeq3/5$ at intermediate distances. We cannot resolve the asymptotic long distance dependence of $g$ upon $x$. For the system sizes considered we find $g(L/2)\propto W\sim L^{2\chi}$ with $\chi\simeq2/3$. The exponent $\eta\simeq3/5$ is in agreement with the Random Manifold exponent obtained from replica calculations and the exponent $\chi\simeq2/3$ is consistent with an exact solution for the chain with {\it transverse} displacements ($u(x)$ perpendicular to $x$).The distribution of nearest distances between pinning wells and chain-particles is found to develop forbidden regions.Comment: 19 pages of LaTex, 6 postscript figures available on request, submitted to Journal of Physics A, MAJOR CHANGE

Tangled Nature: A model of emergent structure and temporal mode among co-evolving agents

Understanding systems level behaviour of many interacting agents is challenging in various ways, here we'll focus on the how the interaction between components can lead to hierarchical structures with different types of dynamics, or causations, at different levels. We use the Tangled Nature model to discuss the co-evolutionary aspects connecting the microscopic level of the individual to the macroscopic systems level. At the microscopic level the individual agent may undergo evolutionary changes due to mutations of strategies. The micro-dynamics always run at a constant rate. Nevertheless, the system's level dynamics exhibit a completely different type of intermittent abrupt dynamics where major upheavals keep throwing the system between meta-stable configurations. These dramatic transitions are described by a log-Poisson time statistics. The long time effect is a collectively adapted of the ecological network. We discuss the ecological and macroevolutionary consequences of the adaptive dynamics and briefly describe work using the Tangled Nature framework to analyse problems in economics, sociology, innovation and sustainabilityComment: Invited contribution to Focus on Complexity in European Journal of Physics. 25 page, 1 figur

How a spin-glass remembers. Memory and rejuvenation from intermittency data: an analysis of temperature shifts

The memory and rejuvenation aspects 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. Invoking the concept of marginal stability, this theory links irreversible `intermittent' events, or `quakes' to thermal fluctuations of record magnitude. The pivotal idea is that the largest energy barrier $b(t_w,T)$ surmounted prior to $t_w$ by thermal fluctuations at temperature $T$ determines the rate $r_q \propto 1/t_w$ of the intermittent events occurring near $t_w$. The idea leads to a rate of intermittent events after a negative temperature shift given by $r_q \propto 1/t_w^{eff}$, where the `effective age' $t_w^{eff} \geq t_w$ has an algebraic dependence on $t_w$, whose exponent contains the temperatures before and after the shift. The analytical expression is verified by numerical simulations. Marginal stability suggests that a positive temperature shift $T \to T'$ could erase the memory of the barrier $b(t_w,T)$. The simulations show that the barrier $b(t_w,T') \geq b(t_w,T)$ controls the intermittent dynamics, whose rate is hence $r_q \propto 1/t_w$. Additional `rejuvenation' effects are also identified in the intermittency data for shifts of both signs.Comment: Revised introduction and discussion. Final version to appear in Journal of Statistical Mechanics: Theory and Experimen