On the Gaussian approximation for master equations

We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen's expansion approach (the fact that the probability distribution is Gaussian at first order). We analyze the scaling of the error with a large parameter of the system and compare it with van Kampen's method. Our theoretical analysis and the study of several examples shows that the Gaussian approximation turns out to be more accurate. This could be specially important for problems involving stochastic processes in systems with a small number of particles

Exact solution of a stochastic protein dynamics model with delayed degradation

We study a stochastic model of protein dynamics that explicitly includes delay in the degradation. We rigorously derive the master equation for the processes and solve it exactly. We show that the equations for the mean values obtained differ from others intuitively proposed and that oscillatory behavior is not possible in this system. We discuss the calculation of correlation functions in stochastic systems with delay, stressing the differences with Markovian processes. The exact results allow to clarify the interplay between stochasticity and delay

On the effect of heterogeneity in stochastic interacting-particle systems

We study stochastic particle systems made up of heterogeneous units. We introduce a general framework suitable to analytically study this kind of systems and apply it to two particular models of interest in economy and epidemiology. We show that particle heterogeneity can enhance or decrease the collective fluctuations depending on the system, and that it is possible to infer the degree and the form of the heterogeneity distribution in the system by measuring only global variables and their fluctuations

Ordering dynamics in the voter model with aging

The voter model with memory-dependent dynamics is theoretically and numerically studied at the mean-field level. The `internal age', or time an individual spends holding the same state, is added to the set of binary states of the population, such that the probability of changing state (or activation probability $p_i$) depends on this age. A closed set of integro-differential equations describing the time evolution of the fraction of individuals with a given state and age is derived, and from it analytical results are obtained characterizing the behavior of the system close to the absorbing states. In general, different age-dependent activation probabilities have different effects on the dynamics. When the activation probability $p_i$ is an increasing function of the age $i$, the system reaches a steady state with coexistence of opinions. In the case of aging, with $p_i$ being a decreasing function, either the system reaches consensus or it gets trapped in a frozen state, depending on the value of $p_\infty$ (zero or not) and the velocity of $p_i$ approaching $p_\infty$. Moreover, when the system reaches consensus, the time ordering of the system can be exponential ($p_\infty>0$) or power-law like ($p_\infty=0$). Exact conditions for having one or another behavior, together with the equations and explicit expressions for the exponents, are provided

Divergent Time Scale in Axelrod Model Dynamics

We study the evolution of the Axelrod model for cultural diversity. We consider a simple version of the model in which each individual is characterized by two features, each of which can assume q possibilities. Within a mean-field description, we find a transition at a critical value q_c between an active state of diversity and a frozen state. For q just below q_c, the density of active links between interaction partners is non-monotonic in time and the asymptotic approach to the steady state is controlled by a time scale that diverges as (q-q_c)^{-1/2}.Comment: 4 pages, 5 figures, 2-column revtex4 forma

Non-universal results induced by diversity distribution in coupled excitable systems

We consider a system of globally coupled active rotators near the excitable regime. The system displays a transition to a state of collective firing induced by disorder. We show that this transition is found generically for any diversity distribution with well defined moments. Singularly, for the Lorentzian distribution (widely used in Kuramoto-like systems) the transition is not present. This warns about the use of Lorentzian distributions to understand the generic properties of coupled oscillators.We acknowledge financial support by the MICINN (Spain) and FEDER (EU) through project FIS2007-60327. L.F.L. is supported by the JAEPredoc program of CSIC.Peer reviewe

Non-universal results induced by diversity distribution in coupled excitable systems

We consider a system of globally coupled active rotators near the excitable regime. The system displays a transition to a state of collective firing induced by disorder. We show that this transition is found generically for any diversity distribution with well defined moments. Singularly, for the Lorentzian distribution (widely used in Kuramoto-like systems) the transition is not present. This warns about the use of Lorentzian distributions to understand the generic properties of coupled oscillators

Coherence Resonance in Chaotic Systems

We show that it is possible for chaotic systems to display the main features of coherence resonance. In particular, we show that a Chua model, operating in a chaotic regime and in the presence of noise, can exhibit oscillations whose regularity is optimal for some intermediate value of the noise intensity. We find that the power spectrum of the signal develops a peak at finite frequency at intermediate values of the noise. These are all signatures of coherence resonance. We also experimentally study a Chua circuit and corroborate the above simulation results. Finally, we analyze a simple model composed of two separate limit cycles which still exhibits coherence resonance, and show that its behavior is qualitatively similar to that of the chaotic Chua systemComment: 4 pages (including 4 figures) LaTeX fil