Long delay times in reaction rates increase intrinsic fluctuations

In spatially distributed cellular systems, it is often convenient to represent complicated auxiliary pathways and spatial transport by time-delayed reaction rates. Furthermore, many of the reactants appear in low numbers necessitating a probabilistic description. The coupling of delayed rates with stochastic dynamics leads to a probability conservation equation characterizing a non-Markovian process. A systematic approximation is derived that incorporates the effect of delayed rates on the characterization of molecular noise, valid in the limit of long delay time. By way of a simple example, we show that delayed reaction dynamics can only increase intrinsic fluctuations about the steady-state. The method is general enough to accommodate nonlinear transition rates, allowing characterization of fluctuations around a delay-induced limit cycle.Comment: 8 pages, 3 figures, to be published in Physical Review

A new modelling framework for statistical cumulus dynamics

We propose a new modelling framework suitable for the description of atmospheric convective systems as a collection of distinct plumes. The literature contains many examples of models for collections of plumes in which strong simplifying assumptions are made, a diagnostic dependence of convection on the large-scale environment and the limit of many plumes often being imposed from the outset. Some recent studies have sought to remove one or the other of those assumptions. The proposed framework removes both, and is explicitly time-dependent and stochastic in its basic character. The statistical dynamics of the plume collection are defined through simple probabilistic rules applied at the level of individual plumes, and van Kampen's system size expansion is then used to construct the macroscopic limit of the microscopic model. Through suitable choices of the microscopic rules, the model is shown to encompass previous studies in the appropriate limits, and to allow their natural extensions beyond those limits

A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model

As well known, cumulant expansion is an alternative way to moment expansion to fully characterize probability distributions provided all the moments exist. If this is not the case, the so called escort mean values (or q-moments) have been proposed to characterize probability densities with divergent moments [C. Tsallis et al, J. Math. Phys 50, 043303 (2009)]. We introduce here a new mathematical object, namely the q-cumulants, which, in analogy to the cumulants, provide an alternative characterization to that of the q-moments for the probability densities. We illustrate this new scheme on a recently proposed family of scale-invariant discrete probabilistic models [A. Rodriguez et al, J. Stat. Mech. (2008) P09006; R. Hanel et al, Eur. Phys. J. B 72, 263268 (2009)] having q-Gaussians as limiting probability distributions

Momentum conservation and correlation analyses in heavy-ion collisions at ultrarelativistic energies

Global transverse-momentum conservation induces correlations between any number of particles, which contribute in particular to the two- and three-particle correlations measured in heavy-ion collisions. These correlations are examined in detail, and their importance for studies of jets and their interaction with the medium is discussed.Comment: 5 pages, 2 figures. v2: corrected typos and added a paragrap

On-off intermittency over an extended range of control parameter

We propose a simple phenomenological model exhibiting on-off intermittency over an extended range of control parameter. We find that the distribution of the 'off' periods has as a power-law tail with an exponent varying continuously between -1 and -2, at odds with standard on-off intermittency which occurs at a specific value of the control parameter, and leads to the exponent -3/2. This non-trivial behavior results from the competition between a strong slowing down of the dynamics at small values of the observable, and a systematic drift toward large values.Comment: 4 pages, 3 figure

Analytical results for a Fokker-Planck equation in the small noise limit

We present analytical results for the lowest cumulants of a stochastic process described by a Fokker-Planck equation with nonlinear drift. We show that, in the limit of small fluctuations, the mean, the variance and the covariance of the process can be expressed in compact form with the help of the Lambert W function. As an application, we discuss the interplay of noise and nonlinearity far from equilibrium.Comment: 5 pages, 4 figure

Statistics of 3-dimensional Lagrangian turbulence

We consider a superstatistical dynamical model for the 3-d movement of a Lagrangian tracer particle embedded in a high-Reynolds number turbulent flow. The analytical model predictions are in excellent agreement with recent experimental data for flow between counter-rotating disks. In particular, we calculate the Lagrangian scaling exponents zeta_j for our system, and show that they agree well with the measured exponents reported in [X. Hu et al., PRL 96, 114503 (2006)]. Moreover, the model correctly predicts the shape of velocity difference and acceleration probability densities, the fast decay of component correlation functions and the slow decay of the modulus, as well as the statistical dependence between acceleration components. Finally, the model explains the numerically [P.K. Yeung and S.B. Pope, J. Fluid Mech. 207, 531 (1989)] and experimentally observed fact [B.W. Zeff et al., Nature 421, 146 (2003)] that enstrophy lags behind dissipation.Comment: 5 pages, 3 figures. Replaced by final version accepted by Phys. Rev. Let

Transient rectification of Brownian diffusion with asymmetric initial distribution

In an ensemble of non-interacting Brownian particles, a finite systematic average velocity may temporarily develop, even if it is zero initially. The effect originates from a small nonlinear correction to the dissipative force, causing the equation for the first moment of velocity to couple to moments of higher order. The effect may be relevant when a complex system dissociates in a viscous medium with conservation of momentum

Relaxation of finite perturbations: Beyond the Fluctuation-Response relation

We study the response of dynamical systems to finite amplitude perturbation. A generalized Fluctuation-Response relation is derived, which links the average relaxation toward equilibrium to the invariant measure of the system and points out the relevance of the amplitude of the initial perturbation. Numerical computations on systems with many characteristic times show the relevance of the above relation in realistic cases.Comment: 7 pages, 5 figure

How accurate are the non-linear chemical Fokker-Planck and chemical Langevin equations?

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order $\Omega^{-3/2}$ for reaction systems which do not obey detailed balance and at least accurate to order $\Omega^{-2}$ for systems obeying detailed balance, where $\Omega$ is the characteristic size of the system. Hence the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order $\Omega^{-1/2}$ and variance estimates accurate to order $\Omega^{-3/2}$. This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.Comment: 39 pages, 3 figures, accepted for publication in J. Chem. Phy