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Three-dimensional Monte Carlo simulations provide a striking confirmation to a recent theoretical prediction: the Brownian non-Gaussian diffusion of critical self-avoiding walks. Although the mean square displacement of the polymer center of mass grows linearly with time (Brownian behavior), the initial probability density function is strongly non-Gaussian and crosses over to Gaussianity only at large time. Full agreement between theory and simulations is achieved without the employment of fitting parameters. We discuss simulation techniques potentially capable of addressing the study of anomalous diffusion under complex conditions like adsorption- or Theta-transition. [ABSTRACT FROM AUTHOR]

A definition of time based on the assumption of scale invariance may enhance and simplify the analysis of historical series with cyclically recurrent patterns and seasonalities. By enforcing simple-scaling and stationarity of the distributions of returns, we identify a successful protocol of time definition in finance, functional from tens of minutes to a few days. Within this time definition, the significant reduction of cyclostationary effects allows analyzing the structure of the stochastic process underlying the series on the basis of statistical sampling sliding along the whole time series. At the same time, the duration of periods in which markets remain inactive is properly quantified by the novel clock, and the corresponding returns (e.g., overnight or weekend) can be consistently taken into account for financial applications. The method is applied to the S&P500 index recorded at a 1 min frequency between September 1985 and June 2013. [ABSTRACT FROM AUTHOR]

Active particles may happen to be confined in channels so narrow that they cannot overtake each other (single-file conditions). This interesting situation reveals nontrivial physical features as a consequence of the strong interparticle correlations developed in collective rearrangements. We consider a minimal two-dimensional model for active Brownian particles with the aim of studying the modifications introduced by activity with respect to the classical (passive) single-file picture. Depending on whether their motion is dominated by translational or rotational diffusion, we find that active Brownian particles in single file may arrange into clusters that are continuously merging and splitting (active clusters) or merely reproduce passive-motion paradigms, respectively. We show that activity conveys to self-propelled particles a strategic advantage for trespassing narrow channels against external biases (e.g., the gravitational field). [ABSTRACT FROM AUTHOR]

We present and discuss a stochastic model of financial assets dynamics based on the idea of an inverse renormalization group strategy. With this strategy we construct the multivariate distributions of elementary returns based on the scaling with time of the probability density of their aggregates. In its simplest version the model is the product of an endogenous auloregressive component and a random rescaling factor designed to embody also exogenous influences. Mathematical properties like increments' stationarity and ergodicity can be proven. Thanks lo the relatively low number of parameters, model calibration can be conveniently based on a method of moments, as exemplified in the case of historical data of the S&P500 index. The calibrated model accounts very well for many stylized facts, like volatility clustering, power-law decay of the volatility autocorrelation function, and multiscaling with time of the aggregated return distribution. In agreement with empirical evidence in finance, the dynamics is not invariant under time reversal, and, with suitable generalizations, skewness of the return distribution and leverage effects can be included. The analytical tractability of the model opens interesting perspectives for applications, for instance, in terms of obtaining closed formulas for derivative pricing. Further important features are the possibility of making contact, in certain limits, with autoregressive models widely used in finance and the possibility of partially resolving the long- and short-memory components of the volatility, with consistent results when applied to historical series. [ABSTRACT FROM AUTHOR]

In the thermodynamic limit, the time evolution of isolated long-range interacting systems is properly described by the Vlasov equation. This equation admits nonequilibrium dynamically stable stationary solutions characterized by a zero order parameter. We show that the presence of external noise sources, such as a heat bath, can reduce their lifetime and induce at a specific time a dynamical phase transition marked by a nonzero order parameter. This transition may be used as a distinctive experimental signature of the temporary existence of nonequilibrium Vlasov-stable states. In particular, we present evidence of a regime characterized by an order parameter pulse. Our analytical results are corroborated by numerical simulations of a paradigmatic long-range model. [ABSTRACT FROM AUTHOR]

We compare simulations performed using the Nosé-Hoover and the Langevin thermostats with the Hamiltonian dynamics of a long-range interacting system in contact with a reservoir. We find that while the statistical mechanics equilibrium properties of the system are recovered by all the different methods, the Nosé-Hoover and the Langevin nonequilibrium results differ from the Hamiltonian ones. [ABSTRACT FROM AUTHOR]

Abstract: We review recent results obtained for the dynamics of incipient chaos. These results suggest a common picture underlying the three universal routes to chaos displayed by the prototypical logistic and circle maps. Namely, the period doubling, intermittency, and quasiperiodicity routes. In these situations the dynamical behavior is exactly describable through infinite families of Tsallis’q-exponential functions. Furthermore, the addition of a noise perturbation to the dynamics at the onset of chaos of the logistic map allows to establish parallels with the behavior of supercooled liquids close to glass formation. Specifically, the occurrence of two-step relaxation, aging with its characteristic scaling property, and subdiffusion and arrest is corroborated for such a system. [Copyright &y& Elsevier]

We discuss the relation between ensemble and time averages for quasistationary states of low-dimensional symplectic maps that present remarkable analogies with similar states detected in many-body long-range-interacting Hamiltonian systems. [Copyright &y& Elsevier]

We present a comparative study of several dynamical systems of increasing complexity, namely, the logistic map with additive noise, one, two and many globally coupled standard maps, and the Hamiltonian mean field model (i.e., the classical inertial infinitely ranged ferromagnetically coupled XY spin model). We emphasize the appearance, in all of these systems, of metastable states and their ultimate crossover to the equilibrium state. We comment on the underlying mechanisms responsible for these phenomena (weak chaos) and compare common characteristics. We point out that this ubiquitous behavior appears to be associated to the features of the nonextensive generalization of the Boltzmann–Gibbs statistical mechanics. [Copyright &y& Elsevier]

We address a simple connection between results of Hamiltonian non-linear dynamical theory and thermostatistics. Using a properly defined dynamical temperature in low-dimensional symplectic maps, we display and characterize long-standing quasi-stationary states that eventually cross over to a Boltzmann–Gibbs-like regime. As time evolves, the geometrical properties (e.g., fractal dimension) of the phase space change sensibly, and the duration of the anomalous regime diverges with decreasing chaoticity. The scenario that emerges is consistent with the non-extensive statistical mechanics one. [Copyright &y& Elsevier]

In this paper, we study the behavior of a Brownian particle in the overdamped regime in the presence of a harmonic potential, assuming its diffusion coefficient to randomly jump between two distinct values. In particular, we characterize the probability distribution of the particle position and provide detailed expressions for the mean square displacement and the kurtosis. We highlight non-Gaussian behavior even within the long-term limit carried over with an excess of probability both in the central part and in the distribution tails. Moreover, when one of the two diffusion coefficients assumes the value zero, we provide evidence that the probability distribution develops a cusp. Most of our results are analytical, and corroborated by numerical simulations. [ABSTRACT FROM AUTHOR]

We develop a theoretical description of the topological disentanglement occurring when torus knots reach the ends of a semiflexible polymer under tension. These include decays into simpler knots and total unknotting. The minimal number of crossings and the minimal knot contour length are the topological invariants playing a key role in the model. The crossings behave as particles diffusing along the chain and the application of appropriate boundary conditions at the ends of the chain accounts for the knot disentanglement. Starting from the number of particles and their positions, suitable rules allow reconstructing the type and location of the knot moving on the chain Our theory is extensively benchmarked with corresponding molecular dynamics simulations and the results show a remarkable agreement between the simulations and the theoretical predictions of the model. [ABSTRACT FROM AUTHOR]

After recalling key phenomenological properties of glass formation, we point out that similar features are exhibited by the dynamical properties of the noise-perturbed iterates of the logistic map at the onset of chaos. The analysis includes two-step relaxation, aging, subdiffusion and arrest, as well as an expression analogous to the Adam–Gibbs relation connecting dynamical and thermodynamic properties of a glass former. The dynamical properties of the logistic map in the presence of external noise are seen to be comparable to those of a supercooled liquid above a glass transition temperature, whereas the noiseless attractor displays typical nonequilibrium aspects like loss of time translation invariance (aging). Reference is made to connections between the noiseless dynamics at the chaos threshold and the nonextensive formalism. [ABSTRACT FROM AUTHOR]

After a brief review of the present status of nonextensive statistical mechanics, we present a conjectural scenario where mixing (characterized by the entropic index qmix⩽1) and equilibration (characterized by the entropic index qeq⩾1) play central and inter-related roles, and appear to determine a priori the values of the relevant indices of the formalism. Boltzmann–Gibbs statistical mechanics is recovered as the qmix=qeq=1 particular case. [Copyright &y& Elsevier]

Single-file diffusion is a ubiquitous physical process exploited by living and synthetic systems to exchange molecules with their environment. It is paramount to quantify the escape time needed for single files of particles to exit from constraining synthetic channels and biological pores. This quantity depends on complex cooperative effects, whose predominance can only be established through a strict comparison between theory and experiments. By using colloidal particles, optical manipulation, microfluidics, digital microscopy, and theoretical analysis we uncover the self-similar character of the escape process and provide closed-formula evaluations of the escape time. We find that the escape time scales inversely with the diffusion coefficient of the last particle to leave the channel. Importantly, we find that at the investigated microscale, bias forces as tiny as 10-15??N determine the magnitude of the escape time by drastically reducing interparticle collisions. Our findings provide crucial guidelines to optimize the design of micro- and nanodevices for a variety of applications including drug delivery, particle filtering, and transport in geometrical constrictions. [ABSTRACT FROM AUTHOR]

We exactly analyze the vibrational properties of a chain of harmonic oscillators in contact with local Langevin heat baths. Nonequilibrium steady-state fluctuations are found to be described by a set of mode temperatures, independent of the strengths of both the harmonic interaction and the viscous damping. Energy is equally distributed between the conjugate variables of a given mode but differently among different modes, in a manner which depends exclusively on the bath temperatures and on the boundary conditions. We outline how bath-temperature profiles can be designed to enhance or reduce fluctuations at specific frequencies in the power spectrum of the chain length. [ABSTRACT FROM AUTHOR]

Redundancy in biology may be explained by the need to optimize extreme searching processes, where one or few among many particles are requested to reach the target like in human fertilization. We show that non-Gaussian rare fluctuations in Brownian diffusion dominates such searches, introducing drastic corrections to the known Gaussian behavior. Our demonstration entails different physical systems and pinpoints the relevance of diversity within redundancy to boost fast targeting. We sketch an experimental context to test our results: polydisperse systems.

Diffusing diffusivity models, polymers in the grand canonical ensemble and polydisperse, and continuous-time random walks all exhibit stages of non-Gaussian diffusion. Is non-Gaussian targeting more efficient than Gaussian? We address this question, central to, e.g., diffusion-limited reactions and some biological processes, through a general approach that makes use of Jensen's inequality and that encompasses all these systems. In terms of customary mean first-passage time, we show that Gaussian searches are more effective than non-Gaussian ones. A companion paper argues that non-Gaussianity becomes instead highly more efficient in applications where only a small fraction of tracers is required to reach the target.

We demonstrate that size fluctuations close to polymers critical point originate the non-Gaussian diffusion of their center of mass. Static universal exponents γ and ν-depending on the polymer topology, on the dimension of the embedding space, and on equilibrium phase-concur to determine the potential divergency of a dynamic response, epitomized by the center-of-mass kurtosis. Prospects in experiments and stochastic modeling brought about by this result are briefly outlined.

We propose a new look at the heat bath for two Brownian particles, in which the heat bath as a "system" is both perturbed and sensed by the Brownian particles. Nonlocal thermal fluctuations give rise to bath-mediated static forces between the particles. Based on the general sum rule of the linear response theory, we derive an explicit relation linking these forces to the friction kernel describing the particles' dynamics. The relation is analytically confirmed in the case of two solvable models and could be experimentally challenged. Our results point out that the inclusion of the environment as a part of the whole system is important for micron- or nanoscale physics.

On the basis of analytical results and molecular dynamics simulations, we clarify the nonequilibrium dynamics of a long-range interacting system in contact with a heat bath. For small couplings with the bath, we show that the system can first be trapped in a Vlasov quasistationary state, then a microcanonical one follows, and finally canonical equilibrium is reached at the bath temperature. We demonstrate that, even out of equilibrium, Hamiltonian reservoirs microscopically coupled with the system and Langevin thermostats provide equivalent descriptions. Our identification of the key parameters determining the quasistationary lifetimes could be exploited to control experimental systems such as the free-electron laser, in the presence of external noise or inherent imperfections.

We derive a central limit theorem for the probability distribution of the sum of many critically correlated random variables. The theorem characterizes a variety of different processes sharing the same asymptotic form of anomalous scaling and is based on a correspondence with the Lévy-Gnedenko uncorrelated case. In particular, correlated anomalous diffusion is mapped onto Lévy diffusion. Under suitable assumptions, the nonstandard multiplicative structure used for constructing the characteristic function of the total sum allows us to determine correlations of partial sums exclusively on the basis of the global anomalous scaling.

We use Hamiltonian dynamics to discuss the statistical mechanics of long-lasting quasistationary states particularly relevant for long-range interacting systems. Despite the presence of an anomalous single-particle velocity distribution, we find that the central limit theorem implies the Boltzmann expression in Gibbs' Gamma space. We identify the nonequilibrium submanifold of Gamma space characterizing the anomalous behavior and show that by restricting the Boltzmann-Gibbs approach to this submanifold we obtain the statistical mechanics of the quasistationary states.

We introduce a Hamiltonian dynamics for the description of long-range interacting systems in contact with a thermal bath (i.e., in the canonical ensemble). The dynamics confirms statistical mechanics equilibrium predictions for the Hamiltonian mean field model and the equilibrium ensemble equivalence. We find that long-lasting quasistationary states persist in the presence of the interaction with the environment. Our results indicate that quasistationary states are indeed reproducible in real physical experiments.