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The complexity of the primary structure of human DNA is explored using methods from nonequilibrium statistical mechanics, dynamical systems theory, and information theory. A collection of statistical analyses is performed on the DNA data and the results are compared with sequences derived from different stochastic processes. The use of Χ² tests shows that DNA can not be described as a low order Markov chain of order up to r = 6. Although detailed balance seems to hold at the level of a binary alphabet, it fails when all four base pairs are considered, suggesting spatial asymmetry and irreversibility. Furthermore, the block entropy does not increase linearly with the block size, reflecting the long-range nature of the correlations in the human genomic sequences. To probe locally the spatial structure of the chain, we study the exit distances from a specific symbol, the distribution of recurrence distances, and the Hurst exponent, all of which show power law tails and long-range characteristics. These results suggest that human DNA can be viewed as a nonequilibrium structure maintained in its state through interactions with a constantly changing environment. Based solely on the exit distance distribution accounting for the nonequilibrium statistics and using the Monte Carlo rejection sampling method, we construct a model DNA sequence. This method allows us to keep both long- and short-range statistical characteristics of the native DNA data. The model sequence presents the same characteristic exponents as the natural DNA but fails to capture spatial correlations and point-to-point details. [ABSTRACT FROM AUTHOR]

The response of a dynamical system to systematic variations of a control parameter in time in the presence of noise is analyzed by a reduction of the multivariate dynamics to a normal form in the vicinity of bifurcations of the pitchfork and of the limit point type. Mean-field responses, mean values, second- and fourth-order cumulants, probability densities, and entropy-like quantities are evaluated as the system sweeps across the bifurcation point, moving forward toward a multiple state region or moving backward out of this region. Depending on the case stabilization of unstable states, delays and slowing down are found and their signatures on particular observables are identified with an emphasis on the role of noise and on global properties beyond linearized theory. [ABSTRACT FROM AUTHOR]

The theory of stochastic resonance in multistable systems is extended to account for both direct transitions between all stable states present and indirect ones involving intermediate states. It is shown that to satisfy these requirements the dynamics needs to be embedded in phase spaces of dimension equal to at least two. Under well defined conditions, the conjunction of the presence of intermediate states and the multidimensional character of the process leads to an enhancement of the response of the system to an external periodic forcing. [ABSTRACT FROM AUTHOR]

The probabilistic properties of extreme values in multivariate deterministic dynamical systems are analyzed. It is shown that owing to the intertwining of unstable and stable modes the effect of dynamical complexity on the extremes tends to be masked, in the sense that the cumulative probability distribution of typical variables is differentiable and its associated probability density is continuous. Still, there exist combinations of variables probing the dominant unstable modes displaying singular behavior in the form of nondifferentiability of the cumulative distributions of extremes on certain sets of phase space points. Analytic evaluations and extensive numerical simulations are carried out for characteristic examples of Kolmogorov-type systems, for low-dimensional chaotic flows, and for spatially extended systems. [ABSTRACT FROM AUTHOR]

Stochastic resonance is analyzed in an array of nonlinear spatially coupled subsystems. Analytic expressions for the different steady-state solutions, for the rates of transitions between them in the presence of noise, and for the response to a weak external periodic forcing are derived. It is shown that the presence of spatial degrees of freedom modifies considerably the mechanisms of transitions between states and is responsible for a marked sensitivity of the response on the coupling constant and on the system size. [ABSTRACT FROM AUTHOR]

The classical setting of stochastic resonance is extended to account for parameter variations leading to transitions between a unique stable state, bistability, and multistability regimes, across singularities of various kinds. Analytic expressions for the amplitude and the phase of the response in terms of key parameters are obtained. The conditions for optimal responses are derived in terms of the bifurcation parameter, the driving frequency, and the noise strength. [ABSTRACT FROM AUTHOR]

Symbolic sequences arising from the coarse graining of deterministic dynamical systems continuous in phase space are considered. The extent to which signatures of the time irreversibility and of the nonequilibrium constraints at the level of the original system, such as fluxes or dissipation, can be identified at the coarse-grained level is analyzed. The roles of the partition, of the time window, and of time averaging in distinguishing in a clear-cut way the equilibrium versus nonequilibrium character of the sequence are brought out. [ABSTRACT FROM AUTHOR]

The principal signatures of deterministic dynamics in the statistical properties of the range and normalized range of sums are identified. Substantial differences from the classical statistical theory are found. Analytical expressions are derived, compared to those of the classical theory, and applied to generic classes of dynamical systems giving rise to chaotic and intermittent chaotic behavior.

The classical setting of stochastic resonance is extended to account for the presence of an arbitrary number of simultaneously stable steady states. General expressions for the linear response are derived for systems involving one variable. The existence of an optimal value of noise strength and of an optimal number of stable states for which the response is maximized is established.

A dynamical approach to recurrent and extreme events is developed focusing on the role of correlations and memory in the structure of the probability distributions and their low-order moments. The procedure is illustrated on homogeneous first and second order Markov chains, non-Markovian and nonhomogeneous processes and deterministic dynamical systems. Substantial differences with classical statistical theory as applied to independent identically distributed random variables are identified.

The propagation of extreme events in space is analyzed for a class of dynamical systems giving rise to spatiotemporal chaos. It is shown that this process can be mapped into a generalized random walk, whereby the mean square displacement increases linearly in time and there is a nonvanishing probability for jumps beyond first neighbors. The relative roles of the local dynamics and of the spatial coupling are identified.

The extreme value statistics of systems possessing a two-hump probability density of the relevant variable, in which the left peak is more pronounced than the right one, is studied. It is shown that systems of this type display a nontrivial transient behavior in the form of anomalous fluctuations around the mean, for certain (finite) ranges of observational time windows. The results are illustrated on independent identically distributed random variables, systems possessing two locally stable states and subjected to additive white noise, and dynamical systems in the regime of deterministic chaos.

Conditions under which the evolution equations of a multivariate system can be cast in a variational form are identified. A kinetic potential generating both the deterministic part of the evolution and the probabilistic properties in a suitably defined set of variables is derived and compared to the thermodynamic potentials. The results are illustrated on the kinetics of phase transitions involving intermediate metastable phases and chemical reactions giving rise to two or more intermediate species.

The principal signatures of a deterministic dynamics in the statistical properties of extreme events are identified. Explicit expressions are derived for generic classes of dynamical systems giving rise to quasiperiodic, strongly chaotic, and intermittent chaotic behaviors.

The early stages of the microphysics of the electrification process within a cloud are considered using a two-dimensional lattice model. Using insights generated from Monte Carlo simulations and the theory of finite Markov processes, the mean walk length statistics of the particles, the instantaneous electric potential and electric field profiles, the time evolution of electrostatic energy and their dependence on system size are studied. Some unexpected features of the kinetics of electrification and of the statistics of crossings of the threshold for an electric discharge to occur are brought out.

The response of a bistable dynamical system to a deterministic forcing is studied with emphasis on the kinetics of the passage across the barrier separating the two states, and compared to classical Kramers' theory describing the response to a Gaussian white noise forcing. The existence of nontrivial thresholds for the occurrence of transitions is established. Analytic results complemented by numerical simulations are derived for the characteristics of these transitions for periodic and chaotic forcings. The probabilistic properties of the response are finally addressed and some connections are established with the universal stable distributions of probability theory.

We study the dynamics of the first two moments and of threshold crossings by the stochastic trajectory in dichotomous diffusion x=xi(t), where xi(t) is a dichotomous Markov process. The transition rate of the latter is regarded as a control parameter and allowed to have specified time variations. The stabilizing or destabilizing effect of this variation is demonstrated, and qualitative changes in the statistical properties of the system are shown to occur. The analysis is then extended to linear dichotomous flow, and to a generalization of dichotomous diffusion in which x is driven by a multilevel Markov noise.