Feedbacks, nonlinearities and nonequilibria: A thermodynamic perspective.

The feedback circuit approach to nonlinear dynamical systems pioneered by Thomas and coworkers is revisited in a thermodynamical perspective. The role of nonequilibrium conditions and of other types of constraints such as mass action kinetics or microscopic reversibility around thermodynamic equilibrium in the way positive feedback circuits are operating is analyzed. It is shown that the appearance of non-trivial steady-state and time-dependent behaviors necessitates that the strengths of the feedback loops present exceed some well-defined critical values. Illustrations are provided on prototypical systems giving rise to multiple steady states.

Stochastic resonance across bifurcation cascades.

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.

Coupling-enhanced stochastic resonance.

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.

Detailed balance, nonequilibrium states, and dissipation in symbolic sequences.

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.

Dynamical responses to time-dependent control parameters in the presence of noise: a normal form approach.

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.

DNA viewed as an out-of-equilibrium structure.

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 χ^{2} 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.

Complexity measures for the evolutionary categorization of organisms.

Complexity measures are used to compare the genomic characteristics of five organisms belonging to distinct classes spanning the evolutionary tree: higher eukaryotes, amoebae, unicellular eukaryotes and bacteria. The comparisons are undertaken using the full four-letter alphabet and the coarse grained two-letter alphabets AG-CT and AT-CG. We show that the conditional probability matrix for the four-letter and AT-CG alphabet is markedly asymmetric in eukaryotes while it is nearly symmetric in bacterial genomes. Spatial asymmetry is revealed in the four-letter alphabet, signifying that the probability fluxes are nonvanishing and thus the reading sense of a sequence is irreversible for all organisms. Calculations of the block entropy and excess entropy demonstrate that the human genome accommodates better all possible block configurations, especially for long blocks. With respect to point-to-point details and to spatial arrangement of blocks the exit distance distributions from a particular letter demonstrate long distance characteristics in the eukaryotic sequences for all three alphabets, while the bacterial (prokaryotic) genomes deviate indicating short range characteristics. Overall, the conditional probability, the fluxes, the block entropy content and the exit distance distributions can be used as markers, discriminating between eukaryotic and prokaryotic DNA, allowing in many cases to discern details related to finer classes. In all cases the reduction from four letters to two masks some important statistical and spatial properties, with the AT-CG alphabet having higher ability of discrimination than the AG-CT one. In particular, the AT-CG alphabet reduction accentuates the CpG related properties (conditional probabilities w32, long ranged exit distance distribution for A and T nucleotides), but masks sequence asymmetry and irreversibility in all examined organisms.

Extreme events in multivariate deterministic systems.

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.

Transformation properties of entropy production.

The transformation properties of entropy production under phase-space partitioning, lumping, or the elimination of intermediate steps and variables in the presence of widely separated time scales are studied. Conditions are derived under which dissipation remains invariant. In systems subjected to external periodic driving the adiabatic, asymptotic, and transient entropy productions are evaluated and the extent to which they can be separately non-negative is determined.

Modeling the early stages of self-assembly in nanophase materials. II. Role of symmetry and dimensionality.

We study the early stages of self-assembly of elementary building blocks of nanophase materials, considering explicitly their structure and the symmetry and the dimensionality of the reaction space. Previous work [Kozak et al., J. Chem. Phys. 134, 154701 (2007)] focused on characterizing self-assembly on small square-planar templates. Here we consider larger lattices of square-planar symmetry having N = 255 sites, and both hexagonal and triangular lattices of N = 256 sites. Furthermore, to assess the consequences of a depletion zone above a basal layer (λ = 1), we study self-assembly on an augmented diffusion space defined by λ = 2 and λ = 5 stacked layers having the same characteristics as the basal plane. The effective decrease in the efficiency of self-assembly of individual nanophase units when the diffusion space is expanded, by increasing the template size and/or by enlarging the depletion zone, is then quantified. The results obtained reinforce our earlier conclusion that the most significant factor influencing the kinetics of formation of a final self-assembled unit is the number of reaction pathways from one or more precursor states. We draw attention to the relevance of these results to zeolite synthesis and reactions within pillared clays.

Equality governing nonequilibrium fluctuations and its information theory and thermodynamic interpretations.

The master equation is cast in the form of an equality involving the variation, in the course of a transformation, of a quantity playing the role of a generalized potential, weighted with the probability of allowable transformations emanating from different initial states. In the most general case this equality cannot be formulated entirely in terms of thermodynamic variables and state functions. Some conditions under which such a reduction becomes possible are identified and a comparison with fluctuation and work type relationships previously reported in the literature is carried out.

Memory effects in recurrent and extreme events.

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.

Irreversible thermodynamics of multistep transitions.

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.

Modeling the early stages of self-assembly in nanophase materials.

The early stages of self-assembly of the elementary building blocks of nanophase materials are studied. The relative roles of entropic and energetic factors in determining the relative abundance of the final products present is analyzed using both a kinetic mean field model and a mesoscopic approach in which self-assembly is viewed as an encounter-controlled process on a discrete lattice. The relevance of the results in zeolite synthesis in connection with the ordered liquid phases recently discovered in these materials is discussed.

Extreme events in deterministic dynamical systems.

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.

Propagating waves in one-dimensional discrete networks of coupled units.

We investigate the behavior of discrete systems on a one-dimensional lattice composed of localized units interacting with each other through nonlocal, nonlinear reactive dynamics. In the presence of second-order and third-order steps coupling two or three neighboring sites, respectively, we observe, for appropriate initial conditions, the propagation of waves which subsist in the absence of mass transfer by diffusion. For the case of the third-order (bistable) model, a counterintuitive effect is also observed, whereby the homogeneously less stable state invades the more stable one under certain conditions. In the limit of a continuous space the dynamics of these networks is described by a generic evolution equation, from which some analytical predictions can be extracted. The relevance of this mode of information transmission in spatially extended systems of interest in physical chemistry and biology is discussed.

Geometrical effects in protein nucleation.

To understand the importance of protein anisotropy and the influence of translational and rotational degrees of freedom on the nucleation event, we calculate numerically-exact values for the mean encounter time for two non-spherically symmetric molecules to form a cluster, regarded here as a precursor to nucleation. A lattice model is formulated in which the asymmetry of the molecules is accounted for by representing each as a 'dimer' in the sense that each molecule is specified to occupy two lattice sites. The two dimers undergo simultaneously translation and/or rotation, and the mean times for their encounter are determined. Exact numerical results are obtained for small lattices via application of the theory of finite Markov processes, and the results corroborated and extended to large lattices by performing Monte Carlo simulations. These calculations allow one to understand in a detailed way the interplay among geometrical anisotropy, translational and internal (rotational) degrees of freedom and system size in influencing the seminal nucleation event.

Transitions across a barrier induced by deterministic forcings.

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.

Lattice model of the early stages of the electrification of a cloud.

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.

Three-state model for cooperative desorption on a one-dimensional lattice.

We develop a master equation approach to the dynamics of immobile reactants on a one-dimensional lattice, in the presence of two different species undergoing cooperative desorption. A common feature of all the schemes studied is the strong dependence of the final coverage on the initial conditions, associated with the lack of ergodicity of the invariant state. Our approach leads to full agreement with Monte Carlo simulations, both asymptotically and transiently.