Cooperative surmounting of bottlenecks

The physics of activated escape of objects out of a metastable state plays a key role in diverse scientific areas involving chemical kinetics, diffusion and dislocation motion in solids, nucleation, electrical transport, motion of flux lines superconductors, charge density waves, and transport processes of macromolecules, to name but a few. The underlying activated processes present the multidimensional extension of the Kramers problem of a single Brownian particle. In comparison to the latter case, however, the dynamics ensuing from the interactions of many coupled units can lead to intriguing novel phenomena that are not present when only a single degree of freedom is involved. In this review we report on a variety of such phenomena that are exhibited by systems consisting of chains of interacting units in the presence of potential barriers. In the first part we consider recent developments in the case of a deterministic dynamics driving cooperative escape processes of coupled nonlinear units out of metastable states. The ability of chains of coupled units to undergo spontaneous conformational transitions can lead to a self-organised escape. The mechanism at work is that the energies of the units become re-arranged, while keeping the total energy conserved, in forming localised energy modes that in turn trigger the cooperative escape. We present scenarios of significantly enhanced noise-free escape rates if compared to the noise-assisted case. The second part deals with the collective directed transport of systems of interacting particles overcoming energetic barriers in periodic potential landscapes. Escape processes in both time-homogeneous and time-dependent driven systems are considered for the emergence of directed motion. It is shown that ballistic channels immersed in the associated high-dimensional phase space are the source for the directed long-range transport

Aspects of stochastic resonance in reaction-diffusion systems: The nonequilibrium-potential approach

We analyze several aspects of the phenomenon of stochastic resonance in reaction-diffusion systems, exploiting the nonequilibrium potential's framework. The generalization of this formalism (sketched in the appendix) to extended systems is first carried out in the context of a simplified scalar model, for which stationary patterns can be found analytically. We first show how system-size stochastic resonance arises naturally in this framework, and then how the phenomenon of array-enhanced stochastic resonance can be further enhanced by letting the diffusion coefficient depend on the field. A yet less trivial generalization is exemplified by a stylized version of the FitzHugh-Nagumo system, a paradigm of the activator-inhibitor class. After discussing for this system the second aspect enumerated above, we derive from it -through an adiabatic-like elimination of the inhibitor field- an effective scalar model that includes a nonlocal contribution. Studying the role played by the range of the nonlocal kernel and its effect on stochastic resonance, we find an optimal range that maximizes the system's response.Comment: 16 pages, 15 figures, uses svjour.cls and svepj-spec.clo. Minireview to appear in The European Physical Journal Special Topics (issue in memory of Carlos P\'erez-Garc\'{\i}a, edited by H. Mancini

Sharp estimates for the global attractor of scalar reaction-diffusion equations with a Wentzell boundary condition

In this paper, we derive optimal upper and lower bounds on the dimension of the attractor AW for scalar reaction-diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds about the constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor AK; K \in {D;N; P}, of reactiondiffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor AW is of different order than the dimension of AK; for each K \in {D;N; P} ; in all space dimensions that are greater or equal than three.Comment: to appear in J. Nonlinear Scienc

Finite dimensional attractor for a composite system of wave/plate equations with localised damping

The long-term behaviour of solutions to a model for acoustic-structure interactions is addressed; the system is comprised of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are: existence of a global attractor for the dynamics generated by this composite system, as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension -- established in a previous work by Bucci, Chueshov and Lasiecka (Comm. Pure Appl. Anal., 2007) only in the presence of full interior acoustic damping -- holds even in the case of localised dissipation. This nontrivial generalization is inspired by and consistent with the recent advances in the study of wave equations with nonlinear localised damping.Comment: 40 pages, 1 figure; v2: added references for Section 1, submitte

Contributions of plasma physics to chaos and nonlinear dynamics

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016

Manifestations of the onset of chaos in condensed matter and complex systems

We review the occurrence of the patterns of the onset of chaos in low-dimensional nonlinear dissipative systems in leading topics of condensed matter physics and complex systems of various disciplines. We consider the dynamics associated with the attractors at period-doubling accumulation points and at tangent bifurcations to describe features of glassy dynamics, critical fluctuations and localization transitions. We recall that trajectories pertaining to the routes to chaos form families of time series that are readily transformed into networks via the Horizontal Visibility algorithm, and this in turn facilitates establish connections between entropy and Renormalization Group properties. We discretize the replicator equation of game theory to observe the onset of chaos in familiar social dilemmas, and also to mimic the evolution of high-dimensional ecological models. We describe an analytical framework of nonlinear mappings that reproduce rank distributions of large classes of data (including Zipf's law). We extend the discussion to point out a common circumstance of drastic contraction of configuration space driven by the attractors of these mappings. We mention the relation of generalized entropy expressions with the dynamics along and at the period doubling, intermittency and quasi-periodic routes to chaos. Finally, we refer to additional natural phenomena in complex systems where these conditions may manifest.Comment: 20 pages, 7 figures. To be published in European Physical Journal Special Topics. Special Issue: "Nonlinear Phenomena in Physics: New Techniques and Applications

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte