Dichotomous noise appears in a wide variety of physical and mathematical models. It has escaped attention that the standard results for the long time properties (e.g., the asymptotic probability density, the drift velocity) cannot be applied when unstable fixed points are crossed in the asymptotic regime. We show how calculations have to be modified to deal with these cases. The full analytic results are illustrated on the problems of hypersensitive transport and of the rocking ratchet.

The distribution function of the fluctuations in systems displaying Gaussian 1/f power spectra is shown to be one of the extreme value distributions (Gumbel distribution). We derive this result analytically for the case of periodic boundary conditions and demonstrate by simulations that the Gumbel distribution is a good approximation for the case of free boundary conditions as well. Experiments on voltage fluctuations in a GaAs films are analyzed and excellent agreement is found with the theory. Generalization to 1/f^α noise signals will also be discussed. Our results open a new perspective on the data analysis of 1/f^α processes.

To most people it appears obvious that a system at rest, when perturbed by an external static force, responds by moving into the direction of that force. The counterintuitive contrary behavior, namely a permanent motion opposite to the external force, is termed absolute negative mobility (ANM). We consider an experimentally realistic system that exhibits ANM when driven away from thermal equilibrium. The underlying physical mechanisms are illustrated by means of a simple model system for a single Brownian particle.

Theoretical predictions regarding the statistical mechanics of systems driven far from equilibrium have recently been verified in laboratory experiments. I will discuss both the theories and the experiments. A particular theme of the talk will be the insight which these developments provide us, regarding the character of the Second Law of Thermodynamics as applied to microscopic systems.

Chirality is an important aspect of chemistry. Through processes that spontaneously break chiral symmetry, and through chirally selective interactions, it is possible to generate unequal amounts of left- and right-handed molecules or crystals. In crystallization, one can see both spontaneous chiral symmetry breaking and chirally selective interactions in addition to stochastic behavior. A particular system of interest is the crystallization of amino acids in the presence of small amount another closely related amino acid as an impurity. In crystallization from a solution containing equal amounts of L- and D- glutamic acid, the presence of a small amount of L-lysine results in unequal rates of crystallization which in turn generates transient optical activity in the system. The sensitivity of this system to the chiral impurity and the stochastic nature of this process will be presented.

We present a new mechanism for noise-induced pattern formation phenomena. In contrast with known mechanisms for pure noise-induced pattern formation, this mechanism is NOT driven by a short-time instability amplified by collective effects. The phenomenon is analyzed by means of a modulated mean field approximation and numerical simulations.

Almost all systems either in nature or in technology are noisy and nonlinear. Can we infer models of complex nonlinear systems and then predict their behavior? If such an algorithm can be found it will underly further progress in many applications in science and technology, including e.g. learning and control of complex dynamical systems, and early prediction of catastrophic events, characterization of electronic devices. In this work a new algorithm for inferencing complex dynamical models and noise strength from multidimensional time series is described.

It is shown that using this algorithm we can infer a complex dynamical systems with many degrees of freedom in the presence of multidimensional noise including chaotic noisy systems and nonlinear systems with time delay. The role of noise in the inference of complex dynamics is analyzed. Specifically, we discuss the situations in which fluctuations provide the only source of information about system dynamics and discuss the optimization of the inference scheme in the presence of large fluctuations. We show that, quite counter-intuitively, the quality of inference can be improved by neglecting those data points in experimental time series that correspond to the rising part of large deviations.

The algorithm is applied to infer models of the human cardiovascular system. In particular, we infer parameters of nonlinear respiratory oscillations and parameters of Mackey-Glass model of respiratory control system [1]. An application of the algorithm to characterization of vertical cavity surface emitting semiconductor lasers is discussed. We discuss an application of our algorithm for inference of parameters of nonlinear stochastic model that describe polarization fluctuations of these lasers in terms of the motion of three Stokes components on the Poincare sphere [2].

[1] M.C. Mackey, L. Glass, Science, 197, 287/288, (1977)
[2] M. B. Willemsen, M. P. van Exter and J. P. Woerdman, Phys. Rev. Lett., 84, 4337 (2000); H. F. Hofmann and O. Hess, Quantum Semiclass. Opt. 10, 87 (1998)

A new generation of microscopic ratchet systems is currently being developed for controlling the motion of electrons and fluxons, as well as for particle separation and electrophoresis. Virtually all of these use static spatially asymmetric potential energies to control transport properties. Here we propose completely new types of ratchet-like systems that do not require fixed spatially asymmetric potentials in the samples. As specific examples of this novel general class of ratchets, we propose devices that control the motion of flux quanta in superconductors and could address a central problem in many superconducting devices; namely, the removal of trapped magnetic flux that produces noise. This controllable vortex motion - useful to design pumps, diodes and lenses of quantized magnetic flux -- can be achieved for instance by means of: (i) asymmetrically corrugated fluxon channels, where rectification takes place through the collective interaction of the fluxons with the asymmetric boundaries; (ii) asymmetric distributions of otherwise symmetric traps, whose global effect on the fluxon assembly can be explained in terms of an asymmetric mean-field potential; (iii) homogeneous distributions (lattices) of asymmetric traps, where movable fluxons are sensitive to a local asymmetry (rectification as a microscopic mechanism). Changes in the applied driving force produces an exceptionally rich variety of distinct dynamic phases, rectification regimes, vortex trajectories and vortex lattice order.

The response of a nonlinear bistable stochastic system driven by a polychromatic time periodic signal is addressed. Using precise numerical solutions of the Langevin and/or Fokker-Planck equations, we study the behavior of the average, the coherent and incoherent parts of the correlation function, the signal-to-noise ratio and the gain. Our results explicitely demonstrate the limitations of linear response theory, not only for large amplitude drivings, but also, for subthreshold low frequency inputs. In particular, gains larger than unity are obtained for small amplitudes driving terms.

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 non-trivial 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.

Random fluctuations in intracellular processes are inevitable since chemical reactions are probabilistic and many genes, mRNAs and proteins are present in low numbers per cell. Numerous simple reaction networks also operate close to nonequilibrium phase transitions with hypersensitive responses to external changes and large spontaneous fluctuations. By connecting Metabolic Control Analysis to van Kampen's omega-expansion, I show how most previous descriptions of such systems approximately map onto a simple disordered birth-and-death process. I also show how some chemical systems may have evolved noise-suppression-by-noise by exploiting fluctuations to enhance the sensitivity in negative feedback loops.

We show that a nonlinear lattice driven by both ends can be an interesting model system to suggest a method to control the thermal flow in a solid state device and to improve our understanding of the statistical properties of turbulence.

Both of these applications rely on nonlinearity that couples modes, shifts their vibrational frequencies, and, in a lattice, allows for the existence of discrete breathers. An appropriate choice of parameters allows a precise control of the heat flow in the lattice, and can even turn the system into a ``thermal rectifier'' which does not have the same thermal conductivity in the two directions. By introducing a special form of dissipation that mimics viscosity, we can also show that such a nonlinear lattice exhibits properties which are strikingly similar to various experimental results obtained in a turbulent von Karman flow.

By providing a model sufficiently simple to be studied in details the application to turbulence may suggest some clues to understand general features of turbulence, while the possibility to control and direct a thermal flow, although it is still very far from a practical device, could open new possibilities for applications.

We consider effects of time delay on two basic models of nonlinear oscillations:noise-induced oscillations in a bistable system, and noisy self-sustained oscillations. For a bistable system we derive the master equation and find its exact analytic solution enabling to give analytic formulae for the correlation function and the power spectrum. The statistics of interswitch intervals is alsodiscussed. In the case of noisy periodic oscillations with delay we use Gaussian approximation to calculate analytically the phase diffusion. We show that the diffusion constant oscillates with the delay time. Numerical simulations with Lorenz system confirm non-monotonous dependence of the diffusion on the delay, and reveal a possibility to drastically reduce diffusion using the delayed feedback.

Thermostated tethered harmonic lattices provide good illustrations of the phase-space dimensionality loss which occurs in the strange-attractor distributions characterizing stationary nonequilibrium flows. We use time-reversible nonequilibrium molecular dynamics, with two Nose-Hoover thermostats, one hot and one cold, to study a family of square heat-conducting systems. We find a phase-space dimensionality loss which can exceed the dimensionality associated with the two driving Nose-Hoover thermostats by as much as a factor of four. We also estimate the dimensionality loss in the purely Hamiltonian part of phase space. By measuring the projection of the total dimensionality loss there we show that nearly all of the loss occurs in the Hamiltonian part. Thus this loss, which characterizes the extreme rarity of nonequilibrium states, persists in the large-system thermodynamic limit.

A system with on-off intermittency is characterized by power-law amplitude distribution of dynamic variable. When the exponent of this power law is close to unity, the system can become sensitive to ultrasmall slow time-dependent signal, responding by large bursts correlated with the signal. In a Kramers oscillator with strong multiplicative noise the signal gain factor can reach many orders of magnitude and is robust to additive noise. The another simple system with on-off intermittency, the couple of maps or oscillators synchronized by common noise, is hypersensitive to small differential signal, but the sensitivity is destroyed even by small intrinsic noise.

The effective diffusion coefficient for the overdamped Brownian motion in a tilted periodic potential is calculated in closed analytical form. Universality classes and scaling properties for weak thermal noise are identified near the threshold tilt where deterministic running solutions set in. In this regime the diffusion may be greatly enhanced, as compared to free thermal diffusion with, for a realistic experimental setup, an enhancement of up to 14 orders of magnitude.

Liquid crystals are strongly anisotropic materials, characterized by high sensitivity on electromagnetic fields and by large birefringence. Light can change their orientation, leading to nonlinear optical effects. If a small amount of dyes is dissolved in the liquid crystals, light-induced reorientation can appear at much lower light intensity with respect to the pure case, so that dye-doped liquid crystals show an extremely large optical sensitivity, without direct transfer of angular momentum from radiation. Experimental results, showing the salient features of the process, will be presented. Then, the origin of the large optical response will be discussed in relation with the photoisomerization of the dyes. In particular, white light experiments (1) will be in put in relation with a recently proposed interpretation (2), showing that the dye molecules act as light-driven molecular motors, transferring angular momentum to the liquid crystals through an orientational Brownian ratchet mechanism.

- 1 A.Petrossian, S. Residori, "Light Driven Motion of the Nematic Director in Azo-Dye Doped Liquid Crystals", submitted to Opt. Commun. (2002).
- 2 P. Palffy-Muhoray, T. Kosa and E. Weinan, Appl. Phys. A 75, 293 (2002).

There exists no closed equation for mean first passage times of time-inhomogeneous Markov processes, as for example for a process that is described by a Langevin equation with an explicitly time dependent drift. This is in contrast to time homogeneous Markov processes for which the mean first passage time is conveniently determined by an inhomogeneous backward equation first formulated by Pontryagin, Andronov and Vit. The lack of such an equation is the reason why relatively little is known about first passage times in time dependent systems. However, processes showing a slow variation of parameters can be treated in the semiadiabatic approximation on the slow time scale. The passage of particular boundaries then also is amenable within this framework. In this talk we will discuss the waiting time distribution in one of the two wells of a bistable potential that periodically moves in time for driving periods that are long compared to intrawell relaxation times. The resulting mean first passage time and splitting probability which are functions of the starting time and position are compared to results from numerical simulations of the corresponding Langevin equations.