**Current Projects:**

(in collaboration with M. Bandi, OIST, Japan)**Fluctuations in renewable energy harvesting:**- One of the main challenges of renewable energy production is the presence of fluctuations in multiple time scales. The focus of our research is the analysis of fluctuations in wind and solar energy productions at different spatial and temporal scales. The analysis helps us to better understand the origin of the fluctuations and thereby suggest methods for improved forecast and modeling of the energy output. The sources of the energy affect many other natural systems, and the knowledge obtained from the analysis can be used to improve atmospheric and ecosystem dynamics.

(in collaboration with K. Rasmussen, LANL, USA)**Breathers in discrete lattices:**- Driven nonlinear oscillators can oscillate in a different frequency than the frequency of the driving force. Under certain conditions, the oscillator exhibits multi-stability; namely, under the same conditions, the steady state is determined by the initial condition (or the noise). Coupled oscillators are expected to be synchronized, but due to the nonlinear dynamics, we found the co-existence of coupled oscillators in different frequencies. The stability of these breathers and their role in determining the steady state of coupled oscillators are investigated. The research is mostly theoretical but is also relevant to the dynamics of DNA and similar systems were designed in laboratories in order to demonstrate the effects.

(in collaboration with O. Dahan)**Water and solute dynamics in heterogeneous porous media:**- The dynamics of water and solutes is strongly affected by the heterogeneity of the soil. However, there is not enough knowledge about the heterogeneity in the field scale. Using data collected by Prof. Dahan's group using a unique technique, we develop methods to characterize the field scale heterogeneity. The derived information allows the probabilistic prediction of the flux to the underground water and thereby the establishment of methods to reduce the risk of underground water contamination.

**Effects of quenched disorder on pattern-forming systems:**In pattern-forming systems, the internal dynamics/interactions select specific spatial scales. The presence of quenched disorder introduces different spatial scales that may be independent of the interaction mechanisms. The interplay between the effects of the disorder and the effects of the nonlinear dynamics results in different dynamics. The effects are important for the understanding of vegetation dynamics and any other spatially extended pattern-forming system.

**Single molecule spectroscopy:**- In previous research we introduced the generating function technique for calculation of single molecule photon emission statistics in systems governed by multi-level quantum dynamics. This opened up the possibility to study phenomena that are outside the realm of purely stochastic and mixed quantum-stochastic models. In particular, this methodology allows for calculation of photon statistics for photons emitted from a particular transition and which are subject to quantum coherence. Several model calculations illustrate the generality of the technique and highlight quantitative and qualitative differences between quantum mechanical models and related stochastic approximations when they arise. Calculations suggest that studying photon statistics as a function of photon frequency has the potential to reveal more about system dynamics than the usual broadband detection schemes. In order to better connect the theory to measurements, we derived the moment generating function for photon emissions from a single molecule driven by laser excitation where the frequencies of the fluoresced photons are explicitly considered. Calculations were performed for the case of a two level dye molecule, showing that measured photon statistics will display a strong and non-intuitive dependence on detector bandwidth. Moreover, it was demonstrated that the anti-bunching phenomenon, associated with negative values of Mandel’s Q-parameter, results from correlations between photons with well separated frequencies. This study is in the process of being extended to multi-level systems with the promise that this new kind of photon statistics will reveal more information about the studied molecules and their interaction with light.

**Previous Projects:**

*Early indicators for catastrophic regime shifts:*(In collaboration with Hezi Yizhak and Ehud Meron)

The responses of ecosystems to small environmental changes are generally divided into two categories, smooth and reversible, or abrupt and irreversible. Various examples of the latter response have been reported, including sudden loss of transparency and vegetation in shallow lakes subject to human-induced eutrophication, coral reefs overgrown by fleshy macroalgae, and desertification induced by climate changes or human disturbances. These catastrophic regime shifts, as they are called, are detrimental to the ecosystem in that they involve loss of bioproductivity and biodiversity, which, in turn, affect ecosystem function and stability. The high concern about potential ecosystem degradation in a time of global climate change has motivated vigorous research efforts aimed at devising early indicators of impending degradation processes. In this study we focus on identifying early indicators for catastrophic shifts in spatially extended systems. The spatial aspects are particularly relevant to desertification where transitions to bare soil usually take place from spotted vegetation. In this project we combine analytical tools, computer simulations and data analysis.*The distribution of winds and ocean currents and their impact on energy use and climate modeling**:*

In light of the ongoing increase in atmospheric greenhouse gases concentration, there is an urgent need to find alternative sources for clean (“green”) energy. In many countries around the world winds and ocean currents are important sources of green energy. In addition, winds, currents and the interplay between them are important factors when dealing with simulations of state of the art coupled ocean atmosphere general circulation models. In spite of the above importance of winds and currents, the statistical properties of both winds and currents are still not properly quantified, the relation between the probability distributions of surface winds and surface ocean currents is still unknown, and the origin of the probability distributions of winds and currents is still elusive. We study the statistical properties of winds and ocean currents in order to find the relation between the probability distributions of surface winds and currents. Theoretical predictions are tested using oceanic and atmospheric general circulation model simulations. Our study will help to identify locations with potentially high sources of current and wind energy. In addition, our theory will help to improve the performance of general circulation models and thus will help to better predict the future climate.

__Dynamics of disordered vortex matter:__- We considered the dynamics of homogeneous moving vortex matter beyond the linear response. Our framework is the time dependent Ginzburg - Landau equation within the lowest Landau level approximation. Both disorder and thermal fluctuations are included using the Martin-Siggia-Rose formalism. We determined the critical current as a function of magnetic field and temperature. The critical current defines a surface in the current - magnetic field - temperature space which separates between the dissipative moving vortex matter regime (flux flow) and an amorphous vortex ”glass”. Both the thermal depinning and the depinning by a driving force were taken into account. The static irreversibility line, determined by the vanishing critical current, was compared to experiments in layered HTSC and is consistent with the one obtained using the replica approach. The non-Ohmic I-V curve (in the depinned phase) was obtained and compared with an experiment in layered superconductors and thin films.
__Weak ergodicity breaking:__- Single molecule tracking became an essential tool in almost every field of science. Many experiments tracking single molecules have reported anomalous diffusion. In order to extract new and useful information from those experiments, it is important to study the properties of anomalous diffusion models and to find characteristics which can distinguish between the different models. Continuous time random walk (CTRW) models are widely used to model diffusion in condensed matter. There are two classes of such models, distinguished by the convergence or divergence of the mean waiting time. Systems with a finite average sojourn time are ergodic and thus Boltzmann–Gibbs statistics can be applied. We investigated the statistical properties of CTRW models with an infinite average sojourn time; in particular, the occupation time probability density function is obtained. It is shown that in the non-ergodic phase the distribution of the occupation time of the particle on a given lattice point exhibits a bimodal U or trimodal W shape, related to the arcsine law. The key points are as follows: (a) In a CTRW with a finite or infinite mean waiting time, the distribution of the number of visits on a lattice point is determined by the probability that a member of an ensemble of particles in equilibrium occupies the lattice point. (b) The asymmetry parameter of the probability distribution function of occupation times is related to the Boltzmann probability and to the partition function. (c) The ensemble average is given by Boltzmann–Gibbs statistics for either finite or infinite mean sojourn time, when detailed balance conditions hold. (d) A non-ergodic generalization of the Boltzmann–Gibbs statistical mechanics for systems with an infinite mean sojourn time was found. We have also studied the concept of weak ergodicity breaking in the context of deterministic dynamics. We showed that weak ergodicity breaking describes a system whose dynamics is governed by a nonlinear map which generates sub-diffusion deterministically.
Inspired by problems in biochemical kinetics, we studied statistical properties of an over-damped Langevin process whose friction coefficient depends on the state of a similar, unobserved process. Integrating out the latter, we derived the long time behavior of the mean square displacement. Anomalous diffusion is found. Since the diffusion exponent can not be predicted using a simple scaling argument, anomalous scaling appears as well. We also found that the coupling can lead to ergodic or non-ergodic behavior of the studied process. We compared our theoretical predictions with numerical simulations and found an excellent agreement. The findings caution against treating biochemical systems coupled with unobserved dynamical degrees of freedom by means of standard, diffusive descriptions.__Anomalous diffusion in coupled over-damped Langevin processes:__Biochemical processes typically involve huge numbers of individual reversible steps, each with its own dynamical rate constants. For example, kinetic proofreading processes rely upon numerous sequential reactions in order to guarantee the precise construction of specific macromolecules. We studied the transient properties of such systems and fully characterized their completion time distributions. We found that as the system size grows, the completion time behavior simplifies: it becomes either deterministic or exponentially distributed, with a very narrow transition between the two regimes. In both regimes, the dynamical complexity of the full system is trivial compared to its apparent structural complexity. In particular, these findings suggest not only that one may not be able to understand individual elementary reactions from macroscopic observations, but also that such understanding may be unnecessary. We have also studied the dynamical properties of discrete stochastic two branch kinetic proofreading schemes. Using the Laplace transform of the corresponding chemical master equation, we obtained an analytical solution for the completion time distribution. We also showed that, for a wide range of parameters, a process distinguishing between two different products can be reduced to a much simpler three point process. Our results allow for the systematic study of the interplay between specificity and completion times as well as testing the validity of the kinetic proofreading model in biological systems.__Temporal characteristics of kinetic proofreading:__We have studied new possible vortices in p-wave superconductors. Due to the fact that the order parameter is a vector and not a scalar, we showed that it is possible to have a vortex in which the order parameter does not vanish at the center (soft core vortex). We have studied the shape of the new kind of vortices and showed that the kappa parameter (the ratio between the penetration depth and the coherence length) determines the structure of the vortex. We also showed that alternating dia-para magnetic domains appear in a p-wave superconductor which is subject to a magnetic field in opposite directions at its ends. We have studied analytically and numerically the relaxation of a quenched normal spot in a type-II superconductor. Various instabilities accompanying recovery of superconductivity were found. It was shown that the relaxation of the normal spot starts with the appearance of a microscopic instability triggering the creation of vortex clusters.__Topological defects in unconventional superconductors:__