Golan Bel

Weak Ergodicity Breaking in the Continuous-Time Random Walk

The continuous time random walk (CTRW) model exhibits a non-ergodic phase when the average waiting time diverges. Using an analytical approach for the non-biased and the uniformly biased CTRWs, and numerical simulations for the CTRW in a potential field, we obtain the non-ergodic properties of the random walk which show strong deviations from Boltzmann--Gibbs theory. We derive the distribution function of occupation times in a bounded region of space which, in the ergodic phase recovers the Boltzmann--Gibbs theory, while in the non-ergodic phase yields a generalized non-ergodic statistical law.

Gradual regime shifts in spatially extended ecosystems

Ecosystem regime shifts are regarded as abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or by global disturbances. Spatially extended ecosystems, however, can also respond to local disturbances by the formation of small domains of the alternative state. Such a response can lead to gradual regime shifts involving front propagation and the coalescence of alternative-state domains. When one of the states is spatially patterned, a multitude of intermediate stable states appears, giving rise to step-like gradual shifts with extended pauses at these states. Using a minimal model, we study gradual state transitions and show that they precede abrupt transitions. We propose indicators to probe gradual regime shifts, and suggest that a combination of abrupt-shift indicators and gradual-shift indicators might be needed to unambiguously identify regime shifts. Our results are particularly relevant to desertification in drylands where transitions to bare soil take place from spotted vegetation, and the degradation process appears to involve step-like events of local vegetation mortality caused by repeated droughts.

The simplicity of completion time distributions for common complex biochemical 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. In this work, we study the transient properties of such systems and fully characterize their first passage (completion) time distributions. In particular, we provide explicit expressions for the mean and the variance of the completion time for a kinetic proofreading process and computational analyses for more complicated biochemical systems. We find that, for a wide range of parameters, 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. Similar simplicity is likely to arise in the dynamics of many complex multistep biochemical processes. 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 an understanding may be unnecessary.

Regime shifts in models of dryland vegetation.

Drylands are pattern-forming systems showing self-organized vegetation patchiness, multiplicity of stable states and fronts separating domains of alternative stable states. Pattern dynamics, induced by droughts or disturbances, can result in desertification shifts from patterned vegetation to bare soil. Pattern formation theory suggests various scenarios for such dynamics: an abrupt global shift involving a fast collapse to bare soil, a gradual global shift involving the expansion and coalescence of bare-soil domains and an incipient shift to a hybrid state consisting of stationary bare-soil domains in an otherwise periodic pattern. Using models of dryland vegetation, we address the question of which of these scenarios can be realized. We found that the models can be split into two groups: models that exhibit multiplicity of periodic-pattern and bare-soil states, and models that exhibit, in addition, multiplicity of hybrid states. Furthermore, in all models, we could not identify parameter regimes in which bare-soil domains expand into vegetated domains. The significance of these findings is that, while models belonging to the first group can only exhibit abrupt shifts, models belonging to the second group can also exhibit gradual and incipient shifts. A discussion of open problems concludes the paper.

Gradual regime shifts in fairy circles

Significance Combining model and empirical data analyses, we show that transitions between alternative stable states (regime shifts) in spatially extended ecosystems are not necessarily abrupt; cascades of local shifts between a multitude of stable states, composed of patterned and uniform domains, can result in global regime shifts that proceed gradually. In the Namibian fairy circle ecosystem (barren circular gaps in grasslands), such local shifts appear as fairy circle birth or death processes. This mechanism of regime shifts has never been demonstrated in a specific natural context. In addition, the results reported here further support the view of fairy circles as a self-organization phenomenon by providing a new type of evidence based on dynamical processes. Large responses of ecosystems to small changes in the conditions—regime shifts—are of great interest and importance. In spatially extended ecosystems, these shifts may be local or global. Using empirical data and mathematical modeling, we investigated the dynamics of the Namibian fairy circle ecosystem as a case study of regime shifts in a pattern-forming ecosystem. Our results provide new support, based on the dynamics of the ecosystem, for the view of fairy circles as a self-organization phenomenon driven by water–vegetation interactions. The study further suggests that fairy circle birth and death processes correspond to spatially confined transitions between alternative stable states. Cascades of such transitions, possible in various pattern-forming systems, result in gradual rather than abrupt regime shifts.

Single Molecule Photon Counting Statistics for Quantum Mechanical Chromophore Dynamics

We extend the generating function technique for calculation of single molecule photon emission statistics (Zheng, Y.; Brown, F. L. H. Phys. Rev. Lett. 2003, 90, 238305) to systems governed by multi-level quantum dynamics. This opens up the possibility to study phenomena that are outside the realm of purely stochastic and mixed quantum-stochastic models. In particular, the present methodology allows for calculation of photon statistics that are spectrally resolved and 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.

Effects of heterogeneous soil‐water diffusivity on vegetation pattern formation

Many mathematical models have been proposed to explain the emergence of vegetation patterns in arid and semiarid environments, but only a few of them take into account the heterogeneity in the system properties. Here we present a rigorous study of the effects of heterogeneous soil-water diffusivity on vegetation patterns, using two mathematical models. The two models differ in the pattern-forming feedback that they capture; one model captures the infiltration contrast between vegetated and bare-soil domains, whereas the other model captures the increased growth rate of denser vegetation due to an enhanced ability to extract water from the soil. In both models, the most significant effect of the heterogeneity on the soil-water diffusivity is the increased durability of patterned vegetation to a reduced precipitation rate. An additional effect is that the heterogeneity makes the desertification process, namely, the transition from a spotted vegetation pattern to a bare-soil state, more gradual than in the homogeneous system. Our findings suggest that the heterogeneity cannot be neglected in the study of critical transitions in heterogeneous ecosystems and, particularly, in the study of the desertification process due to climate changes or anthropogenic disturbances.

Random walk to a nonergodic equilibrium concept

Random walk models, such as the trap model, continuous time random walks, and comb models, exhibit weak ergodicity breaking, when the average waiting time is infinite. The open question is, what statistical mechanical theory replaces the canonical Boltzmann-Gibbs theory for such systems? In this paper a nonergodic equilibrium concept is investigated, for a continuous time random walk model in a potential field. In particular we show that in the nonergodic phase the distribution of the occupation time of the particle in a finite region of space approaches U- or W-shaped distributions related to the arcsine law. We show that when conditions of detailed balance are applied, these distributions depend on the partition function of the problem, thus establishing a relation between the nonergodic dynamics and canonical statistical mechanics. In the ergodic phase the distribution function of the occupation times approaches a delta function centered on the value predicted based on standard Boltzmann-Gibbs statistics. The relation of our work to single-molecule experiments is briefly discussed.

Theory for wavelength-resolved photon emission statistics in single-molecule fluorescence spectroscopy.

We derive the moment generating function for photon emissions from a single molecule driven by laser excitation. The frequencies of the fluoresced photons are explicitly considered. Calculations are performed for the case of a two-level dye molecule, showing that measured photon statistics will display a strong and nonintuitive dependence on detector bandwidth. Moreover, it is demonstrated that the antibunching phenomenon, associated with negative values of Mandels Q parameter, results from correlations between photons with well separated frequencies.

The effects of psammophilous plants on sand dune dynamics

Psammophilous plants are special plants that flourish in sand moving environments. There are two main mechanisms by which the wind affects these plants: (i) sand drift exposes roots and covers branches--the exposed roots turn into new plants and the covered branches turn into new roots; both mechanisms result in an enhanced growth rate of the psammophilous plant cover of the dunes; (ii) strong winds, often associated with sand movement, tear branches and seed them in nearby locations, resulting in new plants and an enhanced growth rate of the psammophilous plant cover of the dunes. Despite their important role in dune dynamics, to our knowledge, psammophilous plants have never been incorporated into mathematical models of sand dunes. Here, we attempt to model the effects of these plants on sand dune dynamics. We construct a set of three ordinary differential equations for the fractions of surface cover of regular vegetation, biogenic soil crust and psammophilous plants. The latter reach their optimal growth under (i) specific sand drift or (ii) specific wind power. We show that psammophilous plants enrich the sand dune dynamics. Depending on the climatological conditions, it is possible to obtain one, two, or three steady dune states. The activity of the dunes can be associated with the surface cover--bare dunes are active, and dunes with significant cover of vegetation, biogenic soil crust, or psammophilous plants are fixed. Our model shows that under suitable precipitation rates and wind power, the dynamics of the different cover types is in accordance with the common view that dunes are initially stabilized by psammophilous plants that reduce sand activity, thus enhancing the growth of regular vegetation that eventually dominates the cover of the dunes and determines their activity.