# Complex systems

### Upcoming Lectures

Theoretical models are central to how we think of ecosystems, and yet in many aspects remain poorly understood. We identify a small number of parameters that are sufficient to predict the large-scale properties of a wide variety ecological-community models. These parameters thus play a role similar to temperature and pressure in thermodynamics. We go on to study the generic model that emerges, and describe its phases, including a critical phase where all states are marginally stable.

Earths jet streams, Jupiters Great Red Spot and its zonal winds are all examples of persistent

large scale ows, whose dynamics is to a good approximation two-dimensional. These ows are

also highly turbulent, and the interaction between the turbulence and these coherent structures

remains poorly understood. Apart from its geophysical relevance, 2D turbulence is a rich and

beautiful fundamental system|where turbulence takes a counter-intuitive role. Indeed, in 2D,

energy is transferred to progressively larger scales, which can terminate in the self organization of

the turbulence into a large scale coherent structure, a so called condensate, on top of small scale

I will describe a recent theoretical framework in which the prole of this coherent mean

can be obtained, along with the mean momentum ux of the uctuations. I will explain how

and when the relation between the two can be deduced from dimensional analysis and symmetry

considerations, and how it can be derived. Finally, I will show that, to leading order, the velocity

two-point correlation function solves a scale invariant advection equation. The solution determines

the average energy of the uctuations, but does not contribute at this order to the momentum

due to parity + time reversal symmetry. Using analytic expressions for the solutions, matched to

data from extensive numerical simulations, it is then possible to determine the main characteristics

of the average energy. This is the rst-ever self-consistent theory of turbulence-ow interaction.

The RT instability, in which a dense fluid invades a less dense fluid under acceleration such as gravity, is pervasive throughout nature. General RT instabilities lie at the heart of myriad applications and diverse phenomena. For example, RT instabilities occur during liquid impact and atomization, the explosion of supernovae, inertial confinement fusion, and in granular media. More prosaically, the RT instability affects resolution control of ink-jet printers and appears when a bottle of vinegar-and-oil salad dressing is turned upside down.

Experimental work on the RT instability in fluids has, until now, been plagued by jitter during acceleration of the tank that contains the fluids. Here I will discuss our development of an alternate method that obviates this problem, viz., magnetic levitation of the dense fluid above the less dense fluid. Using this approach, we have been able to obtain a dispersion relationship for the instability for not only a two fluid / one-interface system, but multiple layers as well. In the latter case, the multiple interfaces are found to couple and modify the dispersion relationship when the intervening fluid layer is sufficiently thin. I will compare our experimental results with long-standing, but until now never tested, theoretical predictions.

### Previous Lectures

Different laser cooling mechanisms have been rising important questions from thermodynamics and statistical physics point of view ever since the beginning of this research field (over 40 years ago). Sisyphus cooling is especially well known in this respect providing experimentally accessible regime to study deviations from thermal equilibrium. Here we discuss the Doppler cooling mechanism (the most basic laser cooling mechanism which works even for a simple two-level atom) and show that it also supports deviations from Gaussian statistics for a certain parameters range. We study experimentally the Doppler cooling in Lithium and point out an interesting deviation from the simple, two-level theory, namely cooling at resonance. We develop a realistic theory which accounts for all energy levels of lithium atoms and all laser fields and show its successes and failures.

In the usual setting non-demographic noise, emanating, e.g., from environmental variability, is manifested by time-varying reaction rates. In this work we investigate a different type of non-demographic noise in the form of uncertainty in the reaction step-size, and demonstrate that this type of noise can have a dramatic effect on the stability of self-regulating populations. By employing the usual reaction scheme mA->kA, but allowing, e.g., the product number k to be a-priori unknown and sampled from a given distribution, we show that such non-demographic noise can greatly increase the population's stability compared to the case of fixed k. Our analysis is tested against numerical simulations, and by using empirical data of different species, we argue that certain distributions may be more evolutionary beneficial than others.

A number of complex physical systems will be presented in a unified way and the main idea of the SCE of mimicking the complex system by a simple but arbitrary simple system will be outlined. Two very simple problems will be presented as models for the application of the SCE, showing its obvious superiority over conventional treatments. Results for some of the complex systems including KPZ and noise driven Navier-Stokes will be discussed.

Inter-particle forces in amorphous solids such as glasses, colloids and granular material can be used to study phenomena such as jamming and force-chains. So far, no generally applicable methods exist for measuring the forces between each and every particle in the system. Our recently developed methods aim to x this unfortunate situation in both a-thermal and thermal systems, and produce some interesting insights as to the nature of these forces. In the a-thermal case all that is required for nding the force-law are the xed particle positions and the pressure. The method is shown to accurately recover the force-law in simulation. In the thermal case, we are developing a method to extract an eective potential, using the mean positions. This will allow for analysis of thermal systems using tools hitherto reserved for a-thermal ones, and thereby prediction of thermodynamic properties, study of stability, etc. Quite remarkably we observe the emergence of eective many-body interactions, even when the bare interactions are purely 2-body. This resolves the puzzle posed by recent studies that showed a quantitative match between 2D/3D measurements and the innite dimension mean-eld prediction.

As temperature is lowered, motion becomes more sluggish. Below the glass transition temperature, the dynamics of super-cooled liquids becomes so slow that the system falls out of equilibrium. One hypothesis for this dynamic arrest is that it is due to a thermodynamic phase transition with a diverging length scale. However, there is scant evidence of such a length scale appearing in the structure. Motivated by this, we study another amorphous system that undergoes a phase transition: jammed soft repulsive spheres at zero temperature. We have discovered a subtle correlation length, associated with the local coordination of particles, that is not seen in the two-point correlation function, g(r). We argue that this scale plays an important role in determining the local rigidity of the system, and diverges with an exponent 2/(d+1) as the jamming transition is approached.

Positive feedback in biochemical networks can lead to a bifurcation in state space. Universality implies that if molecules are well mixed, this bifurcation should exhibit the critical scaling behavior of the Ising universality class in the mean-field limit. Making this statement quantitative requires the appropriate mapping between the biochemical parameters and the Ising parameters. Here we derive this mapping rigorously and uniquely for a broad class of stochastic birth-death models with feedback, and show that the expected static and dynamic critical exponents emerge. The generality of the mapping allows us to extract the order parameter, effective temperature, magnetic field, and heat capacity from T cell flow cytometry data without needing to know the underlying molecular details. We find that T cells obey critical scaling relations and exhibit critical slowing down, and that the heat capacity determines molecule number from fluorescence data. We demonstrate that critical scaling holds even as our system is driven out of its steady state, via the Kibble-Zurek mechanism for driven critical systems. Our approach places a ubiquitous biological mechanism into a known class of physical systems and is immediately applicable to other biological data.

In this talk I will present an experimental study of the anomalous dynamics of ultra-cold Rb atoms propagating in a 1D, dissipative, Sisyphus-type optical lattice. We find that the width of the cloud exhibits a power-law time dependence with an exponent that depends on the lattice depth. Moreover, the distribution exhibits fractional self-similarity with the same characteristic exponent. The self-similar shape of the distribution is found to be well fitted by a Lévy distribution. I will further present a measurement of the phase-space density distribution (PSDD) of the cloud of atoms. The PSDD is imaged using a direct tomographic method comprised of velocity selection and spatial imaging. We show that the position-velocity correlation function, obtained from the PSDD, decays asymptotically as a function of time with a power-law that we relate to a simple scaling theory involving the power-law asymptotic dynamics of the position and velocity. The generality of this scaling theory is confirmed using Monte-Carlo simulations of two distinct models of anomalous diffusion dynamics.

Phase transitions are of unfading interest. While classical systems in equilibrium present no phase transitions in 1 dimension, they can be manifested in systems driven out of equilibrium. In this talk we will explore the current fluctuations in boundary driven systems. Current fluctuations are explored by the probability to observe an atypical current over a long period of time. We will show a few examples of phase transitions and classify them. For a special kind of phase transitions, we will show a mapping to a single particle evolving under classical Lagrangian mechanics. This mapping provides us with a simple picture of where one could expect such transitions.

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What is the fate of a forager that depletes its environment as it wanders? We investigate this question within the "starving" random walk model, in which the forager starves when it travels S steps without eating. The forager consumes food whenever it is found and becomes fully sated. However, when the forager lands on an empty site, it moves one time unit closer to starvation. We determine the forager lifetime, analytically in one dimension and numerically in higher dimensions. In two dimensions, long-lived walks explore a highly ramified region so as to remain close to food.

We also investigate the role of greed, in which the forager preferentially moves towards food when faced with a choice of hopping to food or to an empty site. Paradoxically, the forager lifetime can have a non-monotonic dependence on greed, with a different sense to the non-monotonicity in one and in two dimensions.

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Small-amplitude fast vibrations and small surface micropatterns affect properties of various systems involving wetting, such as superhydrophobic surfaces and membranes. The mathematical method of averaging the effect of small fast vibrations is known as the method of separation of motions. The vibrations are substituted by effective force or energy terms, leading to vibration-induced phase control. The best known example of that is the stabilizationb of an inverted pendulum on a vibrating foundation (the Kapitza pendulum); however, the method can be applied to a number of various situations including wetting. A similar averaging method can be applied to surface micropatterns leading to surface texture-induced phase control. We argue that the method provides a framework that allows studying such effects typical to biomimetic surfaces, such as superhydrophobicity, membrane penetration and others. Patterns and vibration can effectively jam holes and pores in vessels with liquid, separate multi-phase flow, change membrane properties, result in propulsion, and lead to many other multiscale, non-linear effects. These effects can be used to develop novel materials.

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In a liquid all the particles are mobile, while in a glass only some of them are mobile at any given time. Although overall the structure is amorphous in both cases, the difference is that in glasses there are local structures that inhibit the movement of particles inside them. We investigate these structures by considering the minimum number of particles that need to move before a specific particle can move. By mapping the dynamics of the particles to diffusion of mobile vacancies, we find a general algebraic relation between the mean size of the structures and the mean persistence time, which is the time until a particle moves for the first time. The exponent relating these two quantities depends on the system's properties.

We investigated this relation analytically and numerically in several kinetically-constrained models: the Fredrickson-Andersen, Kob-Andersen and Spiral models. These models are either lattice gas models or Ising-like models, in which a particle can move or a spin can flip only if the local environment satisfies some model-dependent rule. Due the discrete nature of these models and relative simplicity, we were able to analytically find the relation between the structure and the dynamics and found an excellent agreement between our analytical results, our numerical simulations, and the heuristic arguments presented above. In these simple models, the minimum number of particles that need to move before a specific particle can move is easily found by using a culling algorithm, also called bootstrap or threshold percolation.

The recently formulated macroscopic fluctuation theory is a successful description of out of equilibrium diffusive systems. I will focus on current fluctuations of boundary driven systems within the macroscopic theory description, and discuss the relevance of the additivity principle to derive the large deviation function associated with the current fluctuations. Three results will be shown

1) Current fluctuations in boundary driven systems are universal

2) A criterion for the validity of the additivity principle and application to dynamical phase transitions.

3) Relevance of the macroscopic fluctuation theory to transport of disordered quantum systems.

Collective movement patterns appear at all scales from microorganisms to invertebrate and vertebrate animals. In certain cases the individual entities communicate indirectly modifying the environment in which they roam by leaving a trace of their action or passage. This form of interaction has a long tradition in the ecological literature and is called stigmergy. In the context of territorial mammals modification of the environment occurs because of scent deposition and is being exploited to maintain exclusive ownership of certain region of space. By introducing the so-called territorial random walkers, it is possible to study the formation of territorial patterns by modelling the movement and interaction of scent-depositing animals. Territorial random walkers consist of agents that move at random and deposit scent, that is mark the locations they visit using temporal flags that decay over a finite amount of time, and retreat upon encountering a foreign scent. Depending solely on the ratio between the time for which the mark is active and the time it takes for the walker to cover its own territory, the system displays different patterns. Short lived marks produce rapidly morphing, fast traveling territories. A broad range of shapes and territory sizes are observed, and these territories may display ergodic trajectories. Marks that remain active for long times yield slowly moving territories that resemble glassy systems. In such state territories are effectively confined in space and have a more homogeneous shape distribution. I will show how these different regimes emerge based on the population density and the length of time for which marks remain active. I will also present an adiabatic mean-field approximation that allows to describe at short times the dynamics of the walker and that of the territory boundaries through a Fokker-Planck formalism.

Regime shifts in ecosystems are typically understood to be abrupt global transitions from one stable state to an alternative stable state, induced by slow environmental changes or global disturbances. However, spatially extended ecosystems often exhibit patterned states, which allows for more complex dynamics to take place. A bistability of a patterned state and a uniform state can lead to a multitude of stable hybrid states, with small domains of one state embedded in the other state. The response of the system to local disturbances or change in global parameters in these systems can lead to gradual regime shifts, involving the expansion of alternative-state domains by front propagation, rather than a global collapse. Moreover, a regime of periodic perturbations can give rise to step-like gradual shifts with extended pauses at these states. The implications of these scenarios to regime shifts in dryland vegetation will be discussed, focusing on the case of fairy circles in Namibia as a concrete example.

Atomic scale matter, like a particle with spin, can respond to external perturbations in a chiral way: the spin rotates in response to a magnetic field. That is, a vector perturbation gives rise to an angular velocity. The technology of pulsed nuclear magnetic resonance exploits this response to organize and manipulate a sample of initially disordered spins. In this talk we explore the analogs of this principle in the world of colloidal matter—micron-scale solid bodies of irregular shape. Such bodies can respond chirally to external forcing via their hydrodynamic coupling. This chiral response is richer than that of a nuclear spin. As with nuclear spin, this response gives a handle that can bring a randomly-oriented dispersion of colloidal objects into a common orientation. The alignment can be created by phase locking, analogous to pulsed nuclear magnetic resonance. It can also be created by *random* external perturbations. Here the alignment principle is the phenomenon of “noise-induced synchronization” known in dynamical systems.

Both thermal fluctuations and material inhomogeneity/disorder play a major role in many branches of science. This talk will focus on various aspects of the interplay between the two. First, we consider the spatial distribution of thermal fluctuational energy and derive universal bounds for internal-stress-free systems. In addition, we show that in 1D systems the thermal energy is equally partitioned even among coupled degrees of freedom. Applications to severing of actin filaments and protein unfolding are discussed. Then, we consider fluctuations in residually-stressed systems and their coupling to anharmonicity. In the context of glassy systems, we show that thermal energy van be spatially localized and suggest that it might serve as a useful structural diagnostic tool, e.g. for identifying glassy lengthscales and precursors to plastic events under driving forces. Lastly, we consider the continuum approach (Statistical Field Theory) to analyzing fluctuations in inhomogeneous systems, and demonstrate fundamental discrepancies between the continuum and the discrete theories in explicit calculations of some, but not all, fluctuation-induced (Casimir-like) forces.

In 1913 Michaelis & Menten published a seminal paper in which they presented a mathematical model of an enzymatic reaction and demonstrated how it can be utilized for the analysis and interpretation of kinetic data. More than a century later, the work of Michaelis & Menten is considered classic textbook material, and their reaction scheme is widely applied both in and out of its original context. At its very core, the scheme can be seen as one which describes a generic first passage time process that has further become subject to stochastic restart. This context free standpoint is not the standard one but I will explain how it has recently allowed us to treat a wide array of seemingly unrelated processes on equal footing, and how this treatment has unified, altered, and deepened our view on single-molecule enzymology, kinetic proof-reading and complex search processes. Newly opened opportunities for theoretical and experimental research will also be discussed.

Our body is colonized by trillions of microbes, known as the human microbiome,

living with us in a complex ecological system. Those micro-organisms play a crucial

role in determining our health and well-being, and there are ongoing efforts to

develop tools and strategies to control these ecosystems.

In this talk I address a simple but fundamental question: are the microbial ecosystems

in different people governed by the same host-independent ecological

principles, represented by a characteristic (i.e. “universal”) mathematical model?

Answering this question determines the feasibility of general therapies and

control strategies for the human microbiome.

I will introduce our novel methodology that distinguishes between two scenarios: host-independent

and host-specific underlying dynamics. This methodology has been applied to study

different body sites across healthy subjects. The results can

fundamentally improve our understanding of forces and processes shaping human microbial

ecosystems, paving the way to design general microbiome-based therapies.

Brownian motion with time-dependent diffusion coefficient is ubiquitous in nature. It has been observed for the mobility of proteins in cell membranes, motion of molecules in porous environment, water diffusion in brain measured in terms of magnetic resonance imaging and also in media with time-dependent temperature such as free cooling granular materials or melting snow.

We investigate a new type of anomalous diffusion processes governed by an underdamped Langevin equation with time-dependent diffusion and friction coefficients and discuss possible applications to real physical systems such as free cooling granular materials. We show that for certain range of parameter values the overdamped limit for the Langevin equation does not exist.

We study the random bond lattice model in which the strength

of the bond between any two neighboring particles is randomly chosen such

that each of N particles is characterized by N

We investigate the Brownian motion of boomerang colloidal particles confined between two glass plates. Our experimental observations show that the mean displacements are biased towards the center of hydrodynamic stress (CoH), and that the mean-square displacements exhibit a crossover from short-time faster to long-time slower diffusion with the short-time diffusion coefficients dependent on the points used for tracking. A model based on Langevin theory elucidates that these behaviors are ascribed to the superposition of two diffusive modes: the ellipsoidal motion of the CoH and the rotational motion of the tracking point with respect to the CoH.

For over 90 years there has been an unexplained puzzle associated with the viscosity of dilute aqueous salt solutions. More specifically, there is a contribution to the viscosity which is linear in the salt concentration and very ion specific for monovalent salts. This is usually discussed in terms of the Jones-Dole coefficient (*B*) which is the amplitude of the linear term in the concentration. The *A* coefficient was derived by Falkenhagen and Onsager many years ago in terms of the Debye-Hückel theory for electrolytes. We shall discuss our current understanding of the Jones-Dole coefficient in the context of ionic hydration.

Mechanics of cancer cells are directly linked to their metastatic potential (MP), or ability to produce a secondary tumor at a distant site. Metastatic cells can squeeze through blood vessel walls and tissue. Such considerable structural changes rely on rapid remodeling of internal cell structure and mechanics. We perform a comparative study, using particle-tracking to evaluate the intracellular mechanics of living epithelial breast cells with varying invasiveness. Probe-particle transport differs between the cell types, likely relating to their cytoskeleton network-structure and underlying transport. The basic analysis included evaluation of the time-dependent mean square displacement (MSD), the second power of the displacement. Particles in all the evaluated cell lines exhibit anomalous super-diffusion with an MSD scaling exponent of 1.4, at short lag times below 1 second. While indicating active transport within the cells, the MSD alone cannot reveal the underlying mechanisms. Hence, we analyze particle motion through a combination the MSD, other powers of the displacement, and various trajectory and displacement analysis procedures to identify structural and dynamic changes associated with metastatic capabilities of cells.

The dynamic cytoskeleton and especially the molecular motors acting on it provide the cell with its remodeling capabilities and allow active transport within the cell. While active transport in living cells has been well-documented, the underlying mechanisms have not been determined. Here, we systematically target the cytoskeleton, molecular motors, and ATP energy related processes to determine their roles in particle transport. Our results show that particle motion is likely driven by different processes in each cell type. Intracellular transport in high MP cells is suggested to originate from fluctuations of microtubule filaments as well as from direct and indirect interactions between particles and microtubule-associated molecular motors. In the low MP cells we suggest that motion results from direct and indirect interactions between particles and microtubule-associated molecular motors, being transported by them or nudged by passing motors, respectively. The benign cells, however, reveal significant involvement of the acto-myosin network, where particle motion was related to network contractions. Thus, we are able to provide insight into dynamic intracellular structure and mechanics that can support the unique function and invasive capabilities of highly metastatic cells.

We discuss models of anomalous diffusion (mostly subdiffusion) in complex systems on the different levels of description and try to classify different types of the behavior. We moreover introduce statistical tests which allow to distinguish between different classes of such models and between different models within the class on the level of ensembles of the trajectories and of the single trajectories of the corresponding processes.

Inspired by biological systems in which thousands of different types of proteins interact within a cell, we use molecular dynamics simulations in 2d to study multi-component systems in the large number of species limit, i.e., all particles differ from each other (APD systems). All the particles are assumed to be of the same size and interact via the Lennard-Jones (LJ) potential, but their pair interaction parameters are generated at random from a uniform or a peaked distribution. We analyze both the global and the local properties of these systems at temperatures above the freezing transition and find that APD fluids relax into a self-organized state characterized by clustering of particles according to the values of their pair interaction parameters.

I discuss the occupation time statistics in thermal, ergodic continuous-time random walks. While the average occupation time is given by the canonical Boltzmann-Gibbs law from Statistical Physics, the finite-time fluctuations around this mean turn out to be large and nontrivial. They exhibit dual time scaling and distribution laws: the infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations.

Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas.

Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and stable densities of fluctuations.

We study the sequence of particles' collisions in a small system of hard balls. We demonstrate that ergodicity implies quite unusual phenomenon. The particles have preferences over long time intervals during which the particle consistently collides more with certain particles and less with others. Things look like there is effective interaction between the particles. Though the preferences change sooner or later the average waiting time to the change is infinite. The results hold for dilute gas with arbitrary short-range interactions and dense fluids of hard balls.

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- תאריך עדכון אחרון: 5/12/2016