# Complex systems

How do biological networks evolve and expand? We study these questions in the context of the plant collaborative-non-self recognition self-incompatibility system. Self-incompatibility evolved to avoid self-fertilization among plants, which could lead to inviable offspring, (‘inbreeding depression’). Self-incompatibility relies on specific molecular recognition between highly diverse proteins of two families: female and male determinants, such that the combination of genes an individual possesses determines its mating partners. Although a few dozen mating specificities are known from population surveys, previous models struggled to pinpoint the evolutionary trajectories by which new specificities evolved. Here, we construct a novel theoretical framework, that crucially affords interaction promiscuity and multiple distinct partners per protein, as is seen in empirical findings. We find two behavioral regimes of the system. In one regime, we demonstrate spontaneous self-organization of the population into distinct 'classes' with full between-class compatibility and a dynamic long-term balance between class emergence and decay. In the other regime the population is fully self-compatible, namely all individuals are capable of self-fertilization. The choice of either regime depends on the promiscuity and inbreeding depression parameter values.

Our work highlights the importance of molecular recognition promiscuity to network evolvability. Promiscuity was found in additional systems suggesting that our framework could be more broadly applicable.

Maxwell’s demon is a classic thought experiment challenging the statistical origins of the second law of thermodynamics. For over 150 years, it has sharpened our understanding of the links between work, entropy and dissipation, and the thermal cost of keeping information. In recent years, an important physical speed limit has been discovered, where the Fisher information constrains the maximal energy dissipation rate of any system driven away from equilibrium. Here, we introduce a new demon that breaks this speed limit. Observing two chambers, the demon sorts driven particles by their diffusive speed. This sorting creates demonic gradients in heat release and the delivery of power to the system, and in the process violates the Fisher bound. As Maxwell's demon reincarnate, the speed demon not only sharpens our understanding of a recently discovered physical law but also opens broad paths for new applications in stochastic control.

Accumulated evidence of transgenerational inheritance of epigenetic and symbiotic changes begs

the question of whether (and under which conditions) the population can benefit from inheritance

of changes that are acquired during the individuals’ lifetime. To address this question, we introduce

a population epigenetics model of individuals undergoing stochastic and/or induced changes that

are either partially or fully transmitted to the offspring. This model is equally applicable to internal

changes in individuals (e.g. epigenetic and symbiotic variations) as well as niche construction

changes that they make in their environment. Potentially adaptive and maladaptive responses are

represented, respectively, by induced changes that reduce and increase the individuals’ rate of death

(i.e. reduction and increase of selective pressure). We use this framework to investigate how inheritance

of acquired changes affects the long-term dynamics of the population. Analytic solution in a

simple case of a population exposed to environments that change in time shows that inheritance of

changes that transiently alleviate the selective pressure is beneficial even when the offspring environment

differs from that of their parents. The benefit from these changes is even more pronounced

at lower fidelity of inheritance as well as for populations with age-dependent decline in fertility.

We also show that this benefit is essential for preventing population extinction under a range of

successive shifts in the environment. Analysis of long-term influences of various population and

environmental factors reveals a non-trivial landscape of outcomes, including a surprising regime in

which the population benefits from inheritance of changes that increase the selective pressure. Taken

together, these findings show that inheritance of changes that are dynamically acquired within a

generation can have a tremendous benefit for the evolution of the population on timescales that are

much longer than the lifetime of an individual.

We study the diffusion of a tracer particle with stochastic dynamics confined to a narrow channel while a constant external pulling force is applied. We show that for such geometry, the presence of strong quenched disorder leads to a counterintuitive behavior: the mobility of the particle decreases as the channel width grows. Such an effect only arises when the quenched disorder is sufficiently strong and the diffusion is anomalous. We present a theoretical basis that predicts and explains this effect. The mean position in the presence of a strong quenched disorder is found to have a complex non-linear dependence on the force which results in breaking of the Einstein relation.

Cell size is an important factor for survival that microorganisms tune to their environmental needs. Nevertheless, all microorganisms exhibit large heterogeneity in cell size even when experiencing homogenous environmental conditions. This heterogeneity is in part due to the stochastic nature of the complex processes that control cellular growth and division dynamics. It is also limited by cellular mechanisms that operate in concert to prevent size divergence over time. In this talk, I will discuss the contribution of one of these mechanisms, namely the Min proteins, to cell size regulation in the bacterial model organism *E. coli*. The Min proteins interactions and diffusion within the cell create oscillations along the cell’s length that aid in the formation of the septal ring at mid-cell. Previously, it was assumed that the Min proteins’ function is limited to the localization of the septal ring along the cell’s long axis. However, the dynamical pattern of these oscillations is sensitive to the cell length, which allows the Min proteins to coordinate the initiation time of the septal ring formation with the cell length and as a result can help regulate cell size. Our results reveal how the pattern of the Min proteins oscillations change with cell size and how that determines the timing of septal ring formation and ultimately cell size at subsequent divisions. The contribution of the Min proteins to cell size regulation is one that enables a cell born smaller than its sister during the “supposedly” symmetric division, to grow more and compensate for the size difference acquired during division. This ensures that cell size is tightly regulated and maintained within a physiologically viable range.

In this talk, we will introduce new chemical dynamics models and theories useful for a

quantitative investigation into dynamics of stochastic gene expression and signal propagation

processes in living cells [1,2]. Our primary focus will be on the chemical fluctuation theorem

governing gene expression and its pioneering applications to quantitative explanations and

predictions of stochastic gene expression and signal propagation dynamics in and across living

cells. Based on the audience’s preferences, we will showcase applications of our novel

chemical dynamics theory to catalytic reactions of single enzymes and nanocatalytic systems

[3,4], or we will discuss our newly developed transport equation, whose solution provides

unified, quantitative understanding of thermal motion of molecules and ions in various complex

fluids and solid electrolytes [5]. During the second part, we will talk about our recent work on

nuclei seeds formation and their condition-dependent crystallization dynamics. This work shed

light on the thermodynamic origin of stable nuclei formation and provide unified, quantitative

explanation of the time-dependent size distribution and size-dependent growth rate of

nanoparticle systems observed by liquid phase TEM, which cannot be explained by the

classical nucleation theory or other previous theories of nucleation.

Refs:

[1] Park et al., The Chemical Fluctuation Theorem governing gene expression, Nat. Commun. 9, 297 (2018); Lim et al., Quantitative understanding of probabilistic behavior of living cells operated by vibrant reaction networks, Phys. Rev. X 5, 031014 (2015).

[2] Song et al., Frequency spectrum of chemical fluctuation: a probe of reaction mechanism and dynamics, PLoS Comp. Biol. 15, e1007356 (2019); Kang et al., Circuit-guided population acclimation of a synthetic microbial consortium for improved biochemical production, Nat. Commun. 13, 6506 (2022).

[3] Yang et al, Quantitative interpretation of the randomness in single enzyme turnover times, Biophys. J. 101, 519 (2011); Park et al., Nonclassical kinetics of clonal yet heterogeneous enzymes, J. Phys. Chem. Letters 8, 3152 (2017); Jeong et al., Phys. Rev. Letters 119, 099801 (2017).

[4] Kang et al, Stochastic kinetics of nanocatalytic systems, Phys. Rev. Letters 126, 126001 (2021); Kang et al., Real-space imaging of nanoparticle transport and interaction dynamics by graphene liquid cell TEM, Sci. Adv. 7, 49 (2021).

[5] Song et al., Transport Dynamics in Complex Fluids, Proc. Nat. Acad. Scie. U.S.A. 116, 12733 (2019); Poletayev et al., Defect-driven anomalous transport in fast-ion conducting solid electrolytes, Nat. Mater. 21, 1066 (2022).

The global population of SARS-CoV-2 is structured into discrete infections within hosts. How does this affect the evolution of the pandemic? First, I will show how one can use the evolution between pairs of hosts to infer how many virus particles contribute to a new infection. Second, I will discuss how the observed timing and dynamics of the Variants of Concern suggest that rare chronic infections are the key drivers of global infection. Finally, I will briefly touch on a case study of rapid evolution and diversification within a chronic infection.

Life is a multifarious bundle of distinct physical phenomena that are distinctive, but not unique to, living things. Self-replication, energy harvesting, and predictive sensing are three such phenomena, and each can be given a clear physical definition. In this talk, we will report recent progress in understanding what physical conditions are required for the spontaneous emergence of these various lifelike behaviours from assemblages of simple, interacting components.

Shape transition and symmetry breaking driven by active mechanical forces are hallmarks of living matter. How an embryo transforms itself from an initial spherical monolayer of homogeneous cells to a complex 3D shape in a programmed and robust manner is a central question in developmental biology. Motivated by the body plan of hydra, which features a bilayer of orthogonally arranged parallel arrays of muscle-like actin fibres, we investigate a "globe" model geometry with elastic springs arranged along discrete lines of latitude and longitudes. We show with two minimal model ingredients, 1. curvature control of active forces, 2. hydrostatic pressure regulation , that an initially spheroidal elastic shell can autonomously transit to elongated ellipsoidal shapes.

*Floating viscous bubbles whose interior gas is rapidly depressurized exhibit a fascinating instability, whereby radial wrinkles permeate the liquid film in the course of its flattening (Debregeas et al, Science 1998; DaSilviera et al., Science 2000, Oratis et al., Science 2020). We show that this instability emerges from a largely unexplored type of Stokes hydrodynamics, that is geometrically-nonlinear flow of curved films that comprise a viscous, volumetrically-incompressible liquid. This theoretical framework highlights profound similarities and differences between the mechanics of elastic sheets and viscous films, revealing experimental observations as a universal, curvature-driven surface dynamics, imparted by viscous resistance to temporal variations of the surface's Gaussian curvature. This novel surface dynamics has close ties to the kinetics of first-order phase transitions and to ``Jelium physics" in continuum media, where topological defects, akin to localized charges in electrostatic media, spontaneously emerge to screen stress within the film.*

Classical statistical mechanics is pivotal for our understanding of equilibrium systems. Yet it mostly fails to address dynamical observables which are key for both equilibrium and far-from-equilibrium systems. In this talk, I will discuss the importance of three such observables: the activity in glassy systems, the integrated current in transport phenomena, and the entropy production in active matter. I will show how these can be addressed within the perspective of large deviation theory and present our findings of an interesting and novel scenario for dynamical phase transitions. It is characterized by the pairwise meeting of first- and second-order bias-induced phase transition curves at two tricritical points. A simple, general criterion predicts its appearance and is complemented by an exact Landau theory for the tricritical behavior.

The Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken at a finite temperature in dimensions two or less in equilibrium systems with short-range interactions. Far from equilibrium, this restriction need not hold, as revealed by the spontaneous flocking of self-propelled particles with polar alignment in two dimensions as in, e.g. the Vicsek model. Most flock phenomenology can be captured by a simpler, discrete-symmetry model known as the Active Ising Model — an active analogue to the equilibrium Ising Model. In this talk, I will present numerical and analytical results which show that the dynamics of the active Ising flock are far richer than previously thought.

We consider Brownian particles diffusing in an external potential which grows at long distance as x^\alpha, 0<\alpha<1. We show that the relaxation to the Gibbs steady-state is anomalous, with stretched exponential relaxation. We calculate the spatial dependence of this slowly relaxing distribution characterizing the approach to equilibrium.

Recurrence in the dynamics of physical systems is an important phenomenon that has many far-reaching consequences. In classical physics, Poincare's recurrence theorem states that a complex system will return to its initial state within a finite time when left alone. This theorem has been extended to the quantum case and has been observed experimentally. In this work, we investigate quantum recurrence for a quantum system that interacts with periodic measurements. Specifically, we consider interacting spin systems where the measurements are performed on one spin. We ask the question: if the monitored spin is initially prepared in the upstate (for example), how long will it take to measure the spin for the first time in the upstate again? We show that the mean recurrence time is fractionally quantized and characterized by the number of dark states, which are eigenstates of the spin system where the monitored spin and the surrounding bath are not entangled. The mean recurrence time is invariant when changing the sampling rate, and this invariance is topologically protected by the quantized winding number. We also show how the mean recurrence time scales with the size of the system. Our results can potentially be observed in experiments with NV centers or quantum computers.

Excitons in low-dimensional semiconductors diffuse, get trapped and eventually released by defects and recombine with a photon emission (photoluminescence). Modern experimental methods allow good precision measurements of the photoluminescence both locally as well as overall. This allows to plot the distributions of the densities of immobile (non-trapped) excitons as a function of time as well a integral characteristics. The experiments reveal a ew interesting an non-trivial effects including the power-law tails o photoluminescence intensity and observable backward (or negative) diffusion. For the explanation we propose and study the mobile-immobile model which splits the population into two groups, the mobile and the immobile particles. The interplay of these two subpopulations explains varios experiments quite well.

Particle hopping is a common feature in heterogeneous media. We explore such motion by using the widely applicable formalism of the continuous time random walk and focus on the statistics of rare events. Numerous experiments have shown that the decay of the positional probability density function P (X, t), describing the statistics of rare events, exhibits universal exponential decay. We show that such universality ceases to exist once the threshold of exponential distribution of particle hops is crossed. While the mean hop is not diverging and can attain a finite value; the transition itself is critical. The exponential universality of rare events arises due to the contribution of all the different states occupied during the process. Once the reported threshold is crossed, a single large event determines the statistics. In this realm, the big jump principle replaces the large deviation principle, and the spatial part of the decay is unaffected by the temporal properties of rare events.

Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. We show that such a decomposition of real-life data allows to infer nontrivial behavioral predictions in the fields of single particle tracking in living cells and movement ecology. Focusing on the ecological scale, we apply this decomposition technique to foraging flights of avian predators. We show that within-patch movement is non-ergodic and separated from large-scale inter-patch movement.

The emergence of order from disorder is the common counter-intuitive property of complex systems. In mixtures of sterically interacting colloids and nonadsorbing polymers, for instance, colloidal crystals may self-assemble to maximize the polymers’ entropy. In contrast to such equilibrium examples, a basic description of nonequilibrium self-organization remains a challenge. I will present experiments where material organization is governed by stresses that are exerted by mechanically active microscopic components. Active chaotic flows drive the assembly and large-scale dynamics of polymerized membranes in an aqueous environment. Initially, homogeneously and isotropically distributed actin filament bundles condense into a thin layer where they connect to form a porous elastic membrane. The polymerized membranes then exhibit out-of-plane bending fluctuations that exceed thermal motions by orders of magnitude. Active bending endows the fluctuating membranes with in-plane soft degrees of freedom that coarsen into large, millimeter-scale, strain fluctuations. For membranes that are a few millimeters in width, system-size displacement oscillations appear that are coupled to unidirectional flow waves. Active stress is thus an emerging paradigm for the assembly and dynamics of matter. I will discuss future extensions of this principle.

(Itamar is a faculty candidate)

In this first part of the talk, we will discuss the progress in tumor predictions using network based methods, including our recent results on the development of an innovative platform for precision medicine for breast cancer. The second part of the talk summarises our current understanding of crackling noise in materials, reviewing research undertaken in the past 30 years, from the early and influential ideas on self-organized criticality, to more modern studies on disordered systems and glasses.

Cooperation is vital for the survival of a swarm. No single bird is faster than a jet plane, and no single fish is faster than a speed boat — humans beat individual animals in air, land, and sea. But, when animals cooperate and swarm, they beat us since biblical times. The science of swarm cooperation contains many open questions, awaiting the discovery of new principles in disordered, far-from-equilibrium, multi-agent systems. A powerful approach to studying swarms experimentally is to design them bottom-up. This requires us to manufacture active particles in large numbers, as when it comes to swarms — more is different. Artificial swarms on both the micro-scale and the macro-scale still struggle to reproduce the agility of natural swarms, in particular, their fluidity when crowded. In my talk, I will describe both microscopic and macroscopic examples that bridge this gap, by looking at swarm cooperation through the lens of non-equilibrium statistical physics. By insuring robots remain autonomous and active when either dilute or crowded, I will show how a completely decentralized swarm can perform collective tasks including transport, corralling, phototaxis, and collective unsupervised learning.

**Our organs and tissues are made of different cell types that communicate with each other in order to achieve joint functions. However, little is known about the universal principles of these interactions. For example, how do cell interactions maintain proper cellular composition, spatial organization and collective division of labor in tissues? And what is the role of these interactions in tissue-level diseases where the healthy balance in the tissue is disrupted such as excess scarring following injury known as fibrosis? In this talk, I will discuss my work in developing theoretical frameworks that explore the collective behavior of cells that emerges from cell-cell communication circuits. I will present work on the cell circuit that controls tissue repair following injury and how it may lead to fibrosis. I will discuss a new approach to explore how cell interactions can be used to provide symmetry breaking and optimal division of labor in tissues, and how this approach can help to interpret complex patterns in real data. Finally, I will introduce a new concept in complex networks – network hyper-motifs, where we explore how small recurring patterns (network motifs) are integrated within large networks, and how these larger patterns (hyper-motifs) can give rise to emergent dynamic properties.**

How animals navigate and perform directional decision making while migrating and foraging, is an open puzzle. We have recently proposed a spin-based model for this process, where each optional target that is presented to the animal is represented by a group of Ising spins, that have all-to-all connectivity, with ferromagnetic intra-group interactions. The inter-group interactions are in the form of a vector dot product, depending on the instantaneous relative, and deformed, angle between the targets. The deformation of the angle in these interactions enhances the effective angular differences for small angles, as was found by fitting data from several animal species. We expose here the rich variety of trajectories predicted by the mean-field solutions of the model, for systems of three and four targets. We find that depending on the arrangement of the targets the trajectories may have an infinite series of self-similar bifurcations, or have a space-filling property. The bifurcations along the trajectories occur on "bifurcation curves'', that determine the overall nature of the trajectories. The angular deformation that was found to fit experimental data, is shown to greatly simplify the trajectories. This work demonstrates the rich space of trajectories that emerge from the model.

In this talk, at first, I will show a formal equivalence among archetypical models in biologically-inspired neural networks (e.g. the Hopfield model for Hebbian learning) and artificial neural networks typically involved in Machine Learning (e.g. the restricted Boltzmann machine): thresholds for learning, storing and retrieving information will be shown to be the same for both these models (analytically in the random setting and numerically on structured datasets). Ultimately, this happens because these models share the same underlying statistical mechanical picture: once understood this duality, it will be enlarged toward more complex and performing neural architectures with a particular focus on dreaming neural networks.

**Instabilities and Geometry of Growing Tissues **

**Doron Grossman and Jean-Francois Joanny **

We present a covariant continuum formulation of a generalized two-dimensional vertexlike model of epithelial tissues which describes tissues with different underlying geometries, and allows for an analytical macroscopic description. Using a geometrical approach and out-of-equilibrium statistical mechanics, we calculate both mechanical and dynamical instabilities of a tissue, and their dependences on various variables, including activity, and cell-shape heterogeneity (disorder). We show how both plastic cellular rearrangements and the tissue elastic response depend on the existence of mechanical residual stresses at the cellular level. Even freely growing tissues may exhibit a growth instability depending on the intrinsic proliferation rate. Our main result is an explicit calculation of the cell pressure in a homeostatic state of a confined growing tissue. We show that the homeostatic pressure can be negative and depends on the existence of mechanical residual stresses. This geometric model allows us to sort out elastic and plastic effects in a growing, flowing, tissue.

Active materials are driven out of equilibrium by a constant consumption of energy at the microscopic level, which is converted into forces and motion. These include biological objects at different scales, ranging from active molecular motors to groups of animals. The physical theory that I will present is motivated by multicellular migration in tissues, which plays a key role in development, wound healing, and metastasis. We propose to describe it as permeation of an active, polar solvent in a viscoelastic environment. We make use of a thermodynamic framework that, similarly to continuum theories of liquid crystals, describes macroscopic physical properties and flows, relying on conservation laws and symmetries. We formulate an active-gel theory that provides a simple description of the dynamic reciprocity between migrating cells and their environment in terms of distinct relative forces and alignment mechanisms. We make new predictions regarding multicellular migration modes based on these mechanisms. Namely, cellular alignment to elastic strains is expected to drive phase-separation in the cellular concentration and orientation for sufficiently extensile collections of cells. Our theory can serve as basis for the analysis of multicellular migration modes and cancer-cell invasion from tumors into connective tissues.

Some simple RNA viruses have a surprising property: the

complementary strand of their genetic code lacks stop codons,

and could code for a protein.

I shall discuss some observations on the structure of the genetic code which explain

how this 'ambigrammatic' property can evolve, and which can distinguish

different explanations of why it offers a reproductive advantage.

A technique called ribosome profiling shows that the ambigrammatic viral

RNA becomes covered by a string of ribosomes, which do not detach from its end.

I argue that this provides a mechanism to hide viral RNA from host cell

defences, which may be utilised by more complex viruses.

This talk reports collaboration with Hanna Retallack, David Yllanes, Greg Huber,

Amy Kistler and Joe DeRisi.

*Many organisms have an elastic hydrostatic skeleton (shell) that often consists of epithelial cells, and is filled with fluid. Living systems (such as hydra) can regulate both elastic forces and hydrostatic pressure by energy consuming (via ATP/GTP hydrolysis) **active processes such as contraction of supracellular actomyosin fibers, and by osmotic control of the hydrostatic pressure inside the shell, respectively. *

** In this work we use computer simulations to study a simple network of springs model of such systems. We introduce hydrostatic, elastic and frictional forces that act on each of the vertices of the network. We model active deformations by changing the equilibrium lengths and the spring constants of randomly selected springs at periodic or at random time intervals. Inspired by recent experiments of Erez Braun and his group, we study the resulting non-thermal fluctuations of the total surface area A of the system both at constant hydrostatic pressure and in the case when pressure and area are coupled. We elucidate the statistical properties of these fluctuations by computing the distribution of A values at different times, and the relaxation of the autocorrelation function of A. **

In many complex systems the emergence of spatio-temporal patterns depends on the interaction between pairs of individuals, agents or subunits comprising the whole system. Theoretical predictions of such patterns rely upon quantifying when and where interactions event might occur. Even in simple scenarios when the dynamics are Markovian, it has been challenging to obtain estimates of encounter statistics between individuals due to the lack of a mathematical formalism to represent the occurrence of multiple random processes at the same time. In this talk Luca will present such formalism and develop a general theory that allows to quantify the spatio-temporal dynamics of interactions. Spatial discretisation is key to develop such a theory bypassing the need to solve unwieldy boundary value problems, giving predictions that are either fully analytical or semi-analytical. Luca presents applications of the theory for simple interaction processes such as pathogen transmission, thigmotaxis, and molecular binding/unbinding in DNA target search. The formalism can be extended to study the dynamics of an individual in a heterogeneous environment where the heterogeneities may represent areas with different diffusivity, partially permeable or impenetrable barriers or the existence of long-range connections between distant sites. In such context Luca will also show the appearance, in the spatial continuous limit, of new fundamental equations that go beyond the diffusion and the Smoluchowski equation.

Harmful invasive species are spreading worldwide, causing severe damage to biodiversity, agriculture, human health, and infrastructure. Virtually all developing countries attempt to manage and suppress invasions, which includes surveillance and eradication of newly detected populations, and containment to slow the spread of larger, established populations of invasive species. However, attempts to eradicate and contain invasive species often fail due to the lack of well-designed strategies dictating when and where to apply each of the various treatment tactics and technologies. In my talk, I will demonstrate how to use optimal control theory to determine the optimal, most cost-effective strategy to manage invasive species in two study systems. In the first study, we ask how to optimally combine three treatment tactics: pesticide application, mating disruption, and sterile male release. We show that each of the three tactics is most efficient across a specific range of population densities. Furthermore, we show that mating disruption and sterile male release inhibit the efficiency of each other, and therefore, they should not be used simultaneously. However, since each tactic is effective at different population densities, different combinations of tactics should be applied sequentially through time when a multiple-annum eradication program is needed. In the second study, we ask how to stop the spread of a propagating front of an invasive species while minimizing the annual cost of treatment. We show that the optimal strategy often comprises eradication in the yet-uninvaded area, and under certain conditions, it also comprises maintaining a “suppression zone” - an area between the invaded and the uninvaded areas, where treatment reduces the invading population but without eliminating it. We examine how the optimal strategy depends on the demographic characteristics of the species and reveal general criteria for deciding when a suppression zone is cost-effective.

Distinguishability plays a major role in quantum and statistical physics. When particles are identical their wave function must be either symmetric or antisymmetric under permutations and the number of microscopic states, which determines entropy, is counted up to permutations. When the particles are distinguishable, wavefunctions have no symmetry and each permutation is a different microstate. This binary and discontinuous classification raises a few questions: one may wonder what happens if particles are almost identical, or when the property that distinguishes between them is irrelevant to the physical interactions in a given system. I will sketch a general answer to these questions. For any pair of non-identical particles there is a timescale, $\tau_d$ required for a measurement to resolve the differences between them. Below $\tau_d$, particles seem identical, above it - different, and the uncertainty principle provides a lower bound for $\tau_d$. Thermal systems admit a conjugate temperature scale, $T_d$. Above this temperature the system appears to equilibrate before it resolves the differences between particles, below this temperature the system identifies these differences before equilibration. As the physical differences between particles decline towards zero, $\tau_d \to \infty$ and $T_d \to 0$.

Spherical colloids that catalyze the interconversion reaction between solute molecules A and B whose concentration at infinity is maintained away from equilibrium, interact due to the nonuniform fields of solute concentrations. We show that this long range 1/r interaction is suppressed via a mechanism that is superficially reminiscent but qualitatively very different from electrostatic screening: catalytic activity drives the concentrations of solute molecules towards their equilibrium values and reduces the chemical imbalance that drives the interaction between the colloids. The imposed nonequilibrium boundary conditions give rise to a variety of geometry-dependent scenarios; while long range interactions are suppressed (except for a finite penetration depth) in the bulk of the colloid solution in 3D, they can persist in quasi-2D geometry in which the colloids but not the solutes are confined to a surface, resulting in the formation of clusters or Wigner crystals, depending on the sign of the interaction between colloids.

The survival of natural populations may be greatly affected by environmental conditions that vary in space and time. We look at a population residing in two locations (patches) coupled by migration, in which the local conditions fluctuate in time. We report on two findings. First, we find that, unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path. We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atypical averages over a long time. It provides a detailed description of the statistics of histories leading to the rare event and the mean time to extinction. The framework applies to rare events in a broad class of systems driven by non-Gaussian noise. Second, applying this framework to the population-dynamics model, we find a phase transition in its extinction behavior. Strikingly, a patch which is a sink (where individuals die more than are born) can nonetheless reduce the probability of extinction, even if it lowers the average population's size and growth rate.

As witnessed by numerous applications, the power spectral density (PSD) embodies a wealth of informations about the time evolution of a stochastic process. In a series of recent works (including { inter alia { [1, 2]) it has been pointed out that the commonly used notion of PSD has certain intrinsic limitations which make its determination/interpretation rather dicult. In particular, in many experimental situations it is not possible to achieve statistically adequate sampling required in order to perform ensemble averages. Moreover, it is not possible to monitor the trajectory for innitely long time, as demanded by the standard definition. The notion of single-trajectory spectral density (STSD) shows how to go beyond the aforementioned conceptual and practical limitations. In the rst part of the talk I will summarize some of the achievements obtained within the single-trajectory analysis of stochastic processes via power spectra; the discussion encompasses the standard diffusion [1] and the fractional Brownian motion [2]. In the second part of the talk I will analyse the spectral content of the so-called active Brownian particle (ABP), which is one of the simplest models of a particle undergoing active motion; e.g., a chemically-active Janus colloid. It is known that the ABP behaves as a standard Brownian particle at large time scaled but nonetheless its spectral content is not known; this gap has been lled only recently in [3]. Firstly, I evaluate the standardly-dened spectral density, i.e. the STSD averaged over a statistical ensemble of trajectories in the limit of an innitely long observation time T. Then, I will present results for the nite-T behavior for the power spectral density and for the coecient of variation of the STSD distribution. The cross correlation between spatial components of the STSD provides an additional spectral ngerprint which is computed. Finally I will address the effects of translational diffusion on the functional forms of spectral densities. The exact expressions that are obtained unveil many distinctive features active Brownian motion compared to its passive counterpart, which allow to distinguish betweenthese two classes based solely on the spectral content of individual trajectories.

References

[1] D. Krapf, E. Marinari, R. Metzler, G. Oshanin, X. Xu, and A. Squarcini, Power spectral density of a single Brownian trajectory: what one can and cannot learn from it, New J. Phys. 20, 023029 (2018).

[2] D. Krapf, N. Lukat, E. Marinari, R. Metzler, G. Oshanin, C. Selhuber-Unkel, A. Squarcini, L. Stadler, M. Weiss, and X. Xu, Spectral content of a single non-Brownian trajectory, Phys. Rev. X 9, 011019 (2019).

[3] A. Squarcini, A. Solon, and G. Oshanin, Power spectral density of trajectories of an active Brownian particle, New Journal of Physics (2021) (in press). [arXiv:2109.03883]

Phase separation of liquids, as illustrated by everyday oil and water mixtures, provides the basis for creating diverse soft materials, from lipid vesicles to emulsions.** **These states of matter are typically formed upon the addition of surface adsorbing agents that stabilize liquid interfaces. I will show that active matter provides a new way for controlling phase separation. Active fluids contain microscopic energy consuming objects that drive far-from-equilibrium large-scale flows. In phase separating mixtures of an active isotropic fluid and a passive fluid, bulk active flows produce interfacial waves whose amplitude is controlled by activity. Motility of the phase separated domains leads to finite-size steady states, where coalescence of smaller droplets is balanced by the break-up of larger ones. When interfaces meet a solid boundary, active waves also drive non-equilibrium wetting transitions. These results demonstrate the promise of mechanically-driven interfaces for creating a new class of soft active matter.

The Holy Grail of quantum thermodynamics is finding a concrete quantum machine that performs better than classical ones. Some early suggestions indicate that quantum resources such as quantum coherences^{1} or quantum reservoir engineering^{2} provide heat machines with unique features compared to their classical counterparts. Nevertheless, once all the preparation costs^{3} and non-equilibrium sources^{4} have been properly accounted for, quantum and classical heat machines are essentially the same. But physicists don’t give up easily, so the quest for a quantum heat machine that is fundamentally different from its classical counterpart still goes on.

In this talk, I will show that a basic quantum property - energy quantization - allows quantum heat machines to operate even with incompressible working fluids, which would forbid work extraction for classical heat machines^{5,6}.

I will discuss how to experimentally measure this effect by realizing the same heat machine operating in the classical and in the quantum limit. This research opens up the possibility for experimentally studying the difference between classical and quantum systems well beyond the realm of heat machines.

[1] Scully, Marlan O., et al. "Extracting work from a single heat bath via vanishing quantum coherence." *Science* 299.5608 (2003): 862-864.

[2] Roßnagel, Johannes, et al. "Nanoscale heat engine beyond the Carnot limit." *Physical review letters* 112.3 (2014): 030602.

[3] Zubairy, M. Suhail. "The Photo‐Carnot Cycle: The Preparation Energy for Atomic Coherence." *AIP Conference Proceedings*. Vol. 643. No. 1. American Institute of Physics, 2002.

[4] Alicki, Robert, and **David Gelbwaser-Klimovsky**. "Non-equilibrium quantum heat machines." *New Journal of Physics* 17.11 (2015): 115012.

[5]** Gelbwaser-Klimovsky, David,** et al. "Single-atom heat machines enabled by energy quantization." *Physical review letters* 120.17 (2018): 170601.

[6] Levy, Amikam, and** David Gelbwaser-Klimovsky**. "Quantum features and signatures of quantum thermal machines." *Thermodynamics in the Quantum Regime*. Springer, Cham, 2018. 87-126.

Confluent epithelial tissues can be viewed as soft active solids. These out of equilibrium substrates are made of active units (the cells) that are self-driven and act autonomously (crawl, contract, etc) in response to local conditions. In parallel, the substrate also supports continuum chemical and mechanical fields, which are capable of creating feedback loops. Perhaps surprisingly, little is known about the emergent dynamic patterns and mechanical properties in such materials, and there is no unified theory for these active solids. In this talk I will present my experimental work on the epithelium of *T. adhaerens, *an understudied organism, considered “the simplest living animal”. Made almost entirely of basal epithelium, with no neurons and muscles, it is yet capable of complex behavior (collective locomotion, taxis, external digestion, reproduction by fission). I will show the extreme contractile dynamics we discovered in the live epithelium, and present a generic model for the propagating of contraction pulses in such tissues, as seen in *T. adhaerens *and in other epithelial systems. These pulses are active-acoustic solitons, that do not attenuate despite the overdamped conditions. The model is based solely on the well-known single-cell response of contraction under tension, and it predicts another emergent phenomenon: enhanced rip resistance via homogenous distribution of external loads. Since keeping integrity is at the heart of epithelium function, the model is relevant to many epithelial tissues, especially such that are prone to repeated mechanical tensions (gut, airway, vasculature, bladder) in different medical contexts (from leaky gut to asthma but also in tissue engineering and implants). Finally, the model may inspire engineering of synthetic materials with enhanced resistance to rupture.

Ribosome biogenesis is an efficient and complex assembly process that has not been reconstructed outside a living cell so far, yet is a critical step for establishing a self-replicating artificial cell.

We developed a platform to reproduce the autonomous synthesis and assembly of a ribosomal subunit from synthetic genes immobilized on the surface of a chip. The genes were spatially organized in the form of dense DNA brushes in contact with a macroscopic reservoir of cell-free minimal gene expression system. We showed that the transcription-translation machinery actively self-organized on DNA brushes, forming local and quasi-2D sources for nascent RNA and proteins.

We recreated the biogenesis of Escherichia coli’s small ribosomal subunit by synthesizing and capturing all its ribosomal proteins and RNA on the chip. Surface confinement provided favorable conditions for autonomous step-wise assembly of new subunits, spatially segregated from original intact ribosomes. Our real-time fluorescence measurements revealed hierarchal assembly, cooperative interactions, unstable intermediates, and specific binding to large ribosomal subunits.

Using only synthetic genes, our methodology is a crucial step towards creation of an autonomous self-replicating artificial cell.

Knowing the position of the Earth today will not enable us to predict its position 10 Myrs from now, yet, the planetary orbits in the Solar System are stable for the next 5 Gyrs [1]. This is a typical feature of classical systems whose Hamiltonian slightly differs from an integrable one --- their Lyapunov time is orders of magnitude shorter than their ergodic time. This puzzling fact may be understood by considering the simple situation of an integrable system perturbed by a weak, random drive: there is no Kolmogorov–Arnold–Moser (KAM) regime and the Lyapunov instability can be shown to happen almost tangent to the invariant tori. I will extend this analysis to the quantum case, and show that the discrepancy between Lyapunov and ergodicity times still holds, where the quantum Lyapunov exponent is defined by the growth rate of the 4-point Out-of-Time-Order Correlator (OTOC) [2]. Quantum mechanics limits the Lyapunov regime by spreading wavepackets on a torus. Still, the system is a relatively good scrambler in the sense that the ratio between the Lyapunov exponent and kT/\hbar is finite, at a low temperature T [3]. The essential characteristics of the problem, both classical and quantum, will be demonstrated via a simple example of a rotor that is kicked weakly but randomly.

[1] J. Laskar. Chaotic diffusion in the solar system. Icarus, 196:1, 2008.

[2] T. Goldfriend and J. Kurchan. Phys. Rev. E 102:022201, 2020.

[3] J. Maldacena, S. H. Shenker, and D. Stanford. A bound on chaos. J. High Energy Phys. 2016:106, 2016.

Ribosomes generate all proteins in a cell, including their own ribosomal proteins. Previous work on bacteria showed that ribosomal protein production imposes a bound on cellular growth rates, since cell doubling requires a commensurate doubling of ribosomes. However, ribosomes are made not only of protein, but also of ribosomal RNA (rRNA). We obtain a new speed limit on cell growth which originates in the generation of ribosomal RNA [1]. A comparison with *E. coli* data reveals that the bacterial ribosome's 1:2 protein-to-RNA mass ratio uniquely maximizes cellular growth rates as permitted by both bounds. This observation leads to a growth-law involving RNA polymerases, and an invariant of bacterial growth. Similar arguments for Eukarya lead to several new growth-laws [2]. Despite the greater complexity of that domain of life, the predictions are consistent thus far with available data for the model organism *S. cerevisiae.*

References:

[1] S. Kostinski and S. Reuveni, "Ribosome composition maximizes cellular growth rates in *E. coli*," Phys. Rev. Lett. 125, 028103 (2020).

[2] S. Kostinski and S. Reuveni, "Growth-laws and invariants from ribosome biogenesis in lower Eukarya," arXiv:2008.11697 (2020).

**Coordinated collective motion of self-driven entities is fundamental in living systems, ranging from flocks of birds, schools of fish, groups of bacteria, down to cell tissues and the cytoskeleton. Emergent collective dynamics cannot be easily traced back to the properties of individual cells, as the knowledge of car mechanics is insufficient for understanding of traffic jams.**

**Fascinating example of collective behaviour is self-shaping of living tissues (morphogenesis). Recent experiments have shown that organization and dynamics of completely different multicellular systems, could be explained within the same physical framework of active liquid crystals. Cells in bidimensional cultures tend to align together and form well-oriented domains of finite size separated by nematic defects of charge ± 1/2. However, unlike passive liquid crystals cells move creating complex dynamic patterns [1,2,3]. I will show spontaneously emerging turbulence in epithelial cell cultures that occurs at low Reynolds numbers when topological defects serve as a template for turbulent-like flows [2]. In another example, I will show emergence of chiral edge currents in fibrosarcoma cancer cells, which are fuelled by layers of +1/2 topological defects, orthogonally anchored at the channel walls and acting as local sources of chiral active stress [3].**

**[1] Nature Physics, (2018), 14(7): 728-732**

**[2] Physical Review Letters, (2018), 120(20): 208101**

**[3] Nature Physics, in submission**

Over the past few decades, there have been spectacular experimental developments in manipulating cold atoms (bosons or fermions) [1, 2], which allow one to probe quantum many-body physics, both for interacting and noninteracting systems. In this talk we focus on the noninteracting Fermi gas, for which a general theoretical framework has been developed over the recent years [3,4].

We consider a generic model of N non-interacting spinless fermions in d dimensions confined by a general trapping potential (we assume a central potential for d>1), in the ground-state. In d=1, for specific potentials, this system is related to classical random matrix ensembles. We develop a theoretical framework for studying the quantum fluctuations of the number of fermions N_D in a domain D of macroscopic size in the bulk of the Fermi gas (in d>1 we assume that D is a spherical domain). We show that the variance of N_D grows as N^((d-1)/d) * (A log(N) + B) for N>>1, and obtain the explicit dependence of A,B on the potential. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, which agrees with exact results for d=1.

The talk is based on the recent work [5].

[1] I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. 80 885 (2008).

[2] S. Giorgini, L. P. Pitaevski and S. Stringari, Rev. Mod. Phys. 80 1215 (2008).

[3] D. S. Dean, P. Le Doussal, S. N. Majumdar, G. Schehr, Phys. Rev. A 94, 063622 (2016).

[4] D. S. Dean, P. Le Doussal, S. N. Majumdar, G. Schehr, J. Phys. A: Math. Theor. 52 144006 (2019).

[5] N. R. Smith, P. Le Doussal, S. N. Majumdar, G. Schehr, arXiv:2008.01045.

It is well known that strange attractors are characterised

by their fractal dimensions, which quantify the mass

clustered into a small ball. Recent work, using statistical

approaches, has revealed other generic properties of

chaotic systems.

The fractal dimensions characterise the dense regions

of the attractor using a power-law, but the distribution of

density in the sparse regions is also characterised by a

power-law, which we term the 'lacunarity exponent'.

The fractal dimension describes the mass of the attractor

contained in small regions, but it is also possible to study the

shape of clusters of points which sample the attractor. The

statistics of the shape of these clusters is characterised by

power laws. The exponents of these lower-laws are found to

exhibit phase transitions.

Physical applications of these phenomena will also

be discussed, including particles advected in fluid flows

and ray trajectories in random media.

We present a method to derive analytically the growths exponents of a surface of 1 + 1 dimensions whose dynamics is ruled by cellular automata. Starting from the automata, we write down the time evolution for the height's average and height's variance (roughness). We discuss the existence of a Probability distribution for the congurations. We apply the method to the etching model[1,2] than we obtain the dynamical exponents, which perfectly match the numerical results obtained from simulations. Those exponents are exact and they are the same as those exhibited by the KPZ model[3] for this dimension. Therefore, it shows that the etching model and KPZ belong to the same universality class[4]. Moreover, we proof that in the continuous limit the majors terms leads to KPZ [5].

In aggregation, masses coalesce irreversibly to form a cluster, and in frag-

mentation, clusters break up into smaller ones. These processes occur in a

wide range of phenomena varying from tissue development in cell biology

to the formation of planetary rings. Our work is motivated by astro-

physical applications, in particular Saturn rings. A model describing such

systems considers binary collision reactions between masses, and upon

collision, the two clusters can either aggregate or fragment completely

(shatter) into the smallest constituent masses. At the mean-field level,

the time evolution of the cluster sizes is described by the deterministic

Smoluchowski equation.

We have studied the effects of stochasticity and finite total mass in these

models. In such a setting, the Smoluchowski equation does not give the

correct answer for the mass distribution. We therefore use Monte Carlo

simulation to study the system. The key question that we address is

whether such a system shows a phase transition, and if so, what is the

nature of the phases? There is indeed an active-absorbing phase transition

in the system; in the absorbed phase, all masses aggregate into a single

cluster. The active phase, depending upon the locality of the reaction

Kernel shows temporal intermittency. Details of these aspects will be

given during the talk.

Identifying the drivers of ecosystem and population dynamical behavior is a fundamental aspect of ecology. In a spatially explicit context, the basic ingredients to consider are the spatial structure of the landscape, the local dynamics at play, and the dispersal and diffusion which mediates between the former two. Numerous studies has looked at each of these components separately, but little is known on the interplay between them. Missing has been a more integrative approach, able to map and identify the possible dynamical regimes in the system, and in particular its response to perturbations.

I will focus my talk on a simple, yet relatively general, scenario: the recovery of a homogeneous metapopulation from a single, spatially localized pulse disturbance. We find that the response can take one of three forms, each representing one of three dynamical regimes: Isolated, Interplay and Mixing. Using dimensional analysis we can predict the transition points between these regimes, and how these change with basic system properties such as its total area and the nonlinearities of local dynamics. This enables us to address pertinent issues in ecology, such as habitat fragmentation, synchrony-induced extinctions, and mechanisms of biomass productivity in metacommunities.

I will finish the talk by briefly presenting a few extensions of this work. In particular, a possible indicator of bistability based on the spatial extent of disturbances, the spatial aggregation of disturbances when their frequency is high, and the spatial patterns of ecosystem engineers along an environmental gradient.

The rapid accumulation of genome sequences from diverse organisms presents an opportunity and a challenge for theoretical research: is it possible to derive quantitative laws of genome evolution and an underlying theory? Microbes have small genomes with tightly packed protein-coding genes, and the different functional classes of genes (such as information processing, metabolism, or regulation) show distinct scaling exponents with the genome size. The compactness of microbial genomes is traditionally explained by genome streamlining under selection for high replication rate but so far, there has been no general theoretical model to account for the observed universal laws of genome content scaling. We developed a model for microbial genome evolution within the framework of population genetics and tested it against extensive data from multiple genome comparisons. The analyses indicate that the evolution of genome size is not governed by streamlining but rather, reflects the balance between the benefit of additional genes and the intrinsic preference for DNA deletion over acquisition. These results explain the observation that, contrary to the common belief, microbes with large genomes are subject to stronger selection than small genomes. Employing this model to recover the differential scaling of functional gene classes in bacterial genomes allowed us to identify the underlying factors that govern the evolution of the genome content. A key factor that we termed genome plasticity shapes genome evolution and provides a simple mathematical representation of evolvability, a central but elusive concept in evolutionary biology. These findings demonstrate that key aspects of genome evolution can be captured by general population genetics models, and pave the way for further theoretical analyses of fundamental evolutionary mechanisms.

Particles in biological and soft matter systems undergo Brownian dynamics: their deterministic motion, induced by forces, competes with random diffusion due to thermal noise. More broadly, Brownian dynamics is a generic and simple model for dynamical systems with noise. Provided only with the time-series of positions of such a system, i.e a trajectory in phase space, it is challenging to infer what force field had produced it. At the same time, this is the key information about the dynamical system, which would allow to characterize it completely. I will show that there is an information-theoretic bound on the rate at which information about the force field can be extracted from a trajectory, quantified by a channel capacity. I will discuss the relation between this capacity and the entropy production rate, as defined in stochastic thermodynamics. I will then present a practical method, Stochastic Force Inference, that uses the information contained in a trajectory to approximate force fields. This technique also permits the evaluation of out-of-equilibrium currents and entropy production. It thus makes it possible to quantify subtle time-irreversibility in biological systems at the mesoscale, and opens the door to an understanding of the importance of time- irreversibility.

We show that a non-linear measure of dependence called the codifference is a useful tool in studying ergodicity breaking and non-Gaussianity. Codifference was previously studied mainly in the context of stable and infinitely divisible processes. We extend its range of applicability to random parameter and diffusing diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible analysing covariance and correlation. At the same time the differences between the covariance and codifference can be used to analyse non-Gaussianity.

Various animals, mammals in particular, display some form of territorial behaviour for which they make their presence conspicuous to others claiming exclusive ownership of regions of space. The signals employed to perform this form of spatial exclusion may be visual, auditory or olfactory depending on the species and the environment. When the mechanism of territorial exclusion occurs via marks deposited on the terrain (olfactory cues), one talks about stigmergy, a form of environment-mediated interaction often encountered in social insect societies.

To study the emergence of spatial segregation in stigmergic systems I have introduced a new type of collective animal movement model where alignment of the agents does not play any role. It is called the territorial random walker model as agents move freely as random walkers on a lattice, scent-marking the terrain wherever they go. As deposited marks remain active for a finite amount of time, each walker retreats upon encountering an active foreign scent. The emerging spatio-temporal dynamics of the system can be quite rich and can be studied at the meso-scale (the territories) as well as at the micro-scale (the agents).

At the meso-scale short-lived marks produce rapidly morphing and highly mobile territories, while long-lived marks yield slow territories with a narrowly defined shape distribution. More importantly the full dependence in territory mobility as a function of the time for which individual marks remain active is accompanied by a liquid-hexatic-solid transition akin to the Kosterlitz-Thouless melting scenario, apparently the first ecological model to predict such a transition.

The dynamics at the micro-scale is in general non-Markovian, but when population density is sufficiently large some mean-field analytic approaches have proved useful. By considering localized walls to mimic the sharp (retreat) interaction when an animal encounters a foreign scent, it is possible to represent via a Fokker-Planck formalism an animal roaming within neighbouring territorial boundaries. Application of this analytic model to movement data from a red fox population in Bristol, UK, is also shown.

Inspired by the findings on territorial dynamics, it is natural to ask whether it is possible to devise a swarm of independent and decentralised territorial robots. Given that building robots with actual marker reading and writing mechanisms is quite difficult in practice, inspiration comes from the behaviour of territorial birds which detect each other presence at a given location by chirping a challenge which is then countered. Rather than broadcasting a scent signal detectable by any individual passing by, the signalling occurs only between two individuals nearby. While the exclusion mechanism is not stigmergic anymore, it can still be exploited to segregate partially the robot population and limit spatial oversampling in search tasks.

References

[1] A. Heiblum-Robles and L. Giuggioli, Phase transitions in stigmergic territorial systems, accepted.

[2] L. Giuggioli, I. Ayre, A. Heiblum Robles and G.A. Kaminka, From ants to birds: a novel bio-inspired approach to on-line area coverage, in Groß R et al. (eds) Distributed Autonomous Robotic Systems, Springer Proceedings in Advanced Robotics, vol 6, pp. 31-43 (2018).

[3] L. Giuggioli and V.M. Kenkre, Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality, Move. Ecol. 2(1), 20 (2014).

[4] L. Giuggioli, J.R. Potts, D.I. Rubenstein and S.A. Levin, Stigmergy, collective actions and animal social spacing, Proc. Natl. Acad. Sci. USA 110(42):16904-9 (2013).

[5] J.R. Potts, S. Harris and L. Giuggioli, Quantifying behavioral changes in territorial animals caused by sudden population declines, Am. Nat. 182:e73-e82 (2013).

[6] L. Giuggioli, J.R. Potts and S. Harris, Predicting oscillatory dynamics in the movement of territorial animals, J. Roy. Soc. Interface 9(72):1529-43 (2012).

[7] J.R. Potts, S. Harris and L. Giuggioli, Territorial dynamics and stable home range formation for central place foragers, PLoS ONE 7(3):e34033 (2012).

[8] L. Giuggioli, J.R. Potts and S. Harris, Brownian walkers within subdiffusing territorial boundaries, Phys. Rev. E 83:061138/1-11 (2011).

[9] L. Giuggioli, J.R. Potts and S. Harris, Animal interactions and the emergence of territoriality, PLoS Comp. Biol. 7(3):e1002008/1-9 (2011).

Macromolecular phase separation and droplet formation have long been proposed as key elements in the formation of protocells during the origin of life. A simple model of a protocell consists of a droplet, where droplet material is produced outside the droplet, and chemical reactions inside the droplet play the role of a simple metabolism. Our theoretical study shows that such chemically active droplets can have a flux-driven shape instability that leads to a symmetric droplet division. We analyze the dependence of the instability on the droplet viscosity and parameters that characterize the metabolism and material production. Our work provides a physical mechanism for the division of early protocells before the appearance of membranes.

Energy consuming processes are important in determining the large-scale organization of chromatin. Experiments indicate that chromosomes are organized in a non-random manner and occupy specific regions of a nucleus, called chromosome territories (CTs), with gene rich regions (euchromatin) more centrally positioned than gene-poor (heterochromatin) regions. Further, chromosomes are largely seen to be positioned radially by gene density, although positioning by chromosomes size is also seen. Our model for large-scale nuclear architecture incorporates the effects of non-equilibrium processes driven by the consumption of ATP, associated to cell-type specific transcriptional processes that are inhomogeneous within and across chromosomes. It yields predictions which compare favorably to experimental data including statistics of positional distributions, shapes and overlaps of each chromosome. Our simulation also reproduce common organizing principles underlying large-scale nuclear architecture across interphase human cell nucleus. These include the differential positioning of two X chromosomes in female cells, the territorial organisation of chromosomes including both gene-density-based and size-based chromosome radial positioning schemes, statistics of the shape of chromosomes, and contact probabilities of individual chromosomes. We proposed that biophysical consequences of the distribution of transcriptional activity across chromosomes should be central to any chromosome positioning code.

Entropy and free-energy estimation are key in thermodynamic characterization of simulated systems ranging from spin models through polymers, colloids, protein structure, and drug-design. Current techniques suffer from being model specific, requiring abundant computation resources and simulation at conditions far from the studied realization. In this talk, I will present a novel universal scheme to calculate entropy using lossless compression algorithms and validate it on simulated systems of increasing complexity. Our results show accurate entropy values compared to benchmark calculations while being computationally effective. In molecular-dynamics simulations of protein folding, we exhibit unmatched detection capability of the folded states by measuring previously undetectable entropy fluctuations along the simulation timeline. Such entropy evaluation opens a new window onto the dynamics of complex systems and allows efficient free-energy calculations.

Most populations are spread over spatial ranges that are far larger than individuals typically disperse. How does this affect how quickly they can adapt, and what kinds of patterns of neutral genetic diversity do we expect? We find that spatial structure creates a large gap in adaptibility between populations which are totally asexual and those that occasionally recombine. We also find that adaptation creates a kind of effective long-range dispersal, increases relatedness between spatially distant individuals.

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.

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.

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.

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.

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.

TBA

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.

TBA

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.