On the tight connection between collective and tagged particle motion in singe file dynamics

Single file dynamics is a generic term for the dynamics of one dimensional systems in which neighboring particles cannot pass each other. A relation first proposed by Alexander and Pincus [1] connects the mean square displacement of a tagged particle in such a system to the time evolution of the collective density. In many cases this leads to a tagged particle mean square displacement proportional to the square root of the MSD of collective excitations; e.g. in the case of regular diffusion for the latter, the tagged particle MSD grows as t 1/2 with time. But in cases where collective excitations and tagged particle motion have different average drift velocities the tagged particle always exhibits regular diffusion about its average drift. For hamiltonian systems with short ranged interactions the tagged particle MSD is dominated by a regular diffusion term proportional to t, due to sound mode contributions to the dynamics of the collective density (in other words, the Brillouin peaks). In addition there is a contribution proportional to t 3/5 , due to the heat mode contribution (the Rayleigh peak). Largely because of the one dimensional structure, finite size effects are strong. Taking these into account one finds very good agreement between theoretical predictions and computer simulations [2]. [1] S. Alexander and P. Pincus, Phys. Rev. B 18, 2011 (1978). [2] H. Posch, private communications.

The KPZ equation, its universality and multi-component KPZ

The KPZ equation is a well-known model equation for describing surface growth. There are renewed interests in the equation itself and applications because of recent theoretical and experimental developments. In this talk we explain the exact solutions of the one-dimensional KPZ equation and its universality. We will also discuss the multi-component KPZ equation.

Macroscopic fluctuations in non-equilibrium mean-field diffusions

I shall discuss the application of the macroscopic fluctuation theory to non-equilibrium diffusions with mean field interactions, in particular, to a model of active rotators that exhibits 2nd order phase transitions between stationary and periodic phases. The stress will be on the behavior of the current fluctuations near such transitions. Based on joint work with Freddy Bouchet and Cesare Nardini.

Emergence of global patterns in bacterial growth: from single cells to communities

Understanding how phenomenological behaviors observed in biological systems emerge from molecular interactions of many individual unit and how these interactions shape the response of living systems to a changing environment are challenging questions which lie at the interface between multiple disciplines. In this talk I will draw an example from the human gut microbiome, the full consortium of microbes living in association with the human gut. Recent developments in DNA sequencing have made it possible to monitor how the compositions of microbial species change in time. Analysis of healthy adults under antibiotic treatment showed that the gut microbiota could take several weeks to recover after treatment cessation. This suggests that the combination of inter-species and host-microbe interactions and external perturbations could lead to hysteresis phenomena. We investigate this possibility and propose an out of equilibrium stochastic model able to explain this phenomenon in terms of a first-order phase transition scenario. Our study reveals the importance of noise-activated dynamics in the recovery from antibiotic-perturbed states.

Searching for transition paths

Abstract: Transition paths are the stochastic paths taken by a system between two states. The Langevin dynamics which describes these paths can be represented by a “path integral” which is a weighted sum over all paths joining the two states. In the first part of the talk, we show how one can compute the dominant paths (paths with largest weight) and how one can calculate dynamical quantities (such as rates or transition path times) from these paths. The method and its limitations are illustrated on some simple examples. In a second part of the talk, we show how the Langevin dynamics can be modified to obtain a stochastic equation which samples directly and efficiently the transition paths. This modified Langevin equation is illustrated on some simple examples.

Singular Large Deviation Functionals.

he talk will present recent progress in understanding density large deviation functional in non-equilibrium systems. First, a general overview of the methods for carrying out such calculations will be outlined. Then the analytical structure of the large deviation functionals, will be contrasted with that of equilibrium systems. Particular emphasis will be placed on cases where the functional become singular. Time permitting a Casimir effect in diffusive systems will also be discussed.

Irreversible work, large deviations, critical Casimir effect, and universality in quantum quenches

Recent experimental progresses in the physics of ultracold atomic gases have revived the interest in the behavior of thermally isolated quantum statistical systems, especially after sudden changes (quenches) of their control parameters. Considering the quench as a thermodynamic transformation, we focus on the probability distribution of the irreversible work done on the system. The large deviations of this intensive work turn out to be surprisingly connected to the physics of a classical system confined in a film geometry. If the quench occurs close to a (quantum) critical point, these large deviations acquire universal features dictated by the critical Casimir effect in the corresponding classical system. In systems of bosons, the statistics of the work may additionally display a transition which is reminiscent of the equilibrium Bose-Einstein condensation.

Why do complex systems look critical?

The study of complex systems is limited by the fact that only few variables are accessible for modeling and sampling, which are not necessarily the most relevant ones to explain the systems behavior. In addition, empirical data typically under sample the space of possible states. We study a generic framework where a complex system is seen as a system of many interacting degrees of freedom, which are known only in part, that optimize a given function. We show that the underlying distribution with respect to the known variables has the Boltzmann form, with a temperature that depends on the number of unknown variables. In particular, when the influence of the unknown degrees of freedom on the known variables is not too irregular, the temperature decreases as the number of variables increases. This suggests that models can be predictable only when the number of relevant variables is less than a critical threshold. Concerning sampling, we argue that the information that a sample contains on the behavior of the system is quantified by the entropy of the frequency with which different states occur. This allows us to characterize the properties of \"maximally informative samples\": Within a simple approximation, the most informative frequency size distributions have power law behavior and Zipf\'s law emerges at the crossover between the under sampled regime and the regime where the sample contains enough statistics to make inference on the behavior of the system. These ideas are illustrated in some applications, showing that they can be used to identify relevant variables or to select most informative representations of data, e.g. in data clustering.

Explosive condensation in one-dimensional particle systems

We study a system of interacting particles, hopping between sites of a one-dimensional lattice with a rate which increases with the number of particles at interacting sites.Condensation is the phenomenon whereby on a system of size L containing M = ρ L particles a finite fraction of the particles are typically found at a single site in the limit of large L.We show that condensation occurs in this system and the condensate moves rapidly through the system, in contrast to previously studied condensation. We show that,the relaxationtime to the stationary state decreases is an inverse power of ln L so that condensation is instantenous for L >> 1. This provides a first example of instantaneous gelation in a spatially extended system.

High-Energy Tail of the Velocity Distribution of Driven Inelastic Maxwell Gases.

A model of homogeneously driven dissipative system, consisting of a collection of N particles that are characterized by only their velocities, is considered. Adopting a discrete time dynamics, at each time step, a pair of velocities is randomly selected. They undergo inelastic collision with probability p. With probability (1−p), energy of the system is changed by changing the velocities of both the particles independently according to v → −r w v+η, where η is a Gaussian noise drawn independently for each particle as well as at each time steps. For the case r w = −1, although the energy of the system seems to saturate (indicating a steady state) after time steps of O(N), it grows linearly with time after time steps of O(N 2 ), indicating the absence of an eventual steady state. For −1 < r w ≤ 1, the system reaches a steady state, where the average energy per particle and the correlation of velocities are obtained exactly. In the thermodynamic limit of large N, an exact equation is obtained for the moment generating function. In the limit of nearly elastic collisions and weak energy injection, the velocity distribution is shown to be a Gaussian. Otherwise, for |r w | < 1, the high-energy tail of the velocity distribution is Gaussian, with a different variance, while for r w = +1 the velocity distribution has an exponential tail.

Distinct sites, common sites and maximal displacement of N random walkers .

For a collection of random walkers, the number of distinctly visited sites S(t) and commonly visited sites W(t) provides useful information about their intersection properties. The statistics of these two quantities are in general non-trivial and are relevant in various physical situations. I will first focus on the distributions of S(t) and W(t) for N independent random walkers in one dimension. For this case a useful mapping between these two quantities and the extreme displacements allows us to compute their distributions exactly for any N at large times. In the second part of the talk I will discuss possible generalizations to the case of N non-intersecting Brownian walkers (vicious walkers).

Entropy production and Fluctuation theorems with odd-parity variables

Total entropy production is conventionally separated into system entropy change and heat flowing into a thermal reservoir. In the presence of odd-parity variables under the time reversal, however, we explicitly find an extra entropy-like term whose physical origin is still in mystery. We also show that the separation of the total entropy production into the housekeeping (adiabatic) and its complementary functionals respectively holding the fluctuation theorems is not generic. This is due to the non-transient housekeeping contribution caused by the asymmetry of the steady-state distribution with respect to the odd- parity variables.

Asymmetric Exclusion Processes with Constrained Dynamics

In this talk, I will introduce a generalization of the asymmetric exclusion process in which the transition probabilities of a particle depend on the occupancy state of the nearby sites. I will discuss the unusual transport properties such systems exhibit at high particle density, including directed motion and negative resistance, rheological-like behaviour and jamming. Finally the statistics of current fluctuations and their symmetry under time-reversal is addressed. The observation of non-Gaussian current fluctuations and deviations from the standard fluctuation relation is related to the existence of growing amorphous correlations and heterogeneous anomalous diffusion regimes.

Statistical mechanics of fitness landscapes

A fitness landscape is a mapping from the space of genetic sequences, commonly modeled as a binary hypercube of dimension L, to the real numbers. The concept was introduced in evolutionary biology more than 80 years ago, but only recently have empirical studies started to offer glimpses into the structure of real fitness landscapes. Motivated by this biological context we consider random models of fitness landscapes where fitness values are assigned according to some probabilistic rule. We study properties such as the mean number of local maxima and the statistics of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. I will focus specifically on recent exact results obtained for the block model, a modular version of Kauffman\'s NK-model, and for the rough Mount Fuji model, which is equivalent to a random energy model in an external field. The talk is based on joint work with Jasper Franke, Johannes Neidhart, Stefan Nowak, Benjamin Schmiegelt and Ivan Szendro.

Nonequilibrium Markov processes conditioned on large deviations

I will discuss in this talk recent works with Raphael Chetrite (Phys. Rev. Lett. 111, 120601, 2013) on Markov processes conditioned on rare events - for example, Brownian motion conditioned on having a given asymptotic velocity or on being positive for a very long time. I will show, using elements of large deviation theory, that a process conditioned on a large deviation event can be represented by a conditioning- free Markov process, called the driven or equivalent process, which has the same typical states as the conditioned process. I will discuss some consequences of this result for the study of nonequilibrium systems and the simulation of large deviations.

Turbulent liquid crystals unveil universal fluctuation properties of growing interfaces

I will present a series of experimental results on universal fluctuation properties of growing interfaces, along with an experimental realization found in turbulent regimes of liquid crystal convection. Measuring interface fluctuations of growing turbulent domains, we found not only the scaling exponents of the Kardar-Parisi-Zhang (KPZ) universality class, but also particular distribution and correlation functions that were previously derived for solvable models in the KPZ class. Interestingly, these statistical properties have direct yet non-trivial relation to random matrix theory, and depend on the global geometry of the interfaces (whether the interfaces are curved or not). In other words, the KPZ class splits into a few universality subclasses. Besides these solved statistical properties, we can also measure some analytically unsolved yet important properties, in particular time correlation, from the experimental data. The results show that curved and flat interfaces are different in many aspects, even qualitatively, yet universal in each subclass. This underpins rich and powerful universality of the KPZ class, ruling detailed statistical properties beyond the scaling exponents in such systems out of equilibrium.

Information engine with a physically realized feedback loop

We propose a minimal two-state model equipped with a demon in form of a two-state memory. The model evolves in a controlled cycle consisting of measurement, feedback, and thermal relaxation. In contrast to previous approaches, all phases of the cycle are realized stochastically, i.e., not only the relaxation but also measurement and feedback are implemented as stochastic Markov processes. This allows us to clearly define entropy production and extracted work. Our analysis suggests that it is not sufficient to restrict our attention to the work extraction during feedback, rather different work contributions during the whole cycle have to be taken into account. This allows us to evaluate the total cost to operate an information engine, suggesting possible ways to compute its efficiency.

Alignment vs noise: minimal but non-trivial framework for active matter and collective motion

The collective properties of self-propelled particles interacting solely via some kind of effective alignment constitute one of simplest framework for studying active matter. I will first argue that despite its minimality, it is not devoid of experimental relevance, and then proceed to provide a synthetic account of our current understanding of the problem. I will in particular stress that the problem of the onset of collective motion/orientational order is better understood as a liquid-gas transition, but one with unusual liquid and coexistence phases. I will also stress the remarkable success of hydrodynamic theories in describing the large-scale properties of these active particle models.

Hard Core Exclusion Models on Lattices:Rods, Rectangles and Discs

Systems of particles with only excluded volume interaction are the simplest systems to exhibit phase transitions. Well known examples are the liquid-solid transition in hard sphere systems, the isotropic- nematic transitions in hard rod systems, and the disordered-sublattice transition in the hard hexagon model. In this talk, I will summarise recent numerical and analytical progress on lattice models of hard rods, rectangles and discretised discs. In particular, it will be argued that with increasing density, these systems undergo multiple entropy driven transitions.

Transport Coefficients in Diffusive Lattice Gases

In this talk I will discuss how to compute density-dependent transport coefficients in dense diffusive lattice gases. I\'ll focus on the diffusion coefficient, and I\'ll briefly discuss the mobility and the self- diffusion coefficient. I will present a number of lattice gases where such computations are feasible. Many of these gases, e.g. repulsion processes and the KLS model, are generalizations of the symmetric exclusion process which additionally include interactions between the particles in neighboring sites; in other examples a site can be occupied by more than one particle. I\'ll also describe calculations for exclusion processes with avalanches.

Anharmonic chains and nonlinear fluctuating hydrodynamics

We argue that for one-dimensional systems with conservation laws in fluctuating hydrodynamics the Euler currents have to be expanded up to second order. We thereby arrive at predictions for the large scale asymptotic of equilibrium time correlation functions, in particular their anomalous spreading. Our predictions are compared with numerical simulations.

The prioritising exclusion process

The prioritising exclusion process is a particle hopping model with dynamically varying lattice length. The model acts as a stochastic scheduling mechanism for a priority queueing system, with the varying lattice length representing the length of the queue of customers. The exact stationary distribution for an unbounded queue is described by domain wall dynamics and can be derived from the microscopic transition rules. The structure of the unbounded queue carries over to bounded queues where, although no longer exact, we find the domain wall theory is in very good agreement with simulation results. Within this approximation we calculate average waiting times for queueing customers.

Weak noise limit of systems driven by non-Gaussian noise

A fundamental understanding of the dynamics of systems under the influence of thermal fluctuations is provided by investigating its large deviation properties in the limit of weak noise strength. This approach has provided, for example, theories of activated escape in low temperature regimes and is also intimately linked to the description of quantum mechanical systems within a semiclassical approximation. However, many complex systems are driven by non-thermal (active) fluctuations with non-Gaussian characteristics. Here, we investigate the weak-noise limit of systems driven by Lévy noise using a path- integral approach. We apply this approach to obtain the large deviation asymptotics of activated escape and of time-integrated observables in simple non-equilibrium models.

Experimental investigation of two dimensional Anderson localization of light in the presence of a nonlocal nonlinearity

We report on the observation of two dimensional disorder-induced spatially localized states in a disordered optical fiber. We show evidence of a nonlinear effect of thermal origin, and measure the localization length versus the input power. We also report on the nonlocal interaction involving various localizations and the resulting collective dynamics. Results are interpreted in terms of a two-dimensional nonlinear Schroedinger equation with a random potential and a nonlocal nonlinearity.

Finite time thermodynamics

In a transformation between stationary states (equilibrium or nonequilibrium) a system necessarilygoes through deviations from stationarity which are small if the transformation is quasi-static. Classical thermodynamics is unable to describe this intrinsically dynamic aspect. In thermodynamic books quasi-static transformations are introduced in a descriptive way appealing to intuition without any mathematics. We show that it is possible to analyse mathematically these transformations by expanding thermodynamic variables in powers of 1/T where T is the time taken by the transformation. In this way we obtain equations relating changes of thermodynamic functionals to solutions of hydrodynamics slightly deviating from a stationary state. Interesting optimization problems arise in this context.

Current Fluctuations in the Asymmetric Exclusion Process

Non-equilibrium systems are often characterized by the transport of a macroscopic observable, such as the current of particles through a wire. We shall explain how, for the one-dimensional asymmetric simple exclusion process, exact results for the statistics of the current can be derived by using integrability techniques, for various boundary conditions. The results obtained agree with the predictions of Macroscopic Fluctuation Theory in some suitable limits.

Condensation and intermittency in an aggregation-fragmentation model

We study real space condensation in an aggregation-fragmentation model in which the total mass is not conserved, but can fluctuate due to the exit of clusters of all sizes. Our 1D model includes diffusion, aggregation and chipping of single units in the bulk, along with influx and outflux of masses at the boundaries. With increasing influx, there is a transition to a condensate phase, in which the condensate, and concomitantly the total mass, shows steady state fluctuations of the order of the mean mass. The time series of the total mass shows quantitative features of intermittency: the flatness diverges as the scaled time approaches zero. These features are established analytically in the limit of zero chipping and by numerical simulations for non-zero chipping. (Work done in collaboration with Himani Sachdeva and Madan Rao)

Relation between classical dynamics and quantum Hamiltonians

I will generalize the correspondence between non-equilibrium dynamics of the Ising model and quantum Hamiltonians by Castelonovo et al. This correspondence allows us to analyse classical dynamics in terms of quantum-mechanical methods, from which I will draw lessons on the relation between simulated annealing and quantum annealing.

Exactly Solvable Mean-Field Monomer-Dimer Models

he seminar will introduce the mean-field monomer-dimer models starting from a review of the deterministic case in the complete graph solved by Heilmann and Lieb. Two new rigorous solutions will be illustrated. The first for the quenched diluted version on locally tree-like graphs where the exact formula, found within the replica symmetric cavity approach, will be proven. The second for the deterministic case with an attractive interaction. In this case the exact solution, proved with a suitable variational principle for the free energy, displays a first-order phase transition along a coexistence curve.

Joint works with Diego Alberici and Emanuele Mingione.

Large deviation in single-file diffusion

Single-file diffusion is referred to the motion of many particles in narrow channel where particles can not pass each other. As a consequence of the forbidden mutual passage the motion of individual particles is sub-diffusive. I will apply the macroscopic fluctuation theory to analyze the probability distribution of position of a tracer particle in a large class of single-file system. For Brownian point particles with hard-core repulsion this macroscopic approach leads to a parametric expression of the large deviation function of tracer position. I will compare the results with that obtained in an exact microscopic analysis. I will emphasize the dependence of the tracer statistics on the initial state.

Stochastic predation-prey competition: biodiversity and species extinction

Since Lotka and Volterra\'s pioneering contributions, the modeling of interacting, competing species has been at the center of a multitude of studies in biology, ecology, mathematics, and physics. In this talk I consider in some detail stochastic systems that involve four or more species. I discuss both well-mixed systems, i.e. without spatial structure, and spatial systems in one and two dimensions. For four or more species one can observe the formation of alliances. For the cyclic four-species system without spatial structure mean-field theory predicts a complex time dependence and that the surviving partner-pair is the one with the larger product of their strengths (rates of consumption). Beyond mean-field much richer behaviour is revealed, including complicated extinction probabilities and non-trivial distributions of the population ratio in the surviving pair. For the lattice systems, I investigate the growth of domains and the related extinction events. I close by a discussion of more complex food networks where in specific cases cyclic dominance within coarsening clusters yields a peculiar coarsening behaviour with intriguing pattern formation.

Anomalous diffusion of deterministic walks on a square lattice

A walker moves on a two dimensional square lattice, the landscape. At every site of the lattice there is an obstacle that determines the walker’s steps. The obstacle has two possible orientations, say left and right, and the walker alters the landscape by changing the orientation of the obstacle as he passes. There are two possibilities for the obstacles [1]. Starting with a small fraction p of right obstacles chosen at random we find anomalous diffusion . [1] H. F. Meng, E. G. D. Cohen, Phys. Rev. E 50 2482 (1994), X. P. Kong, E. G. D.Cohen, J. Stat. Phys. 62 1153 (1991).

Stationary non equilibrium states from a microscopic and a macroscopic point of view

I will discuss the statistical behavior of stationary non equilibrium states for stochastic interacting particle systems. From a microscopic point of view I will discuss combinatorial representations of solvable models and the role of relative entropy. From a macroscopic point of view I will discuss the large deviations rate function for the empirical measure. Models to be analysed are the boundary drive totally asymmetric exclusion process and the totally asymmetric exclusion process with first and second class particles.

Superdiffusive modes in two-species driven diffusive systems

Using mode coupling theory and dynamical Monte-Carlo simulations we investigate the scaling behaviour of the dynamical structure function of a two-species asymmetric simple exclusion process, consisting of two coupled single-lane asymmetric simple exclusion processes. We demonstrate the appearence of a superdiffusive mode with dynamical exponent z=5/3 in the density fluctuations, along with a KPZ mode with z=3/2 and argue that this phenomenon is generic for short-ranged driven diffusive systems with more than one conserved density. When the dynamics is symmetric under the interchange of the two lanes a diffusive mode with z=2 appears instead of the non-KPZ superdiffusive mode.

Magnets: even a bit of disorder can make a great difference.

After reviewing the off-equilibrium evolution taking place in simple clean magnets after a quench from high temperature to below the critical temperature I will discuss the behavior of magnets with quenched disorder. These are systems where the disorder strength is sufficiently small not to alter the basic equilibrium phase structure nor to introduce frustration. Nevertheless, I will show that even such a small amount of disorder is sufficient to change dramatically the dynamical properties.

Information processing in genetic regulatory networks

The processing of information in genetic networks involves several levels of regulation, including the transcriptional regulation of gene expression as well as post-transcriptional regulation by small non- coding RNA\'s. The interplay between these different levels of regulation will be discussed, using dynamical modelling and simulations of several functional network modules. It will be shown that the combination of different regulation mechanisms enables fast, efficient and reliable processing of information. The effects of competition, feedback and fluctuations will be considered using deterministic and stochastic methods.

Survival of a target in a gas of diffusing particles with exclusion

Let an infinite lattice gas of constant density, 0<n0<1, described by the Symmetric Simple Exclusion Process, be brought in contact with a target: a spherical absorber of radius R. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability P(T) that no particle hits the target until long but finite time T. We also find the most likely particle density history conditional on the non-hitting. The results depend on the dimension of space d and on the rescaled parameter l=R/(DT) 1/2 , where D is the gas diffusivity. For small l and d>2, P(T) is determined by exact stationary solutions of the MFT equations that we find. For large l, and for any l in one dimension, the relevant MFT solutions are non-stationary. In this case ln P(T) scales differently with relevant parameters, and also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of non-interacting random walkers.