# Abstracts

Is Boltzmann’s equation reversible? A new large deviation perspective on the irreversibility paradox.

In this talk we will review the classical heuristic understanding of the irreversibility paradox for Boltzmann’s equation, and the related Lanford’s program. We will then propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann’s chaotic hypothesis, we derive a large deviation rate function, or action, that describes asymptotically for large N, the stochastic process for the empirical distribution. This action has the entropy as a quasi-potential, as should be expected. While Boltzmann’s equation appears as the most probable evolution (corresponding to a law of large numbers), the action describes a genuine reversible stochastic process, in agreement with the microscopic reversibility. As a consequence, this not only gives the expected meaning to Boltzmann’s equation, but also quantifies the probability of any dynamical evolution departing from solutions of Boltzmann’s equation. This picture, fully compatible with the heuristic classical view, makes it much more precise in various ways. It gives its full dynamical meaning to the entropy in relation with recent fluctuation theorems. Moreover, our approach clarifies several important points ; for instance the irreversibility of Boltzmann’s equation is not a consequence of the chaotic hypothesis (stosszahlansatz), as often stated, but rather a consequence of confusing the evolution of the average of the empirical distribution with the evolution of the empirical density itself. Proving the validity of this action for describing the large deviation of the empirical distribution of Hamiltonian dynamics remains a challenge for the future.

Aggregation and fragmentation models. Application to Planetary Rings

Simple models of ballistic aggregation and fragmentation are studied. The models are characterized by two energy thresholds, Eagg and Efrag , which demarcate collisions with different outcome -- aggregation, rebound or fragmentation. We analyze different fragmentation models, including the model of complete decomposition into monomers, a fragmentation with a power-law debris size-distribution and collisions with erosion, when only a small fraction of a particle mass is chipped off. We start from Enskog-Boltzmann equation for the mass-velocity distribution function and derive Smoluchowski-like equations for concentrations of particles of different mass and their partial granular temperatures [1,2]. We analyze these equations analytically and numerically and study in detail the steady states of such systems. We show that the steady-state distribution of particles sizes is universal, that is, it does not depend on the particular form of the size distribution of debris in the inter-particles collisions, provided that it is steep enough. The steady-state distribution is not also sensitive to the properties of erosive collisions. Application of our theory to the Saturn Rings demonstrates a perfect agreement with the observation data. One of the important conclusions of our study is the universality of the steady-state size distribution of particles in planetary rings in general [2].

[1] N. Brilliantov, A. Bodrova, P. Krapivsky, J. Stat. Mech. (2009) 06/P06011

[2] N. Brilliantov, P. Krapivsky, A. Bodrova, F. Spahn, H. Hayakawa, V. Stadnichuk and J. Schmidt, PNAS 112 (2015) 9536-9541.

**Zdzislaw Brzezniak
**Large deviations principle for invariant measures for the 2-D stochastic Navier-Stokes Equations

Based on a recent work with S Cerrai and M Freidlin on the quasipotential for the 2-D stochastic Navier-Stokes Equations (to appear in PTRF 2015) we prove the Large deviations principle for invariant measures for these equations driven by an additive nuclear gaussian noise and we identify the action functional. This talk is based on a joint works with S Cerrai.

**Maria Cameron
**Methods for spectral analysis of stochastic networks

Stochastic networks (continuous-time Markov chains) with pairwise transition rates containing a small parameter arise in modeling natural processes. Time-reversible processes include the dynamics of clusters of interacting particles, conformal changes in molecules, and protein folding, while time-irreversible ones are exemplified e.g. by walks of molecular motors. The spectral decomposition of the generator matrix of the stochastic network gives a key to understanding its dynamics, the extraction of quasi-invariant sets, and building coarse-grained models. However, its direct calculation is exceedingly difficult if the matrix is large and its entries vary by tens of orders of magnitude. I will introduce a single-sweep algorithm for computing the asymptotic spectral decomposition and discuss continuation techniques. This approach allows us to easily interpret the results and largely avoid the issues associated with the floating point arithmetic. An application to the Lennard-Jones cluster of 75 atoms will be discussed.

Instantaneous gelation and explosive condensation in non-equilibrium cluster growth

The kinetics of various mechanisms of non-equilibrium cluster growth such as aggregation or exchange-driven growth are characterised by an interaction kernel, K(x,y), which specifies the average rate of interaction of particles having sizes x and y respectively. If the kernel increases quickly enough as a function of cluster size, then the second moment of the cluster size distribution can diverge in a finite time. This singularity, known as the gelation transition, is interpreted as signifying the formation of clusters of infinite size within a finite time. It has been known for some time that there exists a subclass of kernels for which the gelation transition occurs instantaneously. It is not the case that such behaviour is a mathematical pathology since there exist physically reasonable models which exhibit this behaviour such as coagulation driven by differential settling of liquid droplets in the Stokes regime. It was considered unlikely however that such behaviour could survive in spatially extended systems. A counter example was given by Waclaw and Evans in 2012 in which a total asymmetric version of a mass transport model on a one-dimensional lattice in the spirit of the zero-range process was shown to exhibit condensation of all of the particles onto a single site in a time which vanishes as the system size grows, a phenomenon known as "explosive condensation". In this talk I will discuss the relationship between instantaneous gelation and explosive condensation in the light of what is known about cluster growth models such as aggregation and exchange-driven growth. I will also show that a symmetric variant of Waclaw and Evans' model can exhibit the same behaviour provided that the rate of particle exchange is high enough. The fact that the model is spatially extended allows a second regime to exist for lower rates of particle exchange in which clusters grow algebraically by clustering, an regime which is absent at mean-field level.

Joint work with S. Grosskinsky and Y.-X. Chau

**Arghya Dutta
**Modelling aggregation and fragmentation phenomena using the Smoluchowski equation

In nature, there are a number of physical phenomena whose dynamics are dominated by transport, aggregation and fragmentation. Examples include formation of rain drops, polymerization and the formation of the planetary rings. In this talk, I will present results from an analysis of a model with collision dependent fragmentation, based on the Smoluchowski equation. For a general class of collision kernels, I will derive the scaling limits of the mass distribution using moment and singularity analysis of the generating functions, and exact solutions for special cases. We will identify a new regime (relevant for ballistic collision) where the exponents depend non-trivially on the kernel. The results will be compared with the data obtained from the mass distribution of the particles constituting the Saturn rings.

**Grisha Falkovich
**Turbulent flow

A review of an ongoing work to develop a consistent theory which aspires to derive both the mean flow profile and the correlation functions of turbulent fluctuations.

**Anna Frishman
**Permanence and Time irreversibility for particles in turbulence

Particles in turbulent flows, unlike e.g Brownian particles, are driven by an out-of-equilibrium fluctuating medium. In this talk, I will discuss the dynamics of tracer particles following a turbulent velocity field. Focusing on manifestations of the time irreversibility of turbulence, I will first connect the existence of an energy flux through scales in turbulent flows to the time irreversibility of pairs of particles. I will show that the related time irreversible quantity becomes discontinuous in compressible flows, modelled by the one dimensional Burgers equation, due to the presence of shocks. I will briefly review time irreversibility for single particles. It was recently shown that the latter can be measured in incompressible turbulence through energy differences along a trajectory, with particles gaining energy slowly and losing it quickly. I will present the application of this idea to one dimensional Burgers turbulence, where some analytical results can be obtained. Finally, I will describe pair dispersion in a random, linear velocity field, corresponding to turbulent flows at small scales. For incompressible and statistically isotropic velocities I will describe the existence of an all time statistical conservation law. In two dimensional or Hamiltonian flows, this law is extended to a symmetry of the probability distribution function of the finite-time Lyapunov exponent, quantifying the imbalance between the phase space for particle dispersion and convergence.

The Arcetri model(s): exactly solvable spherical model(s) of interface growth

Building on an analogy between the ageing behaviour of magnetic systems and growing interfaces, the Arcetri models,

a set of new exactly solvable models for growing interfaces, are introduced,bwhich share many properties with the kinetic spherical model. The long-time behaviour of the interface width and of the two-time correlators and responses is analysed.

In the first model, for all dimensions $d\ne 2$, universal characteristics distinguish the Arcetri model from the Edwards-Wilkinson model, although for $d>2$ all stationary and non-equilibrium exponents are the same. For $d=1$ dimensions, the first Arcetri model is equivalent to the $p=2$ spherical spin glass. For $2<d<4$ dimensions, its relaxation properties are related to the ones of a particle-reaction model, namely a bosonic variant of the diffusive pair-contact process.

Preliminary results on the logarithmic breaking of dynamic scaling in the second Arcetri model will also be discussed.

**Ostap Hrniv
**Long term behaviour in a model of microtubule growth

We introduce, and study probabilistically, an interacting particle system inspired by the model of microtubule growth proposed in Antal etal (PRE76, 041907). We show that the long term behaviour of the system can be described in terms of the velocity of the active end of the microtubule, and explore its dependence on the parameters of the model.

**Rainer Klages
**Fluctuation relations for anomalous stochastic dynamics

Fluctuation relations (FRs) emerged as a key concept for assessing fluctuations very far from equilibrium [1]. For stochastic processes generating normal diffusion they have been found to exhibit a characteristic large deviation form. We test them for stochastic processes generating anomalous diffusion [2]. We first consider Gaussian stochastic dynamics with memory by using a Langevin approach with two different types of additive noise: (i) internal noise where the fluctuation-dissipation relation of the second kind (FDR II) holds, and (ii) external noise without FDR II. For internal noise the existence of FDR II implies the existence of the fluctuation-dissipation relation of the first kind (FDR I), which in turn leads to conventional (normal) forms of transient work FRs. For systems driven by external noise we obtain violations of normal FRs [3]. We then study non-Gaussian stochastic dynamics generated by three different types of time-fractional Fokker-Planck equations.

By considering both sub- and superdiffusive processes we recover again normal FRs if FDR I holds but observe violations if FDR I is broken [4]. Similar violations of FRs are observed in computer simulations of glassy dynamics and in experiments on biological cell migration.

[1] R.Klages, W.Just, C.Jarzynski (Eds.), Nonequilibrium Statistical Physics of Small Systems: Fluctuation relations and beyond, Wiley-VCH (2013)

[2] R. Klages, G.Radons, I.M.Sokolov (Eds.), Anomalous transport: Foundations and applications, Wiley-VCH (2008)

[3] A.V.Chechkin, F.Lenz, R.Klages, J.Stat.Mech. L11001 (2012)

[4] P.Dieterich, R.Klages, A.V.Chechkin, New J. Phys. 17, 075004 (2015)

**Alexei Kritsuk
**Dual cascades and energy condensation in compressible 2D turbulence

Numerical simulations of compressible 2D turbulence demonstrate that under reasonably weak random pumping at an intermediate scale, an inverse energy cascade develops and produces box-size coherent vortices hosting shock waves. Shock dissipation limits the energy capacity of the condensate and the system ends up in a statistical steady state regime. With stronger pumping or in larger boxes, the energy saturates before condensate vortices form. Compressibility stops the inverse cascade, when shocks start feeding enough energy back to small scales. We argue that direct and inverse cascades cannot be really separated in such a case.

**Jason Laurie
**Universal Mean Flow Profiles in Two-Dimensional Turbulence

An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a large-scale condensate or mean flow--for a square aspect ratio this is a pair of coherent system-size vortices. We present a new theoretical analysis based upon momentum and energy exchanges between the mean flow and the underlying turbulent fluctuations. Our results show that the mean velocity profile has an universal internal structure independent on the mechanisms of small-scale dissipation and small-scale forcing. We verify the theoretical predictions through extensive numerical simulations of the two-dimensional Navier-Stokes equations. We begin our analysis by investigating the square periodic box before studying non-unity aspect ratios and the structure of zonal mean flows.

**Yen Ting Lin
**Effects of bursting noise in gene regulation networks

Including short-lived mRNA populations in models of gene regulation networks introduces both the transcriptional bursting (i.e., transcription occurs at random times) and translational bursting (i.e., random amounts of proteins are translated by each mRNA). In this talk, I will present a coarse-graining method to construct mesoscopic models for such type of dynamical systems, which fully accounts for the bursting noise. We systematically compare different levels of modeling, ranging from individual-molecule-based models including mRNA populations, over protein-only individual-based models to mesoscopic models such as diffusion-type models and our proposed model. We show that the proposed mesoscopic model outperforms conventional diffusion-type models. In a one-dimensional autoregulated network, we present closed-form analytic solutions for both the stationary distribution of protein expression as well as first-passage times of the dynamical system. We present numerical solutions for higher-dimensional gene regulation networks, in which case we also carry out analysis in the weak-noise limit.

The implications of the study are multi-faceted. Bursting noise is a ubiquitous feature of many biological systems. From a modeling perspective, our proposed method provides an alternative way to analyze dynamical systems with bursting noise. Biologically, the study presents quantitative evidences, the first to our knowledge, showing that bursting noise is the predominant form of intrinsic noise in gene regulation networks. Finally, from a mathematical point of view, the proposed model belongs to a class of stochastic processes named piecewise deterministic Markov processes, and our analysis on first-passage times and weak-noise limit of the process may inspire more rigorous analytic investigations in the future.

References: The talk is based on the following preprints:

• arXiv:1508.02945 [q-bio.MN]

• arXiv:1508.00608 [q-bio.MN]

**Bernhard Mehlig
**Confined polymers in the extended de Gennes regime

We show that the problem of describing the conformations of a semiflexible polymer confined to a channel can be mapped onto an exactly solvable model in the so-called extended de Gennes regime. This regime (where the polymer is neither weakly nor strongly confined) has recently been studied intensively experimentally and by means of computer simulations. The exact solution predicts precisely how the conformational fluctuations depend upon the channel width and upon the microscopic parameters characterising the physical properties of the polymer.

**Mauro Mobilia
**The influence of zealotry on the nonlinear q-voter model

The importance of relating "micro-level" interactions with "macro-level" phenomena in modelling social dynamics is well established, and the study of parsimonious models like the variants of the voter model, commonly used in statistical mechanics, has received a growing interest in the last decades. In this talk, after a brief overview of some properties of the paradigmatic "linear" voter model and some sociological considerations, I will discuss the properties of the nonlinear q-voter model with inflexible zealots in a finite well-mixed population. In this system, each individual supports one of two parties and is either a susceptible voter or an inflexible zealot. At each time-step, a susceptible adopts the opinion of a neighbour if this belongs to a group of q>1 neighbours all in the same state, whereas inflexible zealots never change their opinion. In the presence of zealots of both parties, the model is characterized by a fluctuating stationary state and, below a critical zealotry density, the distribution of opinions is bimodal. The long-time dynamics is thus driven by fluctuations and after a characteristic time, most susceptibles become supporters of the party having more zealots with an asymmetric opinion distribution. When the number of zealots of both parties is the same, the distribution of opinions is symmetric and, in the long run, susceptibles endlessly swing from the state where they all support one party to the opposite state. Above the critical zealotry density, when there is an unequal number of zealots of each type, the probability distribution is single-peaked and non-Gaussian. I will also briefly discuss how a consensus is reached when there is only one type of zealots. Ref.: Physical Review E 92, 012803 (2015); e-print: http://arxiv.org/abs/1506.04911

**Nick Moloney
**Thresholding a birth-death process

When analysing time series it is common to apply thresholds. For example, this could be to eliminate noise coming from the resolution limitations of measuring devices, or to focus on extreme events in the case of high thresholds. We analyse the effect of applying a threshold to the duration time of a birth-death process. This toy model allows us to work out the form of the duration time density in full detail. We find that duration times decay with random walk exponent -3/2 for 'short' times, and birth-death exponent -2 for 'long' times, where short and long are characterised by a threshold-imposed timescale. For sparse data the ultimate -2 exponent of the underlying (multiplicative) process may never be observed. This may have implications for real-world data in the interpretation of threshold-specific decay exponents.

(in collaboration with F. Font-Clos, G. Pruessner, A. Deluca)

**Carlos Perez-Espigares
**The spatial fluctuation theorem

For non-equilibrium systems of interacting particles and for interacting diffusions in d-dimensions, a novel fluctuation relation is derived. The theorem establishes a quantitative relation between the probabilities of observing two current values in different spatial directions. The result is a consequence of spatial symmetries of the microscopic dynamics, generalizing in this way the Gallavotti–Cohen fluctuation theorem related to the time-reversal symmetry. This new perspective opens up the possibility of direct experimental measurements of fluctuation relations of vectorial observables.

**Marc Pradas
**Noise-induced critical transitions in multiscale systems

External or internal random fluctuations are ubiquitous in many physical systems and can play a key role in their dynamics often inducing a wide variety of complex spatio-temporal phenomena, including noise-induced spatial patterns and noise-induced phase transitions. Examples can be found in several fields: from biology, climate modelling and technological applications to fluid dynamics and granular media. Many of these phenomena and applications can be modelled by noisy multiscale systems which are characterized by the presence of a wide range of different scales interacting with each other.

In this talk I will present a study of the effects of additive noise on different multiscale systems, including spatially extended systems that are close to the instability onset (and hence there is a clear time scale separation), and Brownian motion in multiscale potentials which are characterized by a clear length scale separation. I will show that in these systems, noise may induce several state transitions which can be characterized in terms of universal critical exponents.

**R. Rajesh
**Aggregation of Charged Polymers

Charged polymers are ubiquitous in biological systems. Examples include DNA, F-Actin and microtubules. Though similarly charged, such polymers attract each other in the presence of oppositely charged counterions in solution. How does the kinetics of aggregation depend on the valency of the counterions and on the long-ranged electrostatic interactions? In this talk, results from extensive molecular dynamics simulations of rigid charged polymers will be presented. The numerical results for the temporal decay of number of aggregates may be obtained from the solution of the Smoluchowski equation for aggregation of particles with short ranged attraction and a suitably chosen kernel. It is thus argued that aggregation is driven by an effective short-ranged attraction, though the bare interactions are long-ranged.

**Gunter Schütz
**The Fibonacci family of dynamical universality classes

We use the universal nonlinear fluctuating hydrodynamics approach to study anomalous one- dimensional transport far from thermal equilibrium in terms of the dynamical structure function. Generically for more than one conservation law mode coupling theory is shown to predict a discrete family of dynamical universality classes with dynamical exponents which are consecutive ratios of neighboring Fibonacci numbers, starting with z = 2 (corresponding to a diffusive mode) or z = 3/2 (Kardar-Parisi-Zhang (KPZ) mode). If neither a diffusive nor a KPZ mode are present, all Fibonacci modes have as dynamical exponent the golden mean z=(1+\sqrt5)/2. The scaling functions of the Fibonacci modes are asymmetric Lévy distributions which are completely fixed by the macroscopic current-density relation and compressibility matrix of the system. The theoretical predictions are confirmed by Monte-Carlo simulations of a three-lane asymmetric simple exclusion process.

**Alexandra Tzella
**FKPP fronts in cellular flows: the large-Peclet regime

We investigate the propagation of chemical fronts arising in Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type models in the presence of a steady cellular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large Péclet number, $Pe \gg 1$) and arbitrary reaction rate (arbitrary Damköhler number $Da$).

We identify three regimes corresponding to the distinguished limits $Da = O(Pe^{-1})$, $Da=O\left((\log Pe)^{-1}\right)$ and $Da = O(Pe)$ and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of $Pe$ and $Da$. Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on $Da$ and $Pe$ and valid in intermediate regimes, are deduced as limiting cases. Taken together, our asymptotic results provide a complete description of the complex dependence of the front speed on $Da$ for $Pe \gg 1$. They are confirmed by numerical solutions of the eigenvalue problem determining the front speed, and illustrated by a number of numerical simulations of the advection-diffusion-reaction equation.

(Joint work with Jacques Vanneste, Edinburgh)

The Geometry of Rare Events

**Lutz Warnke
**Explosive percolation?

Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology). One of their most interesting features, both mathematically and in terms of applications, is the "phase transition": as the ratio of the number of edges to vertices increases past a certain critical point, the global structure changes radically, from only small components to a single macroscopic ("giant") component plus small ones.

An Achlioptas process is any one of a family of variants of the classical Erdos-Renyi process: starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. One might expect these processes to behave not too differently from the Erdos-Renyi one (which only adds random edges).

However, simulations of Achlioptas, D'Souza and Spencer suggested that for certain rules the percolation phase transition has a radically different form: more or less as soon as the macroscopic component appears, it is already extremely large. This phenomenon is known as 'explosive percolation'.

In this talk I will explain this striking and unusual phenomenon, and discuss some rigorous mathematical results obtained in joint work with Oliver Riordan; there are also many open questions.

**Michael Wilkinson
**Large deviations, rain showers and planet formation

Rainfall from ice-free cumulus clouds requires collisions of large numbers of microscopic droplets to create every raindrop. The onset of rain showers can be surprisingly rapid, much faster than the mean time required for a single collision. Large-deviation theory is used to explain this observation.

I shall also discuss applications of these results to planet formation. If planets grow by accretion, of dust particles must happen very rapidly because objects of roughly metre size spiral into the star over a timescale of less than a thousand years. Large deviation theory can be used to estimate the probability of a body growing sufficiently rapidly to avoid this fate.

**Oleg Zaboronski
**What is the probability that a large random matrix has no real eigenvalues?

In this talk I will discuss the calculation of the probability p(n) that a large real matrix with independent normal entries has no real eigenvalues. I will show how to guess the answer using the link between the law of real eigenvalues and a certain interacting particle system. I will then outline the steps of the proof of the result based on the determinantal expression for p(n).