ICTS

Energy current fluctuations and large deviation functions in harmonic crystals

I will discuss an exact method for computing the large deviation function for energy current in an arbitrary harmonic system connected to two thermal reservoirs at different temperatures. Some applications will be give.
 

Analytical methods for extreme fluctuations in generalised exclusion processes

The simple exclusion process is a well loved and well studied model for various reasons, one of them being that it is exactly solvable, and even sometimes exactly solved (depending on the physical objects of interest). In particular, the large deviation function of the current of particles flowing through the system can be obtained exactly from integrability techniques, and shows a few interesting dynamical phase transitions, including one from a hydrodynamic phase to a highly correlated one. Unfortunately, the methods used to analyse that phase transition are specific and limited to integrable models, and adding any extra feature to the simple exclusion process (such as a nearest-neighbour interaction, or site-dependent rates) breaks its integrability. After a brief introduction to the topic, the model, and the mathematical tools appropriate to analyse large deviations, I will show how one can still obtain analytical expressions of the large deviations of the current in generalised totally asymmetric exclusion processes (with site-dependent rates and extra interactions), in the limits of very large deviations. For the current going to infinity, the system behaves like free Fermions (more precisely, an XX spin chain), and the large deviation function of the current is extensive in the system size. For the current going to 0, the typical states of the system are similar to anti-shocks, and the large deviation function of the current can be shown to not depend on the size of the system. The different scaling behaviours indicate the existence of a dynamical phase transition similar to the one observed for the standard TASEP. If time allows, we will see how those methods apply to the large deviations of other observables, such as the average dynamical activity.

Asymptotic properties of the volatility estimator from high-frequency data modeled by mixed fractional Brownian motion

Properties of mixed fractional Brownian motion has been discussed by Cheridito (2001) and Zili (2006). We have proposed an estimator of volatility parameter for a model driven by MFBM. In our article, we have shown that the estimator has some desirable asymptotic properties.
 

Large deviations in non-equilibrium quantum statistical physics: From irreversible work to transport

Recent experimental advances 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. In this talk, I focus on the large deviations of the irreversible work done upon quenching a system and on the large deviations of the energy current which is generated when two systems at different initial temperatures are joined together. In particular, I show that the statistics of the work is unexpectedly connected with the physics of a classical system confined in a film geometry and that, in bosonic systems, a condensation transition may occur in the large deviations, while I discuss the energy current in the paradigmatic case of the quantum Ising chain.
 

Breaking of ensemble equivalence in complex networks

In this talk I will speak about the phenomenon of breaking of ensemble equivalence in complex networks. For many system in statistical physics the microcanonical and canonical ensemble are equivalent in the thermodynamic limit, but not for all. The goal is to classify for which classes of complex networks with topological constraints, breaking of ensemble equivalence occurs. We consider the simple case in which we fix the number of links and then we move to the configuration model (we fix the degree of each vertex). Then we study a more general setting with an arbitrary number of intra-connected and inter-connected layers, thus allowing modular graphs with a multi-partite, multiplex, time-varying, block-model or community structure. We give a full classication of ensemble equivalence in the sparse regime, proving that break- down occurs as soon as the number of constrained degrees is extensive in the number of nodes, irrespective of the layer structure. In addition, we derive an explicit formula for the specic relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.
 

Exact extremal statistics in the classical 1d Coulomb gas

In the last decade, a new universal law called the Tracy-Widom distribution has emerged, which describes the typical fluctuations of the extremum in several strongly interacting systems. But how sensitive is this universality against the pair interactions ? In this talk, I will address this question in the context of extreme statistics in a many-body system interacting via the one dimensional linear Coulomb repulsive potential. We found that the extreme statistics is this system is different from that described by the Tracy-Widom distribution however, the large deviation functions still predict a third order transition.

 

The circular law for sparse non-Hermitian random matrices

Sparse matrices are abundant in statistics, neural network, financial modeling, electrical engineering, and wireless communications. In the regime of sparse non-Hermitian random matrices, I will describe our work that establishes the celebrated circular law conjecture. The conjecture states that the empirical spectral distribution (ESD) of a (properly scaled) matrix with i.i.d.~entries of zero mean and unit variance converges to the uniform measure on the unit disk in the complex plane, as $n$, the dimension of the matrix increases. In the dense regime, after a series of partial results, the conjecture was established in a seminal work by Tao-Vu.

For sparse random matrices, such as the matrix with i.i.d.~Ber$(p_n)$ entries, where $np_n$ grows at a rate sub-polynomial in $n$, the method of Tao-Vu fails due to the presence of a large number of zeros. In case of the adjacency matrix of a $d_n$-regular directed random graph on $n$ vertices, where $d_n=o(n)$ there is an additional difficulty of dependencies within the entries. I will describe new approaches to handle the sparsity and the dependency thereby yielding the circular law limit for the ESDs of these matrices.

This talk is based on joint works with Nicholas Cook, Mark Rudelson, and Ofer Zeitouni. 

Modeling of time evolution of temperature profile in a stochastic momentum conserving system through fractional diffusion equation

We model the temperature evolution in stocastic momentum exchange model in 1D classical harmonic chain. We find a description based on fractional diffusion gives a good agreement with simulation of microscopic dynamics . We construct eigenspectrum of fractional operator and describe time evolution with it. If time permits, I will describe how we can use these ideas to study long range correlations in the system.
 

A disordered open long-range exclusion process

We will consider a long-range exclusion process with site-disorder on a directed tree (i.e. an arborescence) where particles enter at the leaves, make their way on the unique path to the root, and exit at the root. We will consider the simultaneous currents at every single bond, and obtain a closed form expression for the joint cumulant generating function. We will also consider the system when atypical currents are observed at these bonds, and prove properties for the quasistationary process leading to such currents. We will invoke results obtained jointly with A. Shilling, N. Thiery and B. Steinberg.
 

Extreme value statistics in a gas of 2d charged particles

We study a system of N charged particles in two dimensions with Coulomb logarithmic repulsion and confined in an external symmetric potential. At the inverse temperature of interest β = 2, the positions of the charge form a 2d determinantal process. In the case of a quadratic potential, there is a mapping between the positions of these charges and the eigenvalues of complex Ginibre matrices. We focus on the extremal statistics of the positions of the charges and in particular we highlight a new universal regime (with respect to a large class of confining potentials) which had been overlooked before. It allows to solve a puzzle of matching between the typical regime of fluctuations and the large deviation regime. Finally we also considered potentials that deviates from this universality class and computed the extremal statistics in these cases.

Joint work with: Aur ́elien Grabsch, Satya N. Majumdar and Gr ́egory Schehr.
 

Dynamical fluctuations in classical and quantum systems far from equilibrium

In this talk we will present some results derived from the study of dynamical fluctuations in non-equilibrium diffusive systems. We will explain how by applying the Macroscopic Fluctuation Theory, interesting properties such as dynamical phase transitions can be described. In addition, we will show that the dynamics of a class of open quantum systems can be described in terms of fluctuating hydrodynamics, where fermionic open chains display a dynamical phase transition similar to that of classical exclusion processes with open boundaries. This transition is manifested in a singular change in the structure of trajectories: while typical trajectories are diffusive, rare trajectories associated to atypical currents are ballistic and hyperuniform in their spatial structure.
 

Diffusion versus Superdiffusion in a stochastic Hamiltonian lattice field model

We consider a Hamiltonian lattice field model that we perturb by a conserving stochastic noise. Depending on the form of the noise and of the interaction potential, superdiffusion or diffusion is expected. I will discuss how we can interpolate different regimes by tuning the parameters of the model.
 

Non-equilibrium free energy not too far from equilibrium: three examples from passive and active matter

This talk will be divided in two parts. In the first, I will discuss a general theory that allows to compute perturbatively in a small parameter the non-equilibrium free energy of many-body systems, for example when the system is not too far from equilibrium. I will then apply it to two cases: mean-field interacting diffusions and a minimal model of self-propelled particles. In this second case, our approach allows to show that the system is described by a non Botzmann-Gibbs steady state but, at the same time, entropy production vanishes. In the second part of the talk, I will concentrate on field theories which describe phase-separation in systems composed of self-propelled particles, such as bacteria or active colloids. These field theories are minimal modification of Model B (stochastic Cahn-Hilliard equation). I will discuss that even a small amount of activity, i.e. a minimal perturbation from an equilibrium dynamics, can strongly modify the nucleation dynamics.
 

Properties of the Adaptive Multilevel Splitting algorithm

The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile method for the simulation of rare events, such as metastable transitions of diffusion processes. The approach is not well-known, however I will present recent mathematical results concerning its consistency and efficiency, as well as several algorithmic variants. In particular, I will discuss how Large Deviations techniques may be used to analyze the efficiency of the method, with rigorous results in a specific case.

References:

1. Large deviations principle for the Adaptive Multilevel Splitting Algorithm in an idealized setting, CE Bréhier, ALEA, 2015.
2. Unbiasedness of some generalized Adaptive Multilevel Splitting algorithms, CE Bréhier, M Gazeau, L Goudenège, T Lelièvre, M Rousset, The Annals of Applied Probability, 2016.
   
    

Renewal processes with long tails

This is a review of past and recent works on renewal processes when the distribution of time (or space) intervals between renewals have a power-law tail. Such renewal processes are natural generalisations of the excursions of Brownian motion (or of the Brownian bridge). The distributions of the occupation time, of th e longest interval, and correlations are known exactly.
 

Quenched large deviations for random motions in degenerate random media

We prove a quenched large deviation principle for a simple random walk on percolation models. The models under interest include classical Bernoulli bond and site percolation as well as the models that carry long range correlations, like the random cluster model, random interlacements, vacant sets of random interlacements and the level sets of the Gaussian free field. We take the point of view of the moving particle and prove a quenched LDP for the distribution of the pair empirical measures of the environment Markov chain. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas.

If time permits, we will also discuss results in the continuum, which translates to a homogenization statement for degenerate Hamilton-Jacobi-Bellman equations.

Based on joint works with Noam Berger and Kazuki Okamura, as well as Alexander Mielke.
 

Dirichlet and Thomson principles for nonselfadjoint elliptic operators

We present a general method to derive a reduced description of a Markov chain which exhibits a metastable behavior.
 

Stochastic efficiency of an isothermal work-to-work converter engine

We investigated the efficiency of an isothermal Brownian work-to-work converter engine, composed of a Brownian particle coupled to a heat bath at a constant temperature. The system is maintained out of equilibrium by using two external time-dependent stochastic Gaussian forces, where one is called load force and the other is called drive force. Work done by these two forces are stochastic quantities. The efficiency of this small engine is defined as the ratio of stochastic work done against load force to stochastic work done by the drive force. The probability density function as well as large deviation function of the stochastic efficiency are studied analytically and verified by numerical simulations.
 

Rare events in some fat tailed systems

We investigate rare fluctuations in diffusive systems whose typical bulk behavior is described by long tailed L ́evy statistics. Many diffusive systems exhibit a bi-fractal behavior also called strong anomalous diffusion [1]. Here <|x(t)|^q>i ∼ t^qν(q) and qν(q) is a bi-linear function of q. Using the well-known L ́evy walk model we show how this is related to infinite covariant densities, i.e., non normalised states that describe the rare fluctuations [2]. These infinite densities are complementary to the generalised central limit theorem. Such behaviour is found in several microscopic models including transport of cold atoms in optical lattices [3], the Lorentz gas with infinite horizon, and experiments following active tracers in the live cell.

References
[1] P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, and A. Vulpiani, On strong anomalous diffusion Physica D 134, 75 (1999).
[2] A. Rebenshtok, S. Denisov, P. H ̈anggi, and E. Barkai Non-normalizable densities in strong anomalous diffusion: beyond the central limit theorem Phys. Rev. Letters 112, 110601 (2014). ibid Infinite densities for L ́evy walks Phys. Rev. E. 90, 062135 (2014).
[3] E. Aghion, D. A. Kessler, and E. Barkai Large-fluctuations for spatial diffusion of cold atoms Phys. Rev. Lett. (in press). See: arXiv:1701.03357 [cond-mat.statmech].
 

Infinite-density versus large deviations theory for fat-tailed systems

TBA
 

Large deviations of the cover time for the random walk on the torus and related questions

The cover time is the time needed to visit all points on the lattice of size n. As a maximum of correlated random variables (here, with logarithmic decay) it has interesting asymptotics. In dimension 2, recurrence of the walk induces strong correlations, and the large deviations behavior is affected.
 

Random walk in diffusive random environment

I consider the long time behavior of a random walker evolving in a one-dimensional, diffusive, random environment. If the environment is unbiased, the waker has no drift and the main question is about fluctuations. This problem turns out to be very much puzzling: the correlations of the environment decay slowly in time but their effect on the behavior of the walker is hard to predict. I will present a heuristic theory, supported by numerical results, that leads to predictions for the fluctuations of the walker as well as its differential mobility (response to an infinitesimal external force).
 

Large deviations for the Wiener Sausage

The Wiener sausage is the 1-environment of Brownian motion. It is an important mathematical object because it is one of the simplest non-Markovian functionals of Brownian motion. The Wiener sausage has been studied intensively since the 1970's. It plays a key role in the study of various stochastic phenomena, including heat conduction, trapping in random media, spectral properties of random Schr"odinger operators, and Bose-Einstein condensation.

In these lectures we look at two specific quantities: the volume and the capacity. After an introduction to the Wiener sausage, we show that both the volume and the capacity satisfy a downward large deviation principle. We identify the rate and the rate function, and analyse the properties of the rate function. We also explain how the large deviation principles are proved with the help of the skeleton approach.

Joint work with Michiel van den Berg (Bristol) and Erwin Bolthausen (Zurich).
 

Large deviation theory applied to climate dynamics

We will review some of the recent developments in the theoretical and mathematical aspects of the non-equilibrium statistical mechanics of climate dynamics. At the intersection between statistical mechanics, turbulence, and geophysical fluid dynamics, this field is a wonderful new playground for large deviation theory. It involves large deviation theory for stochastic partial differential equations, homogenization, and rare event algorithms. We will discuss two paradigmatic examples. First, we will study extreme heat waves, as an example of rare events with a huge impact. We will explain how algorithms based on large deviation theory allow to sample rare heat wave at a numerical cost that is several orders of magnitude smaller than a direct numerical simulation. This will improve drastically the study of those events dynamics, in climate models, in relation with climate change.A a second example, we will study rare trajectories that suddenly drive turbulent flows from one attractor to a completely different one, in the stochastic barotropic quasigeostrophic equation. This equation, a generalization of the stochastic two dimensional Navier–Stokes equations, models Jupiter's atmosphere jets. We discuss preliminary steps in the mathematical justification of the use of averaging, derive an effective action for large deviations, compute transition rates through Freidlin–Wentzell theory, and instantons (most probable transition paths). This talk is based on works with Francesco Ragone and Jereon Wouters (heat waves) and Brad Marston, Eric Simonnet, Tomas Tangarife, and Eric Woillez (large deviations for quasi geomorphic turbulence).
 

Current and entropy production large deviations in periodically driven systems

Current fluctuations in periodically driven systems can be robustly mapped to fluctuations in a corresponding time-independent nonequilibrium steady state. Doing so requires, however, the construction of a steady state process in which the average entropy production, as well as its fluctuations, is distinct from the time-periodic process. Nevertheless, the construction reveals a distinction between the physical origin of the entropy production in the two types of systems. We further show that the mapping gives rise to a level-2.5 large deviation function for the time-periodically driven system in the limit of weak driving. As a consequence of the decomposition of entropy production, the current fluctuations in this limit are shown to satisfy a universal bound determined by the steady state entropy production.
 

On the discretization of Feynman-Kac semi-groups for diffusions

Feynman-Kac semi-groups naturally appear in the context of large deviations. In particular, their long time behavior provides the scaled cumulant generating function, which is the dual of the rate function. However, their practical implementation is difficult in practice. Several work address their statistical approximation but, for diffusions, one also has to discretize the process in time. In this context, I will present a framework based on Talay-Tubaro expansion that provides error estimates on the long-time behavior of such discretized processes. We will see that a novelty of this work is to take into account the non-probability conserving feature of the dynamics, which provides an alternative expression for the cumulant function. Our analysis is supported by numerical applications.
 

Large deviations for non-interacting trapped fermions

I will consider the (quantum) spatial fluctuations of N non-interacting fermions in an isotropic d-dimensional trapping potential at zero temperature. I will study the maximal radial distance, $r_{\max}$, of the fermions from the trap center and focus on the large deviations of $r_{\max}$ away from its typical position, both to the right (right tail) and to the left (left tail). In d=1, this question can be studied, in several cases, thanks to an exact mapping to random matrix models, where such large deviations regimes have been well studied in the recent past. I will show that in d>1 the large deviation regime to the left exhibits a quite unusual, and rather universal, intermediate regime. This intermediate regime can be studied in detail using the tools of determinantal point processes.
 

Statistical mechanics of assembly of particles activated by non-Gaussian noise

Large deviation nature is characterized by non-Gaussian statistics. The simplest model to characterize such systems is the motion of particles activated by non-Gaussian white noise. I also stress the effects of roles of non-analytic friction such as Coulombic dry friction. In the former half part, I am going to talk about the relevant condition for the non-Gaussian noise which means the violation of the central limit theorem even in long time limit observation of noises. In the later half part, I will talk on many-body effects of particles activated by non-Gaussian noise, in which effective attractive interaction appears.
 

Large deviations of Markov processes

I will give in these lectures an overview of techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, for Markov processes (jump processes or diffusions) modelling equilibrium and nonequilibrium processes driven by external forces and noise. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This will be covered in the first lecture. In the second lecture, I will discuss more recent research describing how fluctuations of dynamical observables are created in time by means of an 'effective' process, called the driven process. This process can be seen as a process generalisation of the concept of fluctuation paths, introduced by Onsager and Machlup (and later Freidlin and Wentzell) to describe noise-activated transitions in the low-noise limit.
 

Carnot efficiency in an irreversible process

In thermodynamics, there exists a conventional belief that ``the Carnot efficiency is reachable only in the reversible (zero entropy production) limit of nearly reversible processes.'' However, there is no theorem proving that the Carnot efficiency is unattainable in an irreversible process. Here, we show that the Carnot efficiency is reachable in an irreversible process through investigation of the Feynman-Smoluchowski ratchet (FSR). We also show that it is possible to enhance the efficiency by increasing the irreversibility.Our result opens a new possibility of designing an efficient heat engine in a highly irreversible process and also answers the long-standing question of whether the FSR can operate with the Carnot efficiency.
 

Large deviations for Markov processes with resetting

Resetting in the context of stochastic processes is a way to model catastrophes in birth-death processes or the clearing of queues due to failure of a server in queueing theory. Brownian motion which is randomly reset to its starting position is a continuous space analogue and can be used to model search strategies where diffusive searching is combined with randomly returning to the starting point. (e.g. lost keys) In this talk I will introduce the mathematical formulation of resetting and show how this modifies the Fokker-Planck equation. The main result of the paper is a simple formula relating the Laplace transform in time of the generating function of time-additive observables of the process with resetting to the same object for the process without resetting. This formula makes it possible to study the large deviations of the process with resetting using the large deviations of the process without resetting which I will demonstrate for the Ornstein-Uhlenbeck process.
 

Large deviations and stochastic stability

When subjected to small additive stochastic noise, a dynamical system's large deviations are independent of the initial conditions, and the fluctuation theorems are trivial. The limit of zero noise may be delicate, especially when one deals with a strongly driven Hamiltonian system. The "chaotic hypothesis" may in some cases be replaced by the "stochastic stability" one, i.e. the assumption that the deterministic limit is continuous.
 

Large deviations and quantum non-equilibrium

I will review recent developments in applying dynamical large deviations (LD) to quantum systems. I will consider in particular open quantum systems - quantum systems interacting with an environment - which in many cases can be described in terms of quantum Markovian dynamics. LD methods provide a "thermodynamic" framework for understanding the statistical properties of dynamics, revealing the existence of dynamical phases and phase transitions, often associated to intermittent emission patterns. Problems of interest include interacting atomic ensembles, quantum glasses, and systems where there is an interplay between coherent transport and dissipation. I will describe concepts relating to quantum trajectory ensemble equivalence, prediction and "retrodiction", matrix product states, and quantum Doob transforms. Time permitting, I will also discuss the connection to ideas about slow quantum relaxation and metastability.
 

Regenerative sequences and processes and MCMC

A sequence of random variables is said to be regenerative if it can be broken in to iid components.Similarly for continuous time processes. In this talk we shall produce many examples of such sequences. Also we shall prove a law of large numbers for such sequences.
We shall give a new proof of Robbins - Kallianpur ergodicthem for one dimensional SBM We shall also introduce a new Monte Carlo procedure for estimating on tegrals on  infinite measure spaces.
 

Experimental Determination of Dynamical Lee-Yang Zeros

Conventional phase transitions involve abrupt changes of a macroscopic system in response to small variations of an external control parameter. This exceptional behavior can be understood from the complex zeros of the partition function of the finite-sized system: in the thermodynamic limit, these Lee-Yang zeros, which correspond to logarithmic singularities of the free energy, approach the critical value of the control parameter on the real axis. This general scheme also applies to dynamical phase transitions in non-equilibrium systems. The partition function is thereby replaced with the moment-generating function of a stochastic process with the counting field playing the role of the external control parameter. Here, we demonstrate that the corresponding dynamical Lee-Yang zeros are not only a theoretical concept but physical observables, which encode remarkable information on the long-time statistics and the dynamical fluctuations of the system [1]. To this end, we analyze a stochastic process involving Andreev-tunneling events in a mesososcopic device consisting of a normal-state island and two superconducting leads. From measurements of the dynamical activity, we extract the Lee-Yang zeros, which reveal a smeared dynamical phase transition outside the range of direct observations. Being obtained only from short-time data, this information allows us to predict the large-deviation statistics of the dynamical activity at long times, which is otherwise difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.
 

Topological localization in out-of-equilibrium dissipative systems

We extend notions of topological protection to stationary configurations that are driven far from equilibrium by active, dissipative processes. We show this for physically two disparate cases : stochastic networks governed by microscopic single particle dynamics as well as collections of driven, interacting particles described by coarse-grained hydrodynamic theory. We show this by mapping to a well-known one-dimensional model for electrons on a lattice (the SSH model) and by exploiting the resulting correspondence between the value of a bulk topological winding number and the steady state solutions localized at the boundary. In both cases, the presence of dissipative couplings to the environment that break time reversal symmetry are crucial to ensuring topological protection. These examples constitute proof of principle that notions of topological protection, established in the context of electronic and mechanical systems, can extend to processes that operate out of equilibrium. Such topologically robust boundary modes have implications for both biological and synthetic systems.
 

Macroscopic Fluctuations, In and Out of Equilibrium

Many natural systems are far from thermodynamic equilibrium and keep on exchanging matter, energy or information with their surroundings. These exchanges produce currents, or fluxes, that break time-reversal invariance. Such systems lie beyond the realm of traditional thermodynamics and the principles of equilibrium statistical mechanics do not apply to them. In fact, there exists no general conceptual framework a la Gibbs-Boltzmann to describe these systems from first principles.

The last two decades, however, have witnessed remarkable progress. The aim of this lecture is to explain some recent developments, such as the Macroscopic Fluctuation Theory (MFT) of Bertini, Gabrielli, Da Sole, Jona-Lasinio and Landim. This theory represent the first steps towards a unifi ed approach to non-equilibrium behaviour.

After reviewing the approach of Onsager and Machlup of fluctuations in the vicinity of equilibrium, we shall explain the basis of the MFT from a "user's approach". We shall give examples of lattice gases for which the transport coefficients can be exactly evaluated and describe some applications to the calculations of large deviation functions of currents and density profiles.
 

States with non-uniform temperature profiles in conformal field theory

I shall discuss a simple exact description of non-equilibrium states with non-uniform temperature (or chemical potential) profiles in (1+1)D conformal field theory.
 

Canonical structures and orthogonality of forces and currents in irreversible Markov cha

We consider dynamical large deviations in Markov chains.  The associated rate functionals are not quadratic, leading to non-linear flux-force relations. We discuss a canonical decomposition of the rate function based on conjugate forces and fluxes, and we also split the `force' into a time-reversal-symmetric (equilibrium) part, and an anti-symmetric (non-equilibrium) part [1].  These two parts of the force satisfy a generalised orthogonality condition. We discuss the implications of these results, and their connections to Macroscopic Fluctuation Theory [2] and to previous work by Maes and co-workers [3].

References:
[1] Kaiser, Marcus, Robert L. Jack, and Johannes Zimmer, arXiv:1708.01453 (2017).
[2] Bertini, Lorenzo, et al., Reviews of Modern Physics 87.2 (2015): 593.
[3] Maes, Christian, and Karel Netočný, EPL (Europhysics Letters) 82.3 (2008): 30003.
 

Breaking of Ensemble Equivalence in dense random graphs

In this talk I will consider a random graph on which topological restrictions are imposed, such as con- straints on the total number of edges, wedges, and triangles. I will consider dense graphs, in which the number of edges per vertex scales proportionally to the number of vertices n. My goal is to compare the micro-canonical ensemble (in which the constraints are satised for every realisation of the graph) with the canonical ensemble (in which the constraints are satised on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as n grows large, where two ensembles are said to be equivalent in the dense regime if this relative entropy divided by n^2 tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints satisfy a certain terseness condition. Examples are provided for three dierent choices of constraints.
 

Order and symmetry-breaking in the fluctuations of driven systems

Dynamic phase transitions (DPTs) in the space of trajectories are one of the most intriguing phenomena of nonequilibrium physics, but their nature in realistic high-dimensional systems remains puzzling. Here we observe for the first time a DPT in the current vector statistics of an archetypal two-dimensional ($2d$) driven diffusive system, and characterize its properties using macroscopic fluctuation theory. The complex interplay among the external field, anisotropy and vector currents in $2d$ leads to a rich phase diagram, with different symmetry-broken fluctuation phases separated by lines of $1^{\text{st}}$- and $2^{\text{nd}}$-order DPTs. Remarkably, different types of $1d$ order in the form of jammed density waves emerge to hinder transport for low-current fluctuations, revealing a connection between rare events and self-organized structures which enhance their probability.
 

Large deviations in the context of heavy tails

This series of lectures will focus on large deviations for random variables with power law (or more generally, regularly varying) distributions. Because of polynomial decay of the tail, rare events happen in many cases due to the large value of one random variable. This phenomenon is called one large jump principle, which will be explained in these lectures. Major difference with classical large deviation theory for light-tailed distributions will also be discussed.
 

Memory effects in Kardar Parisi Zhang growth: exact results via the replica Bethe ansatz

We review recent progress in describing the statistics of height fluctuations in 1D Kardar Parisi Zhang (KPZ) growth, focusing on the KPZ equation and its integrability properties via the mapping onto the Lieb Liniger model of impenetrable bosons. We recall the replica Bethe Ansatz method and how it allows to calculate one time probabilities, and shows the emergence of the Tracy Widom distributions of random matrix theory. We then study the two-time problem, the so called aging problem, which is still outstanding: the aim is to obtain the joint probability distribution of heights at time t and t', in the limit of large times with fixed ratio t/t'>1. We provide a partial solution of this problem, exact in some limits. In particular we derive the exact form of the persistent correlations in the limit t/t' large, which quantifies the memory effect in the time evolution, also called ergodicity breaking. Comparison with experiments and numerics shows a very nice agreement.
Most is joint work with J. de Nardis and K. Takeuchi.
 

Velocity statistics of Granular gases

Granular gases are less dense systems of Granular particles. Though well studied, one is yet to understand the nature of the velocity distributions in these systems when driven. In this talk, I will discuss some of our recent works in this direction and how these could be helpful in understanding generic features in the velocity statistics of driven granular gases
 

Exact stationary state solution of one-dimensional heat transport model with two conserved quantities

In this talk, I will present the exact stationary state solution of stochastic harmonic heat transport model with two conserved quantities in one-dimensional. I will show the calculation for the temperature profile in the stationary state which unusually show asymmetric behavior. I will also present the exact expression for energy current which decays with system size as $N^{-1/2}$.
 

Queues and large deviations in stochastic models of gene expression

In several biological systems, phenotypic variations are seen even among genetically identical cells in homogeneous environments.  Recent research indicates that such `non-genetic individuality' can be driven by rare events arising from the intrinsic stochasticity of gene expression. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression.

In this talk, I will discuss analytical approaches1 developed by my group for stochastic models of gene expression.  By developing a mapping to systems analyzed in queueing theory2, we derive analytical results characterizing mRNA and protein distributions for general kinetic schemes of gene expression. We combine approaches from queueing theory and non-equilibrium statistical mechanics to characterize large deviations and driven processes for general models of gene expression3. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.

References:

1. H. Pendar, T. Platini, and R. V. Kulkarni. “Exact protein distributions for stochastic models of gene expression using partitioning of Poisson processes” Phys Rev E, 87, 042720 (2013)
2. T. Jia and R.V. Kulkarni “Intrinsic noise in stochastic models of gene expression with molecular memory and bursting” Phys. Rev. Lett., 106, 058102 (2011)
3. J. Horowitz and R. V. Kulkarni “Stochastic gene expression conditioned on large deviations” Phys. Biol. Lett. (accepted for publication) (2017)
   
    

Regular Variation and Branching Random Walk

Branching random walk with regularly varying displacements is a heavy-tailed random field indexed by a Galton-Watson tree. We shall connect it to stable point processes introduced and characterized by Davydov, Molchanov and Zuyev (2008), who showed that such point processes can always be represented as a scale mixture of iid copies of one point process with the scaling points coming from an independent Poisson random measure. We obtain such a point process as a weak limit of a sequence of point processes induced by a branching random walk with regularly varying displacements. In particular, we show that a prediction of Brunet and Derrida (2011), remains valid in this setup, and recover a slightly improved version of a result of Durrett (1983). This talk is based on joint works with Ayan Bhattacharya and Parthanil Roy.
 

On a local Lyapunov function for the McKean-Vlasov dynamics

The McKean-Vlasov dynamics is a large system limit of the evolution of empirical measures of a weakly interacting system of particles. The empirical measure creates a field which then influences the evolution of any single particle. The process of empirical measures is itself a Markov process which has relative entropy with respect to the invariant measure as a natural Lyapunov function. By scaling this Lyapunov function and taking the infinite particle limit, we propose to arrive at a local Lyapunov function for the McKean-Vlasov dynamics. The approach is conceptually simpler than a prior approach, naturally identifies the desired Lyapunov function, and clarifies why the Freidlin-Wentzell quasipotential associated with the interacting system is a local Lyapunov function. (The talk is based on ongoing work with Laurent Miclo.).
 

On Gibbs-Shannon Entropy

In this talk, I will speak on second principle and Landauer bound. Despite some old and classical formulations, some points are still source of interest and debates. By example the physical contents of the Gibbs-Shannon entropy outside equilibrium. In peculiar, I will explain the theoritical background under recent experimental measure of the Gibbs-Shannon Entropy associated to a bit by Gavrilov-Chetrite-Bechhoefer.
 

Dynamical fluctuations in systems with and without detailed balance

We compare dynamical fluctuations in stochastic processes with and without detailed balance. For general Markov chains, there are explicit rate functions for the joint large deviations of the empirical current and density, which can be derived either in a long-time limit (so-called level-2.5) or by considering many copies of the Markov chain (an ensemble) [1]. For some Markov processes, such as exclusion processes, the behaviour on large length and time scales can be described by macroscopic fluctuation theory (MFT) [2]. This theory also provides formulae for rate functions of the current and density, as well as a decomposition of the current into two orthogonal contributions that are symmetric and anti-symmetric under time reversal. In the context of sampling by Markov chain Monte Carlo (MCMC), a number of recent results indicate that systems where detailed balance is broken generically converge more rapidly to their steady states, leading to improved sampling efficiency [3]. We show how this result is related to the existence of two orthogonal currents within MFT [4]. Then, we show how a similar pair of currents can be defined in general Markov chains, together with their conjugate forces [5]. This last result shows how some aspects of MFT are already present within generic Markov chains, as well as providing new insight into the improved sampling efficiency that is available on breaking detailed balance.

[1] Maes, Netocny, Wynants, Markov Proc. Rel. Fields 14, 445 (2008)
[2] Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim, Rev Mod Phys 87, 593 (2016)
[3] Rey-Bellet, Spiliopoulos J Stat Phys 164, 472 (2016)
[4] Kaiser, Jack, Zimmer, J Stat Phys 168, 259 (2017)
[5] Kaiser, Jack, ZImmer, in preparation.
 

Current fluctuations and dynamical phase transitions

I will present some recent results on current fluctuations in non-equilibrium stochastic processes, focusing particularly on the way in which temporal correlations can lead to dynamical phase transitions corresponding to qualitatively different mechanisms for rare current events. In this context, I will emphasize analogies with known equilibrium phenomenology such as the case of dynamics subject to intermittent reset where it turns out phase transitions can be characterized via a mapping to a fifty-year-old model of DNA denaturation [joint work with H. Touchette]. I will also discuss more briefly a many-particle "queueing" model where temporal correlations promote the condensation of particles [joint work with M. Cavallaro and R. Mondragón].
 

The use of Large Deviation Theory in Non-equilibrium statistical Mechanics

Large deviation theory can be an important tool in statistical mechanics, particularly in the transition from microscopic to macroscopic description of the evolution of a large system over time. We will examine some examples.
 

Percolation, spatial minimal spanning trees, and Stein's method

Kesten and Lee [Ann. Appl. Probab. (1996)] proved that the total length of a minimal spanning tree on certain random point configurations in \mathbb{R}^d satisfies a central limit theorem. They also raised the question: how to make these results quantitative? Error estimates in central limit theorems satisfied by many other standard functionals studied in stochastic geometry are known, but techniques employed to tackle the problem for those functionals do not apply directly to the minimal spanning tree. Thus, the problem of determining the convergence rate in the central limit theorem for Euclidean minimal spanning trees had remained open. We discuss a general technique for approaching this problem and establish bounds on the convergence rate, thus answering the question of Kesten and Lee. We also discuss a way of quantifying the classical Burton-Keane argument for uniqueness of the infinite open percolation cluster, which plays a crucial role in our approach. Based on joint work with Sourav Chatterjee.
 

Selecting the best population using large deviations and multi-armed bandit methods

Consider the problem of finding a population amongst many with the smallest mean when these means are unknown but population samples can be generated. Typically, by selecting a population with the smallest sample mean, it can be shown that the false selection probability decays at an exponential rate. Lately researchers have sought algorithms that guarantee that this probability is restricted to a small d in order log(1/d) computational time by estimating the associated large deviations rate function via simulation. We show that such guarantees are misleading. Enroute, we identify the large deviations principle followed by the empirically estimated large deviations rate function that may also be of independent interest. Further, we show a negative result that when populations have unbounded support, any policy that asymptotically identifies the correct population with probability at least 1-d for each problem instance requires more than O(log(1/d)) samples in making such a determination in any problem instance. This suggests that some restrictions are essential on populations to devise O(log(1/d)) algorithms with 1-d correctness guarantees. We note that under restriction on population moments, such methods are easily designed. Further, under similar restrictions, sequential methods from multi-armed bandit literature can also be adapted to devise such algorithms.
 

Fluctuation theorem for entropy production of a partial system in the weak coupling limit

Small systems in contact with a heat bath evolve by stochastic dynamics. We show that, when one such small system is weakly coupled to another one, it is possible to infer the presence of such weak coupling by observing the violation of the steady state fluctuation theorem for the partial entropy production of the observed system. We give a general mechanism due to which the violation of the fluctuation theorem can be significant, even for weak coupling. We analytically demonstrate on a realistic model system that this mechanism can be realized by applying an external random force to the system. In other words, we find a new fluctuation theorem for the entropy production of a partial system, in the limit of weak coupling.
 

Large deviation principle and beyond in a single-electron box

We measure the full distribution of current fluctuations in a single-electron box with a controllable bistability. The conductance switches randomly between two levels due to the tunneling of single electrons in a separate single-electron box. The electrical fluctuations are detected over a wide range of time scales and excellent agreement with theoretical predictions is found. For long integration times, the distribution of the time-averaged current obeys the large-deviation principle.  We formulate and verify a fluctuation relation for the bistable region of the current distribution.
 

Some Remarks on non-local Schrodinger Operators

In this talk we shall introduce non-local operators that come from jump diffusions and prove a Harnack Inequality for the Schrodinger operator. We shall present some open problems related to large deviations. This is joint work with Koushik Ramachandran.
 

(Very) Large Deviations for the Totally Asymmetric Exclusion Process​

This is a joint work with Li-Cheng Tsai (Columbia University). We prove a large deviation principle from the hydrodynamic limit of the TASEP, for deviations that have log-probability of order $-N^2$. The rate function is null on sub-solutions of the Burgers equation.
 

Large deviation study of random field disorder in magnets

The effect of quenched disorder on the behavior of a system, whose pure version undergoes a first order transition is well understood by now for d<=2. For two dimensions, an infinitesimal amount of disorder is known to change the order of transition to a continuous transition. The situation in higher dimensions is not so clear. In higher dimensions, it was conjectured that there exist a impurity threshold beyond which the transition should change its character[2]. We solve $p$ spin interaction model and Blume Capel model in the presence of random field disorder on fully connected graphs using large deviation theory. For $p$ spin interaction model, we find that for p>= 3, the first order transition continues to be a first order for all strengths of disorder. While for Blume Capel model with bimodal random crystal field, we find the behavior is modified: for very weak disorder, the phase diagram is similar to the pure model with regions of first and second order transitions separated by a tricritical point. With increasing strength of disorder, the transition soon becomes a continuous transition.
 

Dissipation-based uncertainty bounds for currents

Markov jump processes can generate steady-state probability currents at the expense of dissipation. I will discuss how the dissipation also constraints the fluctuations in those currents. A small deviations corollary proves a "thermodynamic uncertainty principle"---to reduce the uncertainty in the estimate of nonequilibrium currents, a process must dissipate more. I will further discuss corresponding dissipation-based uncertainty bounds for first passage time fluctuations.
 

Algorithms for computational statistical physics: a mathematical viewpoint

Computational statistical physics consists in, starting from a microscopic model of matter, evaluating macroscopic properties as ensemble averages. There are two types of quantities: (i) thermodynamic quantities are averages with respect to probability measures over the configurational space (canonical measure, microcanonial measure, etc...) ; (ii) dynamical quantities are obtained by sampling trajectories. Both are evaluated using probabilistic methods, as averages over stochastic processes. In terms of numerical computations, the difficulty to estimate these quantities is that the underlying stochastic processes are metastable: they remain trapped for very long times in some regions of the phase space, called metastable regions. Transitions between metastable regions are rare events. We will explain how this problem is circumvented in practice by appropriate numerical methods, and how to analyze mathematically these methods using various mathematical tools: entropy techniques, quasi-stationary distribution, semi-classical analysis, etc...
 

Long-range correlations in non-equilibrium diffusive systems

Systems driven out of equilibrium often reach stationary state which under generic conditions exhibit long-range correlation. Such correlations have been observed both in experiment in fluid medium as well as in theoretical models. I shall present a fluctuating hydrodynamics description of interacting particles in which both spatial and temporal correlation can be obtained systematically to arbitrary order. The formulation is an application of the Macroscopic Fluctuation Theory. The long-range nature of the correlation can be seen in an electrostatic analogy where the correlation is potential due to localized charge which is non-zero when detailed balance is violated. To verify the hydrodynamic results, I shall derive spatial and temporal correlations in an exact solution of the symmetric exclusion process. Our exact solution also presents a direct verification the fluctuating hydrodynamics equation. As an application of our results, I shall discuss a generalization of the fluctuation dissipation relation in non-equilibrium stationary state where the response function is expressed in terms of two-time correlations.

References:

1. Tridib Sadhu and Bernard Derrida, Correlations of the density and of the current in non-equilibrium diffusive systems, J Stat Mech (2016) 113202.
2. Tridib Sadhu, Satya N Majumdar, and David Mukamel, Long-range correlations in a locally driven exclusion process, Phys. Rev. E. 90, (2014) 012109.
   
    

Small noise limits in the stationary regimes

This talk will describe a control theoretic approach to large deviations associated with small noise limits in the stationary regimes. The two cases discussed will be stationary diffusions in the whole space and McKean-Vlasov type mean field limits for interacting continuous time Markov chains.
 

Finite-size scaling of a first-order dynamical phase transition: adaptive population dynamics and an effective model

We analyze large deviations of the time-averaged activity in the one-dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multicanonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts. Work in collaboration with Takahiro Nemoto, Robert L. Jack.
 

Phase transitions and symmetry breaking in current distributions of diffusive systems

The talk will discuss singularities in current large deviation functions of boundary-driven diffusive systems. It will be shown that even when the underlying system is in equilibrium singularities can emerge. The phase transitions, which can be of first and second order, are characterized by an exact Landau theory which applies both to equilibrium and close to equilibrium systems.
 

Path-wise analysis for the age-structured population dynamics

The large deviation theory describing rare events forms a mathematical basis for statistical physics. In equilibrium situations, the rate function estimating ``rareness” gives the entropy function. On the other hand, in nonequilibrium situations, we can evaluate the entropy production by calculating the rate function on the time series describing the time evolution of nonequilibrium dynamics. Beyond the field of physics, it has been recently suggested that the large deviation theory can be applied to analysis of growing cell populations. To be more precise, the stationary population growth rate (also called ``fitness”) can be evaluated by the Legendre transform of the rate function on the cell lineage. Owing to this structure, we can calculate responses of the population growth with respect to external environmental changes. Furthermore, it is elucidated that such responses are also evaluated by statistics on a lineage obtained by retrospectively tracing ancestors. In this talk, we demonstrate the above framework on an age-structured population model, which is asexual population dynamics with age- and type-dependent growth rate. In a calculation for this population model, the explicit form of the rate function for the semi-Markov process plays an essential role. In the first half of this talk, we briefly show the explicit form of the rate function that is an extension of the ``Level 2.5” rate function on Markov processes to semi-Markov situations. In the second half, by using the explicit form, we demonstrate the path-wise analysis for the age-structured population model.