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## Über dieses Buch

Stemming from the IHP trimester "Stochastic Dynamics Out of Equilibrium", this collection of contributions focuses on aspects of nonequilibrium dynamics and its ongoing developments.

It is common practice in statistical mechanics to use models of large interacting assemblies governed by stochastic dynamics. In this context "equilibrium" is understood as stochastically (time) reversible dynamics with respect to a prescribed Gibbs measure. Nonequilibrium dynamics correspond on the other hand to irreversible evolutions, where fluxes appear in physical systems, and steady-state measures are unknown.

The trimester, held at the Institut Henri Poincaré (IHP) in Paris from April to July 2017, comprised various events relating to three domains (i) transport in non-equilibrium statistical mechanics; (ii) the design of more efficient simulation methods; (iii) life sciences. It brought together physicists, mathematicians from many domains, computer scientists, as well as researchers working at the interface between biology, physics and mathematics.

The present volume is indispensable reading for researchers and Ph.D. students working in such areas.

## Inhaltsverzeichnis

### Stochastic Mean-Field Dynamics and Applications to Life Sciences

Abstract
Although we do not intend to give a general, formal definition, the stochastic mean-field dynamics we present in these notes can be conceived as the random evolution of a system comprised by N interacting components which is: (a) invariant in law for permutation of the components; (b) such that the contribution of each component to the evolution of any other is of order $$\frac{1}{N}$$. The permutation invariance clearly does not allow any freedom in the choice of the geometry of the interaction; however, this is exactly the feature that makes these models analytically treatable, and therefore attractive for a wide scientific community.
Paolo Dai Pra

### Alignment of Self-propelled Rigid Bodies: From Particle Systems to Macroscopic Equations

Abstract
The goal of these lecture notes is to present in a unified way various models for the dynamics of aligning self-propelled rigid bodies at different scales and the links between them. The models and methods are inspired from [17, 18], but, in addition, we introduce a new model and apply on it the same methods. While the new model has its own interest, our aim is also to emphasize the methods by demonstrating their adaptability and by presenting them in a unified and simplified way. Furthermore, from the various microscopic models we derive the same macroscopic model, which is a good indicator of its universality.
Pierre Degond, Amic Frouvelle, Sara Merino-Aceituno, Ariane Trescases

### Fluctuations in Stochastic Interacting Particle Systems

Abstract
We discuss fluctuations in stochastic lattice gas models from a microscopic and mesoscopic perspective by using techniques from algebra, in particular the use of symmetries and time-reversal. First we present a generic method to derive rigorously duality functions. As applications we obtain detailed information about density fluctuations in the symmetric simple exclusion process on any graph and about the microscopic structure and fluctuations of shocks in the one-dimensional asymmetric simple exclusion process. Then we use time reversal to prove a general current fluctuation theorem from which celebrated fluctuation relations such as the Jarzynski relation and the Gallavotti-Cohen symmetry arise as corollaries and which can be straightforwardly generalized to derive other fluctuation relations. Finally, going beyond rigorous results, we describe briefly how nonlinear fluctuating hydrodynamics yields the Fibonacci family of dynamical universality classes which has the diffusive and Kardar-Parisi-Zhang universality classes as its first two members.
Gunter M. Schütz

### Hydrodynamics for Symmetric Exclusion in Contact with Reservoirs

Abstract
We consider the symmetric exclusion process with jumps given by a symmetric, translation invariant, transition probability $$p(\cdot )$$. The process is put in contact with stochastic reservoirs whose strength is tuned by a parameter . Depending on the value of the parameter $$\theta$$ and the range of the transition probability $$p(\cdot )$$ we obtain the hydrodynamical behavior of the system. The type of hydrodynamic equation depends on whether the underlying probability $$p(\cdot )$$ has finite or infinite variance and the type of boundary condition depends on the strength of the stochastic reservoirs, that is, it depends on the value of $$\theta$$. More precisely, when $$p(\cdot )$$ has finite variance we obtain either a reaction or reaction-diffusion equation with Dirichlet boundary conditions or the heat equation with different types of boundary conditions (of Dirichlet, Robin or Neumann type). When $$p(\cdot )$$ has infinite variance we obtain a fractional reaction-diffusion equation given by the regional fractional laplacian with Dirichlet boundary conditions but for a particular strength of the reservoirs.
Patrícia Gonçalves

### Stochastic Solutions to Hamilton-Jacobi Equations

Abstract
In this expository paper we give an overview of the statistical properties of Hamilton-Jacobi Equations and Scalar Conservation Laws. The first part (Sects. 24) is devoted to the recent proof of Menon-Srinivasan Conjecture. This conjecture provides a Smoluchowski-type kinetic equation for the evolution of a Markovian solution of a scalar conservation law with convex flux. In the second part of the paper (Sects. 5 and 6) we discuss the question of homogenization for Hamilton-Jacobi PDEs and Hamiltonian ODEs with deterministic and stochastic Hamiltonian functions.
Fraydoun Rezakhanlou

### On Optimal Decay Estimates for ODEs and PDEs with Modal Decomposition

Abstract
We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the “optimal” Lyapunov functional to quantify the convergence to the unique normalized steady state. The Lyapunov functional is optimal in the sense that it yields decay estimates in $$L^2$$-norm with the sharp exponential decay rate and minimal multiplicative constant. The modal decomposition of the Goldstein-Taylor model leads to the study of a family of 2-dimensional ODE systems. Therefore we discuss the characterization of “optimal” Lyapunov functionals for linear ODE systems with positive stable diagonalizable matrices. We give a complete answer for optimal decay rates of 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.
Franz Achleitner, Anton Arnold, Beatrice Signorello

### Adaptive Importance Sampling with Forward-Backward Stochastic Differential Equations

Abstract
We describe an adaptive importance sampling algorithm for rare events that is based on a dual stochastic control formulation of a path sampling problem. Specifically, we focus on path functionals that have the form of cumulate generating functions, which appear relevant in the context of, e.g. molecular dynamics, and we discuss the construction of an optimal (i.e. minimum variance) change of measure by solving a stochastic control problem. We show that the associated semi-linear dynamic programming equations admit an equivalent formulation as a system of uncoupled forward-backward stochastic differential equations that can be solved efficiently by a least squares Monte Carlo algorithm. We illustrate the approach with a suitable numerical example and discuss the extension of the algorithm to high-dimensional systems.
Omar Kebiri, Lara Neureither, Carsten Hartmann

### Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force

Abstract
We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted $$L^{\infty }$$ spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force.
Benedict Leimkuhler, Matthias Sachs

### Exit Event from a Metastable State and Eyring-Kramers Law for the Overdamped Langevin Dynamics

Abstract
In molecular dynamics, several algorithms have been designed over the past few years to accelerate the sampling of the exit event from a metastable domain, that is to say the time spent and the exit point from the domain. Some of them are based on the fact that the exit event from a metastable region is well approximated by a Markov jump process. In this work, we present recent results on the exit event from a metastable region for the overdamped Langevin dynamics obtained in [22, 23, 56]. These results aim in particular at justifying the use of a Markov jump process parametrized by the Eyring-Kramers law to model the exit event from a metastable region.
Tony Lelièvre, Dorian Le Peutrec, Boris Nectoux

### Collisional Relaxation and Dynamical Scaling in Multiparticle Collisions Dynamics

Abstract
We present the Multi-Particle-Collision (MPC) dynamics approach to simulate properties of low-dimensional systems. In particular, we illustrate the method for a simple model: a one-dimensional gas of point particles interacting through stochastic collisions and admitting three conservation laws (density, momentum and energy). Motivated from problems in fusion plasma physics, we consider an energy-dependent collision rate that accounts for the lower collisionality of high-energy particles. We study two problems: (i) the collisional relaxation to equilibrium starting from an off-equilibrium state and (ii) the anomalous dynamical scaling of equilibrium time-dependent correlation functions. For problem (i), we demonstrate the existence of long-lived population of suprathermal particles that propagate ballistically over a quasi-thermalized background. For (ii) we compare simulations with the predictions of nonlinear fluctuating hydrodynamics for the structure factors of density fluctuations. Scaling analysis confirms the prediction that such model belong to the Kardar-Parisi-Zhang universality class.
Stefano Lepri, Hugo Bufferand, Guido Ciraolo, Pierfrancesco Di Cintio, Philippe Ghendrih, Roberto Livi

### A Short Introduction to Piecewise Deterministic Markov Samplers

Abstract
The use of velocity jump Markov processes in MCMC algorithms have recently drawn attention in various fields, such as statistical physics or Bayesian statistics. The aim of this paper is to introduce these processes and to give a few justifications on their interest.
Pierre Monmarché

### Time Scales and Exponential Trend to Equilibrium: Gaussian Model Problems

Abstract
We review results on the exponential convergence of multidimensional Ornstein-Uhlenbeck processes and discuss notions of characteristic time scales by means of concrete model systems. We focus, on the one hand, on exit time distributions and provide explicit expressions for the exponential rate of the distribution in the small-noise limit. On the other hand, we consider relaxation time scales of the process to its equilibrium measure in terms of relative entropy and discuss the connection with exit probabilities. Along these lines, we study examples which illustrate specific properties of the relaxation and discuss the possibility of deriving a simulation-based, empirical definition of slow and fast degrees of freedom which builds upon a partitioning of the relative entropy functional in connection with the observed relaxation behaviour.
Lara Neureither, Carsten Hartmann

### Stochastic Models of Blood Vessel Growth

Abstract
Angiogenesis is a complex multiscale process by which diffusing vessel endothelial growth factors induce sprouting of blood vessels that carry oxygen and nutrients to hypoxic tissue. There is strong coupling between the kinetic parameters of the relevant branching - growth - anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. A hybrid mesoscale tip cell model involves stochastic branching, fusion (anastomosis) and extension of active vessel tip cells with reaction-diffusion growth factor fields. Anastomosis prevents indefinite proliferation of active vessel tips, precludes a self-averaging stochastic process and ensures that a deterministic description of the density of active tips holds only for ensemble averages over replicas of the stochastic process. Evolution of active tips from a primary vessel to a hypoxic region adopts the form of an advancing soliton that can be characterized by ordinary differential equations for its position, velocity and a size parameter. A short review of other angiogenesis models and possible implications of our work is also given.
Luis L. Bonilla, Manuel Carretero, Filippo Terragni

### Survival Under High Mutation

Abstract
We consider a stochastic model for an evolving population. We show that in the presence of genotype extinctions the population dies out for a low mutation probability but may survive for a high mutation probability. This turns upside down the widely held belief that above a certain mutation threshold a population cannot survive.
Rinaldo B. Schinazi

### Particle Transport in a Confined Ratchet Driven by the Colored Noise

Abstract
In this paper, we study particle transport in a confined ratchet which is constructed by combining a periodic channel with a ratchet potential under colored Gaussian noise excitation. Due to the interaction of colored noise and confined ratchet, particles host remarkably different properties in the transporting process. By means of the second-order stochastic Runge-Kutta algorithm, effects of the system parameters, including the noise intensity, colored noise correlation time and ratchet potential parameters are investigated by calculating particle current. The results reveal that the colored noise correlation time can lead to an increase of particle current. The increase of noise intensity along the horizontal or vertical direction can accelerate the particle transport in the corresponding direction but slow down the particle transport when there are the same noise intensities in both directions. For potential parameters, an increase of the slope parameter results into an increase of particle currents. The interactions of potential parameters and correlation time can induce complex particle transport phenomena, i.e. particle current increases with the increase of the potential depth parameter for a smaller asymmetric parameter and non-zero correlation time, while the tendency changes for a larger asymmetric parameter. Accordingly, suitable system parameters can be chosen to accelerate the particle transport and used to design new devices for particle transport in microscale.
Yong Xu, Ruoxing Mei, Yongge Li, Jürgen Kurths

### Long-Time Dynamics for a Simple Aggregation Equation on the Sphere

Abstract
We give a complete study of the asymptotic behavior of a simple model of alignment of unit vectors, both at the level of particles, which corresponds to a system of coupled differential equations, and at the continuum level, under the form of an aggregation equation on the sphere. We prove unconditional convergence towards an aligned asymptotic state. In the cases of the differential system and of symmetric initial data for the partial differential equation, we provide precise rates of convergence.
Amic Frouvelle, Jian-Guo Liu

### Tracy-Widom Asymptotics for a River Delta Model

Abstract
We study an oriented first passage percolation model for the evolution of a river delta. This model is exactly solvable and occurs as the low temperature limit of the beta random walk in random environment. We analyze the asymptotics of an exact formula from [13] to show that, at any fixed positive time, the width of a river delta of length L approaches a constant times $$L^{2/3}$$ with Tracy-Widom GUE fluctuations of order $$L^{4/9}$$. This result can be rephrased in terms of particle systems. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.
Guillaume Barraquand, Mark Rychnovsky

### Hydrodynamics of the N-BBM Process

Abstract
The Branching Brownian Motion (BBM) process consists of particles performing independent Brownian motions in $$\mathbb R$$, and each particle creating a new one at rate 1 at its current position. The newborn particles’ increments and branchings are independent of the other particles. The N-BBM process starts with N particles and, at each branching time, the left-most particle is removed so that the total number of particles is N for all times. The N-BBM process has been originally proposed by Maillard, and belongs to a family of processes introduced by Brunet and Derrida. We fix a density $$\rho$$ with a left boundary $$\sup \{r\in \mathbb R: \int _r^\infty \rho (x) d x=1\}>-\infty$$, and let the initial particles’ positions be iid continuous random variables with density $$\rho$$. We show that the empirical measure associated to the particle positions at a fixed time t converges to an absolutely continuous measure with density $$\psi (\cdot ,t)$$ as $$N\rightarrow \infty$$. The limit $$\psi$$ is solution of a free boundary problem (FBP). Existence of solutions of this FBP was proved for finite time-intervals by Lee in 2016 and, after submitting this manuscript, Berestycki, Brunet and Penington completed the setting by proving global existence.
Anna De Masi, Pablo A. Ferrari, Errico Presutti, Nahuel Soprano-Loto

### 1D Mott Variable-Range Hopping with External Field

Abstract
Mott variable-range hopping is a fundamental mechanism for electron transport in disordered solids in the regime of strong Anderson localization. We give a brief description of this mechanism, recall some results concerning the behavior of the conductivity at low temperature and describe in more detail recent results (obtained in collaboration with N. Gantert and M. Salvi) concerning the one-dimensional Mott variable-range hopping under an external field.
Alessandra Faggionato

### Invariant Measures in Coupled KPZ Equations

Abstract
We discuss coupled KPZ (Kardar-Parisi-Zhang) equations. The motivation comes from the study of nonlinear fluctuating hydrodynamics, cf. [11, 12]. We first give a quick overview of results of Funaki and Hoshino [6], in particular, two approximating equations, trilinear condition (T) for coupling constants $$\varGamma$$, invariant measures and global-in-time existence of solutions. Then, we study at heuristic level the role of the trilinear condition (T) in view of invariant measures and renormalizations for 4th order terms. Ertaş and Kardar [2] gave an example which does not satisfy (T) but has an invariant measure. We finally discuss the cross-diffusion case.

### Reversible Viscosity and Navier–Stokes Fluids

Abstract
Exploring the possibility of describing a fluid flow via a time-reversible equation and its relevance for the fluctuations statistics in stationary turbulent (or laminar) incompressible Navier-Stokes flows.
Giovanni Gallavotti

### On the Nonequilibrium Entropy of Large and Small Systems

Abstract
Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid.
Sheldon Goldstein, David A. Huse, Joel L. Lebowitz, Pablo Sartori

### Marginal Relevance for the -Stable Pinning Model

Abstract
We investigate disorder relevance for the pinning of a renewal when the law of the random environment is in the domain of attraction of a stable law with parameter $$\gamma \in (1,2)$$. Assuming that the renewal jumps have power-law decay, we determine under which condition the critical point of the system is altered by the introduction of a small quantity of disorder. In an earlier study of the problem [20] we have shown that the answer depends on the value of the tail exponent $$\alpha$$ associated to the distribution of renewal jumps: when $$\alpha >1-\gamma ^{-1}$$ a small amount of disorder shifts the critical point whereas it does not when $$\alpha <1-\gamma ^{-1}$$. The present paper is focused on the boundary case $$\alpha =1-\gamma ^{-1}$$. We show that in this case, the critical point is shifted, and obtain an estimate for the intensity of this shift.
Hubert Lacoin

### A Rate of Convergence Result for the Frederickson-Andersen Model

Abstract
Our paper [2] considers the Frederickson-Andersen model (FA model) on a general class of graphs of bounded degree.
Thomas Mountford, Glauco Valle

### Stochastic Duality and Eigenfunctions

Abstract
We start from the observation that, anytime two Markov generators share an eigenvalue, the function constructed from the product of the two eigenfunctions associated to this common eigenvalue is a duality function. We push further this observation and provide a full characterization of duality relations in terms of spectral decompositions of the generators for finite state space Markov processes. Moreover, we study and revisit some well-known instances of duality, such as Siegmund duality, and extract spectral information from it. Next, we use the same formalism to construct all duality functions for some solvable examples, i.e., processes for which the eigenfunctions of the generator are explicitly known.
Frank Redig, Federico Sau
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