# Stochastic Dynamics Out of Equilibrium

3 April 2017 to 7 July 2017
Europe/Paris timezone

## Young researchers' seminar

The speakers of this seminar are PhD students and post-doctoral fellows. It is organized by Luisa Andreis (Univ. Padova), Ofer Busani (Bar Ilan University) and Federico Sau (TU Delft).

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Ofer Busani (Bar Ilan University), July 6th
Time Changed Continuous Time Random Walk

In this talk we consider the following Continuous Time Random Walks (CTRW); imagine a particle waiting W_1 seconds before performing a jump in space of size J_1 then it waits another W_2 seconds before performing a jump of J_2 etc.  In the case where W_i are i.i.d with finite mean and J_i are i.i.d with finite variance, zooming out properly in space and time, one sees trajectories of what is known as a Brownian Motion.  In 1965, motivated by long binding times of particles to substrates, Montroll and Weiss introduced a generalization of this classical CTRW by allowing the mean of W_i to be infinite. In many daily phenomena, this model seems to better describe the dynamics of a plume of particles diffusing in a “sticky” environment. Scaling this CTRW in space and time one obtain a subdiffusion. We will show that subdiffusive CTRWs are essentially time changed diffusive CTRW and therefore can be thought of as CTRW in random environment.

Matthias Sachs (University of Edinburgh), July 6th
Asymptotic properties and numerical discretisation of quasi-Markovian generalised Langevin equations

The generalised Langevin equation is the non-Markovian extension of the underdamped Langevin equation. It provides an accurate description of the stochastic dynamics of systems derived via the Mori-Zwanzig projection formalism when no scale separation between the resolved and not resolved degrees of freedom is present. It can be also used for the design of approximate Markov chain Monte Carlo methods with enhanced sampling properties. In this talk I will present criteria for the ergodicity of quasi-Markovian generalised Langevin equations and results on the limiting dynamics for certain scalings of the memory kernel. I will present a class of numerical integrators which we proved to preserve ergodicity under some technical assumptions and behave consistently in the respective parameter limits.

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Gerardo Barrera Vargas (Research Center in Mathematics, CIMAT), June 29th
Cut-off phenomenon for stochastic small random perturbations of hyperbolic dynamical systems

Our main goal is the study of the convergence to equilibrium for a family of stochastic small perturbations of a given dynamical system.
We study the so-called cut-off phenomenon. The term “cut-off” was introduced by D. Aldous and P. Diaconis in the early eighties to describe the phenomenon of abrupt convergence of Markov chains modeling card shuffling.
We focus in a semi-flow of a deterministic differential equation with a unique hyperbolic fixed point. We add to the deterministic dynamics a Brownian motion of small variance. Assuming that the vector field is strongly coercive, we prove that the family of perturbed dynamical system always presents  cut-off phenomena in the total variation distance.
This is a joint work with Milton Jara.

Venanzio di Giulio (University of L'Aquila), June 29th
Surface tension in a model with diffuse interface

See here the summary.

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Chiara Franceschini  (University of Modena and Reggio Emilia), June 22nd
Stochastic duality and orthogonal polynomials

Duality theory is a powerful tool to deal with stochastic Markov processes from which information on a given process can be extracted from another process, its dual. The link between the two processes is provided by a set of so-called duality functions. For a series of Markov processes a stochastic duality relation is proved: duality functions are in terms of the polynomials orthogonal with respect to the stationary measure of the initial process and the result follows from the structural properties of these hypergeometric family of polynomials (joint work with C. Giardina').

Max Fathi (CNRS & University Toulouse), June 22nd
Gradient flows in spaces of probability measures and hydrodynamic limits

I will discuss how some concepts from optimal transport theory, such as gradient flows in the Wasserstein space, can be used to study scaling limits for (reversible) interacting particle systems. The talk will be mostly focused on interacting diffusion processes, and if time allows I will briefly discuss other situations, such as non-gradient models and the symmetric exclusion process. Partly based on joint work with Marielle Simon (INRIA Lille).

Grégoire Ferré (Ecole des Ponts & Inria Paris), June 22nd [slides]
On the discretization of Feynman-Kac semi-groups

In this presentation, I will present a framework for the numerical analysis in the discretization time step of Feynman-Kac semigroups. I will first explain why these semigroups naturally appear in several fields, in particular large deviation theory and Diffusion Monte Carlo (DMC). Then, I will present elements of analysis of the error on the invariant measure that is done when discretizing a stochastic differential equation. Finally, I will show how we extended these results to the case of Feynman-Kac semi-groups, and why this analysis is interesting in general to discretize non probability-conserving processes. Our analysis is supported by relevant numerical applications.

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Chiranjib Mukherjee (WIAS Berlin), June 8th
Large deviations for random walks on supercritical percolation clusters

We prove a quenched large deviation principle for a simple random walk on supercritical percolation clusters, including long-range correlations. The models under interest include classical Bernoulli bond and site percolation as well as the random cluster model, the random interlacement, its vacant set and the level sets of the Gaussian free field. We take the point of view of the moving particle and first 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.

Joint work with Noam Berger and Kazuki Okamura.

Lara Neureither (FU Berlin), June 8th [slides]
Different notions of timescales in molecular dynamics

A phenomenon which has not yet been understood in molecular dynamics is  single point mutations. These are mutations which consist of exchanging a single atom in a molecule and potentially lead to dramatic changes in the molecules' properties. Simulationwise molecular dynamics  can be modelled via overdamped Langevin equations where the potential is given by the intramolecular forces. Mathematically speaking, a single point mutation then refers to changes of parameters in the potential. Trying to understand what is happening from the point of the root model here, we look at the timescales of the process as properties we wish to study under certain parameter changes. These timescales are given by convergence rates (convergence of the process to its stationary distribution) or mean first exit times of the process. I will review some results on convergence rates and mean first exit times for reversible processes. Further I will introduce entropy production which should be viewed as a prospective extension of convergence rates for irreversible processes. I will discuss its behaviour for linear but possibly irreversible processes.

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Nauhel Soprano Loto (Gran Sasso Science Institute), June 2nd
Turing instability in a model with two interacting Ising lines: hydrodynamic limit

In [1], the author introduces a reaction-diffusion system to model the pattern formation phenomena present in morphogenesis. Under the assumption that the reaction part of the system is stable around an equilibrium point, he finds condiditions over the diffusion coefficients under which the hole system is unstable due to the amplification of non-zero Fourier modes.  This phenomena is known as Turing instability.

In this talk, we introduce an interacting particle system at which the later phenomena is present. The system is a continuous-time Markov process that has two coupled discrete toruses with Ising spins as state-space. The evolution in each torus responds to macroscopic ferromagnetic Kac's potentials,
while the spins in different toruses interact in a local attractive-repulsive way. About this model, we prove hydrodynamic limit and find conditions that guarantee the occurence of Turing instability.

[1] A. M. Turing, The chemical basis of morphogenesis.

Monia Capanna (Univ. del'Aquila), June 2nd
Turing instability in a model with two interacting Ising lines: fluctuations

We continue with the analysis of the model introduced in the previous talk. In the Turing instability regime, we analyze the fluctuations of the density fields around the equilibrium point (0,0) by studying the limiting behaviour of the discrete Fourier modes of the system. More precisely, we prove that, at a time at which the process is infinitesimal, and under the proper spatial scaling, the unstable Fourier modes converge to a normal distribution while the rest of the modes vanish. We finally give a result about pattern formation at a time that converges to the critical one at which the process starts to be finite.

Daniele Tovazzi (University of Padova), June 2nd
Macroscopic oscillations in an Ising model with dissipation

Self-sustained macroscopic rhythm may appear in complex systems in which single units tend to cooperate with each other but their interaction energy is dissipated over time. This mechanism dumps the influence of interaction when no transition occurs for a long time and, as a result, macroscopic observables of the system will oscillate between magnetized states rather than polarize in a fixed point. Recently, existence of periodic collective behaviors has been proven for some classes of mean-field systems derived as perturbation of classical reversible ferromagnetic models by adding a dissipation term in the interaction energy.

In this talk we try to go beyond the mean-field hypothesis and we aim at understanding whether the mechanism of dissipation is capable to produce macroscopic oscillations even in a short-range interaction setting. To this purpose, we consider a 1-dimensional Ising model and we modify the classical Glauber dynamics by introducing a dissipation in each local field: we will show that a time-scaled version of the total magnetization converges to an oscillating process in a suitable large volume - low temperature regime. This is a joint work with Raphael Cerf, Paolo Dai Pra and Marco Formentin.

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Federico Sau (TU Delft), May 11th
A generating function approach to duality

Besides the wide applicability of duality for Markov processes, effort has been employed also in unveiling all possible dualities between two processes via rather algebraic techniques (e.g. [1] in population genetics, [2,3] in the context of particle systems).

In this talk, we present an alternative road map to duality suitable for interacting particle systems and diffusions with stationary product measures. Typical examples are independent random walkers, symmetric exclusion/inclusion processes and their continuum counterparts - all used as microscopic models of non-equilibrium phenomena such as heat conduction or mass transport.

Starting from a relation between the stationary product measures and the duality functions - the objects that "link" the two dual processes - we obtain the full list of possible dualities in factorized form. Here, orthogonal polynomials w.r.t. the marginals of the stationary measure (also obtained in [4] via direct computation) appear.

At last, by passing to generating functions, we obtain equivalent formulations of duality: in one shot, we gain new dualities for the continuum processes and prove all dualities previously characterized. Joint work with Frank Redig (TU Delft).

[1] Möhle, M., The concept of duality and applications to Markov processes arising in neutral population genetics models, Bernoulli, Volume 5, Number 5 (1999).
[2] Giardinà, C,; Kurchan, J.; Redig, F.; Vafayi, K., Duality and hidden symmetries in interacting particle systems. J. Stat. Phys., Vol. 135, (2009), no. 1, pp. 25-55.
[3] Schütz, G.; Sandow, S., Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems, Phys. Rev. E 49, 2726 (1994).
[4] Franceschini, C; Giardinà, C., Stochastic Duality and Orthogonal Polynomials, available at https://arxiv.org/abs/1701.09115 (2017).

Luisa Andreis (University of Padova), May 11th
Ergodicity of a system of interacting random walks with asymmetric interaction

We consider a model of interacting random walks on nonnegative integers. We provide every particle with an intrinsic dynamics given by a biased random walk reflected in zero and with an asymmetric interaction that pushes each particle towards the origin and depends only on the fraction of particles below its position. We focused on the critical interaction strength above which the N particle system and its corresponding nonlinear limit have a stationary measure, balancing the tendency of the biased random walks to escape to infinity. Similar models have been studied in the continuous with diffusive dynamics, where there is a system of particles interacting through their cumulative distribution function. The discrete model we consider displays a peculiar difference: the particles can form large clusters on a single site and, according to our description, they cannot interact. This gives rise to non-trivial expression for the critical interaction strength, unexpected from the analysis of the continuum model. This is a joint work with Amine Asselah and Paolo Dai Pra.

Julien Roussel (Ecole des Ponts and Inria Paris), May 11th
Variance reduction for non-equilibrium systems: A control variate approach for Langevin dynamics

Transport properties in materials, such as the thermal conductivity in atom chains, can be studied by using the linear response of the system to an external perturbation. To speed up the empirical means estimated on the simulated trajectories, standard variance reduction techniques cannot be used. Indeed the invariant measure of the non-equilibrium system is unknown, so straightforward stratification or importance sampling techniques are impossible. We propose here a control variate technique and present some illustrative numerical results.

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Xiaocheng Shang (Brown University), May 4th
Efficient numerical methods for molecular and particle simulation

We discuss the construction of state-of-the-art numerical methods for molecular dynamics, focusing on the demands of soft matter simulation. The purposes include sampling and dynamics calculations both in and out of equilibrium. We discuss the characteristics of different algorithms, including essential issues such as their conservation properties, the convergence of averages,  and the accuracy of numerical discretizations. Formulations of the equations of motion which are suited to both equilibrium and nonequilibrium simulation include Langevin dynamics, dissipative particle dynamics (DPD) and the more recently proposed pairwise adaptive Langevin'' (PAdL) method, which, unlike Langevin dynamics, conserves momentum and better matches the relaxation rate of orientational degrees of freedom. PAdL is easy to code and suitable for a variety of problems in nonequilibrium soft matter modelling; our simulations of polymer melts indicate that this method can also provide dramatic improvements in computational efficiency.   Moreover we show that PAdL gives excellent control of the relaxation rate to equilibrium. In the nonequilibrium setting, we further demonstrate that PAdL allows the recovery of accurate shear viscosities at much higher shear rates than are possible using the DPD method.

Nicolò Defenu (University of Heidelberg), May 4th [slides]
Criticality in Quantum Long Range Systems

The interest in long range interactions has increased in the last years due to the recent simulations with cold atomic devices. Our investigation focus on the derivation of the phase diagram and the critical exponents of weak long range interacting N components quantum rotor models (the N = 1 case being the quantum Ising model). Our picture is generic, non perturbative and it is valid for any spin components number N, spatial dimension $d$ and decay exponent $sigma$. The phase diagram as a function of the decay exponent σ is rich and the correlation functions are strongly anisotropic in the spatial and time coordinates. Our results are in very good agreement with Monte Carlo simulations and Density matrix renormalization group calculations pursued on quantum Ising and XY spin chains and also with spin systems coupled with an anomalous Bosonic bath.
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Nikolas Nüsken (Imperial College London), April 20th
Optimal perturbations of Langevin dynamics to sample from probability distributions

Computing expectations with respect to high-dimensional probability distributions is a recurring task both in statistical methodology and molecular dynamics simulations. A powerful and general approach is to consider long-time averages of a Markov chain that is ergodic with respect to a given target measure. Finding appropriate Markovian dynamics that are optimal in terms of convergence characteristics (spectral gap and asymptotic variance) therefore represents an interesting and challenging inverse problem. I will discuss recent results on optimal perturbations of overdamped and underdamped Langevin dynamics.

Wei Zhang (FU Berlin), April 20th [slides]
Model reduction of ergodic diffusion process along reaction coordinate and related algorithmic issues

Projecting a high-dimensional stochastic dynamics on the reaction coordinate space has attracted considerable attentions in the literature. In this talk, I will discuss the model reduction problem of an ergodic diffusion process along certain given reaction coordinate. The main focus will be the properties which are inherited from the original dynamics, as well as error estimates of the timescales comparing to those of the full dynamics. Some algorithmic issues will be discussed as well.

Aritra Kundu (ICTS Bangalore), April 20th [slides]
Equilibrium dynamical correlations in the Toda chains in 1D

I am going to discuss about the equilibrium spatio-temporal correlation functions of conserved quantities in an integrable Toda chain. In the special limiting choices of parameter values, for which the Toda chain tends to either the harmonic chain or the equal mass hard-particle gas and obtain correlation functions exactly and find excellent match with numerics.  I will briefly describe going to normal modes and fluctuating hydrodynamics predictions for non-integrable system.  Finally we investigate differences in “normal mode” correlations between Toda chain (integrable) and truncated Toda chain (non - integrable). See Phys. Rev. E 94, 062130.

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Clément Erignoux (IMPA), April 13th
Coupling method for hydrodynamic limits

(work in progress with Claudio Landim) The modern method to derive hydrodynamic limits from microscopic particle systems on a lattice usually relies on one of the variants of the entropy method, which requires some control over the entropy of the measure of the process with respect to some equilibrium product measure. In some cases however, for example for boundary driven particle systems, such control cannot be obtained, and the entropy does not seem to be of the right order. In this talk, I will briefly describe a coupling method, which allows to estimate directly the density and the correlations of the system in order to derive hydrostatic and hydrodynamic limits. I will also try and give and overview of the types of models it can apply to.

Deepat Bhak (ICTS Bangalore), April 13th
Dynamics of a piston pushed by a single particle gas as a microscopic model for Szilard engine

See the summary (pdf file).

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