Anton Arnold (Vienna University of Technology) [slides]
Entropy method for hypocoercive & non-symmetric Fokker-Planck equations with linear drift
In the last 15 years the entropy method has become a powerful tool for analyzing the large-time behavior of the Cauchy problem for linear and non-linear Fokker-Planck type equations (advection-diffusion equations, kinetic Fokker-Planck equation of plasma physics, e.g.). In particular, this entropy method can be used to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy.
For degenerate parabolic equations, the entropy dissipation may vanish for states other than the equilibrium. Hence, the standard entropy method does not carry over.
For hypocoercive Fokker-Planck equations (with drift terms that are linear in the spatial variable) we introduce an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. We show that the obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with non-quadratic potential) and non-degenerate, non-symmetric Fokker-Planck equations. For the latter examples the "hypocoercive entropy method" yields the sharp global decay rate (as an envelope for the relative entropy function), while the standard entropy method only yields the sharp local decay rate.
- A. Arnold, J. Erb. Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, arXiv 2014.
- F. Achleitner, A. Arnold and D. Stürzer: Large-time behavior in non-symmetric Fokker-Planck equations; Rivista di Matematica della Università di Parma 6 (2015) 1-68.
Abhishek Dhar (ICTS Bangalore) [slides]
Coupled rotors and classical spin chains: predictions from fluctuating hydrodynamics and numerical tests
The talk will discuss the predictions of nonlinear fluctuating hydrodynamics for two classical one-dimensional models, namely that of a chain of coupled rotors and a spin chain described by the XXZ Hamiltonian. The emergence and role of extra "almost" conserved quantities at low temperatures will be emphasized. Numerical results checking the predictions of the theory will be presented.
Tobias Grafke (Courant Institute) [slides]
Non-equilibrium self-organization of motile bacteria with fluctuating population and speed
Active materials can self-organize in many more ways than their equilibrium counterparts. For example, self-propelled particles with density dependend motility can display motility-induced phase separation (MIPS), leading to pattern formation. In this talk it is shown how internal fluctuations in the population size and swimming speed of motile bacteria have a significant impact on the way they self-organize. Two nontrivial regimes are identified, depending on the population carrying capacity. Below a certain threshold, the fluctuations make bacteria clusters appear and disappear periodically in time at random locations in space, with a period that is roughly independent of the noise amplitude. Above the threshold, bacteria organize in metastable clusters, and fluctuations lead to transitions between those at random times that are exponentially long in the noise amplitude, following specific out-of-equilibrium pathways. Both in the quasi-periodic and the metastable regimes, these findings can be explained by combining tools from large deviation theory with a bifurcation analysis in which the mean bacteria density, assumed to vary slowly via birth and death, plays the role of control parameter.
Haoxue Han [slides]
Probing interfacial heat and mass transfer at the molecular scale
We study by molecular dynamics (MD) simulations the molecular processes during evaporation of submicron-size droplets. To this end, a new variety of the reverse non-equilibrium molecular dynamics (RNEMD) method has been developed, which allows the evaporation of the droplet to proceed in a non-equilibrium steady state. Evaporated molecules are located in the vapour phase and are re-inserted into the droplet, so its size remains constant. In this way, the statistics at each drop size can be improved. In the second part, we use the thermal relaxation method and non-equilibrium Green's function to probe heat transport at various interfaces of nanomaterials: (1) the interface between a non-polar solid surface and a complex liquid. (2) a graphene-aminosilane molecular junction.
Carsten Hartmann (BTU Cottbus) [slides]
Importance sampling of rare events using cross-entropy minimization
Cross-entropy (CE) minimization is a versatile Monte Carlo method for combinatorial optimization and sampling of rare events, which goes back to work by Reuven Rubinstein and co-workers. I will report on recent algorithmic extensions of the CE method to diffusions that can be used to design efficient importance sampling strategies for computing the rare events statistics of equilibrated systems. The approach is based on a Legendre-type duality between path functionals of diffusion processes associated with certain sampling and control problems that can be reformulated in terms of CE minimization. The method will be illustrated with several numerical examples and discussed along with algorithmic issues and possible extensions of the method to high-dimensional multiscale systems.
Corinna Kollath (Universität Bonn)
Using matrix product states to investigate dissipative systems
Atomic gases cooled to Nanokelvin temperatures are a new exciting tool to study a broad range of quantum phenomena. In particular, an outstanding and rapid control over the fundamental parameters, such as interaction strength, spin composition, and the coupling to environments. In my talk, I will address how matrix product state methods can be used in order to study the influence of a coupling to an environment on the system dynamics of bosonic optical lattice gases. The interplay between the interaction, the kinetic energy and the coupling to the environment causes interesting dynamics such as a crossover in the behaviour of correlations.
Werner Krauth (ENS Paris) [slides]
Lifting and non-reversible Markov chains for particle systems and spin models
The Markov-chain Monte Carlo method is an outstanding computational tool in science. Since its origins, it has relied on the detailed-balance condition and the Metropolis algorithm to solve general computational problems under the conditions of thermodynamic equilibrium with zero probability flows.
In this talk, I discuss a class of “Beyond-Metropolis” algorithms that violate detailed balance, yet satisfy global balance (the Markov chains are irreversible). Equilibrium is reached as a steady state with non-vanishing probability flows. The Metropolis acceptance criterion based on the change in the energy is replaced by a consensus rule originating in a new factorized Metropolis algorithm. The system energy is not computed, providing a fresh perspective for long-range interactions. Moves are infinitesimal and persistent, implementing the lifting concept (Diaconis et al, 2000).
I will specifically discuss recent progress in setting up and in analyzing these algorithms for continuum particle systems and for models of continuum spins (XY model, Heisenberg model), which clearly show reduction in the dynamical scaling exponent (the scaling of the inverse-gap with system size). I will also discuss prospects for exact solution of non-reversible Glauber-type dynamics.
Vivien Lecomte (LiPhy Grenoble) [slides]
Population dynamics method: systematic errors and feedback control
Atypical or rare events can play a pivotal role, depending on their consequences. We discuss a class of numerical procedures that uses population dynamics in order to estimate large deviation functions associated to the distribution of time-averaged observables in Markov processes. The idea of such procedures is to simulate a large number of copies of a system, subjected to selection rules that favour atypical histories of the original system, and allow ones to study the properties of rare events.
It happens that systematic errors in the results of such algorithms can be large in some circumstances, particularly for systems with weak noise, with many degrees of freedom, or close to dynamical phase transitions. To study analytically these errors, we explicitly devise a stochastic birth-death process that describes the evolution of the population probability. From this formulation, we derive that the systematic errors of the algorithm decrease proportionally to the inverse of the population size. Based on this observation, we propose a simple interpolation technique for a better estimation of large deviation functions. Besides, we show how these errors can be mitigated by introducing control forces within the algorithm. These forces are determined by an iteration-and-feedback scheme, inspired by multicanonical methods in equilibrium sampling. We demonstrate substantially improved results in a simple model, and we discuss potential applications to more complex systems.
Work done in collaboration with Esteban Guevara (Institut Jacques Monod), Takahiro Nemoto (ENS Paris) and Rob L Jack (Bath University).
Tony Lelièvre (Ecole des Ponts) [slides]
Efficient simulation of metastable trajectories
I will present two types of numerical methods to efficiently simulate metastable trajectories. These techniques are in particular well adapted for the simulation of non equilibrium systems. In the first part of the talk, I will concentrate on accelerated dynamics techniques, which have been introduced by A.F. Voter in the 90's to efficiently simulate metastable dynamics. These numerical methods use an underlying kinetic Monte Carlo model to efficiently simulate a molecular dynamics trajectory. In the second part of the talk, I will discuss the adaptive multilevel splitting technique, which has been originally introduced by F. Cérou and A. Guyader as a rare event simulation, and was then adapted to the simulation of reactive paths and transition rates.
Stefano Lepri (ISC Firenze) [slides]
Anomalous dynamical scaling in 1D systems with multiparticle collisions
We study the anomalous dynamical scaling of equilibrium correlations in one dimensional systems. Two different models are compared: the Fermi-Pasta-Ulam chain with cubic and quartic nonlinearity and a gas of point particles interacting stochastically through the multiparticle collision dynamics. For both models -that admit three conservation laws- by means of detailed numerical simulations we verify the predictions of nonlinear fluctuating hydrodynamics for the structure factors of density and energy fluctuations at equilibrium. Despite this, violations of the expected scaling in the currents correlation are found in some regimes, hindering the observation of the asymptotic scaling predicted by the theory. In the case of the gas model this crossover is clearly demonstrated upon changing the coupling constant.
Pierre Monmarché (Ecole des Ponts) [slides]
Piecewise deterministic sampling and annealing
We will introduce a non-reversible Markov sampler, obtained as a scaling limit of persistant walks, both kinetic an piecewise deterministic, and study its long-time behaviour. We will be especially interested in the small temperature regime, namely in the limit T->0 when the target invariant measure is the Gibbs measure with density exp(-U(x)/T) with a multi-modal potential U.
Gary Morriss (UNSW Australia) [slides]
Heat conduction in atomic systems
We consider heat conduction for hard and soft disks systems. For hard disks confined to a narrow channel with periodic boundary conditions (a Quasi-One-Dimensional system) we construct a local macroscopic description for mass, momentum and energy densities and make the connection with the microscopic representation. In this way we discuss the continuity equation for energy both the kinetic and potential contributions. The thermal conductivity for this system is anomalous and we observe a crossover between two different forms of anomalous behaviour at a particular system size.
For particles interacting via a continuous pair potential the equations of motion have to be integrated to generate the microscopic dynamics. This is more difficult and less accurate than hard disk dynamics. For soft disk systems the conductivity has been calculated in a system with constant temperature gradient. At higher temperature gradients the homogeneity of the system is lost and a soliton carries the energy. Here we explore the use of a constant heat current ensemble which may suppress the formation of a soliton.
Michela Ottobre (Herriott Watt)
Sampling with non-reversible dynamics
The observation that irreversible processes may have better convergence properties than their reversible counterparts has sparked in recent years a significant amount of research to exploit irreversibility within Markov Chain Monte Carlo sampling schemes. Indeed, while the Metropolis-Hastings algorithm gives a simple and general algorithmic prescription to generate reversible chains with a given invariant distribution, no algorithmic framework as (simple and) general as M-H is available at the moment to routinely produce non-reversible chains. Nonetheless, a lot of progress has been recently made in this field and the problem has been tackled by various standpoints.
In this talk we will (i) give an overview of the different sampling and simulation approaches taken so far; (ii) present some of the available non-reversible MCMC algorithms; (iii) make further considerations on how (not) to use irreversibility within MCMC.
Work in collaboration with N. Pillai (Harvard), A. Stuart (Caltech), F. Pinski (Cincinnati) and K. Spiliopoulos (Boston)
G. Pavliotis (Imperial College London)
Spectral methods for Langevin dynamics with a multiscale structure
In this talk we will present recently developed techniques for the numerical solution of stochastic differential equations with a multiscale structure. We consider SDEs for which a coarse-grained (homogenized) model exists and we calculate the drift and diffusion coefficients in the coarse-grained model by solving an appropriate Poisson equation using a spectral method. This spectral method is based either on the expansion of the solution to the Poisson equation in Hermite polynomials or in the calculation of the eigenvalues and eigenfunctions of a particular Schrodinger operator. Spectral convergence is proved under suitable assumptions. Our methodology is illustrated in two particular examples: fast/slow systems of SDEs for which the fast process is a reversible diffusion and the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction.
Luc Rey-Bellet (University of Massachussetts)
Uncertainty quantification for non-equilibrium extended stochastic systems
Based on new information inequalities involving the relative entropy and the relative Renyi entropy we develop a systematic theory of uncertainty quantification for stochastic systems including systems in a long-time steady state regime and extended systems (in the thermodynamic limit). The basic framework is that a system is described by some probability measure known only approximately and specified up to an error which we quantify using information-theoretic quantities. Our techniques provides then uncertainty bounds for a variety statistical estimators such as expected values, variance of observables, moment generating functions, and rare event probabilities. Such bounds provide a natural approach to model-uncertainty but can also be used to develop an information-theoretic approach to numerical analysis.
Billy Todd (Swinburne University) [slides]
Temperature control and measurement of flow enhancement for confined nonequilibrium fluids
As the science of nanotechnology transitions from fantasy to fact, the need for computational methodologies to make physically realistic predictions to aid in the development of nano-devices needs serious attention. Of considerable recent interest has been the flow of liquids in either nanotubes or nanochannels. Theory, simulation and experiment all give conflicting results and there are multiple sources of potential error in each realm.
With regards to computational modelling of nonequilibrium fluids under high spatial confinement, both the measurement of flow enhancement for high slip systems (e.g. water flowing in carbon nanotubes or graphene channels) and the control of fluid temperature are significant problems that require careful simulation design and analysis of data. This talk will focus of nonequilibrium molecular dynamics (NEMD) simulations of such highly confined fluids. It will be demonstrated how careless design of the simulation can lead to physically misleading results, a number of which are quoted in the literature. Methodologies to construct careful NEMD simulations will be presented and results reported to demonstrate their validity.
Hugo Touchette (Stellenbosch) [slides]
Large deviation simulations: Equilibrium vs nonequilibrium systems
I will give in this talk a brief overview of several numerical approaches, based on importance sampling, control theory, variational principles and spectral methods, that can be used to estimate large deviation functions of integrated quantities (e.g., work, current, or heat exchanged) of equilibrium and nonequilibrium processes in stationary states. In discussing these methods, I will draw attention to simplifications and difficulties that arise when applying them to equilibrium or nonequilibrium systems and will also discuss their relation with recent work on Markov processes conditioned on large deviations, done in collaboration with Raphael Chetrite.
Jonathan Weare (University of Chicago)
Trajectory stratification for rare event simulation
I will outline a general mathematical framework for trajectory stratification of stochastic processes based on the non-equilibrium umbrella sampling method. Trajectory stratification involves decomposing trajectories of the underlying process into fragments limited to restricted regions of state space (strata), computing averages over the distributions of the trajectory fragments within the strata with minimal communication between them, and combining those averages with appropriate weights to yield averages with respect to the original underlying process. The result is an efficient and robust scheme for very general rare event simulation problems. Our framework reveals the full generality and flexibility of trajectory stratification, and it illuminates a common mathematical structure shared with the highly successful, equilibrium umbrella sampling method for free energy calculations.
L'évolution à long terme du système solaire et le comportement de la matière à l'échelle moléculaire peuvent être décrits à l'aide des mêmes outils mathématiques: des modèles d'équations différentielles (équations qui mélangent une fonction et ses dérivées) avec certaines propriétés géométriques similaires. Leur complexité rend leur solution exacte inaccessible, et nécessite donc une approximation par des méthodes numériques adaptées.
Alors que la méthode d'Euler, première méthode numérique, fête maintenant son 250è anniversaire, la construction et l'analyse de méthodes numériques efficaces et fiables pour les équations différentielles reste un défi majeur des mathématiques appliquées et du calcul scientifique. Dans cet exposé, nous présentons à travers ces deux exemples de la mécanique céleste et de la dynamique moléculaire quelques enjeux de l'analyse numérique pour ce type de problème d'évolution en temps long, où préserver la géométrie du problème est essentiel, y compris dans un cadre aléatoire (équations différentielles stochastiques).