Molecular Dynamics, Thermostats

and Convergence to Equilibrium

A Workshop in Edinburgh, November 12-14, 2008

Programme (5.11.2008 REVISED)

Contents

1. Theme

2. Schedule

3. Titles and Abstracts

4. Meeting Venue and Map

5. Telephones

6. Websites

7. Travel Information

1. Theme of Meeting

The purpose of this meeting is to bring together groups from mathematics and physics to try to gain a better understanding of the question of how molecular systems (and molecular models) reach equilibrium, schemes for accelerating or manipulating this process of both deterministic and stochastic type, and numerical techniques for computational treatment of these schemes. Of particular interest are techniques to analyse or quantify the convergence to equilibrium in molecular simulations.

2, Schedule

The meeting will take place November 12-14, 2008, starting at 12pm on the 12th and concluding by 3:30 on the 14th.

	12.11	13.11		14.11
9:00		Gulio Casati		Angelo Vulpiani
9:45		Rainer Klages		Oliver Penrose
10:30		coffee		coffee
11:00		Greg Ezra		Alex Samoletov
11:45		Nawaf Bou-Rabee		Li Baowen
12:30	12pm Registration & Buffet Lunch@icms	Lunch (participants fend for themselves)		Buffet Lunch @icms
2:00	Christoph Dellago	Martin Hairer		Michael Tretyakov
2:45	break	break		break
2:50	Tony Lelievre	Carl Dettmann		Ben Leimkuhler
3:35	coffee	coffee		Close of meeting
4:00	Roman Schubert	Florian Theil
4:45	break	break
4:50	Greg Pavliotis	Noorizadeh	Howard
5:35	Close of day	Close of day
	reception@icms	7pm: dinner@Nargile

Talks are (mostly) 40 minutes + 5 minutes for questions, often followed by a 5 minute break

3. Speakers/Talk Titles/Abstracts

(ordered according to appearance in schedule)

Christoph Dellago (Vienna)

Sampling rare chaotic and regular trajectories

Tony Lelievre (ENPC, Paris)

Adaptive methods in molecular dynamics

We will present a mathematical viewpoint on adaptive methods for free energy computations, with an emphasis on the Adaptive Biasing Force (ABF) techniques. An implementation in parallel of the ABF method is proposed, with an interacting particle system with a selection mechnism associated to a birth death process. Applications of the method to sampling problems in Bayesian statistics will be mentioned.

Roman Schubert (Bristol)

Quantum Transition State Theory

Greg Pavliotis (Imperial)

Weak friction asymptotics for the Langevin equation

We study the long-time, weak-friction asymptotics for the one dimensional Langevin equation with a periodic potential. We show that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined weak friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.

Gulio Casati (Como)

Classical and quantum transport: from Fourier law to thermoelectric efficiency

The understanding of the underlying dynamical mechanisms which determines the macroscopic laws of heat conduction is a long standing task of non-equilibrium statistical mechanics. A better understanding of such mechanism may also lead to potentially interesting applications based on the possibility to control the heat flow. Of particular interest is the problem, almost completely unexplored, of the derivation of Fourier law from quantum dynamics. To this end we discuss heat transport in a model of a quantum interacting spin chain and we provide clear numerical evidence that Fourier law sets in above the transition to quantum chaos. In particular a new phenomenon of negative differential conductivity is illustrated. Finally we consider the transport of particles and heat in models of elastically colliding particles and we discuss the conditions under which thermoelectric efficiency can approach the Carnot limit.

Rainer Klages (Queen Mary)

Nosé-Hoover thermostats, active Brownian particles and crater-like velocity distributions

Nosé-Hoover thermostats provide a well-known deterministic version of ordinary Langevin dynamics by ideally generating a canonical velocity distribution in equilibrium situations. Active Brownian particles, on the other hand, refer to a theory that is used in order to model the self-propelled motion of biological entities such as, for example, cells migrating on substrates. As for Nosé-Hoover dynamics, in this case of generalised Langevin dynamics the friction coefficient is assumed to be velocity dependent representing the take-up of energy from some external reservoir and its conversion into kinetic energy. In my talk I will show that these two seemingly different models are quite related to each other. Particularly, I will focus onto the origin of crater-like velocity distributions, which are produced by both types of generalised Langevin dynamics. Starting from molecular dynamics computer simulations for Nosé-Hoover dynamics I will argue that these distributions can be understood in terms of a combination of canonical with microcanonical distributions.

Greg Ezra (Cornell)

Thermostats: almost-Poisson structure, measure-preserving integrators, and mappings between Hamiltonian and non-Hamiltonian systems

We describe the Poisson (in general, almost-Poisson) structure common to most deterministic thermostats proposed to date, and indicate how such structure can be exploited to obtain measure-preserving geometric integration algorithms. We outline the difficulties that arise as a result of the failure of the Jacobi identity, in particular for formal backward error analysis of non-symplectic integration algorithms, but note that such non-symplectic methods can nevertheless perform well in practice.

In the second part of our talk, we discuss some new ideas (developed in collaboration with Wiggins) concerning the study of mappings between Hamiltonian and non-Hamiltonian systems. Application of recent results and methods from the theory of multidimensional Hamiltonian systems, specifically the theory of molecular reaction rates, provides a potentially useful route to the characterization of thermostat dynamics.

Nawaf Bou-Rabee (New York University)

Geometric Langevin Integrators

This talk analyzes generalizations of variational integrators from simple mechanical systems to Langevin processes that are relevant in, e.g., computations of molecular dynamics trajectories. An introduction to variational integrators and discrete mechanics is provided. Special attention is paid towards operating these integrators in multiscale systems at the verge of linear stability where autonomous backward error analysis often breaks down. Against this backdrop Langevin integrators are presented based on composing a variational integrator with a stochastic Ornstein-Uhlenbeck flow. These discretizations of Langevin equations are quite natural, but seem to have only recently been proposed in the literature by Vanden-Eijnden & Ciccotti [2006] and Bussi & Parrinello [2007]. In the context of uniformly Lipschitz Hamiltonian vector fields, it is shown that these splitting methods are geometrically ergodic and yield approximations of the invariant measure dictated by the energy error of the variational integrator. In particular, if the variational integrator admits no energy error, then the scheme exactly samples the invariant measure.

If the Hamiltonian vector field is only locally Lipschitz a.e. (as is often the case in molecular dynamics), rejection sampling becomes essential to guarantee nonlinear stability of the Langevin integrator. By using a modified detailed balance condition, it is shown that the rejection rate of the Metropolis-stabilized scheme is a function of only the energy error of the variational integrator. Geometric ergodicity of the resulting Metropolis-stabilized, discrete Markov chain is demonstrated in the locally Lipschitz a.e. context. Numerics involving conformation changes of butane are presented.

Parts of the talk are based on a collaborative paper with Houman Owhadi (Caltech), and a forthcoming collaborative paper with Christof Schütte (FUB) and Eric Vanden-Eijnden (NYU).

Martin Hairer (Warwick)

How hot can a heat bath get?

We consider one of the simplest possible models of non-equilibrium statistical mechanics: two coupled oscillators in contact with two Langevin heat baths. The twist is that one of the heat baths is at "infinite" temperature in the sense that no friction acts on the corresponding degree of freedom. We explore the question of the existence of a stationary state in this situation and what it looks like if it exists. In particular, we will see that the question "is the corresponding degree of freedom at infinite temperature" can have a surprising variety of answers.

Carl Dettmann (Bristol)

Thermostats, stochastic dynamics and ergodicity

I will present a general scheme for generating thermostat equations from stochastic dynamical systems. Beginning with the Langevin equation, the familiar Gaussian and Nose-Hoover schemes can be recovered. I will then introduce some new thermostats: virial, Smoluchowski and NH-Langevin. Finally I will give some recent examples of non-ergodicity in Lennard-Jones systems, and discuss the interaction between thermostats and ergodicity. This talk is based on work with M Chaplain, A Samoletov, and separately with B Todd and W Zheng.

Florian Theil (Warwick)

Tba

Emad Noorizadeh (Edinburgh)

Highly degenerate diffusions for sampling rare events

Teil Howard (Bristol)

Tba

Angelo Vulpiani (INFN Rome)

Coarse graining of master equations with fast and slow states

We propose a general method for simplifying master equations by eliminating from the description rapidly evolving states. The physical recipe we impose is the suppression of these states and a renormalization of the rates of all the surviving states. In some cases, this decimation procedure can be analytically carried out and is consistent with other analytical approaches, like in the problem of the random walk in a double-well potential. We discuss the application of our method to nontrivial examples: diffusion in a lattice with defects and a model of an enzymatic reaction outside the steady state regime.

Oliver Penrose (Heriot-Watt)

A "gentle" Langevin thermostat

Alex Samoletov (Donetsk)

Thermostats beyond standard applications of the statistical mechanics

I will discuss an application of configurational (Smoluchowski) thermostat to modeling of fluctuations in the Verhulst equation. The aim is a better understanding of our method to improve ergodicity in the deterministic Smoluchowski thermostat. Then I will discuss relevance of the excitable dynamics to this thermostat. Finally I will discuss the intersection between thermostats and multi-time scales dynamics. This talk is based on joint work with C. Dettmann and M. Chaplain.

Li Baowen (Singapore)

Anomalous diffusion and anomalous heat conduction in nano structure: experiment and theory

Michael Tretyakov (Leicester)

A Langevin Thermostat for Rigid Body Dynamics

We present a new method for isothermal rigid body simulations using Langevin dynamics. It can be combined with the traditional Langevin or gradient (Brownian) dynamics for the translational degrees of freedom to correctly sample the NVT distribution in a simulation of rigid molecules. We propose simple, quasi-symplectic second-order numerical integrators and test their performance on the TIP4P model of water. We also investigate the optimal choice of thermostat parameters. The talk is based on a joint work with Ruslan L. Davidchack and Richard Handel (Leicester).

Ben Leimkuhler (Edinburgh)

Exact Distributions from Dynamical and Stochastic Thermostats

Numerical methods for dynamical (e.g. Nose Hoover) and stochastic (e.g Langevin) thermostats introduce bias which may be severe when the stepsize is large but still below the stability threshold. In this talk, I will describe methods to correct that bias. This talk presents joint work with Stephen Bond (Illinois) and Sebastian Reich (Potsdam).

4. Meeting Venue and Map

ICMS 14 India Street, Edinburgh EH3 6EZ

website: http://www.icms.org.uk

for other better quality maps see websites listed below

5. Telephones:

Andrea Dobson: +44 (0) 131 650 5935

Ben Leimkuhler�??s mobile: +44 (0) 7944 982 504

(on during workshop only)

ICMS: +44 (0) 131 220 1777

(for emergencies or if other contacts fail)

6. Websites

Meeting: http://www.maths.bris.ac.uk/~macpd/edinburgh/

ICMS http://www.icms.org.uk/

Ardenlee Guesthouse http://www.ardenlee.co.uk/

Nargile Restaurant http://www.nargile.co.uk/

7. Travel Information

Arrival at Edinburgh Airport: The Airport Express (Airlink) bus will go directly very near to ICMS, and makes only 3 or 4 stops along the way. It costs £3 one way or £5 return. Get off at West End at the start of Princes Street and walk to India Street (see map).