# Main Page

### From Werner KRAUTH

Werner Krauth Laboratoire de Physique École normale supérieure 24 rue Lhomond 75231 Paris Cedex 05 France Tel +33 (0)1 44 32 34 94 werner.krauth@ens.fr Office: LE110 How to access my office

CNRS Research Director (Theoretical Physics).

Adjunct Professor Ecole normale supérieure (professeur attaché à l'ENS).

From January to June 2018, I was the 2018 Martin-Gutzwiller fellow at the Max-Planck-Institute for the Physics of Complex Systems in Dresden (Germany).

In 2018, I was a recipient of the Humboldt research award (Alexander von Humboldt Foundation).

# ICFP Master 2018, Course on Statistical Physics

see this page for tutorials and homeworks, syllabus, and lecture notes.

# Third MOOC Statistical Mechanics: Algorithms and Computations - Now self-paced

The 3rd edition of the Massive open online course (MOOC) on Coursera: Statistical Mechanics: Algorithms and Computations has started on February 29, 2016 (participation is free of charge, and open to everyone). The first edition of the MOOC, in 2014, drew 30,000 registered students from 160 countries. Videos were viewed 250,000 times, there were close to 6000 forum posts, and students had a great time. Look here for an editorial that I wrote after 'coming home from a MOOC'.

The 3rd edition of SMAC comes with two major changes:

- Statistical Mechanics: Algorithms and Computations is now a self-paced course, just like all other courses on Coursera. I will be curious to see how it will turn out, especially whether the individual pace still allows some kind of group experience. In any case, we put in a lot of effort to make our popular course accessible to an even larger community of students. We will continue to be very present on the forum! So let's all have fun with the third edition of SMAC.
- There will be no more certificate, as ENS was unable to keep the certificate free of charge.

# Milestone Research

A paper, on a **first**-order transition in **two** dimensions, by a collaboration on **three** continents (!) that I published a few years ago in Physical Review E together with M. Engel, J. A. Anderson, S. C. Glotzer, M. Isobe, and E. P. Bernard, was chosen as the milestone article for 2013 by the journal's editorial board. This 2013 paper confirmed research published in 2011, in Physical Review Letters, with Etienne Bernard, on what really goes on in two-dimensional melting. See here for the story of the paper.

# Video recordings of research talks

Fast stochastic sampling with irreversible, totally asymmetric, Markov chains (Invited talk at Institute for Pure & Applied Mathematics, UCLA, Los Angeles (USA), 2017)

# Current research

I am deeply interested in statistical and condensed-matter physics, often in connection to computation and algorithms.
Current interests are in hard spheres, mainly the melting transition in two-dimensional disks and in two-dimensional melting, bosons (in collaboration with the experimental groups at ENS), and the theory of convergence and of coupling in Markov chains. Recent work in my research group has led to the redefinition of the dominant Markov-chain Monte Carlo paradigm, namely the Metropolis algorithm. This has already allowed us to propose powerful algorithms for particle systems, continuous spin models and long-range systems, and to obtain important physical results. Research on the *beyond-Metropolis* paradigm, together with applications in classical and quantum physics and its interfaces will likely be a focus of my research activity in the next few years.

## All-atom Coulomb simulations with irreversible Markov chains

In a nutshell, classical molecular-dynamics simulations consist in computing the forces on particles, at discretized time steps, and in moving these particles in accordance with Newton's law of motion, the famous **F**=m**a**. Likewise (in a nutshell), classical Monte Carlo calculations consist in proposing a move, then in computing the change of the total system energy, and then accepting or rejecting the move with a probability given by the Metropolis filter. How to compute the forces (for molecular dynamics) or the energies (for Monte Carlo) is a science in its own right, whenever the interactions are long-ranged, as for the Coulomb potential. Much used elaborate methods go by the names of *PP* (for particle-particle) or *PPPM* (for particle-particle / particle-mesh), or else *particle-mesh* Ewald etc. They have in common that much ingenuity is applied to compute a quantity (force / energy) that, as we claimed a few years ago, is not needed to drive the system forward! For a recent article in Journal of Chemical Physics, I teamed up with Michael Faulkner, Liang Qin, and Anthony C. Maggs, to show how this can be done in practice. In what, internally, we call our 'Confirmation paper', we explicitly show how to set up a highly efficient algorithm to simulate a model of liquid water. We indeed confirm that it is possible to sample the Boltzmann distribution (which involves the Boltzmann weight, and therefore the system energy), without computing the energy. As often, the difference lies in the subtle difference between the concepts of 'sampling' (that is, obtaining examples of a certain distribution) and of 'computing' (for example computing the energy). Technically, we succeed in drawing independent samples with a complexity 'N' log 'N' (just like the best PPPM algorithms but, we think, much faster). Now, of course, after the first excitement of our 'confirmation paper', we are all excited by the forthcoming 'benchmark paper', where we will compare not only complexities, but actual running times.

## Thermodynamic phases in two-dimensional active matter

Active matter (for example the collective dynamics of flocks of birds, of schools of fish, etc) is a very *active* field of research in statistical physics. However, active matter cannot really be described by equilibrium statistical theory where the state of what is called *the system* is fully characterized by two numbers (for example the volume and the pressure), and where the statistical weight of each configuration can be attributed an energy E, and a statistical Boltzmann weight exp(-beta E) which depends on the energy alone. Many active materials are two-dimensional (ranging from sheep on a meadow to bacterial colonies to artificial *Janus particles* on a glass place. As we are so much interested in *regular* two-dimensional particle systems (that are described by equilibrium statistical physics), we posed the question of whether there was some kind of continuous passage between the two types of models. Teaming up with Juliane U. Klamser and Sebastian C. Kapfer, we studied this question in detail. Our conclusions were published, in November 2018, in Nature Communications.

## Irreversible local Markov chains with rapid convergence towards equilibrium

Monte Carlo algorithms, generally satisfy the detailed balance condition, which prescribes that in the limit of infinite times, the*probability flow*from a configuration

**a**to a configuration

**b**equals the flow from

**b**to

**a**. This may seem terrible abstract, but it simply means that if, in a room full of air molecules, each molecule moves to the left and to the right with the same probability (and sometimes does not move at all, because there is already another particle where it wants to go), the density of air will be more or less uniform. In a recent paper with Sebastian Kapfer, in Physical Review Letters, we systematically studied irreversible local Markov chain, that is, Monte Carlo algorithms which only satisfy the global balance condition, but not the detailed balance (in the example of the air-filled room, this corresponds to algorithms where the molecules are much more likely to move in one direction than the other, but where the asymptotic density is still uniform). We considered the case of hard-sphere gases in one spatial dimension with periodic boundary conditions and, to our greatest surprise, came up with Markov chains such as the 'forward Metropolis algorithm' or the 'lifted forward Metropolis algorithm', or even the 'lifted forward Metropolis algorithm with restart' that mix much faster than the usual methods, although they reach exactly the same steady state in the limit of infinite times. We even made contact with the vast research literature on the TASEP (totally asymmetric simple exclusion process), a discrete variant of our Markov chains. We are all the more excited that the algorithms studied are but special versions of the event-chain algorithm, that we used a lot during the last years.

## Cell-veto Monte Carlo algorithm for long-range systems

In a recent paper, together with Sebastian Kapfer, we have presented what we think might be a new start idea for the notoriously difficult simulation of long-ranged systems (such as the Coulomb 1/r interaction). Usually it poses problems, because the evaluation of the energy is so difficult: In a long-ranged system of N particles, the interactions are basically*of everybody with everybody else*. This makes that the evaluation of the energy becomes complicated, and the energy is needed in 99.99% of all simulation algorithms (Monte Carlo or Molecular dynamics). In our new algorithm (an application of the event-chain method), one does not compute the system energy in order to decide on a change of the physical system, but rather looks at all the interactions separately. So, if a particle

**a**(the active particle) wants to move, it has to ask all its partners

**t_1**,

**t_2**, .... (the target particles). If there is only a single veto, the move is rejected. In the cell-veto algorithm (see the right side of the figure), the identification of the rejecting particle is preceeded by that of a veto cell. The advantage of this is that cell vetos can be identified immediately (in a constant number of operations, that is, in O(1)), and then instantly confirmed or infirmed on the particle level.

## Event-chain algorithm for continuous spin systems: XY & Heisenberg models, spin glasses

In past years, several of our key results, for example about two-dimensional melting for hard disks but also the melting scenario for soft-disk systems, have relied on the new*event-chain*algorithm, that applies to both systems, hard-core and soft-core. More recently, we realized that the event-chain algorithm could also be made to work for continuum spin systems. Earlier in 2015, work started with Manon Michel and Johannes Maier, PhD candidate and ENS-ICFP master student, respectively. The first simulations were followed by a period of hectic activity: We had discovered that the event-chain algorithm was about 100 times faster that the local Monte Carlo algorithm. Our findings were written up, during the month of August 2015, in a manuscript entitled Event-chain Monte Carlo for classical continuous spin models. Only six weeks later (!), Manon Michel and I submitted another manuscript, together with our colleagues Yoshihiko Nishikawa and Koji Hukushima, from the University of Tokyo, entitled Event-chain algorithm for the Heisenberg model: Evidence for z ≃ 1 dynamic scaling. This new finding (that still awaits confirmation for larger systems than the ones we could simulate quickly), has had an electrifying effect on us: For the first time, we see the kind of maximum speed-up that can be realized by irreversible Markov chains using the lifting paradigm, if we suppose that the recent mathematical theories apply to the algorithms we have been developing. Of course, we now hope to find the z=1 scaling in Heisenberg spin glasses and related systems and, why not, in the original hard-sphere models, in two dimensions as well as in three.

## Soft-disk melting: From liquid-hexatic coexistence to continuous transitions

By the way: the term**melting of hard disks**does not relate to the irreversible memory loss when your

**computer hard disk**catches fire (left figure), but to a fundamental phase transition in the model of two-dimensional hard spheres, that is, billiard balls without friction and without inner structure (center and right figures). The possibility that two-dimensional systems with continuous degrees of freedom could melt was discovered in 1962, by Alder and Wainwright, but the nature of the transition remained a mystery for several decades (until we solved it).

In a recent paper with Sebastian Kapfer, in Physical Review Letters (2015), we discuss phase transitions in two-dimensional socalled soft-disk systems with repulsive power-law pair interactions ∝r^(−n), using the recent generalization of Event-Chain Monte Carlo to continuous potentials. The recently established melting scenario for hard disks (corresponding to n=∞) is preserved for finite n, and first-order liquid-hexatic and continuous hexatic-solid transitions are identified. The density difference between the coexisting hexatic and liquid is non-monotonous as a function of n. For smaller n, the coexisting liquid shows extremely long orientational correlations, and positional correlations in the hexatic become extremely short. For n≲6, the liquid-hexatic transition is continuous, with correlations consistent with the KTHNY scenario.

The graph on the left provides the main result of our paper (x-axis: density/ pure hexatic; y-axis: power n): We see a large region with liquid-hexatic coexistence.

Continue with Past Research Notices

# Upcoming events

- University of Essen-Duiburg (Germany) 10 April 2019 Department colloquium (invited)
- University of Kent (Great Britain) EMNEQ workshop on "Algorithmic Perspectives on Complex Matter" 30 April - 01 May 2019 (invited talk)
- University of Tokyo, 06 June - 09 June 2019 (invited talk)
- International Congress on Industrial and Applied Mathematics 15-19 July 2019 (Invited talk, mini-symposium on molecular simulation)
- Workshop on Monte Carlo Methods, Cecam, Lausanne, Switzerland, 02 - 04 September 2019 (invited talk)
- Conference: Molecular and materials simulation at the turn of the decade: Celebrating 50 years of CECAM, Lausanne, Switzerland, 09 - 12 September 2019 (invited plenary talk)

Here is the schedule of past events

# Text book

# Interview, Popular story, video conference

2012 interview at Ecole normale supérieure (in French)

CNRS special on our work on two-dimensional melting (June 2013) (in French) in Japanese (!)

2012 Conference on time's arrow (video, in French) in the framework of the Festival "acceleration" Sacre Doctoral school

Video presentation of the Massive Open Online course at ENS

# A picture book of algorithms