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École normale supérieure École normale supérieure
24 rue Lhomond 24 rue Lhomond
- 75231 Paris Cedex 05+ 75005 Paris
France France
- Tel +33 (0)1 44 32 34 94 
werner.krauth@ens.fr werner.krauth@ens.fr
- Office: GH216  
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 +CNRS Research Director - Theoretical Physics (Directeur de recherche CNRS - classe exceptionnelle).
 +Visiting Professor in Physics - University of Oxford (2023 - 26).
 +Keeley Visiting Fellow - [https://www.wadham.ox.ac.uk Wadham College] (Oxford, UK) (October 2023 - March 2024).
-CNRS Research Director - Theoretical Physics (Directeur de recherche CNRS - classe exceptionnelle).+Distinguished Visiting Professor, Simons Center for Computational Physical Chemistry, New York University, New York (NY, USA) - April, May 2024
From January to June 2018, I was the [https://www.pks.mpg.de/fileadmin/user_upload/MPIPKS/Research/Gutzwiller_Fellows/laudatio_Krauth.pdf 2018 Martin-Gutzwiller fellow] at the [https://www.pks.mpg.de/ Max-Planck-Institute for the Physics of Complex Systems] in Dresden (Germany). From January to June 2018, I was the [https://www.pks.mpg.de/fileadmin/user_upload/MPIPKS/Research/Gutzwiller_Fellows/laudatio_Krauth.pdf 2018 Martin-Gutzwiller fellow] at the [https://www.pks.mpg.de/ Max-Planck-Institute for the Physics of Complex Systems] in Dresden (Germany).
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In 2018, I was a recipient of the Humboldt research award (Alexander von Humboldt Foundation). In 2018, I was a recipient of the Humboldt research award (Alexander von Humboldt Foundation).
- 
-=Markov-chain Monte Carlo: A modern primer= 
- 
-This is the title of a [[KingsCollege_Masterclass_2022| Masterclass]] that I gave from 28 February through 3 March 2022 at the [https://www.kcl.ac.uk/noneqsys/seminars-and-videos/past-seminars-and-videos Faculty of Natural & Mathematical Sciences at King's College London] (Great Britain). The Masterclass was organized into six separate lectures that treated everything from the foundations of MCMC (Markov-chain Monte Carlo) to modern lifted Markov chains, the concepts of perfect sampling (with practical applications), meta algorithms and, to wrap up: consensus sampling. The wonderful week at King's and in central London was my first physics trip since February 2020. While at King's College London, I also gave a talk [http://www.lps.ens.fr/%7Ekrauth/images/1/1c/KingsSeminar.pdf Fast non-reversible Markov chains in statistical physics], that included some of our most recent work. 
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 +=Oxford Lectures 2025=
-=Workshop on ECMC and related subjects=+Since 2023, I have enjoyed myself tremendously as Visiting Professor in Physics at the University of Oxford where, again this year, I spend several months. I'm heavily engaged in research within an incredibly active and collaborative environment. I also give a set of public lectures on [https://www.physics.ox.ac.uk/events/algorithms-and-computations-theoretical-physics-set-lectures Algorithms and Computations in theoretical physics] (see the announcement for a syllabus). [[Oxford_Lectures_2025|Lecture notes are made available]] step by step, or rather [https://www.collinsdictionary.com/dictionary/french-english/au-fil-de-l-eau "with the flow of the water"], as we say in French.
-An informal [[Workshop_ECMC_11_May_2021|workshop on ECMC]] and related subjects took place on Tuesday 11 May 2021 8:30am - 12:30pm Paris time. 30 researchers participated, from +I am very grateful to the Department and the University to have arranged for me to lecture on one of my favorite subjects, and also to have made these lectures "public". Anyone can attend: Students at the Oxford Physics department (or not), postdocs (or not), young and old, from all walks of life. This year's lectures started on 21 January 2025, and they take place (or rather: they took place) every Tuesday afternoon at 2pm until 11 March 2025.
-the University of Dortmund (Germany), FAU-University of Erlangen (Germany), University of Bonn (Germany), University of Tokyo (Japan), Nagoya Institute of Technology (Japan), University Clermont-Auvergne (France), ESPCI Paris (France) and ENS Paris (France).+[[Image:OxfordLectures2025.png|100px|right|frame|Public Lectures at University of Oxford, Tuesdays 2-4pm, 21 January through 11 March 2025 ]]
 +<br clear="all" />
-=Join our group=+=Research over the ages=
-Please check out [[Group_WK|my group page]] if you are interested in joining my group.+A (big) number of months ago, in 1988-89 (sic!), we were fascinated by a certain problem in the then new field of neural-networks research, namely the binary perceptron. The question was about the storage capacity of the now famous Hopfield model that, in a certain continuous case, had long been proven to be 2N, a result due to Thomas M.
- +Cover (1965) and reinterpreted by E. Gardner (1987). We found this result depressing, as it stated that you needed (with, here on the website, a few shortcuts) N real numbers in order to store 2N single bits 0 and 1. This is like paying N gold coins (floats) to buy 2N copper dimes (single bits), clearly a bad deal. So we were very interested in finding out what would happen in the '''binary''' perceptron where, instead of N real numbers (the weights of one row of the Hopfield net) you had N bits (the weights of the binary Hopfield net). Together with Manfred Opper, who then visited ENS (where I did my thesis) from the University of Giessen in Germany, [https://iopscience.iop.org/article/10.1088/0305-4470/22/11/012 we wrote a computational paper], which quite clearly showed that the capacity was reduced from 2 (Cover's result) to a strange number very close to 0.82. At the time, we were quite proud of the two tricks which made our approach work namely, on the one hand, that we approximated the binary patterns with Gaussians patterns (which greatly improved the finite-size scaling) and, on the other hand, that we used [https://en.wikipedia.org/wiki/Gray_code the Gray code] for a certain enumeration job. A few months later, again in 1989, with Marc Mézard, [https://doi.org/10.1051/jphys:0198900500200305700 we conjectured], based on a one-step replica-symmetry-breaking calculation, that the critical capacity had to be equal to 0.833, a known function. In the conclusion of our 1989 paper, we speculated that our solution might well be exact
-=Fast irreversible Markov chains in statistical physics=+[[Image:KrauthMezard1989RED.png|100px|right|frame|... conclusions of a 1989 paper...]]
-This was the title of an invited plenary talk ([http://www.lps.ens.fr/%7Ekrauth/images/b/be/CECAM50Krauth.pdf slides]) that I gave on September 10, 2019 at the [https://cecam50.cecam.org CECAM50 conference] celebrating the 50 years of this European Center for atomic and molecular computing. +
- +
-=Third MOOC Statistical Mechanics: Algorithms and Computations - Now self-paced=+
-[[Image:Poster Statistical Mechanics 2016.jpg|left|frame|Announcement poster of SMAC2016 [http://www.lps.ens.fr/%7Ekrauth/images/0/03/POSTER_Statistical_Mechanics_2016.pdf Click here for a High-definition version]]]+
- +
-The 3rd edition of the [http://en.wikipedia.org/wiki/Massive_open_online_course Massive open online course (MOOC)] on [https://www.coursera.org Coursera]: [https://www.coursera.org/course/smac 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. [[Krauth_2014|Look here]] for an editorial that I wrote after 'coming home from a MOOC'.+
- +
-The 3rd edition of SMAC comes with two major changes: +
-<ul>+
-<li>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.</li> +
-<li>There will be no more certificate, as ENS was unable to keep the certificate free of charge. </li>+
-</ul>+
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 +As I mentioned above, this was in 1989, before the bicentennial celebration of the French Revolution and many other events. For a fair number of years (months, I should say), the question of the critical capacity of the binary perceptron fascinated physicists, mathematicians and computer scientists, as it had first fascinated us. But it always appeared that "some more work" was "needed". Then finally, early in 2024, [https://arxiv.org/abs/2404.18902 a keystone paper] was written by Brice Huang, a MIT graduate student in Mathematics. He completed an enormous body of previous works, actually showed that our 0.833 had been correct all along. Brice was awarded the very prestigious [https://lids.mit.edu/news-and-events/news/brice-huang-receives-machtey-award-best-student-paper-focs-2024 Machtey award for best student paper in 2024]. Bravo, congratulations, and celebrations, Brice --- the long wait was worth it and, indeed, "'''some''' more analytical work had been required..."!
=Milestone Research= =Milestone Research=
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=Video recordings of research talks= =Video recordings of research talks=
-[https://www.youtube.com/watch?v=b55shxMUFv0 Hard-disk packings and two phase transitions of two-dimensional particle systems] (Invited talk at the workshop "Optimal Point Configurations on Manifolds", Erwin Schrödinger International Institute for Mathematics and Physics, University of Vienna, Vienna (Austria), 2021 (online talk)+[https://www.youtube.com/watch?v=b55shxMUFv0 Hard-disk packings and two phase transitions of two-dimensional particle systems] Invited talk at the workshop "Optimal Point Configurations on Manifolds", [https://www.esi.ac.at/ Erwin Schrödinger International Institute for Mathematics and Physics], University of Vienna, Vienna (Austria), 2021 (online talk)
[http://www.ipam.ucla.edu/abstract/?tid=14269&pcode=ELWS2 Fast stochastic sampling with irreversible, totally asymmetric, Markov chains] (Invited talk at Institute for Pure & Applied Mathematics, UCLA, Los Angeles (USA), 2017) [http://www.ipam.ucla.edu/abstract/?tid=14269&pcode=ELWS2 Fast stochastic sampling with irreversible, totally asymmetric, Markov chains] (Invited talk at Institute for Pure & Applied Mathematics, UCLA, Los Angeles (USA), 2017)
=Current research= =Current research=
- +[[PastResearchNotices| Go to Past Research Notices]]
I am deeply interested in statistical and condensed-matter physics, often in connection to computation and algorithms. 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. 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.
-==Event-Chain Monte Carlo: Foundations, Applications, and Prospects==+==Fast, approximation-free molecular simulation of the SPC/Fw water model using non-reversible Markov chains==
-[[Image:Event chain movie small.gif|100px|left|frame|Event-chain Monte Carlo algorithm for hard spheres and related systems (see [[Bernard Krauth Wilson 2009|article with E. P. Bernard and D. B. Wilson]]). +Many fields of computational science concern the sampling of configurations x from a distribution pi(x) which can often be written as pi(x) = exp[-beta E(x)] (that is, as a Boltzmann distribution). The configuration x then refers to the positions of thousands or millions of atoms with complicated, often long-ranged, interactions. Over the years, I have been interested in sampling methods which do not evaluate the energy E (or the difference of energies, or the gradient of E) in order to sample pi(x) = exp[-beta E(x)]. This is possible because of our use of the factorized Metropolis filter within the framework of the event-chain Monte Carlo algorithm. In July 2024, finally, the manuscript [[https://doi.org/10.1038/s41598-024-66172-0 "Fast, approximation-free molecular simulation ... "]] on which I had worked together with Philipp Höllmer and A. C. Maggs was published in the Journal "Scientific Reports". It actually proves that one can simulate large water systems (in our case, the SPC/Fw water model) without any approximation. We generate millions of samples, but absolutely do not know what is the energy of our configurations.
-]]+
-In the last decade, I've been very interested in non-reversible Markov chains, and in the event-chain Monte Carlo. What started (for us) as an ad-hoc Monte Carlo algorithm for hard spheres has morphed into a general method, with the conceptual underpinnings coming out clearer with time. In June 2021, [[Krauth_2021| my review ''Event-Chain Monte Carlo: Foundations, Applications, and Prospects'']] was published in Frontiers in Physics. Access is ''open''.+
 +The remarkable efficiency of our simulation method is rooted in three paradoxes.
 +First, the Markov process is non-reversible (that is, effectively out-of-equilibrium), yet its steady state coincides
 +with the equilibrium Boltzmann distribution. Second, the Boltzmann distribution exp(−βU) is sampled
 +without any approximation and with great efficiency although the total potential U and its derivatives, the forces,
 +are never evaluated. This sidesteps all the problems with limited-precision calculations of energies and forces.
 +The third paradox is the bundling of O(N) independent decisions to interrupt the straight-line trajectory of the
 +piecewise-deterministic Markov process into an expression that can be evaluated in constant time. The paper is openly accessible, and even all the computer code has been rendered open-source.
 +==Fast, approximation-free molecular simulation of the SPC/Fw water model using non-reversible Markov chains==
-==Fast sequential Markov chains== 
-[[Image:ConfigurationDipole.png|left|600px|border|Dipole with two atoms.]] 
-As many people know, the Markov-chain Monte Carlo method in the form of the Metropolis algorithm for reversible Markov chains, was discovered in 1953, by Metropolis, Rosenbluth, Rosenbluth (sic), Teller, and Teller (sic) in a [https://doi.org/10.1063%2F1.1699114 landmark paper]. But the authors not only discovered reversible Markov chains, they also implemented the first non-reversible Markov chain! The difference may be a bit difficult to understand here, on a website, but for a reversible Markov chain a move and its inverse must always be likely. This is realized if, in a system of N particles, at each time step, one picks a random particle and moves it through a random displacement (see my book for an explanation). But Metropolis et al. didn't implement it this way: At time 1, they moved particle 1, then at time 2, they moved particle 2, and so on until, at time N+1, the first particle is moved again. This is manifestly non-reversible, but still samples the Boltzmann distribution. In later research, over the decades, it was found that whether one uses the sequential algorithm or chooses particles randomly doesn't make a great deal of a difference. No big deal! 
-In June 2020, Liang Qin (Laboratoire de Physique, ENS), Philippe Hoellmer (University of Bonn) and I looked into a variant of the sequential Markov chain algorithm, but instead of going sequentially through particle indices, we go sequentially through the directions. In practice, all particles first move (for a long time) along the x-axis, then with an inclination of 1 degree, then 2 degrees, etc (see the above figure). We studied this not for hard disks (as Metropolis et al in 1953), but a simple model of an elastic dipole. What we found was really surprising: By sequentially going through the directions, one induces very persistent counter-clockwise rotation but then, after a while, another very persistent clockwise rotation follows. We were able to prove rigorously that if we replace the 1 degree change by an angle Delta Phi, then in the limit of small Delta Phi, the rolled-out angle diverges! However, we could also prove that the net rotation of the clockwise and counter-clockwise rotations adds up to exactly zero. Finally, we showed numerically that the sequential Markov-chain algorithm is much faster than what we did before. The story is told in [[Qin_Hoellmer_Krauth_2020| this paper]].+==Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains==
 +In recent years, my colleagues and I have worked on a multitude of computational algorithms which improve on the classical methods. Specifically, we have worked on Monte Carlo algorithms based on non-reversible Markov chains. Such algorithms have had successes in applications but are generally difficult to analyze, resulting in a scarcity of exact results. In a recent manuscript [[Essler_Krauth_2023|Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains]], now [https://doi.org/10.1103/PhysRevX.14.041035 published in Physical Review X] with my colleague [https://www.physics.ox.ac.uk/our-people/essler Fabian Essler (Oxford)], we introduce the “lifted” TASEP (totally asymmetric simple exclusion process) as a paradigm for non-reversible Markov chains. Our model can be viewed as a second-generation lifting of the reversible Metropolis algorithm on a one-dimensional lattice and is exactly solvable by an unusual kind of coordinate Bethe ansatz. We establish the integrability of the model and present strong evidence that the lifting leads to relaxation on shorter timescales than in the KPZ (Kardar–Parisi–Zhang) universality class.
-==Multithreaded event-chain Monte Carlo with local times==+Here the "Popular Summary" that accompanied our paper:
-[[Image:Coincidence_table_small.jpg|left|600px|border|Coincidence table for two threads.]]+
-In the last few years, our delocalized group has worked on a new paradigm of Markov-chain Monte Carlo methods that break detailed balance yet sample the equilibrium Boltzmann distribution. The algorithms go by the name of 'event-chain Monte Carlo', and as the name indicates, they are event-driven (this means: they hop from event to event, rather than from femto-second timestamp to femto-second timestamp). One of the problem of event-driven algorithms is that they were never successfully parallelized. Looking at today's computer architectures, which are massively parallel (either on GPUs or on shared-memory CPUs), this long provided a bleak outlook for the practical applications of ECMC.+
-However, in a [[Li_Todo_Maggs_Krauth_2020| paper]] that recently appeared in Computer Physics Communications, Botao Li (from ENS), Synge Todo (from University of Tokyo), A. C. Maggs (from ESCPI Paris) and myself were able to present a multithreaded event-chain Monte Carlo algorithm for hard spheres. Threads synchronize at infrequent breakpoints and otherwise scan for local horizon violations. On x86 and ARM processors, a C++ (OpenMP) implementation that uses compare-and-swap primitives for data access achieves considerable speed-up with respect to single-threaded code. The paper appeared in a computer physics journal, and there is a lot of computer-nerdy content in it (everything is open-source and all our C++ and Python programs are open-source and available on github) . However, there is also a lot of math in the paper, including real lemmas and their proofs, which at the beginning were not at all clear to us. We present a total of six algorithm (the final one goes really fast), and one of them (Algorithm 5) is rigorously proven to be without bugs, as we can map it onto an absorbing Markov chain with 3670 states, that we can analyze exactly (the picture gives a visual impression of the proof). +Markov-chain Monte Carlo (MCMC) algorithms formulate the sampling problem for complex probability distributions as a simulation of fictitious physical systems in equilibrium, where all motion is diffusive and time reversible. But nonreversible algorithms can, in principle, sample distributions much more efficiently. In recent years, a class of “lifted” Markov chains has implemented this idea in practice, but the resulting algorithms are extremely difficult to analyze. In this work, we introduce an exactly solvable paradigm for nonreversible Markov chains.
-PS: This is the first paper of Botao Li, first-year (now second-year) graduate student at ENS, and we are all very happy.+Our paradigm, which we term the lifted totally asymmetric simple exclusion process (TASEP), describes a particular type of nonreversible dynamics for particles on a one-dimensional lattice. We show that this dynamics allows for polynomial speedups in particle number compared to the famous Metropolis MCMC algorithm. The lifted-TASEP dynamics is, in fact, faster than that of any other known class of models. To arrive at our conclusions, we combine exact methods from the theory of integrable models with extensive numerical simulations. In particular, we prove that the lifted TASEP is integrable and determine the scaling of its relaxation and mixing times with system size.
-==JeLLyFysh-Version1.0 -- a Python application for all-atom event-chain Monte Carlo==+Our work opens the door to obtaining mathematically rigorous results for speedups of nonreversible MCMC algorithms, and more generally, of lifted Markov chains arising in interacting many-particle systems.
-[[Image:JeLLyFysh.png|left|600px|border|TheJeLLyFysh logo.]]+
-For the last few years, the development of irreversible-Markov-chain methods for physics applications has been at a focus of our group's interest. Although we have constantly thought about algorithms (in particular about the event-chain algorithm), no general code was available. This changed during the week of 29 July 2019 where, on Monday, we posted a [[Hoellmer Qin Faulkner Maggs Krauth 2019| 50-page manuscript, that has since been accepted by Computer Physics Communications]], and on Thursday (1 August 2019), when we pushed to first Version of the associated open-source Python application to GitHub (see https://github.com/jellyfysh, some 15,000 lines of Python3 code). Both works are authored by Philipp Höllmer, Liang Qin, Michael F. Faulkner, A. C. Maggs, and me, and we are all quite proud to have finished. The code is 100% open access (it suffices to go onto the GitHub website and to fork it by clicking on a button). By the way, the program's name is JeLLyFysh (because we think that it will be very helpful to treat systems mostly with water, and some other stuff).+
-==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 [[Kapfer_Krauth_2016| we claimed a few years ago]], is not needed to drive the system forward! [[Faulkner_Qin_Maggs_Krauth_2018|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 'Proof-of-Concept 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, [[Klamser Kapfer Krauth 2018|in Nature Communications]]. 
-==Irreversible local Markov chains with rapid convergence towards equilibrium== 
-[[Image:Figure1_Kapfer_Krauth_2017a.jpg|left|600px|border|Mixing time scales for local Markov chains in 1d]] 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 [[Kapfer_Krauth_2017a|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 [[Bernard_Krauth_Wilson_2009|event-chain algorithm]], that we used a lot during the last years. 
-<br clear="all" /> 
[[PastResearchNotices| Continue with Past Research Notices]] [[PastResearchNotices| Continue with Past Research Notices]]
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=Upcoming events= =Upcoming events=
 +* Dresden (Germany), [https://www.pks.mpg.de/ Max Planck Institute for the Physics of Complex Systems], [https://www.pks.mpg.de/odcd25 One Dimension, Countless Directions, International Focus Workshop] 20 - 21 November 2025
 +* Trieste (Italy), [https://www.ictp.it/ International Centre for Theoretical Physics], [https://indico.ictp.it/event/10860 School on Quantum Dynamics of Matter, Light and Information] 30 August - 5 September 2025 (Invited lecture series "The second Markov-chain revolution)
 +* Roscoff (France), CECAM workshop "[https://www.cecam.org/workshop-details/crystallization-and-self-assembly-from-soft-matter-to-pharmaceuticals-to-biomineralisation-1420 Crystallization and Self-Assembly: from Soft Matter to Pharmaceuticals to Biomineralisation]" 5 - 7 May 2025 (Invited talk: [https://www.lps.ens.fr/%7Ekrauth/images/0/03/Roscoff.pdf Markov-chain sampling for long-range systems without evaluating total energies / forces])
 +* University of Florence (Italy), [https://www.ggi.infn.it/ Galileo Galilei Institute for Theoretical Physics], Winter School [https://www.ggi.infn.it/showevent.pl?id=493 SFT 2025 Lectures on Statistical Field Theories] 10 - 14 February 2025 (Lecture course: The second Markov-chain revolution).
 +* University of Tokyo, Komaba (Japan) 9th Workshop on Physics between ENS and UTokyo, 11-12 December 2024 (Co-organizer, Talk: Hamiltonian MC vs. event-chain MC: Sampling strategies beyond the diffusive regime)
 +* University of Cambridge (UK), Isaac Newton Institute for Mathematical Sciences, Workshop: [https://www.newton.ac.uk/event/ssdw04/ Monte Carlo sampling: beyond the diffusive regime] 25 November - 29 November 2024 (Invited talk: Hamiltonian Monte Carlo vs. event-chain Monte Carlo: Synopsis, benchmarks, prospects).
 +* University of Oxford (UK), Research stay 15 October - 11 December 2024
 +* Beg Rohu, Brittany (France), Summer School Concepts and Methods of Statistical Physics" 3 - 15 June 2024 (Lecture course: [[BegRohu_Lectures_2024|The second Markov-chain revolution]])
 +* New York, NY (USA), D. E. Shaw Research, 28 May 2024 (Invited Seminar: [http://www.lps.ens.fr/%7Ekrauth/images/6/6a/DEShaw.pdf Lifted Markov chains---from solvable models to applications in chemical physics])
 +* New York University (USA), Simons Center for Computational Physical Chemistry, 21 May 2024 (Seminar: [http://www.lps.ens.fr/%7Ekrauth/images/1/1a/NYU_Simons.pdf Lifted Markov chains---from solvable models to applications in chemical physics])
 +* Flatiron Institute, Center for Computational Mathematics (New York, NY (USA) 10 May 2024 (Seminar: [http://www.lps.ens.fr/%7Ekrauth/images/3/3e/FlatIronSeminar.pdf Lifted Markov chains---from solvable models to applications in chemical physics]).
 +* Stony Brook University (Stony Brook (NY) (USA) Institute for Advanced Computational Science 26 April 2024 (Invited Seminar [http://www.lps.ens.fr/%7Ekrauth/images/9/9d/StonyBrookSeminar.pdf Mixing, stopping, coupling, lifting, and other keys to the second Markov-chain revolution])
 +* New York University (USA), Simons Center for Computational Physical Chemistry, 1 April - 31 May 2024 (Distinguished Visiting Professor).
 +* King's College London (UK), Department of Physics, 5 February 2024 (Invited talk: Thermodynamic Integration, fermion sign problem, and real-space renormalization).
 +* University of Cambridge (UK), Department of Physics, 1 February 2024 (Blackboard talk: Thermodynamic Integration, fermion sign problem, and real-space renormalization).
 +* University of Cambridge (UK), Department of Physics, 31 January 2024 ( [http://www.lps.ens.fr/%7Ekrauth/images/1/1e/ColloquiumCambridge.pdf Cavendish Quantum Colloquium: Mixing, stopping, coupling, lifting, and other keys to the second Markov-chain revolution]).
 +* University of Bristol (UK), Department of Mathematics, 25-26 January 2024 ( [https://www.bristolmathsresearch.org/seminar/werner-krauth/ Invited seminar: Lifted TASEP, an integrable example of non-reversible Markov chains]).
 +* University of Bonn (Germany), Fachgruppe Physik/Astronomie, 8 December 2023 (Physikalisches Kolloquium [http://www.lps.ens.fr/%7Ekrauth/images/c/c3/PhysikKolloquium.pdf Mixing, stopping, coupling, lifting, and other keys to the second Monte Carlo revolution]).
 +* University of Bonn (Germany), Institute for applied Mathematics, 7 December 2023 (Invited "Ober"seminar) [http://www.lps.ens.fr/%7Ekrauth/images/3/3b/OberseminarBonn.pdf "The lifted TASEP, an integrable example of non-reversible Markov chains"].
 +* University of Oxford (UK), Department of Physics, CMT Forum, 8 November 2023 [http://www.lps.ens.fr/%7Ekrauth/images/9/9c/Oxford_CMT.pdf Mixing, stopping, lifting, and other keys to the second Markov-chain revolution].
 +* [https://qft.physik.hu-berlin.de/solving-quantum-field-theories-matthiasfest/ Solving Quantum Field theories] (Humboldt University, Berlin) 13-14 September 2023 (Invited talk "Integrability in the PGB Triangle: Souvenirs and Projects")
 +* [https://www.cecam.org/workshop-details/1256 CECAM Conference: "Computational statistical physics in the 21st century: The legacy of Kurt Binder"], Mainz (Germany), 11-13 September 2023 (Participation).
 +* IUPAP Conference on Computational Physics (CCP2023, Kobe, Japan) (Invited keynote talk).
 +* Nogoya Institute of Technology (Nagoya, Japan), Seminar
 +* University of Tokyo (Tokyo, Japan) (Research visit within the NOREMIA Research project framework).
* The University of Melbourne (Australia) ([https://www.matrix-inst.org.au/about-us/ MATRIX Institute], Creswick) (26 June-7 July, 2023) Program: [https://www.matrix-inst.org.au/events/monte-carlo-algorithms-in-statistical-mechanics/ Monte Carlo Algorithms in Statistical Mechanics] (Invited participation). * The University of Melbourne (Australia) ([https://www.matrix-inst.org.au/about-us/ MATRIX Institute], Creswick) (26 June-7 July, 2023) Program: [https://www.matrix-inst.org.au/events/monte-carlo-algorithms-in-statistical-mechanics/ Monte Carlo Algorithms in Statistical Mechanics] (Invited participation).
 +* CIRM Marseille
 +* Rutgers University, Piscataway, NJ (USA) 123rd (18-20 December 2022) [https://cmsr.rutgers.edu/news-events-cmsr/statistical-mechanics-conference/icalrepeat.detail/2022/12/18/1959/-/123rd-statistical-mechanics-conference 123rd Statistical Mechanics Conference] (invited talk: Lifted non-reversible Markov chains: From solvable models to real-life applications)
 +* University of Kent, Canterbury (UK) (14-17 November 2022) Invited Lecture series: "Markov-chain Monte Carlo, a modern primer"
* International Centre for Theoretical Physics (ICTP), Trieste (Italy) (04-10 September 2022) Summer school "Quantum Dynamics: From Electrons to Qbits" (Invited lecture series "Markov-chain Monte Carlo: A modern primer") * International Centre for Theoretical Physics (ICTP), Trieste (Italy) (04-10 September 2022) Summer school "Quantum Dynamics: From Electrons to Qbits" (Invited lecture series "Markov-chain Monte Carlo: A modern primer")
* Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford (UK) (01 July 2022) Seminar [http://www.lps.ens.fr/%7Ekrauth/images/6/69/Oxford_Phys_Talk.pdf Fast non-reversible Markov chains for one-dimensional particle systems] * Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford (UK) (01 July 2022) Seminar [http://www.lps.ens.fr/%7Ekrauth/images/6/69/Oxford_Phys_Talk.pdf Fast non-reversible Markov chains for one-dimensional particle systems]
* London Mathematical Society, Clay Mathematics Institute and Heilbronn Institute for Mathematical Research, London (UK) 27 June-1 July 2022) Summer school "Point Configurations: Deformations and Rigidity" (distinguished plenary lecture: [http://www.lps.ens.fr/%7Ekrauth/images/c/c0/LondonMath1.pdf Markov-chain Monte Carlo and the hard-disk Universe]). * London Mathematical Society, Clay Mathematics Institute and Heilbronn Institute for Mathematical Research, London (UK) 27 June-1 July 2022) Summer school "Point Configurations: Deformations and Rigidity" (distinguished plenary lecture: [http://www.lps.ens.fr/%7Ekrauth/images/c/c0/LondonMath1.pdf Markov-chain Monte Carlo and the hard-disk Universe]).
-* CECAM, Lausanne (Switzerland) 10-13 May 2022, [https://members.cecam.org/workshops/1124/view Machine Learning Augmented Sampling for the Molecular Sciences] (Invited talk [http://www.lps.ens.fr/%7Ekrauth/images/2/22/CECAM_Tutorial1.pdf Markov-chain Monte Carlo: A modern primer Part 1] [http://www.lps.ens.fr/%7Ekrauth/images/a/ab/CECAM_Tutorial2.pdf Part 2]M+* CECAM, Lausanne (Switzerland) 10-13 May 2022, [https://members.cecam.org/workshops/1124/view Machine Learning Augmented Sampling for the Molecular Sciences] (Invited talk [http://www.lps.ens.fr/%7Ekrauth/images/2/22/CECAM_Tutorial1.pdf Markov-chain Monte Carlo: A modern primer Part 1] [http://www.lps.ens.fr/%7Ekrauth/images/a/ab/CECAM_Tutorial2.pdf Part 2]
* King's College London (UK) (28 February-3 March 2022) Masterclass: "Markov-chain Monte Carlo: A modern primer" * King's College London (UK) (28 February-3 March 2022) Masterclass: "Markov-chain Monte Carlo: A modern primer"
-* Erwin Schrödinger International Institute for Mathematics and Physics, University of Vienna (Austria) (17-21 January 2022) workshop "Optimal Point Configurations on Manifolds", (Invited talk Hard-disk packings and two phase transitions of two-dimensional particle systems (online talk) +* [https://www.esi.ac.at/ Erwin Schrödinger International Institute for Mathematics and Physics], University of Vienna (Austria) (17-21 January 2022) workshop "Optimal Point Configurations on Manifolds", (Invited talk [http://www.lps.ens.fr/%7Ekrauth/images/b/b0/ViennaVirtual.pdf Hard-disk packings, fast Markov chains, and the two phase transitions of two-dimensional particle systems] (online talk)
[[Past_Events |Here is the schedule of past events]] [[Past_Events |Here is the schedule of past events]]
Line 139: Line 148:
-[[Image:Boson trap fast.gif|100px|right|frame|Direct-sampling algorithm for ideal bosons in a trap (see [[Holzmann Krauth 1999|article with M. Holzmann]]). Adapted for interacting bosons, this algorithm was used in a variety of articles.]]+[[Image:Boson trap fast.gif|100px|right|frame|[[Direct N bosons.py|Direct-sampling algorithm for ideal bosons in a trap]] (see [[Holzmann Krauth 1999|article with M. Holzmann]]). Adapted for interacting bosons, this algorithm was used in a variety of articles.]]
[[Image:Event chain movie small.gif|100px|left|frame|Event-chain Monte Carlo algorithm for hard spheres and related systems (see [[Bernard Krauth Wilson 2009|article with E. P. Bernard and D. B. Wilson, including Python implementation]]). This (fantastic) algorithm, about two orders of magnitude faster than local Monte Carlo, was used in our [[Bernard Krauth 2011|discovery of the first-order liquid-hexatic phase transition in hard disks]]. The method can be generalized to [[Michel_Kapfer_Krauth_2013|continuous potentials]], and we used it to [[Kapfer_Krauth_2014|map out the phase diagrams of soft-disk systems]]. Look [[Event chain.py|here for an implementation of the event-chain algorithm]] ]] [[Image:Event chain movie small.gif|100px|left|frame|Event-chain Monte Carlo algorithm for hard spheres and related systems (see [[Bernard Krauth Wilson 2009|article with E. P. Bernard and D. B. Wilson, including Python implementation]]). This (fantastic) algorithm, about two orders of magnitude faster than local Monte Carlo, was used in our [[Bernard Krauth 2011|discovery of the first-order liquid-hexatic phase transition in hard disks]]. The method can be generalized to [[Michel_Kapfer_Krauth_2013|continuous potentials]], and we used it to [[Kapfer_Krauth_2014|map out the phase diagrams of soft-disk systems]]. Look [[Event chain.py|here for an implementation of the event-chain algorithm]] ]]

Current revision

50px
Werner Krauth
Laboratoire de Physique
École normale supérieure
24 rue Lhomond
75005 Paris
France
werner.krauth@ens.fr


CNRS Research Director - Theoretical Physics (Directeur de recherche CNRS - classe exceptionnelle).

Visiting Professor in Physics - University of Oxford (2023 - 26).

Keeley Visiting Fellow - Wadham College (Oxford, UK) (October 2023 - March 2024).

Distinguished Visiting Professor, Simons Center for Computational Physical Chemistry, New York University, New York (NY, USA) - April, May 2024

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).


Contents

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Oxford Lectures 2025

Since 2023, I have enjoyed myself tremendously as Visiting Professor in Physics at the University of Oxford where, again this year, I spend several months. I'm heavily engaged in research within an incredibly active and collaborative environment. I also give a set of public lectures on Algorithms and Computations in theoretical physics (see the announcement for a syllabus). Lecture notes are made available step by step, or rather "with the flow of the water", as we say in French.

I am very grateful to the Department and the University to have arranged for me to lecture on one of my favorite subjects, and also to have made these lectures "public". Anyone can attend: Students at the Oxford Physics department (or not), postdocs (or not), young and old, from all walks of life. This year's lectures started on 21 January 2025, and they take place (or rather: they took place) every Tuesday afternoon at 2pm until 11 March 2025.

Public Lectures at University of Oxford, Tuesdays 2-4pm, 21 January through 11 March 2025
Public Lectures at University of Oxford, Tuesdays 2-4pm, 21 January through 11 March 2025


[edit]

Research over the ages

A (big) number of months ago, in 1988-89 (sic!), we were fascinated by a certain problem in the then new field of neural-networks research, namely the binary perceptron. The question was about the storage capacity of the now famous Hopfield model that, in a certain continuous case, had long been proven to be 2N, a result due to Thomas M. Cover (1965) and reinterpreted by E. Gardner (1987). We found this result depressing, as it stated that you needed (with, here on the website, a few shortcuts) N real numbers in order to store 2N single bits 0 and 1. This is like paying N gold coins (floats) to buy 2N copper dimes (single bits), clearly a bad deal. So we were very interested in finding out what would happen in the binary perceptron where, instead of N real numbers (the weights of one row of the Hopfield net) you had N bits (the weights of the binary Hopfield net). Together with Manfred Opper, who then visited ENS (where I did my thesis) from the University of Giessen in Germany, we wrote a computational paper, which quite clearly showed that the capacity was reduced from 2 (Cover's result) to a strange number very close to 0.82. At the time, we were quite proud of the two tricks which made our approach work namely, on the one hand, that we approximated the binary patterns with Gaussians patterns (which greatly improved the finite-size scaling) and, on the other hand, that we used the Gray code for a certain enumeration job. A few months later, again in 1989, with Marc Mézard, we conjectured, based on a one-step replica-symmetry-breaking calculation, that the critical capacity had to be equal to 0.833, a known function. In the conclusion of our 1989 paper, we speculated that our solution might well be exact

... conclusions of a 1989 paper...
... conclusions of a 1989 paper...


As I mentioned above, this was in 1989, before the bicentennial celebration of the French Revolution and many other events. For a fair number of years (months, I should say), the question of the critical capacity of the binary perceptron fascinated physicists, mathematicians and computer scientists, as it had first fascinated us. But it always appeared that "some more work" was "needed". Then finally, early in 2024, a keystone paper was written by Brice Huang, a MIT graduate student in Mathematics. He completed an enormous body of previous works, actually showed that our 0.833 had been correct all along. Brice was awarded the very prestigious Machtey award for best student paper in 2024. Bravo, congratulations, and celebrations, Brice --- the long wait was worth it and, indeed, "some more analytical work had been required..."!

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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.

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Video recordings of research talks

Hard-disk packings and two phase transitions of two-dimensional particle systems Invited talk at the workshop "Optimal Point Configurations on Manifolds", Erwin Schrödinger International Institute for Mathematics and Physics, University of Vienna, Vienna (Austria), 2021 (online talk)

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

[edit]

Current research

Go to Past Research Notices

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.

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Fast, approximation-free molecular simulation of the SPC/Fw water model using non-reversible Markov chains

Many fields of computational science concern the sampling of configurations x from a distribution pi(x) which can often be written as pi(x) = exp[-beta E(x)] (that is, as a Boltzmann distribution). The configuration x then refers to the positions of thousands or millions of atoms with complicated, often long-ranged, interactions. Over the years, I have been interested in sampling methods which do not evaluate the energy E (or the difference of energies, or the gradient of E) in order to sample pi(x) = exp[-beta E(x)]. This is possible because of our use of the factorized Metropolis filter within the framework of the event-chain Monte Carlo algorithm. In July 2024, finally, the manuscript ["Fast, approximation-free molecular simulation ... "] on which I had worked together with Philipp Höllmer and A. C. Maggs was published in the Journal "Scientific Reports". It actually proves that one can simulate large water systems (in our case, the SPC/Fw water model) without any approximation. We generate millions of samples, but absolutely do not know what is the energy of our configurations.

The remarkable efficiency of our simulation method is rooted in three paradoxes. First, the Markov process is non-reversible (that is, effectively out-of-equilibrium), yet its steady state coincides with the equilibrium Boltzmann distribution. Second, the Boltzmann distribution exp(−βU) is sampled without any approximation and with great efficiency although the total potential U and its derivatives, the forces, are never evaluated. This sidesteps all the problems with limited-precision calculations of energies and forces. The third paradox is the bundling of O(N) independent decisions to interrupt the straight-line trajectory of the piecewise-deterministic Markov process into an expression that can be evaluated in constant time. The paper is openly accessible, and even all the computer code has been rendered open-source.

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Fast, approximation-free molecular simulation of the SPC/Fw water model using non-reversible Markov chains

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Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains

In recent years, my colleagues and I have worked on a multitude of computational algorithms which improve on the classical methods. Specifically, we have worked on Monte Carlo algorithms based on non-reversible Markov chains. Such algorithms have had successes in applications but are generally difficult to analyze, resulting in a scarcity of exact results. In a recent manuscript Lifted TASEP: a Bethe ansatz integrable paradigm for non-reversible Markov chains, now published in Physical Review X with my colleague Fabian Essler (Oxford), we introduce the “lifted” TASEP (totally asymmetric simple exclusion process) as a paradigm for non-reversible Markov chains. Our model can be viewed as a second-generation lifting of the reversible Metropolis algorithm on a one-dimensional lattice and is exactly solvable by an unusual kind of coordinate Bethe ansatz. We establish the integrability of the model and present strong evidence that the lifting leads to relaxation on shorter timescales than in the KPZ (Kardar–Parisi–Zhang) universality class.

Here the "Popular Summary" that accompanied our paper:

Markov-chain Monte Carlo (MCMC) algorithms formulate the sampling problem for complex probability distributions as a simulation of fictitious physical systems in equilibrium, where all motion is diffusive and time reversible. But nonreversible algorithms can, in principle, sample distributions much more efficiently. In recent years, a class of “lifted” Markov chains has implemented this idea in practice, but the resulting algorithms are extremely difficult to analyze. In this work, we introduce an exactly solvable paradigm for nonreversible Markov chains.

Our paradigm, which we term the lifted totally asymmetric simple exclusion process (TASEP), describes a particular type of nonreversible dynamics for particles on a one-dimensional lattice. We show that this dynamics allows for polynomial speedups in particle number compared to the famous Metropolis MCMC algorithm. The lifted-TASEP dynamics is, in fact, faster than that of any other known class of models. To arrive at our conclusions, we combine exact methods from the theory of integrable models with extensive numerical simulations. In particular, we prove that the lifted TASEP is integrable and determine the scaling of its relaxation and mixing times with system size.

Our work opens the door to obtaining mathematically rigorous results for speedups of nonreversible MCMC algorithms, and more generally, of lifted Markov chains arising in interacting many-particle systems.



Continue with Past Research Notices

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Upcoming events

Here is the schedule of past events

[edit]

Text book

Cover of a book I wrote in 2006 Here is the book's website
Cover of a book I wrote in 2006 Here is the book's website


[edit]

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

Editorial "Coming home from a MOOC", about teaching a Massive Open Online Course (MOOC) (October 2014)

Grande conférence scientifique "Du déterminisme au stochastique : du hasard classique à l'aléatoire quantique", for the incoming science students at ENS (in French, September 2015)

"The largest Lecture Hall in the world", Article in "Physik Journal" on MOOCs, and in particular on my own MOOC (in German, March 2017)

[edit]

A picture book of algorithms

Direct-sampling algorithm for ideal bosons in a trap (see article with M. Holzmann). Adapted for interacting bosons, this algorithm was used in a variety of articles.
Direct-sampling algorithm for ideal bosons in a trap (see article with M. Holzmann). Adapted for interacting bosons, this algorithm was used in a variety of articles.
Event-chain Monte Carlo algorithm for hard spheres and related systems (see article with E. P. Bernard and D. B. Wilson, including Python implementation). This (fantastic) algorithm, about two orders of magnitude faster than local Monte Carlo, was used in our discovery of the first-order liquid-hexatic phase transition in hard disks. The method can be generalized to continuous potentials, and we used it to map out the phase diagrams of soft-disk systems. Look here for an implementation of the event-chain algorithm
Event-chain Monte Carlo algorithm for hard spheres and related systems (see article with E. P. Bernard and D. B. Wilson, including Python implementation). This (fantastic) algorithm, about two orders of magnitude faster than local Monte Carlo, was used in our discovery of the first-order liquid-hexatic phase transition in hard disks. The method can be generalized to continuous potentials, and we used it to map out the phase diagrams of soft-disk systems. Look here for an implementation of the event-chain algorithm


Exact diagonalization algorithm for Dynamical mean field theory (see article with M. Caffarel). This algorithm has been instrumental in our discovery of a first-order Mott transition in the Hubbard model in infinite dimensions. Much of our early work in the field is written up in our review with Georges, Kotliar, and Rozenberg
Exact diagonalization algorithm for Dynamical mean field theory (see article with M. Caffarel). This algorithm has been instrumental in our discovery of a first-order Mott transition in the Hubbard model in infinite dimensions. Much of our early work in the field is written up in our review with Georges, Kotliar, and Rozenberg
Rejection-free cluster algorithm for dimers (see article with R. Moessner). This algorithm was used for our discovery of a critical phase in three-dimensional dimer models (paper with Huse, Sondhi, and Moessner). Note that dimers flip about a symmetry axis between one valid configuration and another.
Rejection-free cluster algorithm for dimers (see article with R. Moessner). This algorithm was used for our discovery of a critical phase in three-dimensional dimer models (paper with Huse, Sondhi, and Moessner). Note that dimers flip about a symmetry axis between one valid configuration and another.
Alder and Wainwright's event-driven Molecular Dynamics algorithm (1957). (Animation by Maxim Berman).
Alder and Wainwright's event-driven Molecular Dynamics algorithm (1957). (Animation by Maxim Berman).


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