Werner Krauth (Ecole Normale Supérieure)
Abstract. 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 notorious 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). The resulting general class of fast algorithms overcomes the Markov-chain Monte Carlo algorithm’s limitations of the detailed-balance condition and goes beyond hybrid Monte Carlo.
As an application I discuss our solution of the 2-d melting problem for hard disks and general potentials, and present the cell-veto algorithm for treating long-range systems without cutoffs nor Ewald summations. I will then discuss some open mathematical and algorithmic problems associated with the Beyond-Metropolis paradigm.
E. P. Bernard, W. Krauth, D. B. Wilson Phys.Rev. E 80 056704 (2009).
E. P. Bernard, W. Krauth Phys. Rev. Lett. 107, 155704 (2011).
M. Michel, S. C. Kapfer, W. Krauth J. Chem. Phys. 140 54116 (2014).
S. C. Kapfer, W. Krauth Phys. Rev. Lett. 114, 035702 (2015).
S. C. Kapfer, W. Krauth, arXiv:1606.06780 (2016).