## Two-dimensional melting : New algorithms, new insights

The hard-disk model has exerted outstanding influence on computational physics and statistical mechanics. Decades ago, hard disks were the first system to be studied by Markov-chain Monte Carlo methods (1) and by molecular dynamics (2). It was in hard disks, through numerical simulations, that a two-dimensional melting transition was first seen to occur (3) even though such systems cannot develop long-range crystalline order. Analysis of the system was made difficult by the absence of powerful simulation methods. In recent years, we developed the powerful event-chain algorithm (4) which allowed us to prove (5) that hard disks melt with a first-order transition from the liquid to the hexatic and a continuous transition from the hexatic to the solid. Subsequent work for soft-sphere potentials results in a generic theory of two-dimensional melting (6).

The event-chain algorithm is a first example of a class of “Beyond-Metropolis” (7) 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 (8). Moves are infinitesimal and persistent, implementing the lifting concept. 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.

1/ N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, J. Chem. Phys. 21 1087 (1953).

2/ B. J. Alder and T. E. Wainwright, J. Chern. Phys. 27,1208 (1957).

3/ B. J. Alder and T. E.Wainwright, Phys. Rev. 127, 359 (1962).

4/ E. P. Bernard, W. Krauth, D. B. Wilson, Phys.Rev. E 80 056704 (2009).

5/ E. P. Bernard, W. Krauth, Phys. Rev. Lett. 107, 155704 (2011).

6/ S. C. Kapfer, W. Krauth, Phys. Rev. Lett. 114, 035702 (2015).

7/ M. Michel, S. C. Kapfer, W. Krauth J. Chem. Phys. 140 54116 (2014).

8/ S. C. Kapfer, W. Krauth, arXiv:1606.06780 (2016) (PRE to appear).