# Bernard Krauth Wilson 2009

E. P. Bernard, W. Krauth, D. B. Wilson Event-chain algorithms for hard-sphere systems Physical Review E 80 056704 (2009)

# Paper

Description: In this paper, we first presented the event-chain algorithm, a fast irreversible Markov-chain (initially) for hard spheres and related systems. In a single move of these rejection-free methods, an arbitrarily long chain of particles is displaced, and long-range coherent motion can be induced. Numerical simulations initially showed that event-chain algorithms outperform the conventional Metropolis method. The method violates detailed balance but satisfied a global-balance conditions.

Further information: We used this algorithm, which is about 100 times faster than the local Metropolis algorithm, for our work on the liquid-hexatic transition in two-dimensional hard disks. See my 2011 paper, with Etienne Bernard, for the generalization of the event-chain algorithm to general, continuous, potentials.

This algorithm can be generalized to arbitrary interaction potentials. A first version was presented in a paper with Etienne Bernard, but the definitive method was worked out in collaboration with Manon Michel and Sebastian Kapfer, in 2014.

Much further research was dedicated to analyzing event-chain Monte Carlo. In later works, with Sebastian Kapfer, we showed that, in one dimensional models (which can be analyzed more fully), ECMC has much better scaling with the system size than regular Monte Carlo. This was even proven mathematically in a special case, together with Ze Lei.

# Illustration

Here is the basic schema of the event-chain algorithm (the added length of the arrows equals a fixed value l. The event-chain algorithm is microreversible and it has no rejections. In our applications, we use a version of the algorithm where particles move

either in the +x or the +y direction. This is also done in the below Python implementation.

# Python Implementation

Here is a basic python implementation of the straight event-chain algorithm for four disks in the version breaking detailed balance. A randomly disk always moves in the +x direction. At each step, the configuration is translated (and flipped) so that the moves starts at $(x,y)=(0.,0.)$.

###========+=========+=========+=========+=========+=========+=========+=
## PROGRAM : event_chain.py
## TYPE    : Python script (python 2.7)
## PURPOSE : Event-chain simulation for 4 particles in a square box
##           with periodic boundary conditions.
## COMMENT : L is a list of tuples, move is always in +x direction
##           Flip_conf exchanges x and y (effective move in +y direction)
## VERSION : 10 NOV 2011
##========+=========+=========+=========+=========+=========+=========+
from random import uniform, randint, choice, seed
import math, pylab, sys, cPickle
def dist(x,y):
"""periodic distance between two two-dimensional points x and y"""
d_x= abs(x[0]-y[0])%box
d_x = min(d_x,box-d_x)
d_y= abs(x[1]-y[1])%box
d_y = min(d_y,box-d_y)
return  math.sqrt(d_x**2 + d_y**2)
def Flip_conf(L,rot):
L_flip = []
for (a,b) in L:
if rot == 1: L_flip.append((box - b,a))
else: L_flip.append((b,box - a))
return L_flip,-rot
#=========================== main program starts here =========================================
box = 4.0
data = []
L = [(1.,1.),(2.,2.),(3.,3.),(3.,1.)] # initial condition
rot = 1
ltilde = 0.9
for iter in xrange(1000000):
if iter%10000 == 0: print iter
i = randint(0,3)
j = (i + randint(1,3))%4
data.append(dist(L[i],L[j]))
if randint(0,1) < 1: L,rot = Flip_conf(L,rot)
Zero_distance_to_go = ltilde
next_particle = choice(L)
while Zero_distance_to_go > 0.0:
#
#     this iteration will stop when the Zero_distance_to_go falls to zero
#
L.remove(next_particle)
L = [((x[0]-next_particle[0])%box,(x[1]-next_particle[1])%box) for x in L]
next_position,next_particle = (float("inf"),(float("inf"),0.0))
Current_position = 0.0
for x in L:
x_image = list(x)
if x[0]> box/2: x_image[0] = x[0] - box
if x[1]> box/2: x_image[1] = x[1] - box
if abs(x_image[1]) < 1.0:
x_dummy = x_image[0] - math.sqrt(1.0 - x_image[1]**2)
if x_dummy > 0.0 and x_dummy < min(Zero_distance_to_go,next_position):
next_position,next_particle = (x_dummy,x)
distance_to_next_event = next_position - Current_position
if Zero_distance_to_go  < distance_to_next_event:
Current_position += Zero_distance_to_go
L.append((Current_position,0.0))
break
else:
Current_position += distance_to_next_event
Zero_distance_to_go -= distance_to_next_event
L.append((Current_position,0.0))
pylab.hist(data,bins=40,normed=True,alpha=0.5,cumulative=True)
pylab.title("Event chain for four hard disks, l = "+str(ltilde))
pylab.xlabel("Periodic pair distance $r_{ij}$")
pylab.ylabel("Integrated probabilities $\Pi(r_{ij})$")
pylab.savefig('Event_chain'+str(ltilde)+'.png')
pylab.show()


# Comparison of output with direct-sampling algorithm

Here is a comparison between the event-chain algorithm and a perfect-sampling approach for four disks of size 1/2 in an $L\times L$ square box of length $L=4$.