In molecular simulations, efficient methods for investigating equilibration and slow relaxation in dense systems are crucial yet challenging. This study focuses on the diffusional characteristics of monodisperse hard disk systems at equilibrium, comparing novel methodologies of event-chain Monte Carlo variants, specifically the Newtonian event-chain and straight event-chain algorithms. We systematically analyze both event-based and CPU time-based efficiency in liquid and solid phases, aiming to elucidate the microscopic mechanisms underlying structural relaxation. Our results demonstrate how chain length or duration, system size, and phase state influence the efficiency of diffusion dynamics, including hopping motion. This work provides insights into optimizing simulation techniques for highly packed systems and has the potential to improve our understanding of diffusion dynamics even in complex many-body systems.

1.
W.
Krauth
,
Statistical Mechanics: Algorithms and Computations
(
Oxford University Press
,
Oxford
,
2006
).
2.
M. P.
Allen
and
D. J.
Tildesley
,
Computer Simulation of Liquids
, 2nd ed. (
Oxford University Press
,
Oxford
,
2017
).
3.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation
, 3rd ed. (
Academic Press
,
San Diego
,
2023
).
4.
D. C.
Rapaport
,
The Art of Molecular Dynamics Simulation
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
2004
).
5.
N.
Metropolis
,
A. W.
Rosenbluth
,
M. N.
Rosenbluth
,
A. H.
Teller
, and
E.
Teller
, “
Equation of state calculations by fast computing machines
,”
J. Chem. Phys.
21
,
1087
1092
(
1953
).
6.
B. J.
Alder
and
T. E.
Wainwright
, “
Phase transition for a hard sphere system
,”
J. Chem. Phys.
27
,
1208
1209
(
1957
).
7.
W. W.
Wood
and
J. D.
Jacobson
, “
Preliminary results from a recalculation of the Monte Carlo equation of state of hard spheres
,”
J. Chem. Phys.
27
,
1207
1208
(
1957
).
8.
B. J.
Alder
and
T. E.
Wainwright
, “
Studies in molecular dynamics. I. General method
,”
J. Chem. Phys.
31
,
459
466
(
1959
).
9.
M.
Isobe
, “
Simple and efficient algorithm for large scale molecular dynamics simulation in hard disk system
,”
Int. J. Mod. Phys. C
10
,
1281
1293
(
1999
).
10.
Y.
Hiwatari
and
M.
Isobe
(Ed.), “
The 50th anniversary of The Alder Transition—Recent progress on computational statistical physics
,”
Prog. Theor. Phys. Suppl.
178
(Kyoto,
2009
).
11.
M.
Isobe
, “
Hard sphere simulation in statistical physics—Methodologies and applications
,”
Mol. Simul.
42
,
1317
1329
(
2016
).
12.
B.
Li
,
Y.
Nishikawa
,
P.
Höllmer
,
L.
Carillo
,
A. C.
Maggs
, and
W.
Krauth
, “
Hard-disk pressure computations—A historic perspective
,”
J. Chem. Phys.
157
,
234111
(
2022
).
13.
B. J.
Alder
and
T. E.
Wainwright
, “
Phase transition in elastic disks
,”
Phys. Rev.
127
,
359
361
(
1962
).
14.
E. P.
Bernard
and
W.
Krauth
, “
Two-step melting in two dimensions: First-order liquid-hexatic transition
,”
Phys. Rev. Lett.
107
,
155704
(
2011
).
15.
M.
Engel
,
J. A.
Anderson
,
S. C.
Glotzer
,
M.
Isobe
,
E. P.
Bernard
, and
W.
Krauth
, “
Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods
,”
Phys. Rev. E
87
,
042134
(
2013
).
16.
J. M.
Kosterlitz
and
D. J.
Thouless
, “
Ordering, metastability and phase transitions in two-dimensional systems
,”
J. Phys. C: Solid State Phys.
6
,
1181
(
1973
).
17.
B. I.
Halperin
and
D. R.
Nelson
, “
Theory of two-dimensional melting
,”
Phys. Rev. Lett.
41
,
121
124
(
1978
).
18.
D. R.
Nelson
and
B. I.
Halperin
, “
Dislocation-mediated melting in two dimensions
,”
Phys. Rev. B
19
,
2457
2484
(
1979
).
19.
A. P.
Young
, “
Melting and the vector Coulomb gas in two dimensions
,”
Phys. Rev. B
19
,
1855
1866
(
1979
).
20.
K. J.
Strandburg
, “
Two-dimensional melting
,”
Rev. Mod. Phys.
60
,
161
207
(
1988
).
21.
E. P.
Bernard
,
W.
Krauth
, and
D. B.
Wilson
, “
Event-chain Monte Carlo algorithms for hard-sphere systems
,”
Phys. Rev. E
80
,
056704
(
2009
).
22.
W.
Krauth
, “
Event-chain Monte Carlo: Foundations, applications, and prospects
,”
Front. Phys.
9
,
229
(
2021
).
23.
G.
Tartero
and
W.
Krauth
, “
Concepts in Monte Carlo sampling
,”
Am. J. Phys.
92
,
65
77
(
2024
).
24.
M.
Michel
,
S. C.
Kapfer
, and
W.
Krauth
, “
Generalized event-chain Monte Carlo: Constructing rejection-free global-balance algorithms from infinitesimal steps
,”
J. Chem. Phys.
140
,
054116
(
2014
).
25.
S. C.
Kapfer
and
W.
Krauth
, “
Irreversible local Markov chains with rapid convergence towards equilibrium
,”
Phys. Rev. Lett.
119
,
240603
(
2017
).
26.
M.
Klement
and
M.
Engel
, “
Efficient equilibration of hard spheres with Newtonian event chains
,”
J. Chem. Phys.
150
,
174108
(
2019
).
27.
P.
Höllmer
,
N.
Noirault
,
B.
Li
,
A. C.
Maggs
, and
W.
Krauth
, “
Sparse hard-disk packings and local Markov chains
,”
J. Stat. Phys.
187
,
31
(
2022
).
28.
M.
Isobe
and
W.
Krauth
, “
Hard-sphere melting and crystallization with event-chain Monte Carlo
,”
J. Chem. Phys.
143
,
084509
(
2015
).
29.
M.
Klement
,
S.
Lee
,
J. A.
Anderson
, and
M.
Engel
, “
Newtonian event-chain Monte Carlo and collision prediction with polyhedral particles
,”
J. Chem. Theory Comput.
17
,
4686
(
2021
).
30.
D.
Chandler
and
J. P.
Garrahan
, “
Dynamics on the way to forming glass: Bubbles in space-time
,”
Annu. Rev. Phys. Chem.
61
,
191
217
(
2010
).
31.
L.
Berthier
and
G.
Biroli
, “
Theoretical perspective on the glass transition and amorphous materials
,”
Rev. Mod. Phys.
83
,
587
645
(
2011
).
32.
G.
Biroli
and
J. P.
Garrahan
, “
Perspective: The glass transition
,”
J. Chem. Phys.
138
,
12A301
(
2013
).
33.
C. P.
Royall
and
S. R.
Williams
, “
The role of local structure in dynamical arrest
,”
Phys. Rep.
560
,
1
75
(
2015
).
34.
S.
Torquato
and
F. H.
Stillinger
, “
Jammed hard-particle packings: From Kepler to Bernal and beyond
,”
Rev. Mod. Phys.
82
,
2633
2672
(
2010
).
35.
C.
Scalliet
,
B.
Guiselin
, and
L.
Berthier
, “
Thirty milliseconds in the life of a supercooled liquid
,”
Phys. Rev. X
12
,
041028
(
2022
).
36.
M.
Isobe
and
B. J.
Alder
, “
Generalized bond order parameters to characterize transient crystals
,”
J. Chem. Phys.
137
,
194501
(
2012
).
37.
H.
Banno
,
D.
Mugita
, and
M.
Isobe
, “
Diffusional characteristics of a Newtonian event-chain Monte Carlo in hard disk systems
,”
J. Phys.: Conf. Ser.
2207
,
012011
(
2022
).
38.
P.
Höllmer
,
A. C.
Maggs
, and
W.
Krauth
, “
Hard-disk dipoles and non-reversible Markov chains
,”
J. Chem. Phys.
156
,
084108
(
2022
).
39.
C.-H.
Lam
, “
Repetition and pair-interaction of string-like hopping motions in glassy polymers
,”
J. Chem. Phys.
146
,
244906
(
2017
).
40.
C.-T.
Yip
,
M.
Isobe
,
C.-H.
Chan
,
S.
Ren
,
K.-P.
Wong
,
Q.
Huo
,
C.-S.
Lee
,
Y.-H.
Tsang
,
Y.
Han
, and
C.-H.
Lam
, “
Direct evidence of void-induced structural relaxations in colloidal glass formers
,”
Phys. Rev. Lett.
125
,
258001
(
2020
).
41.
H.
Miyagawa
,
Y.
Hiwatari
,
B.
Bernu
, and
J. P.
Hansen
, “
Molecular dynamics study of binary soft-sphere mixtures: Jump motions of atoms in the glassy state
,”
J. Chem. Phys.
88
,
3879
3886
(
1988
).
42.
A. S.
Keys
,
L. O.
Hedges
,
J. P.
Garrahan
,
S. C.
Glotzer
, and
D.
Chandler
, “
Excitations are localized and relaxation is hierarchical in glass-forming liquids
,”
Phys. Rev. X
1
,
021013
(
2011
).
43.
T.
Speck
and
D.
Chandler
, “
Constrained dynamics of localized excitations causes a non-equilibrium phase transition in an atomistic model of glass formers
,”
J. Chem. Phys.
136
,
184509
(
2012
).
44.
M.
Isobe
,
A. S.
Keys
,
D.
Chandler
, and
J. P.
Garrahan
, “
Applicability of dynamic facilitation theory to binary hard disk systems
,”
Phys. Rev. Lett.
117
,
145701
(
2016
).
45.
S. S.
Schoenholz
,
E. D.
Cubuk
,
D. M.
Sussman
,
E.
Kaxiras
, and
A. J.
Liu
, “
A structural approach to relaxation in glassy liquids
,”
Nat. Phys.
12
,
469
471
(
2016
).
46.
D.
Mugita
and
M.
Isobe
, “
Non-equilibrium response and slow equilibration in hard disk systems
,”
EPJ Web Conf.
249
,
14004
(
2021
).
47.
B. J.
Alder
and
T. E.
Wainwright
, “
Velocity autocorrelations for hard spheres
,”
Phys. Rev. Lett.
18
,
988
990
(
1967
).
48.
B. J.
Alder
and
T. E.
Wainwright
,
J. Phys. Soc. Jpn.
26
(
Suppl
),
267
(
1969
), http://adsabs.harvard.edu/cgi-bin/nph-data_query?link_type=ABSTRACT&bibcode=1969JPSJS..26..267A.
49.
B. J.
Alder
and
T. E.
Wainwright
, “
Decay of the velocity autocorrelation function
,”
Phys. Rev. A
1
,
18
21
(
1970
).
50.
M.
Isobe
, “
Long-time tail of the velocity autocorrelation function in a two-dimensional moderately dense hard-disk fluid
,”
Phys. Rev. E
77
,
021201
(
2008
).
51.
M.
Isobe
, “
Vortex flows in two dimensions: The origin of hydrodynamic tail
,”
Prog. Theor. Phys. Suppl.
178
,
72
78
(
2009
).
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