Synchronization plays an important role in many physical processes. We discuss synchronization in a molecular dynamics simulation of a single particle moving through a viscous liquid while being driven across a washboard potential energy landscape. Our results show many dynamical patterns as the landscape and driving force are altered. For certain conditions, the particle's velocity and location are synchronized or phase-locked and form closed orbits in phase space. Quasi-periodic motion is common, for which the dynamical center of motion shifts the phase space orbit. By isolating synchronized motion in simulations and table-top experiments, we can study complex natural behaviors important to many physical processes.

1.
A.
Pikovsky
,
M.
Rosenblum
, and
J.
Kurths
,
Synchronization: A Universal Concept in Nonlinear Sciences
(
Cambridge U. P
.,
Cambridge
,
2003
).
2.
K.
Okamoto
,
A.
Kijima
,
Y.
Umeno
, and
H.
Shima
, “
Synchronization in flickering of three-coupled candle flames
,”
Sci. Rep.
6
,
36145
36155
(
2016
).
3.
T.
Arane
,
A. K. R.
Musalem
, and
M.
Fridman
, “
Coupling between two singing wineglasses
,”
Am. J. Phys.
77
,
1066
1067
(
2009
).
4.
J.
Jia
,
Z.
Song
,
W.
Liu
,
J.
Kurths
, and
J.
Xiao
, “
Experimental study of the triplet synchronization of coupled nonidentical mechanical metronomes
,”
Sci. Rep.
5
,
17008
17020
(
2015
).
5.
S.
Portugal
,
T.
Hubel
,
J.
Fritz
,
S.
Heese
,
D.
Trobe
,
B.
Voelkl
,
S.
Hailes
,
A. M.
Wilson
, and
J. R.
Usherwood
, “
Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight
,”
Nature
505
,
399
402
(
2014
).
6.
I.
Aihara
,
T.
Mizumoto
,
T.
Otsuka
,
H.
Awano
,
K.
Nagira
,
H. G.
Okuno
, and
K.
Aihara
, “
Spatio-temporal dynamics in collective frog choruses examined by mathematical modeling and field observations
,”
Sci. Rep.
4
,
3891
3899
(
2014
).
7.
P.
Tranchant
,
D. T.
Vuvan
, and
I.
Peretz
, “
Keeping the beat: A large sample study of bouncing and clapping to music
,”
PLoS ONE
11
(
7
),
e0160178
(
2016
).
8.
G.
Martin Hall
,
Sonya
Bahar
, and
Daniel J.
Gauthier
, “
Prevalence of rate-dependent behaviors in cardiac muscle
,”
Phys. Rev. Lett.
82
,
2995
2998
(
1999
).
9.
W.
Singer
, “
Striving for coherence
,”
Nature
397
,
391
393
(
1999
).
10.
M.
Bennett
,
M. F.
Schatz
,
H.
Rockwood
, and
K.
Wiesenfeld
, “
Huygens’ clocks
,”
Proc. Roy. Soc. A
458
,
563
579
(
2002
).
11.
M. P. N.
Juniper
,
A. V.
Straube
,
R.
Besseling
,
D. G. A. L.
Aarts
, and
R. P. A.
Dullens
, “
Microscopic dynamics of synchronization in driven colloids
,”
Nat. Commun.
6
,
7187
7194
(
2015
).
12.
M. P. N.
Juniper
,
U.
Zimmermann
,
A. V.
Straube
,
R.
Besseling
,
D. G. A. L.
Aarts
,
H.
Löwen
, and
R. P. A.
Dullens
, “
Dynamic mode locking in a driven colloidal system: Experiments and theory
,”
New J. Phys.
19
(
1
),
013010
(
2017
).
13.
A.
Ashkin
, “
Optical trapping and manipulation of neutral particles using lasers
,”
Proc. Natl. Acad. Sci. U. S. A.
94
,
4853
4860
(
1997
).
14.
D. G.
Grier
, “
A revolution in optical manipulation
,”
Nature
424
,
810
816
(
2003
).
15.
A.
Pertsinidis
and
X.
Ling
, “
Equilibrium configurations and energetics of point defects in two-dimensional colloidal crystals
,”
Phys Rev. Lett.
87
,
098303
(
2001
).
16.
C.
Reichhardt
and
C. J.
Olson Reichhardt
, “
Depinning and nonequilibrium dynamic phases of particle assemblies driven over random and ordered substrates: A review
,”
Rep. Prog. Phys.
80
,
026501
(
2017
).
17.
C.
Reichhardt
and
C. J. O.
Reichhardt
, “
Shapiro steps for skyrmion motion on a washboard potential with longitudinal and transverse ac drives
,”
Phys. Rev. B
92
(
22
),
224432
(
2015
).
18.
E. M.
Purcell
, “
Life at low Reynolds numbers
,”
Am. J. Phys.
45
,
3
11
(
1977
).
19.
C.
Reichhardt
,
R. T.
Scalettar
,
G. T.
Zimányi
, and
N.
Grønbech-Jensen
, “
Phase-locking of vortex lattices interacting with periodic pinning
,”
Phys. Rev. B
61
,
R11914
(
2000
).
20.
P.
Bak
, “
The devil’s staircase
,”
Phys. Today
39
(
12
),
38
45
(
1986
).
21.
S.
Shapiro
, “
Josephson currents in superconducting tunneling: The effect of microwaves and other observations
,”
Phys. Rev. Lett.
11
,
80
82
(
1963
).
22.
A. A.
Golubov
,
M. Yu.
Kupriyanov
, and
E.
Ilìchev
, “
The current-phase relation in Josephson junctions
,”
Rev. Mod. Phys.
76
,
411
469
(
2004
).
23.
See supplementary material at https://www.scitation.org/doi/suppl/10.1119/10.0005037 for the molecular dynamics code to generate the figures and solve the problems and <https://github.com/mcddanielle/AJP_project/>.
24.
M.
Newman
,
Computational Physics
(
CreateSpace Independent Publishing Platform
,
South Carolina
,
2012
).
25.
J.
Taylor
,
Classical Mechanics
(
University Science Books
,
California
,
2005
).
26.
G.
Volpe
and
G.
Volpe
, “
Simulation of a Brownian particle in an optical trap
,”
Am. J. Phys.
81
(
3
),
224
230
(
2013
).
27.
IAPWS R12-08
, “
Release on the IAPWS formulation 2008 for the viscosity of ordinary water substance
, September
2008
, <http://www.iapws.org/relguide/viscosity.html>.
28.
A.
Einstein
,
Investigations on the Theory of the Brownian Movement
(
Dover Publications
,
New York
,
1956
).
29.
C. R.
Harris
,
K. J.
Millman
,
S. J.
van der Walt
et al, “
Array programming with NumPy
,”
Nature
585
,
357
362
(
2020
).
30.
S. P.
Benz
,
M. S.
Rzchowski
,
M.
Tinkham
, and
C. J.
Lobb
, “
Fractional giant Shapiro steps and spatially correlated phase motion in 2D Josephson arrays
,”
Phys. Rev. Lett.
64
,
693
696
(
1990
);
[PubMed]
D.
Domínguez
and
J. V.
José
, “
Giant Shapiro steps with screening currents
,”
Phys. Rev. Lett.
69
,
514
517
(
1992
).
[PubMed]

Supplementary Material

AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.