It has been recently shown that 2D systems can exhibit crystalline phases with long-range translational order showcasing a striking violation of the Hohenberg–Mermin–Wagner (HMW) theorem, which is valid at equilibrium. This is made possible by athermal driving mechanisms that inject energy into the system without exciting long wavelength modes of the density field, thereby inducing hyperuniformity. However, as thermal fluctuations are superimposed on the non-equilibrium driving, long-range translational order is inevitably lost. Here, we discuss the possibility of exploiting non-equilibrium effects to suppress arbitrarily large density fluctuations even when a global thermal bath is coupled to the system. We introduce a model of a harmonic crystal driven both by a global thermal bath and by a momentum conserving noise, where the typical observables related to density fluctuations and long-range translational order can be analytically derived and put in relation. This model allows us to rationalize the violation of the HMW theorem observed in previous studies through the prediction of large-wavelength phonons, which thermalize at a vanishing effective temperature when the global bath is switched off. The conceptual framework introduced through this theory is then applied to numerical simulations of a hard-disk solid in contact with a thermal bath and driven out-of-equilibrium by active collisions. Our numerical analysis demonstrates how varying driving and dissipative parameters can lead to an arbitrary enhancement of the quasi-long-range order in the system regardless of the applied global noise amplitude. Finally, we outline a possible experimental procedure to apply our results to a realistic granular system.
REFERENCES
1.
P. C.
Hohenberg
, “Existence of long-range order in one and two dimensions
,” Phys. Rev.
158
(2
), 383
(1967
).2.
N.
David Mermin
and H.
Wagner
, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models
,” Phys. Rev. Lett.
17
(22
), 1133
(1966
).3.
B. I.
Halperin
, “On the Hohenberg–Mermin–Wagner theorem and its limitations
,” J. Stat. Phys.
175
(3–4
), 521
–529
(2019
).4.
N. D.
Mermin
, “Crystalline order in two dimensions
,” Phys. Rev.
176
(1
), 250
(1968
).5.
B.
Julian Alder
and T. E.
Wainwright
, “Phase transition for a hard sphere system
,” J. Chem. Phys.
27
(5
), 1208
–1209
(1957
).6.
J.
Fröhlich
and C.
Pfister
, “On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems
,” Commun. Math. Phys.
81
, 277
–298
(1981
).7.
D. R.
Nelson
and K. J.
Strandburg
, Bond-Orientational Order in Condensed Matter Systems
, Partially Ordered Systems
(Springer
, New York
, 2012
).8.
U.
Gasser
, “Crystallization in three- and two-dimensional colloidal suspensions
,” J. Phys.: Condens. Matter
21
(20
), 203101
(2009
).9.
A.
Cavagna
, A.
Cimarelli
, I.
Giardina
, G.
Parisi
, R.
Santagati
, F.
Stefanini
, and M.
Viale
, “Scale-free correlations in starling flocks
,” Proc. Natl. Acad. Sci. U. S. A.
107
(26
), 11865
–11870
(2010
).10.
T.
Vicsek
, A.
Czirók
, E.
Ben-Jacob
, I.
Cohen
, and O.
Shochet
, “Novel type of phase transition in a system of self-driven particles
,” Phys. Rev. Lett.
75
(6
), 1226
(1995
).11.
J.
Toner
and Y.
Tu
, “Long-range order in a two-dimensional dynamical XY model: How birds fly together
,” Phys. Rev. Lett.
75
(23
), 4326
(1995
).12.
J.
Toner
and Y.
Tu
, “Flocks, herds, and schools: A quantitative theory of flocking
,” Phys. Rev. E
58
(4
), 4828
(1998
).13.
J.
Toner
, Y.
Tu
, and S.
Ramaswamy
, “Hydrodynamics and phases of flocks
,” Ann. Phys.
318
(1
), 170
–244
(2005
).14.
E.
Bertin
, M.
Droz
, and G.
Grégoire
, “Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis
,” J. Phys. A: Math. Theor.
42
(44
), 445001
(2009
).15.
C. A.
Weber
, T.
Hanke
, J.
Deseigne
, S.
Léonard
, O.
Dauchot
, E.
Frey
, and H.
Chaté
, “Long-range ordering of vibrated polar disks
,” Phys. Rev. Lett.
110
(20
), 208001
(2013
).16.
K. E.
Bassler
and Z.
Rácz
, “Existence of long-range order in the steady state of a two-dimensional, two-temperature XY model
,” Phys. Rev. E
52
(1
), R9
(1995
).17.
M. D.
Reichl
, C. I.
Del Genio
, and K. E.
Bassler
, “Phase diagram for a two-dimensional, two-temperature, diffusive XY model
,” Phys. Rev. E
82
(4
), 040102
(2010
).18.
L.
Giomi
, J.
Toner
, and N.
Sarkar
, “Long-ranged order and flow alignment in sheared p-atic liquid crystals
,” Phys. Rev. Lett.
129
(6
), 067801
(2022
).19.
H.
Nakano
, Y.
Minami
, and S.-i.
Sasa
, “Long-range phase order in two dimensions under shear flow
,” Phys. Rev. Lett.
126
(16
), 160604
(2021
).20.
Y.
Minami
and H.
Nakano
, “Origin of long-range order in a two-dimensional nonequilibrium system under laminar flows
,” arXiv:2212.06390 (2022
).21.
Y.
Minami
, H.
Nakano
, and Y.
Hidaka
, “Rainbow Nambu-Goldstone modes under a shear flow
,” Phys. Rev. Lett.
126
(14
), 141601
(2021
).22.
H.
Ikeda
, “Scaling theory of continuous symmetry breaking under advection
,” arXiv:2401.01603 (2024
).23.
H.
Ikeda
and Y.
Kuroda
, “Does spontaneous symmetry breaking occur in periodically driven low-dimensional non-equilibrium classical systems?
,” arXiv:2304.14235 (2023
).24.
H.
Ikeda
, “Correlated noise and critical dimensions
,” Phys. Rev. E
108
(6
), 064119
(2023
).25.
B.
Benvegnen
, O.
Granek
, S.
Ro
, R.
Yaacoby
, H.
Chaté
, Y.
Kafri
, D.
Mukamel
, A.
Solon
, and J.
Tailleur
, “Metastability of discrete-symmetry flocks
,” Phys. Rev. Lett.
131
, 218301
(2023
).26.
A.
Maitra
and S.
Ramaswamy
, “Oriented active solids
,” Phys. Rev. Lett.
123
, 238001
(2019
).27.
Q.-L.
Lei
, M.
Pica Ciamarra
, and R.
Ni
, “Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations
,” Sci. Adv.
5
(1
), eaau7423
(2019
).28.
C.
Huang
, L.
Chen
, and X.
Xing
, “Alignment destabilizes crystal order in active systems
,” Phys. Rev. E
104
, 064605
(2021
).29.
X.-q.
Shi
, F.
Cheng
, and H.
Chaté
, “Extreme spontaneous deformations of active crystals
,” Phys. Rev. Lett.
131
, 108301
(2023
).30.
S.
Dey
, A.
Bhattacharya
, and S.
Karmakar
, “Enhanced long wavelength Mermin-Wagner fluctuations in two-dimensional active crystals and glasses
,” arXiv:2402.10625 (2024
).31.
S.
Ramaswamy
, R. A.
Simha
, and J.
Toner
, “Active nematics on a substrate: Giant number fluctuations and long-time tails
,” Europhys. Lett.
62
(2
), 196
(2003
).32.
V.
Narayan
, S.
Ramaswamy
, and N.
Menon
, “Long-lived giant number fluctuations in a swarming granular nematic
,” Science
317
(5834
), 105
–108
(2007
).33.
L.
Galliano
, M. E.
Cates
, and L.
Berthier
, “Two-dimensional crystals far from equilibrium
,” Phys. Rev. Lett.
131
, 047101
(2023
).34.
S.
Torquato
, “Hyperuniform states of matter
,” Phys. Rep.
745
, 1
–95
(2018
).35.
J.
Kim
and S.
Torquato
, “Effect of imperfections on the hyperuniformity of many-body systems
,” Phys. Rev. B
97
(5
), 054105
(2018
).36.
Y.
Kuroda
, T.
Kawasaki
, and K.
Miyazaki
, “Long-range translational order and hyperuniformity in two-dimensional chiral active crystal
,” arXiv:2402.19192 (2024
).37.
Y.
Kuroda
and K.
Miyazaki
, “Microscopic theory for hyperuniformity in two-dimensional chiral active fluid
,” J. Stat. Mech.: Theory Exp.
2023
(10
), 103203
(2023
).38.
R.
Maire
, A.
Plati
, M.
Stockinger
, E.
Trizac
, F.
Smallenburg
, and G.
Foffi
, “Interplay between an absorbing phase transition and synchronization in a driven granular system
,” Phys. Rev. Lett.
132
(23
), 238202
(2024
).39.
R.
Kubo
, “The fluctuation-dissipation theorem
,” Rep. Prog. Phys.
29
(1
), 255
(1966
).40.
U.
Marini Bettolo Marconi
, A.
Puglisi
, L.
Rondoni
, and A.
Vulpiani
, “Fluctuation–dissipation: Response theory in statistical physics
,” Phys. Rep.
461
(4-6
), 111
–195
(2008
).41.
D.
Villamaina
, A.
Baldassarri
, A.
Puglisi
, and A.
Vulpiani
, “The fluctuation-dissipation relation: How does one compare correlation functions and responses?
,” J. Stat. Mech.: Theory Exp.
2009
(07
), P07024
.42.
A.
Puglisi
, A.
Gnoli
, G.
Gradenigo
, A.
Sarracino
, and D.
Villamaina
, “Structure factors in granular experiments with homogeneous fluidization
,” J. Chem. Phys.
136
(1
), 014704
(2012
).43.
E.
Mikhailovich Lifshitz
and L.
Petrovich Pitaevskii
, Statistical Physics: Theory of the Condensed State
(Elsevier
, 2013
), Vol. 9
.44.
T. P. C.
van Noije
, M. H.
Ernst
, E.
Trizac
, and I.
Pagonabarraga
, “Randomly driven granular fluids: Large-scale structure
,” Phys. Rev. E
59
(4
), 4326
(1999
).45.
G.
Gradenigo
, A.
Sarracino
, D.
Villamaina
, and A.
Puglisi
, “Fluctuating hydrodynamics and correlation lengths in a driven granular fluid
,” J. Stat. Mech.: Theory Exp.
2011
(08
), P08017
.46.
U.
Marini Bettolo Marconi
, L.
Caprini
, and A.
Puglisi
, “Hydrodynamics of simple active liquids: The emergence of velocity correlations
,” New J. Phys.
23
(10
), 103024
(2021
).47.
P. C.
Hohenberg
and B. I.
Halperin
, “Theory of dynamic critical phenomena
,” Rev. Mod. Phys.
49
(3
), 435
(1977
).48.
C.
Nardini
, É.
Fodor
, E.
Tjhung
, F.
van Wijland
, J.
Tailleur
, and M. E.
Cates
, “Entropy production in field theories without time-reversal symmetry: Quantifying the non-equilibrium character of active matter
,” Phys. Rev. X
7
(2
), 021007
(2017
).49.
U. C.
Täuber
, “Field-theory approaches to nonequilibrium dynamics
,” in Ageing and the Glass Transition
(Springer
, 2007
), pp. 295
–348
.50.
H.
Ikeda
, “Harmonic chain far from equilibrium: Single-file diffusion, long-range order, and hyperuniformity
,” arXiv:2309.03155 (2023
).51.
P.
Glorioso
, J.
Guo
, J. F.
Rodriguez-Nieva
, and A.
Lucas
, “Breakdown of hydrodynamics below four dimensions in a fracton fluid
,” Nat. Phys.
18
(8
), 912
–917
(2022
).52.
D.
Das
, R.
Vink
, and M.
Krüger
, “Friction of a driven chain: Role of momentum conservation, goldstone and radiation modes
,” J. Phys.: Condens. Matter
36
, 215707
(2024
).53.
B.
Cui
, J.
Yang
, J.
Qiao
, M.
Jiang
, L.
Dai
, Y.-J.
Wang
, and A.
Zaccone
, “Atomic theory of viscoelastic response and memory effects in metallic glasses
,” Phys. Rev. B
96
(9
), 094203
(2017
).54.
A.
Plati
and A.
Puglisi
, “Long range correlations and slow time scales in a boundary driven granular model
,” Sci. Rep.
11
(1
), 14206
(2021
).55.
L.
Caprini
, U.
Marini Bettolo Marconi
, and A.
Puglisi
, “Spontaneous velocity alignment in motility-induced phase separation
,” Phys. Rev. Lett.
124
, 078001
(2020
).56.
P.
Espanol
and P. B.
Warren
, “Perspective: Dissipative particle dynamics
,” J. Chem. Phys.
146
(15
), 150901
(2017
).57.
R. D.
Groot
and P. B.
Warren
, “Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation
,” J. Chem. Phys.
107
(11
), 4423
–4435
(1997
).58.
C. P.
Lowe
, “An alternative approach to dissipative particle dynamics
,” Europhys. Lett.
47
(2
), 145
(1999
).59.
A.
Castellano
, J. P.
Batista
, and M. J.
Verstraete
, “Mode-coupling theory of lattice dynamics for classical and quantum crystals
,” J. Chem. Phys.
159
(23
), 234501
(2023
).60.
J.
Mabillard
and P.
Gaspard
, “Nonequilibrium statistical mechanics of crystals
,” J. Stat. Mech.: Theory Exp.
2021
(6
), 063207
.61.
K.
Hiura
, “Microscopic derivation of nonlinear fluctuating hydrodynamics for crystalline solid
,” Phys. Rev. E
108
, 054101
(2023
).62.
G.
Szamel
and M. H.
Ernst
, “Slow modes in crystals: A method to study elastic constants
,” Phys. Rev. B
48
(1
), 112
–118
(1993
).63.
G.
Szamel
, “Statistical mechanics of dissipative transport in crystals
,” J. Stat. Phys.
87
(5–6
), 1067
–1082
(1997
).64.
P. D.
Fleming
and C.
Cohen
, “Hydrodynamics of solids
,” Phys. Rev. B
13
(2
), 500
–516
(1976
).65.
P. C.
Martin
, O.
Parodi
, and P. S.
Pershan
, “Unified hydrodynamic theory for crystals, liquid crystals, and normal fluids
,” Phys. Rev. A
6
(6
), 2401
–2420
(1972
).66.
J.
Mabillard
and P.
Gaspard
, “Microscopic approach to the macrodynamics of matter with broken symmetries
,” J. Stat. Mech.: Theory Exp.
2020
(10
), 103203
; arXiv:2005.14012 [cond-mat].67.
L. D.
Landau
, E. M.
Lifshitz
, R. T.
Beyer
et al, “Hydrodynamic fluctuations
,” in Perspectives in Theoretical Physics
(Elsevier
, 1992
), pp. 359
–361
.68.
A.
Cavagna
, J.
Cristín
, I.
Giardina
, and M.
Veca
, “From noise on the sites to noise on the links: Discretizing the conserved Kardar-Parisi-Zhang equation in real space
,” Phys. Rev. E
109
(6
), 064136
(2024
).69.
L.
Caprini
, U.
Marini Bettolo Marconi
, and H.
Löwen
, “Entropy production and collective excitations of crystals out of equilibrium: The concept of entropons
,” Phys. Rev. E
108
(4
), 044603
(2023
).70.
P. M.
Chaikin
and T. C.
Lubensky
, Principles of Condensed Matter Physics
(Cambridge University Press
, 1995
).71.
J.-P.
Hansen
and I. R.
McDonald
, Theory of Simple Liquids: With Applications to Soft Matter
(Academic Press
, 2013
).72.
D.
Chen
, H.
Zhuang
, M.
Chen
, P. Y.
Huang
, V.
Vlcek
, and Y.
Jiao
, “Disordered hyperuniform solid state materials
,” Appl. Phys. Rev.
10
(2
), 021310
(2023
).73.
Q.-L.
Lei
and R.
Ni
, “Hydrodynamics of random-organizing hyperuniform fluids
,” Proc. Natl. Acad. Sci. U. S. A.
116
(46
), 22983
–22989
(2019
).74.
D.
Hexner
and D.
Levine
, “Noise, diffusion, and hyperuniformity
,” Phys. Rev. Lett.
118
(2
), 020601
(2017
).75.
D.
Hexner
and D.
Levine
, “Hyperuniformity of critical absorbing states
,” Phys. Rev. Lett.
114
(11
), 110602
(2015
).76.
A.
Mukherjee
, D.
Tapader
, A.
Hazra
, and P.
Pradhan
, “Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry
,” arXiv:2312.11181 (2023
).77.
T.
Bertrand
, D.
Chatenay
, and R.
Voituriez
, “Nonlinear diffusion and hyperuniformity from Poisson representation in systems with interaction mediated dynamics
,” New J. Phys.
21
(12
), 123048
(2019
).78.
Y.-W.
Li
and M. P.
Ciamarra
, “Accurate determination of the translational correlation function of two-dimensional solids
,” Phys. Rev. E
100
(6
), 062606
(2019
).79.
Y.
Imry
and L.
Gunther
, “Fluctuations and physical properties of the two-dimensional crystal lattice
,” Phys. Rev. B
3
, 3939
–3945
(1971
).80.
U.
Gasser
, C.
Eisenmann
, G.
Maret
, and P.
Keim
, “Melting of crystals in two dimensions
,” ChemPhysChem
11
(5
), 963
–970
(2010
).81.
E. P.
Bernard
and W.
Krauth
, “Two-step melting in two dimensions: First-order liquid-hexatic transition
,” Phys. Rev. Lett.
107
(15
), 155704
(2011
).82.
S. C.
Kapfer
and W.
Krauth
, “Two-dimensional melting: From liquid-hexatic coexistence to continuous transitions
,” Phys. Rev. Lett.
114
, 035702
(2015
).83.
B. I.
Halperin
and D. R.
Nelson
, “Theory of two-dimensional melting
,” Phys. Rev. Lett.
41
(2
), 121
(1978
).84.
X.
Ma
, J.
Pausch
, and M. E.
Cates
, “Theory of hyperuniformity at the absorbing state transition
,” arXiv:2310.17391 (2023
).85.
T.
Song
and H.
Xia
, “Kinetic roughening and nontrivial scaling in the Kardar–Parisi–Zhang growth with long-range temporal correlations
,” J. Stat. Mech.: Theory Exp.
2021
(7
), 073203
.86.
E.
Medina
, T.
Hwa
, M.
Kardar
, and Y.-C.
Zhang
, “Burgers equation with correlated noise: Renormalization-group analysis and applications to directed polymers and interface growth
,” Phys. Rev. A
39
(6
), 3053
(1989
).87.
G.
Grinstein
, D.-H.
Lee
, and S.
Sachdev
, “Conservation laws, anisotropy, and ‘self-organized criticality’ in noisy nonequilibrium systems
,” Phys. Rev. Lett.
64
(16
), 1927
(1990
).88.
P. L.
Garrido
, J. L.
Lebowitz
, C.
Maes
, and H.
Spohn
, “Long-range correlations for conservative dynamics
,” Phys. Rev. A
42
(4
), 1954
(1990
).89.
R.
Aditi Simha
and S.
Ramaswamy
, “Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles
,” Phys. Rev. Lett.
89
(5
), 058101
(2002
).90.
A.
Kundu
, O.
Hirschberg
, and D.
Mukamel
, “Long range correlations in stochastic transport with energy and momentum conservation
,” J. Stat. Mech.: Theory Exp.
2016
(3
), 033108
.91.
H.
Spohn
, “Long range correlations for stochastic lattice gases in a non-equilibrium steady state
,” J. Phys. A: Math. Gen.
16
(18
), 4275
(1983
).92.
J. R.
Dorfman
, T. R.
Kirkpatrick
, and J. V.
Sengers
, “Generic long-range correlations in molecular fluids
,” Annu. Rev. Phys. Chem.
45
(1
), 213
–239
(1994
).93.
G.
Grinstein
, “Generic scale invariance in classical nonequilibrium systems (invited)
,” J. Appl. Phys.
69
(8
), 5441
–5446
(1991
).94.
P.
Bak
, “Self-organized criticality in non-conservative models
,” Physica A
191
(1–4
), 41
–46
(1992
).95.
J. A.
Bonachela
and M. A.
Munoz
, “Self-organization without conservation: True or just apparent scale-invariance?
,” J. Stat. Mech.: Theory Exp.
2009
(09
), P09009
.96.
R. L.
Jack
, I. R.
Thompson
, and P.
Sollich
, “Hyperuniformity and phase separation in biased ensembles of trajectories for diffusive systems
,” Phys. Rev. Lett.
114
(6
), 060601
(2015
).97.
L.
Caprini
, U.
Marini Bettolo Marconi
, A.
Puglisi
, and H.
Löwen
, “Entropons as collective excitations in active solids
,” J. Chem. Phys.
159
(4
), 041102
(2023
).98.
M.
Huang
, W.
Hu
, S.
Yang
, Q.-X.
Liu
, and H. P.
Zhang
, “Circular swimming motility and disordered hyperuniform state in an algae system
,” Proc. Natl. Acad. Sci. U. S. A.
118
(18
), e2100493118
(2021
).99.
B.
Zhang
and A.
Snezhko
, “Hyperuniform active chiral fluids with tunable internal structure
,” Phys. Rev. Lett.
128
(21
), 218002
(2022
).100.
Z.
Ma
and S.
Torquato
, “Random scalar fields and hyperuniformity
,” J. Appl. Phys.
121
(24
), 244904
(2017
).101.
S.
Wilken
, A.
Chaderjian
, and O. A.
Saleh
, “Spatial organization of phase-separated DNA droplets
,” Phys. Rev. X
13
(3
), 031014
(2023
).102.
Y.
Zheng
, M. A.
Klatt
, and H.
Löwen
, “Universal hyperuniformity in active field theories
,” arXiv:2310.03107 (2023
).103.
F.
De Luca
, X.
Ma
, C.
Nardini
, and M. E.
Cates
, “Hyperuniformity in phase ordering: The roles of activity, noise, and non-constant mobility
,” J. Phys.: Condens. Matter
36
, 405101
(2024
).104.
L.
Corte
, P. M.
Chaikin
, J. P.
Gollub
, and D. J.
Pine
, “Random organization in periodically driven systems
,” Nat. Phys.
4
(5
), 420
–424
(2008
).105.
D. J.
Pine
, J. P.
Gollub
, J. F.
Brady
, and A. M.
Leshansky
, “Chaos and threshold for irreversibility in sheared suspensions
,” Nature
438
(7070
), 997
–1000
(2005
).106.
R.
Brito
, D.
Risso
, and R.
Soto
, “Hydrodynamic modes in a confined granular fluid
,” Phys. Rev. E
87
(2
), 022209
(2013
).107.
A.
Plati
, R.
Maire
, E.
Fayen
, F.
Boulogne
, F.
Restagno
, F.
Smallenburg
, and G.
Foffi
, “Quasi-crystalline order in vibrating granular matter
,” Nat. Phys.
20
, 465
(2024
).108.
F.
Smallenburg
, “Efficient event-driven simulations of hard spheres
,” Eur. Phys. J. E
45
(3
), 22
(2022
).109.
J.
Javier Brey
, V.
Buzón
, P.
Maynar
, and M. I.
García de Soria
, “Hydrodynamics for a model of a confined quasi-two-dimensional granular gas
,” Phys. Rev. E
91
(5
), 052201
(2015
).110.
V.
Garzó
, R.
Brito
, and R.
Soto
, “Enskog kinetic theory for a model of a confined quasi-two-dimensional granular fluid
,” Phys. Rev. E
98
(5
), 052904
(2018
).111.
Y.
Komatsu
and H.
Tanaka
, “Roles of energy dissipation in a liquid-solid transition of out-of-equilibrium systems
,” Phys. Rev. X
5
(3
), 031025
(2015
).112.
K. J.
Runge
and G. V.
Chester
, “Monte Carlo determination of the elastic constants of the hard-sphere solid
,” Phys. Rev. A
36
(10
), 4852
(1987
).113.
D.
Frenkel
and A. J. C.
Ladd
, “Elastic constants of hard-sphere crystals
,” Phys. Rev. Lett.
59
(10
), 1169
(1987
).114.
B. B.
Laird
, “Weighted-density-functional theory calculation of elastic constants for face-centered-cubic and body-centered-cubic hard-sphere crystals
,” J. Chem. Phys.
97
(4
), 2699
–2704
(1992
).115.
F. V.
Reyes
and J. S.
Urbach
, “Effect of inelasticity on the phase transitions of a thin vibrated granular layer
,” Phys. Rev. E
78
(5
), 051301
(2008
).116.
J. S.
Olafsen
and J. S.
Urbach
, “Clustering, order, and collapse in a driven granular monolayer
,” Phys. Rev. Lett.
81
, 4369
–4372
(1998
).117.
W.
Losert
, D. G. W.
Cooper
, and J. P.
Gollub
, “Propagating front in an excited granular layer
,” Phys. Rev. E
59
, 5855
–5861
(1999
).118.
A.
Prevost
, P.
Melby
, D. A.
Egolf
, and J. S.
Urbach
, “Nonequilibrium two-phase coexistence in a confined granular layer
,” Phys. Rev. E
70
, 050301
(2004
).119.
B.
Néel
, I.
Rondini
, A.
Turzillo
, N.
Mujica
, and R.
Soto
, “Dynamics of a first-order transition to an absorbing state
,” Phys. Rev. E
89
(4
), 042206
(2014
).120.
M.
Argentina
, M. G.
Clerc
, and R.
Soto
, “van der Waals–like transition in fluidized granular matter
,” Phys. Rev. Lett.
89
, 044301
(2002
).121.
M. G.
Clerc
, P.
Cordero
, J.
Dunstan
, K.
Huff
, N.
Mujica
, D.
Risso
, and G.
Varas
, “Liquid–solid-like transition in quasi-one-dimensional driven granular media
,” Nat. Phys.
4
(3
), 249
–254
(2008
).122.
G.
Castillo
, N.
Mujica
, N.
Sepúlveda
, J. C.
Sobarzo
, M.
Guzmán
, and R.
Soto
, “Hyperuniform states generated by a critical friction field
,” Phys. Rev. E
100
(3
), 032902
(2019
).123.
L.
Lovász
, “Discrete and continuous: Two sides of the same?
,” in Visions in Mathematics: GAFA 2000 Special Volume, Part I
(Springer
, 2010
), pp. 359
–382
.124.
J. L.
Gross
, J.
Yellen
, and M.
Anderson
, Graph Theory and its Applications
(Chapman and Hall/CRC
, 2018
).125.
A.
Cavagna
, J.
Cristín
, I.
Giardina
, T. S.
Grigera
, and M.
Veca
, “Discrete Laplacian thermostat for flocks and swarms: The fully conserved Inertial Spin Model
,” arXiv:2403.07644 (2024
).126.
A.
Cavagna
, J.
Cristín
, I.
Giardina
, and M.
Veca
, “Discrete laplacian thermostat for spin systems with conserved dynamics
,” Phys. Rev. B
107
, 224302
(2023
).127.
K. H.
Michel
, S.
Costamagna
, and F. M.
Peeters
, “Theory of anharmonic phonons in two-dimensional crystals
,” Phys. Rev. B
91
(13
), 134302
(2015
).128.
M. D.
Bustamante
, K.
Hutchinson
, Y. V.
Lvov
, and M.
Onorato
, “Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system
,” Commun. Nonlinear Sci. Numer. Simul.
73
, 437
–471
(2019
).129.
S.
Nazarenko
, Wave Turbulence
(Springer Science & Business Media
, 2011
), Vol. 825
.130.
T.
Harada
and S.-i.
Sasa
, “Equality connecting energy dissipation with a violation of the fluctuation-response relation
,” Phys. Rev. Lett.
95
(13
), 130602
(2005
).131.
A.
Crisanti
, A.
Puglisi
, and D.
Villamaina
, “Nonequilibrium and information: The role of cross correlations
,” Phys. Rev. E
85
(6
), 061127
(2012
).132.
J. S.
Lee
and H.
Park
, “Stochastic thermodynamics and hierarchy of fluctuation theorems with multiple reservoirs
,” New J. Phys.
20
(8
), 083010
(2018
).133.
Y.
Murashita
and M.
Esposito
, “Overdamped stochastic thermodynamics with multiple reservoirs
,” Phys. Rev. E
94
(6
), 062148
(2016
).134.
C.
Van den Broeck
and M.
Esposito
, “Three faces of the second law. II. Fokker-Planck formulation
,” Phys. Rev. E
82
(1
), 011144
(2010
).135.
A.
Puglisi
, P.
Visco
, E.
Trizac
, and F.
van Wijland
, “Dynamics of a tracer granular particle as a nonequilibrium Markov process
,” Phys. Rev. E
73
(2
), 021301
(2006
).136.
A.
Sarracino
, D.
Villamaina
, G.
Costantini
, and A.
Puglisi
, “Granular brownian motion
,” J. Stat. Mech.: Theory Exp.
2010
(04
), P04013
.137.
A.
Plati
, A.
Puglisi
, and A.
Sarracino
, “Thermodynamic bounds for diffusion in nonequilibrium systems with multiple timescales
,” Phys. Rev. E
107
(4
), 044132
(2023
).138.
A.
Plati
and A.
Puglisi
, “Slow time scales in a dense vibrofluidized granular material
,” Phys. Rev. E
102
, 012908
(2020
).139.
N. E.
Glatt-Holtz
, D. P.
Herzog
, S. A.
McKinley
, and H. D.
Nguyen
, “The generalized Langevin equation with power-law memory in a nonlinear potential well
,” Nonlinearity
33
(6
), 2820
(2020
).140.
141.
I.
Goychuk
, “Viscoelastic subdiffusion: From anomalous to normal
,” Phys. Rev. E
80
(4
), 046125
(2009
).142.
R.
Kupferman
, “Fractional kinetics in Kac–Zwanzig heat bath models
,” J. Stat. Phys.
114
, 291
–326
(2004
).143.
M. M.
Dygas
, B. J.
Matkowsky
, and Z.
Schuss
, “A singular perturbation approach to non-Markovian escape rate problems with state dependent friction
,” J. Chem. Phys.
84
(7
), 3731
–3738
(1986
).144.
A.
Puglisi
and D.
Villamaina
, “Irreversible effects of memory
,” Europhys. Lett.
88
(3
), 30004
(2009
).145.
L.
Ma
, X.
Li
, and C.
Liu
, “Fluctuation-dissipation theorem consistent approximation of the Langevin dynamics model
,” Commun. Math. Sci.
15
(4
), 1171
–1181
(2017
).146.
I.
Pagonabarraga
, E.
Trizac
, T. P. C.
van Noije
, and M. H.
Ernst
, “Randomly driven granular fluids: Collisional statistics and short scale structure
,” Phys. Rev. E
65
(1
), 011303
(2001
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
