Starting from a fine-scale dissipative particle dynamics (DPD) model of self-motile point particles, we derive meso-scale continuum equations by applying a spatial averaging version of the Irving–Kirkwood–Noll procedure. Since the method does not rely on kinetic theory, the derivation is valid for highly concentrated particle systems. Spatial averaging yields stochastic continuum equations similar to those of Toner and Tu. However, our theory also involves a constitutive equation for the average fluctuation force. According to this equation, both the strength and the probability distribution vary with time and position through the effective mass density. The statistics of the fluctuation force also depend on the fine scale dissipative force equation, the physical temperature, and two additional parameters which characterize fluctuation strengths. Although the self-propulsion force entering our DPD model contains no explicit mechanism for aligning the velocities of neighboring particles, our averaged coarse-scale equations include the commonly encountered cubically nonlinear (internal) body force density.

1.
Barannyk
,
L. L.
and
Panchenko
,
A.
, “
Optimizing performance of deconvolution closure for large ODE systems
,”
IMA J. Appl. Math.
80
,
1099
1123
(
2015
).
2.
Baskaran
,
A.
and
Marchetti
,
M. C.
, “
Statistical mechanics and hydrodynamics of bacterial suspensions
,”
Proc. Natl. Acad. Sci. U. S. A.
106
,
15567
15572
(
2009
).
3.
Berlyand
,
L.
and
Panchenko
,
A.
, “
Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions
,”
J. Fluid Mech.
578
,
1
34
(
2007
).
4.
Bertin
,
E.
,
Droz
,
M.
, and
Grégoire
,
G.
, “
Boltzmann and hydrodynamic description of self-propelled particles
,”
Phys. Rev. E
74
,
022101
(
2006
).
5.
Boek
,
E. S.
,
Coveney
,
P. V.
,
Lekkerkerker
,
N. H. W.
, and
van der Schoot
,
P.
, “
Simulating the rheology of dense colloidal suspension using dissipative particle dynamics
,”
Phys. Rev. E
55
(
3
),
3124
3133
(
1997
).
6.
Bolintineanu
,
D. S.
,
Crest
,
G. S.
,
Lechman
,
J. B.
,
Pierce
,
F.
,
Plimpton
,
S. J.
, and
Schunk
,
P. R.
, “
Particle dynamics modeling methods for colloid suspensions
,”
Comput. Part. Mech.
1
,
321
356
(
2014
).
7.
Bricard
,
A.
,
Caussin
,
J.-B.
,
Desreumaux
,
N.
,
Dauchot
,
O.
, and
Bartolo
,
D.
, “
Emergence of macroscopic directed motion in populations of motile colloids
,”
Nature
503
,
95
98
(
2013
).
8.
Capriz
,
G.
, “
On ephemeral continua
,”
Phys. Mesomech.
11
,
285
298
(
2008
).
9.
Capriz
,
G.
,
Fried
,
E.
, and
Seguin
,
B.
, “
Constrained ephemeral continua
,”
Rend. Lincei Mat. Appl.
23
,
157
195
(
2012
).
10.
Chuang
,
Y.
,
D’Orsogna
,
M. R.
,
Marthaler
,
D.
,
Bertozzi
,
A. L.
, and
Chayes
,
L. S.
, “
State transitions and the continuum limit for a 2D interacting, self-propelled particle system
,”
Phys. D
232
,
33
47
(
2007
).
11.
Dunkel
,
J.
,
Heidenreich
,
S.
,
Bär
,
M.
, and
Goldstein
,
R. E.
, “
Minimal continuum theories of structure formation in dense active fluids
,”
New J. Phys.
15
,
045016
(
2013
).
12.
Dusenbery
,
D. B.
,
Living at Micro Scale: The Unexpected Physics of Being Small
(
Harvard University Press
,
Cambridge, MA
,
2009
).
13.
Espanol
,
P.
, “
Hydrodynamics from dissipative particle dynamics
,”
Phys. Rev. E
52
,
1734
1742
(
1995
).
14.
Espanol
,
P.
and
Warren
,
P.
, “
Statistical mechanics of dissipative particle dynamics
,”
Europhys. Lett.
30
,
191
196
(
1995
).
15.
Evans
,
D. J.
,
Cohen
,
E. G. D.
, and
Morriss
,
G. P.
, “
Probability of second law violations in shearing steady flows
,”
Phys. Rev. Lett.
71
,
2401
2404
(
1993
).
16.
Gallavotti
,
G.
and
Cohen
,
E. G. D.
, “
Dynamical ensembles in non-equilibrium statistical mechanics
,”
Phys. Rev. Lett.
74
,
2694
2697
(
1995
).
17.
Hardy
,
R. J.
, “
Formulas for determining local properties in molecular-dynamics simulations: Shock waves
,”
J. Chem. Phys.
76
,
622
628
(
1982
).
18.
Hinz
,
D.
,
Panchenko
,
A.
,
Kim
,
T.-Y.
, and
Fried
,
E.
, “
Motility versus fluctuations: Mixtures of self-propelled and passive particles
,”
Soft Matter
10
,
9082
9089
(
2014
).
19.
Hoogerbrugge
,
P. J.
and
Koelman
,
J. M. V. A.
, “
Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics
,”
Europhys. Lett.
19
(
3
),
155
160
(
1992
).
20.
Ibele
,
M.
,
Mallouk
,
T. E.
, and
Sen
,
A.
, “
Schooling behavior of light-powered autonomous micromotors in water
,”
Angew. Chem., Int. Ed.
48
,
3308
3312
(
2009
).
21.
Ihle
,
T.
, “
Kinetic theory of flocking: Derivation of hydrodynamic equations
,”
Phys. Rev. E
83
,
030901(R)
(
2011
).
22.
Irving
,
I.
and
Kirkwood
,
J. G.
, “
The statistical theory of transport processes. IV. The equations of hydrodynamics
,”
J. Chem. Phys.
18
,
817
829
(
1950
).
23.
Koch
,
D. L.
and
Subramanian
,
G.
, “
Collective hydrodynamics of swimming microorganisms: Living fluids
,”
Annu. Rev. Fluid Mech.
43
,
637
659
(
2011
).
24.
Kudrolli
,
A.
, “
Concentration dependent diffusion of self-propelled rods
,”
Phys. Rev. Lett.
104
,
088001
(
2010
).
25.
Kudrolli
,
A.
,
Lumay
,
G.
,
Volfson
,
D.
, and
Tsimring
,
L. S.
, “
Swarming and swirling in self-propelled polar granular rods
,”
Phys. Rev. Lett.
100
,
058001
(
2008
).
26.
Laurati
,
M.
,
Mutch
,
K. J.
,
Koumakis
,
N.
,
Zausch
,
J.
,
Amann
,
C. P.
,
Schofield
,
A. B.
,
Petekidis
,
G.
,
Brady
,
J. F.
,
Horbach
,
J.
,
Fuchs
,
M.
, and
Egelhaaf
,
S. U.
, “
Transient dynamics of dense colloidal suspension under shear: Shear rate dependence
,”
J. Phys.: Condens. Matter
24
,
464104
(
2012
).
27.
Marchetti
,
M. C.
,
Joanny
,
J. F.
,
Ramaswamy
,
S.
,
Liverpool
,
T. B.
,
Prost
,
J.
,
Rao
,
M.
, and
Aditi Simha
,
J.
, “
Hydrodynamics for soft active matter
,”
Rev. Mod. Phys.
85
,
1143
1189
(
2013
).
28.
Murdoch
,
A. I.
, “
A critique of atomistic definitions of the stress tensor
,”
J. Elasticity
88
,
113
140
(
2007
).
29.
Murdoch
,
A. I.
,
Physical Foundations of Continuum Mechanics
(
Cambridge University Press
,
Cambridge
,
2012
).
30.
Murdoch
,
A. I.
and
Bedeaux
,
D.
, “
Continuum equations of balance via weighted averages of microscopic quantities
,”
Proc. R. Soc. A
445
,
157
179
(
1994
).
31.
Murdoch
,
A. I.
and
Bedeaux
,
D.
, “
A microscopic perspective on the physical foundations of continuum mechanics–Part I: Macroscopic states, reproducibility, and macroscopic statistics, at prescribed scales of length and time
,”
Int. J. Eng. Sci.
34
,
1111
1129
(
1996
).
32.
Murdoch
,
A. I.
and
Bedeaux
,
D.
, “
A microscopic perspective on the physical foundations of continuum mechanics–II: A projection operator approach to the separation of reversible and irreversible contributions to macroscopic behaviour
,”
Int. J. Eng. Sci.
35
,
921
949
(
1997
).
33.
Narayan
,
V.
,
Menon
,
N.
, and
Ramaswamy
,
S.
, “
Nonequilibrium steady states in a vibrated-rod monolayer: Tetratic, nematic and smectic correlations
,”
J. Stat. Mech.: Theory Exp.
2006
,
P01005
.
34.
Narayan
,
V.
,
Ramaswamy
,
S.
, and
Menon
,
N.
, “
Long-lived giant number fluctuations in a swarming granular nematic
,”
Science
317
,
105
108
(
2007
).
35.
Noll
,
W.
, “
Die herleitung der grundgleichungen der thermomechanik der kontinua aus der statistischen mechanik
,”
Indiana Univ. Math. J.
4
,
627
646
(
1955
).
36.
Panchenko
,
A.
and
Tartakovsky
,
A.
, “
Discrete models of fluids: Spatial averaging, closure, and model reduction
,”
SIAM J. Appl. Math.
74
,
477
515
(
2014
).
37.
Panchenko
,
A.
,
Barannyk
,
L. L.
, and
Gilbert
,
R. P.
, “
Closure method for spatially averaged dynamics of particle chains
,”
Nonlinear Anal.: Real World Appl.
12
,
1681
1697
(
2011
).
38.
Rabani
,
A.
,
Ariel
,
G.
, and
Beer
,
A.
, “
Collective motion of spherical bacteria
,”
PLoS One
8
(
12
),
e83760
(
2013
).
39.
Saintillan
,
D.
and
Shelley
,
M. J.
, “
Active suspensions and their nonlinear models
,”
C. R. Phys.
14
,
497
517
(
2013
).
40.
Schaller
,
V.
,
Weber
,
C.
,
Semmrich
,
C.
,
Frey
,
E.
, and
Bausch
,
A. R.
, “
Polar patterns of driven filaments
,”
Nature
467
,
73
77
(
2010
).
41.
Toner
,
J.
and
Tu
,
Y.
, “
Long-range order in a two-dimensional dynamical XY model: How birds fly together
,”
Phys. Rev. Lett.
75
,
4326
4329
(
1995
).
42.
Vicsek
,
T.
,
Czirok
,
A.
,
Ben-Jacob
,
E.
,
Cohen
,
I.
, and
Shochet
,
O.
, “
Novel type of phase transition in a system of self-driven particles
,”
Phys. Rev. Lett.
75
,
1226
1229
(
1995
).
43.
Wensink
,
H. H.
,
Dunkel
,
J.
,
Heidenreich
,
S.
,
Drescher
,
K.
,
Goldstein
,
R. E.
,
Lwena
,
H.
, and
Yeomans
,
J. M.
, “
Meso-scale turbulence in living fluids
,”
Proc. Natl. Acad. Sci. U. S. A.
109
,
14308
14313
(
2012
).
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