Kinetics Models of Granular and Active Matter:
Hydrodynamics and Fluctuations
Alessandro Manacorda
Universita degli studi di Roma Sapienza
January 26, 2015
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Non-equilibrium Physics
A physical system is out of equilibrium when its macroscopicdynamics is not invariant under time inversion.
→ Nonequilibrium is ubiquitous!
thermodynamic transformations
chemical reactions
biological systems
weather
financial processes
your breathe in this moment
Nonequilibrium statistical mechanics:→ approach to equilibrium, NonEquilibrium Stationary States(NESS), critical phenomena, hydrodynamics, complex systems,
chaos theory ...
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Non-equilibrium Physics
A physical system is out of equilibrium when its macroscopicdynamics is not invariant under time inversion.
→ Nonequilibrium is ubiquitous!
thermodynamic transformations
chemical reactions
biological systems
weather
financial processes
your breathe in this moment
Nonequilibrium statistical mechanics:→ approach to equilibrium, NonEquilibrium Stationary States(NESS), critical phenomena, hydrodynamics, complex systems,
chaos theory ... granular and active matter
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Granular Matter
A granular systems is made ofgrains:
classical grains
in experiments,N ∼ 103 ≪ NA
dissipative interactions
sand, cereals, icebergs, Saturnrings...
⇓
metastable stationary states
non gaussian velocitydistributions
correlations
patterns, segregation,frustration...
phase transitions
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Active Matter
Active matter takes energyfrom the environment andconverts it in motion→ self-propelling particles
bacteria
fish schools
bird flocks
people?
Granular matter:kinetic energy dissipationActive matter:kinetic energy injection
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Kinetic Theory and Hydrodynamics
Nonequilibrium → no hamiltonian H and equilibrium definitions
Microscopic dynamics
Langevin equation
Noise over initial conditions
Kinetic theory ⇒ phase-space distribution P({r , v} , t), Boltzmannequation
∂tP + v · ∇rP +F
m· ∇vP =
(
∂P
∂t
)
coll
Hydrodynamics
Coarse-graining of the system
Conservation laws → balance equations for conserved fields varying overlong time and space scales
n(r, t) =
∫d3NvP(r, v, t), u(r, t) =
∫d3Nv vP(r, v, t), T (r, t) =
∫d3Nvv2P(r, v, t)
e.g. Navier-Stokes eq.
ρ∂tu+ ρ(u · ∇)u = f −∇P + η∇2u
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Fluctuating Hydrodynamics and Macroscopic Fluctuations
Theory
Fluctuating hydrodynamics
Useful representation for granular and active matter hydrodynamics
Small number of particles → fluctuations become relevant →stochastic hydrodynamic equations
Local equilibrium approximation → local gaussian white noise
Macroscopic fluctuations theory
Recently developed theory (Bertini, De Sole, Gabrielli, Jona-Lasinio,Landim; 2001-2014)
Density and current fluctuations from large deviation theory withoutlocal equilibrium approximation
Developed for non-equilibrium stationary states of driven diffusivesystems
Hard technical challenge
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Lattice Models
Goal:microscopic kinetics ⇒ transport coefficients, current fluctuations,
macroscopic stochastic equations and dynamics
1980: Kipnis-Marchioro-Presutti (KMP) → lattice hydrodynamicalmodel with conserved energy
{ρi}Ni=1 , ρ′i = p(ρi + ρi+1) , ρ′
i+1 = (1− p)(ρi + ρi+1)
2010-2013: Prados, Lasanta and Hurtado → similar system withenergy dissipation ∝ (1− α)(ρi + ρi+1)
{ρi}Ni=1 , ρ′i = pα(ρi + ρi+1) , ρ′
i+1 = (1− p)α(ρi + ρi+1)
⇓
Development of MFT and Fluctuating HydrodynamicsLack of microscopic theory with conserved momentum
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
BBP Model
2002: Baldassarri, Bettolo and Puglisi → microscopic model withvelocities
periodic chain of L particles with transversal velocities
Maxwell molecules
momentum conservation
energy dissipation
Discrete space-time stochastic continuity equation:
vl ,p+1 = vl ,p − jl ,p + jl−1,p
jl ,p = 1+α2 δyp ,l (vl ,p − vl+1,p)
(1)
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Hydrodynamic Equations
Quasi-elastic limit:
L ≫ 1, α→ 1L2(1− α2) = ν <∞
Time-length scaling:
∆x = 1/L∆t = 1/L3
Local equilibrium approximation:
P(vl , vm; p) = Pp(vl)Pp(vm)
=1
2π√
Tl ,pTm,p
exp
[
−(vl − ul ,p)2
2Tl ,p− (vm − um,p)
2
2Tm,p
]
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Hydrodynamic Equations
Quasi-elastic limit:
L ≫ 1, α→ 1L2(1− α2) = ν <∞
Time-length scaling:
∆x = 1/L∆t = 1/L3
Local equilibrium approximation:
P(vl , vm; p) = Pp(vl)Pp(vm)
=1
2π√
Tl ,pTm,p
exp
[
−(vl − ul ,p)2
2Tl ,p− (vm − um,p)
2
2Tm,p
]
⇓Hydrodinamics
∂tu(x , t) = ∂2xu(x , t)
∂tT (x , t) = ∂2xT (x , t)− νT (x , t) + 2 (∂xu(x , t))2
(2)
diffusion sink viscous heating
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Cooling Regimes
Homogeneity ⇒ u(x , t) = ∂x = 0⇓
Homogeneous cooling state:THCS(t) = T0e
−νt (Haff 1983)
Rescaled fields:u = u/
√THCS , T = T/THCS
→ stationary cooling state
Linear analysis:
{
∂t u(k , t) =(
ν/2− k2)
u(k , t)
∂tT (k , t) = −k2T (k , t)
⇒ homogeneous cooling stable forν < νc = 8π2
1e-05
0.0001
0.001
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1
T(t
)
t
L = 1000ν = 10ν = 20ν = 30ν = 40
e-10 t
e-20 t
e-30 t
e-40 t
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.02 0.04 0.06 0.08 0.1 0.12u m
ax /
T1/
2 HC
S (
t)
t
A = 0.1, L = 1000ν = 68ν = 78ν = 79ν = 98
umax (t,68)umax (t,78)umax (t,79)umax (t,88)
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Nonhomogeneity
No symmetry for space translations ⇒ Nonhomogeneous regimeFirst mode → exact solution
u(x , 0) = u0 sin(2πx), T (x , 0) ≡ T0
⇓{
u(x , t) = u0 sin(2πx)e−νc t/2
T (x , t) = T0e−νt +
νcu202 e−νc t
[
1−e−(ν−νc )t
ν−νc+ cos(4πx)1−e−(ν+νc )t
ν+νc
]
.
(3)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u(x,
t)
x
A = 1, ν = 68, L = 1000
t = 2/ν 0.24
0.25
0.26
0.27
0.28
0.29
0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T(x
,t)
x
A = 1, ν = 68, L = 1000
t = 2/ν
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Correlations and Failure of LE
Numerical simulations ⇒ LE good approximation withdiscrepancies
Momentum conservation ⇒ non-zero correlationsC (x , t) = 〈v(0, t)v(x , t)〉Closed set of equations for correlations without LE ⇒diffusion equation with PB and symmetric profile
∂t C (x , t) = 2∂2x C (x , t)+νC (x , t)
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Correlations and Failure of LE
Numerical simulations ⇒ LE good approximation withdiscrepancies
Momentum conservation ⇒ non-zero correlationsC (x , t) = 〈v(0, t)v(x , t)〉Closed set of equations for correlations without LE ⇒diffusion equation with PB and symmetric profile
∂t C (x , t) = 2∂2x C (x , t)+νC (x , t)
⇓Stationary correlation
C∞(x) ≃ −L−1 1
2
√
ν
2
cos[√
ν2
(
x − 12
)]
sin(
12
√
ν2
)
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5
D~ (
x,t)
x
Rescaled spatial correlation, MM
L=250, ν=40νt = 0νt = 5
νt = 10D~
th(x)
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Effect of Correlations on Temperature
Nearest-neighbours collisions ⇒Nearest-neighbour correlations
ψ(t) = limx→0 LC (x , t) counts
1st order cooling equation:
T (t) ≡ T (t)
THCS(t)⇒ ∂tT (t) =
1
Lνψ(t)
Stationary rescaled temperature
T (t) ≃ 1 +1
Lνψ∞t
⇒ violation of Haff’s law
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
0 2 4 6 8 10
T~ (
t)
ν t
Rescaled temperature, MM
L = 250ν=10
m = -2.1912e-03ν=20
m = -1.9e-05ν=30
m = 2.825e-03ν=40
m = 6.964e-03ν=50
m = 1.316e-02ν=60
m = 2.38e-02ν=70
m = 4.08e-02
-2
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70
ν
ψ~
(ν), MM
L = 250ψ~
expψ~
th
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Further Analysis
Maxwell molecules vs hard spheres?
Energy fluctuations σ2E (t) =
〈E 2(t)〉−〈E(t)〉2
〈E(t)〉2 ⇒ Multiscaling?
Nonhomogeneous profile ⇒ momentum and energy currents, j(x , t)and JE (x , t) = 〈(vl,p + vl+1,p) jl,p〉Velocity distribution P({v}) and local equilibrium
Thermostat ⇒ Heat flux? Linear response? Entropy production?Fluctuation-dissipation?
-15
-10
-5
0
5
10
15
0 0.1 0.2 0.3 0.4 0.5
D~(x
)
x
Rescaled correlation profiles
L=250, νt=9ν=40, MMν=40, HSν=70, MMν=70, HS
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
0 2 4 6 8 10
T~(t
)
ν t
Rescaled temperature
L = 250ν=10, MMν=10, HSν=40, MMν=40, HSν=70, MMν=70, HS
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics
Perspectives
Molecular vs lattice models
Active matter have inertia ⇒ theoretical models ofself-propelled particles with inertia
General laws for fluctuations, transport coefficients, FDTbeyond phenomenological approach
Comparison with experiments and patterns for furtherinvestigations
Alessandro Manacorda Kinetics Models of Granular and Active Matter: Hydrodynamics