Using Galilean-Invariance to ResumGreen-Kubo Relations in Stochastic Rotation

Dynamics

T. IhleInstitute for Computer Applications, Stuttgart University

E. Tuzel, and D.M. KrollSupercomputing Institute, University of Minnesota

Motivation

Hydrodynamics of complex liquids

• dynamics of polymers, colloids, emulsions in flow

• wide range of length and time scales

• importance of thermal fluctuations

• physico-chemical effects, phase transitions

−→ Mesoscale particle-based methods promising:solvent modeled by ”fluid-particles” with efficient dynamics

• very robust, no instabilities

• easy to implement

• intrinsic thermal fluctuations

Algorithm

Stochastic Rotation Dynamics(Malevanets-Kapral method)

• Fluid particles with continuous velocities vj and positions rj

• ”Artificial” many-body interaction: rotation of particlevelocities relative to mean velocity u by angle ±α.

&%'$

qqqqq qqq

qqqq

q qqq

qq qqq

~r′j = ~rj + ~vj ∆t

~v′j = ~u+ Ω(~vj − ~u)

Ω =(

cosα sinα− sinα cosα

)

~u =1M

M∑

j=1

~vj

A. Malevanets, R. Kapral, J. Chem. Phys. 110 (1999) 8605

Molecular chaos

Important Limits

sssHH

HY

CCCCCCW

@@@@R

s ss

@@I

mean free path: λ = ∆t√kBT

lattice constant: a

λ a:

• Strong mixing of particles from different cells

• no correlation before collision → Molecular chaos

λ a:

• isolated communities, collision partners stay the same

• correlations before collisions → no Molecular chaos

Analysing and improving SRD

Why is Galilean invariance broken at λ a ?

Let’s impose a homogenous flow ~V :→ particles are swept quickly to other cells→ faster exchange of collision partners, correlations decrease→ transport coefficients depend on ~V

Galilean invariance can be exactly restored!

Trick: Mimicking the effect of a homogenous flow

Grid is shifted before every rotation step with random vector ~b:bx, by ∈ [0,a]

Why exact procedure?

Probability to find a particle at certain distance to a cell border:• is constant now• does not depend on history of particle anymore

Deriving Green-Kubo relations I

Microscopic operators for conserved quantities

Aα(ξ) =N∑

j=1

aα,j f(ξ, rj)

Density : a1,j = 1

x-velocity : a2,j = vx,j

y-velocity : a3,j = vy,j

Energy : a4,j = v2j /2

Weight function: f(ξ, rj)

f(ξ, rj) = θ(a/2− |ξx − rx,j |) θ(a/2− |ξy − ry,j |)

• reflects shape of the cell

• f(ξ, rj) = 1, if particle j is in cell ξ

Deriving Green-Kubo relations II

Fourier transformation of operators

Aα(k) =N∑

j=1

aα,j eikξj

ξj is index of cell which contains particle j

Goal:

• Derive macroscopic equations for the discrete dynamics inthe limit of large length and time scales

• Find microscopic expressions for transport coefficients

• Evaluate these expressions analytically and numerically

Deriving Green-Kubo relations III

Alternative way: equations for correlation functions

Matrix G : Gαβ(k, t) = V −1 〈δA∗(0) δAβ(t)〉

= 〈Aα(0)|Aβ(t)〉

Find linearized hydrodynamics

∂

∂tG + (ikΩ + k2Λ)G = 0

Ω = Frequency matrix, contains Euler terms

Λ = Matrix of transport coefficients

Deriving Green-Kubo relations IV

Discrete evolution of Aα(k, t)

∆tAα = Aα(t+ 1)−Aα(t)

=N∑

j=1

aα,j(t+ 1) eikξj(t+1) − aα,j(t) eikξj(t)

This leads to formal definition of the flux Dα:

∆tAα + ik Dα = 0,

multiplying by Aα(0) and averaging gives

∆t 〈Aα(0)|Aβ(t)〉+ ik〈Aα(0)|Dβ(t)〉 = 0

Deriving Green-Kubo relations V

Outline

• manipulate expression for ∆tGαβ using stationarity:〈A(t)|A(t′)〉 = 〈A(t+ τ)|A(t′ + τ)〉,and conservation laws

• collect contributions in lowest order in k

• perform Laplace transform; go to the limit of k, ω → 0

Microscopic expression of conservation laws

N∑

j=1

eikξSj (t+1) aα,j(t+ 1)− aα,j(t) = 0

It means:Collisions (stochastic rotations) do not change cell densities

Deriving Green-Kubo relations VI

Expression for flux at small k

Dα|k=0 = −N∑

j=1

∆ξj aα,j + ∆aα,j ∆ξSj

eikξj

∆ξj = ξj(t+ 1)− ξj(t), ∆ξSj = ξj(t+ 1)− ξSj (t+ 1)∆aα,j = aα,j(t+ 1)− aα,j(t),

Laplace transform

Aα(k, ω) =∞∑

t=0

Aα(k, t+ 1) e−ωt

Dα(k, ω) =∞∑

t=0

Dα(k, t) e−ωt

Stationarity: 〈A|D(t)〉 = 〈D|A(t+ 1)〉 −→〈A|D(ω)〉 = 〈D|A(ω)〉

Deriving Green-Kubo relations VII

Looking at the poles of the equation for Gαβ at small ω and k

gives matrix of transport coefficients:

Λ = V −1

12〈I|I(0)〉+

∞∑

t=1

〈I|I(t)〉

X−1

X=Susceptibility matrix, Xαβ = 〈Aα(0)|Aβ(0)〉,

Iα(t): Reduced flux

Iα(t) = k

Dα − 〈Dα|Aγ〉X−1γβ Aβ(t)

Iα(t) is component of the flux kDα that is orthogonal to thehydrodynamic variables A.

Examples

Shear viscosity

ν =1

N kBT

12〈σ2xy(0)〉+

∞∑

t=1

〈σxy(0)σxy(t)〉

σxy: off-diagonal element of stress tensor

σxy = σKin + σRot

σKin =N∑

j=1

vy,j ∆ξx,j

σRot =N∑

j=1

∆vy,j ∆ξSx,j

Leads to: ν = νKin + νMix + νRot

→ measurements in equilibrium; long time tails

Evaluating Green-Kubo expressions I

First terms in series for viscosity

ν =1

N kBT

(

12C0 + C1 + C2 + ...

)

C0 =∑

i,j

〈∆ξx,i(0) vy,i(0) ∆ξx,j(0) vy,j(0)〉

C1 =∑

i,j

〈∆ξx,i(0) vy,i(0) ∆ξx,j(1) vy,j(1)〉

Analytical expressions; effect of the cell shape

〈∆ξ2x,j〉 = a2

16

+(

λ

a

)2

−∞∑

n=1

1π2n2

exp(−2π2n2(λ/a)2)

λ = ∆t√kBT : mean free path, a: lattice constant

Evaluating Green-Kubo expressions II

Limits

〈∆ξ2x,j〉 =

16a2 + λ2, λ a

= aλ√

2/π, λ a

Expressions for next term(rotation angle 90 and large particle density)

〈∆ξx,j(0) vy,j(0) ∆ξx,j(1) vy,j(1)〉 =

−kBTaλ2√π

∞∑

n=−∞exp

(

−(an/2λ)2)

Similiar treatment of other terms and at different rotationangles

Resummation

• occurence of ∆ξ in stress correlations

• ∆ξ depends on velocity and position → complicatedcorrelations

• only the first terms in sum could be obtained analytically

Symmetrized expression:

ν =1

2N kBT〈σxy(0)

∞∑

t=−∞σxy(t)〉

Successive cancellation of terms:

.. . ξ(2)− ξ(1)v(1) + v(2)− v(1)ξ(2)− ξS(1)

+ ξ(3)− ξ(2)v(2) + v(3)− v(2)ξ(3)− ξS(2)

+ ξ(4)− ξ(3)v(3) + v(4)− v(3)ξ(4)− ξS(3)

New Green-Kubo relations

using stationarity to further simplify expression:→ new stress tensor in Green-Kubo relation:

σKin =N∑

j=1

vj,y vj,x

σRot =N∑

j=1

vj,y Bj,x

Bj,x(t) = ξSj,x(t+ 1)− ξSj,x(t)−∆t vj,x(t)

New random variables Bj,x are uncorrelated to velocities:

〈Bj,α〉 = 0

〈Bj,α vi,β〉 = 0

B’s are uncorrelated for time lags larger than one time step

〈Bj,α(n)Bi,β(m)〉 =a2

12δαβ(1 + δij)2δnm − δn,m+1 − δn,m−1

Results• transport coefficients contain only two terms,

no mixed contributions:

ν = νRot + νKin

νRot =a2

6d∆t

(

M − 1 + e−M

M

)

1− cosα

DT = DRot +DKin

DRot =a2

3(d+ 2) ∆t1M

(

1− e−M∫ M

0

ex − 1x

dx

)

(1− cosα)

M : average number of particles per cell,

d : dimension, α : rotation angle

• Fluctuation of particle number per box included(Poisson-distribution assumed)

Numerical simulations

Measuring the viscosity in 2D

0 60 120 180Rotation Angle α

0

0.1

0.2

0.3N

orm

aliz

ed S

hear

Vis

cosi

ty

0 100 200 300Time

0.040

0.050

0.060

Shea

r V

isco

sity

ν =a2

12 ∆t

(

1− 1M

)

(1−cosα)+kBT ∆t

2

(

1(1− 1/M) sin2α

− 1)

α: rotation angle

M : average particle number per cell (here M 1)

Density correlations I

• measuring density correlations 〈ρk(t)ρ−k(0)〉 and thestructure factor S(k, ω)

• comparison with theoretical values for bulk viscosity, heatdiffusivity, and speed of sound

-0.75 -0.6 -0.45 -0.3 -0.15 0 0.15 0.3 0.45 0.6 0.75ω

0

0.05

0.1

S(k,

ω)| k=

0.0.

28

ω=-ck ω=ck

SimulationTheory ζ=0Theory ζ=0.5kBT

Structure FunctionL=32 λ/a=1.00 M=10 α=120

Density correlations II

• fitting bulk viscosity and heat diffusivity

• shear viscosity and speed of sound set to theoretical values

0 0.1 0.2 0.3 0.4

k2

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

ζ/τ

k BT

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

DT/τ

kBT

ζDTTheory

L=32,64 λ/a=1.00 M=10 α=120

Beyond Molecular Chaos I

• so far assumption of molecular chaos

• deviations due to correlated collisions start at time=2

• similiar to ring collisions, but here multi-particle collisionsand non-local interaction

Example: stress correlations: 〈vi,x(0) vi,y(0) vj,x(2) vj,y(2)〉

12 1

2

1 23

1 23

chaos

12 1

2

13

2

13

2

t=τ τt=2

no molecular

• analytic expression in limit λ→ 0

Beyond Molecular Chaos II

Measuring stress correlations: 〈vi,x(0) vi,y(0) vj,x(t) vj,y(t)〉

λ/a = 0.01 relative deviation

0 2 4 6 8time

-0.5

0

0.5

1

<σxy

kin (0

) σxy

kin (t

)> /

Nk B

T2

Numerics

0 0.5 1 1.5 2λ/a

0

0.2

0.4

0.6

0.8

Rel

ativ

e co

rrec

tion

to <

σ xyki

n (0) σ

xyki

n (2τ)

>

α=120

α=90

Summary

• restoring Galilean-invariance at small mean free path byrandom shift

• derivation of Green-Kubo relations by means ofprojector-operator technique

• resumming Green-Kubo relations by using properties ofrandom shift

• formulae for transport coefficients assuming molecular chaos

• measuring bulk viscosity and heat diffusivity via structurefactor

• corrections beyond molecular chaos

Long time tails

Back flow effect

Velocity auto-correlation: 〈vj,x(0) vj,x(t)〉 ∼ Atd−1

Stress correlation: 〈σxy(0)σxy(t)〉 ∼ Btd−1

Logarithmic divergence in 2 D: ν(t) = C +B lnt

In 2D we have for example:

A =kBT

8πρ(ν +D)

D: bare self-diffusion constantρ: Mass density