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Mesoscopic model for solvent dynamics

Anatoly Malevanets and Raymond Kapral

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Mesoscopic model for solvent dynamics

Anatoly Malevanets Department of Physics, Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, England

Raymond KapralChemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto M5S 3H6,Canada

Received 18 November 1998; accepted 1 February 1999

Complex fluids such as polymers in solution or multispecies reacting systems in fluid flows often

can be studied only by employing a simplified description of the solvent motions. A stochastic

model utilizing a synchronous, discrete-time dynamics with continuous velocities and local

multiparticle collisions is developed for this purpose. An H theorem is established for the model and

the hydrodynamic equations and transport coefficients are derived. The results of simulations are

presented which verify the properties of the model and demonstrate its utility as a hydrodynamics

medium for the study of complex fluids. © 1999 American Institute of Physics.

S0021-96069952416-1

I. INTRODUCTION

The investigation of the dynamics of complex fluids is a

challenging task. In complex chemically reacting systems

many chemical species dissolved in a solvent may undergo

sequences of reactions leading to oscillations or chemical

patterns.1 The reactive dynamics may depend on the solvent

motions since the fluid flow fields influence the nature of the

mixing of the species. Specific instabilities have been as-

cribed to flow effects, such as differential-flow-induced

chemical instabilities.2 Reactions in porous media constitute

another example. Here one is interested in how a flow field

in a complicated medium influences the reactions that may

themselves change the nature of the medium, for instance,

through dissolution reactions.

3

Similar considerations applyto the study of rheological properties of colloidal suspen-

sions or polymers in solution.4 An especially important class

of problems concerns the conformational dynamics of

biopolymers such as proteins in solution.5

All of the above examples, and others like them, share

the feature that one is interested in the detailed microscopic

dynamics of some degrees of freedom of the system interact-

ing with a solvent whose dynamics is essential for the phe-

nomena but whose detailed properties are not of interest. The

systems are sufficiently complex that a full molecular dy-

namics MD simulation of the system plus solvent is impos-

sible. Furthermore, the dynamics of interest often occurs on

long time scales and over long distances; for example, therelaxation times for large polymers can be very long and the

distance scales for many chemical pattern forming processes

range from mesoscopic to macroscopic scales.

In such circumstances one is led to consider mesoscopic

models for the solvent dynamics that incorporate the essen-

tial dynamical properties, yet are simple enough to be simu-

lated for long times and on long distance scales.6 These me-

soscopic models must not only faithfully reproduce the main

dynamical features of the solvent but must also be micro-

scopic in character to permit coupling between microscopic

degrees of freedom, whose dynamics are of interest, and the

solvent. For instance, one must be able to couple the mono-

mer units of a polymer to the solvent or the reactive colli-

sions between molecules to the flow of the solvent respon-

sible for their mixing.

A variety of mesoscopic models have been constructed

for this purpose ranging from Langevin models that have

been employed as simple ‘‘heat baths’’ in MD studies of

biomolecule dynamics,7 to schemes such as Direct Simula-

tion Monte Carlo DSMC methods,8 lattice Boltzmann

methods,9 and extensions of hydrodynamic lattice gas au-

tomaton models.10

In this paper we present a mesoscopic model for simu-

lating a fluid that has the desirable features mentioned above.

The fluid is modeled by ‘‘particles’’ whose positions andvelocities are treated as continuous variables. The system is

coarse grained onto the cells of a regular lattice and there is

no restriction on the number of particles that may reside in a

cell. The dynamics is carried out synchronously at discrete

time steps. Particle streaming is treated exactly while the

cells are the collision volumes for a multiparticle collision

dynamics that differs from that of DSMC schemes. The dy-

namics satisfies the mass, momentum and energy conserva-

tion laws and we shall show it yields the correct hydrody-

namic equations. Since it is a particle model, collisional

coupling to other microscopic degrees of freedom is easily

incorporated and its application to complex system geom-

etries is straightforward.The outline of the paper is as follows. Section II de-

scribes the streaming and multiparticle collision dynamics

that forms the basis of the mesoscale description of the sys-

tem. The Boltzmann approximation to the full dynamics is

considered in Sec. III; a Boltzmann-type kinetic equation is

derived, an H theorem for the evolution is proved and the

Chapman–Enskog expansion is used to derive the Navier–

Stokes equations and determine the transport coefficients for

a particular collision model. In Sec. IV simulations are car-

ried out to confirm the utility of the model. The velocity

JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 17 1 MAY 1999

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distribution is shown to be Maxwellian and the hydrody-

namic flow patterns for flow past a disk are examined. The

conclusions of the study are presented in Sec. V.

II. STREAMING AND MULTIPARTICLE COLLISIONDYNAMICS

The system we study consists of N particles of unit mass

with position xi and velocity vi coordinates, i1,2,. . . , N .The evolution through a unit time interval is given by suc-

cessive application of streaming and collision steps. During

the streaming step the particle positions change in the stan-

dard way,11

xi→xivi , 1

while in the collision step particle velocities transform ac-

cording to

vi→V ˆ viV, 2

where ˆ is a random rotation from a set and V is the

average velocity of the colliding particles. In this multipar-

ticle collision event, the velocity vector of particle i, relative

to the mean velocity V, is rotated by a randomly chosen

rotation operator to give the post-collision value of the ve-

locity. We shall show below that the mass, momentum and

energy collision invariants are preserved under this multipar-

ticle collision dynamics.

We imagine coarse graining the system into Wigner–

Seitz cells centered on the nodes of a regular lattice L. For a

cubic lattice the Wigner-Seitz cell V is defined by x 1/2, where we introduce the norm xmax( x x , x y , x z) and

denotes a lattice coordinate. These cells define the collision

volumes for the multiparticle collision dynamics. The colli-

sions are simultaneously performed on all particles in aWigner–Seitz cell with the same rotation ˆ , but ˆ may dif-

fer from cell to cell.

The evolution governed by Eqs. 1 and 2 may be writ-

ten in the form of a Liouville equation for the probability

density,

PV N ,X N V

N , t 1C PV N ,X N ,t , 3

where

C PV N ,X N ,t 1

L L

d V̆ N PV̆ N ,X N ,t

i1

N

viV ˆ v̆iV , 4

where X( N )(x1 ,x2 , . . . ,x N ), V N

(v1 ,v2 , . . . ,v N ), V̆( N )

(v̆1 ,v̆2 , . . . , v̆ N ) and, LL is the number of lattice

nodes.

The evolution described by Eqs. 1 and 2 preserves

total energy and momentum of the particles in each cell, as

can be seen from the identities,

ixV

vi ixV

V ˆ viV

and

ixV

vi2

ixV V ˆ viV2. 5

We note that the elementary measure d d xid vi is

invariant with respect to streaming and collision transforma-

tions. For the deterministic streaming operator it follows

from the identity,

Jacobian xi t 1

xi t 1, 6

and for the collision transformation it results from the semi-

detailed balance condition and phase space volume conser-

vation during rotations,

i

d vi ˆ , V̆ N ˆ V̆ N V

N

p( ˆ V̆ N )i

d v̆i

i

d vi ˆ , V̆ N ˆ V̆ N V

N

p ˆ V̆ N , 7

which is valid if the choice of the rotations is independent of

velocities. Here p( ˆ V̆( N )) is the conditional probability of

the rotation ˆ given V̆( N ). In the above equation vi and v̆i are

the post- and precollision velocities, respectively. Assuming

the validity of the ergodic hypothesis, which is supported by

the results of numerical simulations, we conclude that the

stationary distribution is given by the microcanonical en-

semble expression,

PV N ,X N A 2 N i1

N

vi2

d

2

i1

N

viu , 8

where u is the mean velocity of the system. After integrationover the coordinates and velocities of particles with i

2, . . . , N we arrive in the limit of large N at the Maxwell

distribution,

P mv1 ,x11

V

2

d /2

exp v1u2 /2, 9

where (1/ k BT ) , V is the system volume and d is the

dimension. Although the correct distribution results from the

semidetailed balance condition, the computation of the trans-

port coefficients is greatly simplified if we impose detailed

balance conditions on the collision operator. In the current

context this is tantamount to

1

.

III. BOLTZMANN APPROXIMATION

In Sec. II we found that the stationary distribution for the

one-particle velocity distribution was Maxwellian. In order

to study the relaxation to equilibrium we assume that there

are no correlations among the colliding particles in Eq. 3.

In this case the probability distribution P is a product of

identical one-particle probability distributions,

PV N ,X N , t i1

N

P 1vi ,xi , t . 10

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In order to derive the Boltzmann equation we multiply

Eq. 3 by i (vvi) (xxi) and subsequently integrate

over vi and xi , where i1, . . . , N . Focusing on x in cell

and accounting for all possible ways of assigning n particles

to the cell, we obtain

C ( f (t ))n1

N

N

n in1

N

d xid v̆iP 1( v̆i ,xi , t )

( ( xi 1/2)V n

d V̆nd Xn

i1

n

d xid v̆iP 1 v̆i ,xi , t (v,V̆n), 11

where

v,V̆n1

i, ˆ

vV ˆ Vv̆i xxi.

12

In writing this equation we have used the fact that we may

express P 1(vi ,xi , t ) as the sum of two terms,

P 1vi ,xi , t P 1vi ,xi , t xi 1/2

1/2xi , 13

where is the Heaviside function. Particles with coordinates

outside the cell do not contribute to the collision term and

this is accounted for by the Heaviside function in the first

factor of Eq. 11. We use the fact that the prefactor in Eq.

11 may be written as

••• in1

N

d xid v̆iP 1(v̆i ,xi , t )( (xi 1/2)

d xid v̆iP 1(v̆i ,xi , t )( (xi 1/2) N n

1

N

N n

, 14

where is the total particle number density in the Wigner-

Seitz cell . In the large N limit we have ( n N )( 1

( / N )) N n ( N n / n!) e . Incorporating the factor of N

into the definition f(v,x,t ) N P 1(v,x, t ), we obtain the Bolt-

zmann equation,

f v,xv,t 1 C f t , 15

where

C f t n1

e

n!

V n

d V̆nd XnF V̆n,Xn,t v,V̆n,

16

and we have defined

F V̆n,Xn,t i1

n

f v̆i ,xi , t . 17

The prefactor in Eq. 16 shows that the cell particle number

is Poisson distributed.

Equation 15 preserves expectation values of integrals

of motion such as the density, momentum and energy. De-

noting these quantities by J 1(v)1, J 2(v)v, and J 3(v)

12v2, respectively, we have

d vd xJ v f v,x,t 1

d vd xJ

vC f t

i, ,n1

e

n!

V n

d V̆nd XnJ v̆iF V̆n,Xn,t

d vd xJ v f v,x,t , 18

where we used the following identities:

i

1i

1, 19a

i V ˆ Vv˘

i i v˘

i , 19b

i

V ˆ Vv̆i2

i v̆i

2, 19c

for the density, momentum, and energy conservation laws,

respectively.

A. H theorem

The evolution Eq. 3 involves a multiparticle collision

term and it not immediately clear that an H theorem exists.

We define the H functional in terms of the reduced one-particle distribution function,

H t d vd x f v,x, t ln f v,x,t

,nN

e

n!

V nd Vnd Xn

i1

n

f vi ,xi , t ln

i1

n

f vi ,xi , t , 20

where the second equality arises from the representation of

the system in terms of phase space cells and makes use of the

resolution of identity 1n0 ( e x x n / n!). In the Appendix

we show that H ( t ) decreases on each evolution step so that

H t d v̆d x f v̆,x, t 1ln f v̆,x,t 1. 21

For the fixed momentum and energy expectation values

the Maxwell distribution,

f m N

V 1

2 k BT

d /2

evu2

2k BT , 22

provides a lower bound to the H functional. This follows

from the identity,

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d xd v f v,xln f mv d xd v f mvln f mv, 23

and the inequality,

d xd v f v,xln f v,x

f mv

N

d xd v

f mv

N

f v,x

f mv ln

f v,x

f mv 0. 24

Equations 18 show that the expectation values of the den-

sity, momentum and energy are conserved during evolution

under the Boltzmann equation; thus, completing the proof of

the H theorem.

B. Chapman–Enskog asymptotic expansion

We derive hydrodynamical equations by using an expan-

sion of the reduced probability distribution in slowly varying

density fields. This Chapman–Enskog procedure12,13 is based

on the assumption that any relevant functional can be ex-

panded into a series of partial derivatives of the conserved

fields. After scaling x→x and t → t the expansion is or-

dered in powers of .

We rewrite Eq. 15 in terms of the average cell prob-

ability distribution f ̄ and averaged collision operator C ¯ ,

f ̄ v, ,t x 1/2

d x f v,x,t , 25

C ¯ f ̄ x 1/2

d xC f . 26

If f is a smooth function the following relation holds

x 1/2

d xe t v• f v,x,t e t v• f ̄ v, , t , 27

and the evolution of the averaged cell probability distribution

is described by a Galilean invariant, spherically isotropic

equation

e t v• f ̄ C ¯ f ̄ . 28

It is further assumed that the reduced probability distri-

bution function f ̄ is defined by the instantaneous spatial dis-

tribution of local collision invariants J ,

f ¯

v, ,t f ¯

v, ,t n0 n

f ¯

nv, ,t . 29

The density of a local collision invariant is given by the

average value

,t d vJ f ̄ ,v,t J f ̄ ,v, t , 30

and to ensure uniqueness an additional requirement is im-

posed,

J f ̄ n ,v,t 0 for all and n0, 31

where J is the set of the density, momentum, and energy

dynamical variables introduced earlier.

Time evolution of a functional of the conserved quanti-

ties is governed by an operator given as an expansion in the

small parameter ,

t

n0nDn , 32

and the gradient term in Eq. 28 is scaled by . The expan-

sion of the collision operator in a series in is written for-

mally as

C ¯ f ̄ n0

nC ¯ n f ̄ . 33

To ensure the identity J C ¯ ( f ̄ )J f ̄ we constrain the av-

erage of each term of the series by J C ¯ n( f ̄ )J f ̄ n.

By expanding the evolution Eq. 28 in powers of we

arrive at the following set of equations:

C ¯ 0 f ̄ C ¯ f ̄ 0 f ̄ 0 , 34a

C ¯ 1 f ̄ f ̄ 1D0v• f ̄ 0 , 34b

C ¯

2 f ¯ f

¯ 2D1 f

¯ 0D0v

• f

¯ 1 12 D0v

• 2 f

¯ 0 .34c

The solution of Eq. 34a yields a local Maxwellian dis-

tribution,

f ̄ 0 1

2 k BT

d /2

e vu2 /2k BT . 35

The average of the local collision invariants over v com-

mutes with the operator Di so that integration of Eq. 34b

yields

D 0 •J

v f ̄ 0. 36

The average of Eq. 34c gives the second order correc-tion to the Euler equations,

D 1

12 J

D0 •v f ̄ 1C ¯ 1 f ̄ ,

and after transformations Euler equations are dissipation-

less,

D 1

12 •J

v f ̄ 1C ¯ 1 f ̄ , 37

where we used the conditions on J f ̄ 1 and J C ¯ 1( f ̄ ).

C. Navier–Stokes equation

In this section we apply general formulas 34a–34c toa specific collision model. Regardless of the collision model,

the equation for the zeroth expansion term has the same form

and constitutes the Euler equations of compressible flow.

Evaluation of the averages in Eq. 36 yields the following

results for the evolution of the conserved quantities:

t u 0,

t u u u k B T 0, 38

t 12 u2

C v T u

12 u2

C p T 0,

where C vdk B /2 and C pC

vk B . The subscripts and

refer to the Cartesian coordinates and the Einstein summa-



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tion convention is used. We have dropped the subscript to

avoid proliferation of notation. Algebraic manipulations

transform the above system into a set of evolution equations

for ,v, and T ,

t u 0 , 39a

t u u u

k B

T 0 , 39b

C v t T u T k BT u 0 . 39c

With the use of Eqs. 39a–39c we rewrite Eq. 34b in the

following form:

C ¯ 1 f ̄ f ̄ 1 f ̄ 0 log f ̄ 0

v u

log f ̄ 0

u

v u u u

k B

T

log f ̄ 0

T v T u T

k B

C v

T u

.

We substitute the explicit form of f ̄ 0 given by Eq. 35 and,

after algebraic transformation, arrive at the following equa-

tion:

C ¯ 1 f ̄ f ̄ 1 f ̄ 0 c2

2k BT

C p

k Bc log T

1

k BT c c

1

d c2 u , 40

where cvu.

We define the function h 1 by the relation h 1 f ¯

0 f ¯

1. Thecollision operator C ¯ 1 has the following form:

C ¯ 1 f ̄ vn1

n

n! e d Vn

j1

n

vv j d V̆n R n

Vn,V̆nPmV̆n

i1

n

h 1 v̆i, 41

where R (n) is defined by Eq. A2. Using the invariance of

the Maxwell distribution with respect to the collision trans-

formation we obtain the following relation:

C ¯ 1 f ̄

v ,n1

n

n! e

d Vn

j1

n

vv jPmVn

i1

n

h 1 v̆i, 42

where v̆iV ˆ viV.

Equations 42 and 40 constitute a linear integral equa-

tion for h 1, which can be split into two equations, one that

involves terms log T and the other that depends on gradi-

ents of velocity fields.

Explicit calculations for two dimensions. For d 2 and

/2, /2 we may show that the functions,

s1c c

1

d c2 and s 2 c

2

2T C pc ,

are eigenfunctions of the collision operator and through their

eigenvalues we evaluate values of the viscosity and thermal

conductivity coefficients, respectively.

Consider s1. Using Galilean invariance of the Boltz-

mann equation, without loss of generality we can choose a

stationary frame with u0. In this case cv and write

1

ˆ ,i1

n

h 1 v̆i1

ˆ ,i1

n

V V ˆ V ˆ V

ˆ vi ˆ vi 1

2 vi

2 . 43

By using the relations ˆ v x ˆ v yv xv y a nd ( ˆ v x)2

(v y) 2 valid for ˆ /2 we arrive at the following equa-

tion:

1

ˆ ,i1

n

h 1 v̆i

i1

n

2V V V2

vi vi

1

2 vi

2 . 44

We note that the cross terms with different particle index i

vanish upon integration so that we have the following ex-

pression for the eigenvalue problem:

C ¯ 1s1 f ̄ 0s 1 f ¯ 0n1

n1

n! e 2n

21e

s1 f ̄ 0r s 1 f ̄ 0. 45

An analogous calculation for s 2 yields,

C ¯ 1s2 f ̄ 0e

1

s2 f ̄ 0r s 2 f ̄ 0. 46

Using these results in Eq. 37 we may compute the

shear viscosity and thermal conductivity coefficients in terms

of the damping factors r and r as,

k BT 1r

21r k BT

1e

2e 1

, 47

and

k BT 1r

1r k BT

1e

1 e . 48

Evaluation of the averages in Eq. 37 yields the follow-

ing expressions for the second order terms in the Chapman–

Enskog expansion:

D1 0, 49

D1 u , 50

D112 u2

C v T u q , 51



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where is thermal conductivity coefficient and the irrevers-

ible contribution to the pressure tensor and the heat flux are

given by the following expressions:

u u 12 u , 52

q T . 53

In the ,u,T set of variables the resulting hydrodynamic

equations have the following standard form:

t u 0 , 54a

t u u u

1

T

1

, 54b

C v t T u T k BT u

1

u

1

q .

54c

Similar calculations can be carried for other collision

model. For example, in three dimensions, averaging the set

of all random rotations yields,

1

ˆ ,i1

n

h 1 v̆ii1

n

V V 13 V2 , 55

where we used the fact that ( 1/ ) ˆ ˆ v ˆ v

13v2 for the set of random rotations.

Combining Eqs. 55 and 42 we arrive at the following

expressions for the damping factor r and viscosity coeffi-

cient:

r 1e

, 56

k BT 1e

2e

1

. 57

IV. SIMULATIONS OF HYDRODYNAMIC FLOWS

In this section we present results of simulations imple-

menting an algorithm based on the present model. The simu-

lations validate the assumptions used in the derivation of

hydrodynamic equations and show that the model correctly

reproduces fluid flow behavior. Simulations have also been

performed to determine the equilibrium velocity distribution

and the rate of relaxation to the equilibrium see Fig. 1.

The nature of the viscous dissipation was determined by

simulating flow in the rectangular domain of width L30,

density 10 and temperature k BT 1.0. In the y-directionwe impose bounce-back boundary conditions and the flow is

driven by assigning Maxwell-distributed velocities with pro-

file u x( y)1.2( L y) y / L 2 to particles in the region x5.

The results are presented in Fig. 2 and show that the profile

is quadratic to a good approximation. The small deviations

are due to energy dissipation along the channel and the dif-

ference between local equilibrium distribution and the sta-

tionary distribution in the Poiseuille flow.

The model also reproduces hydrodynamic flows on

larger scales. In Fig. 3 we present the results of simulations

of flow past a disk. The system size is 360120 and the disk

diameter 2a32. The flow is induced by assigning

Maxwell-distributed velocities with temperature T 1 and

velocity u x0.5 to particles in the region x8. Periodic

boundary conditions are imposed in y direction. For particle

density 10.0, the kinematic viscosity coefficient 47 is 0.055 and the corresponding Reynolds number Re

2au x / 288.

The flow was sampled on a set of predefined points by

averaging velocities of nearby particles with weights 1/(r 4

100), where r is the distance between a test point and a

particle. After a transient time a flow is established at short

distances from the object; however, a flow tail grows indefi-

nitely until it occupies the entire system length. In experi-

ments on fluid flows for these values of the Reynolds number

the existence of von Karman streets is documented14 see

Fig. 4.12.6 of Batchelor. The density of the fluid is nearly

uniform and velocity randomization at the boundary elimi-

nates the feedback due to the periodic boundary conditions inthe system.

As another illustration of the method in Fig. 4 we

present the results of simulations of the stages in the devel-

opment of the boundary layer at the rear of a disk suddenly

set in motion. The system size is 400400 and the disk

diameter 2a100. The flow velocity at x0, the density of

the system and the Reynolds number are u x0.4, 20.0,

FIG. 1. Velocity distribution: Maxwell velocity distribution solid line;

histogram of v x distribution computed in simulation ( x-step line.

FIG. 2. Poiseuille flow profile: The solid line is a quadratic fit to the com-

puted velocity distribution triangles.



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and Re1520, respectively. The system setup is the same as

in the simulations of von Karman streets. The initially sym-

metric flow separates from the disk and a backflow at the the

far end of the disk develops. At the contact line between the

normal flow and the oppositely-directed backflow, a systemof vortices appears which later expands into the full-scale

boundary layer cf. Fig. 5.9.3. in Ref. 14.

V. CONCLUSION

The multiparticle-collision stochastic model combines

advantageous features of both Direct Simulation Monte

Carlo and lattice gas methods for simulations of fluid flows.

It differs from DSMC and its variants in the nature of the

collision rule. In DSMC collisions in a cell are carried out

sequentially on randomly chosen pairs of particles in the cell.

This rule leads to exponentially distributed collision times,

and the possibility of multiple collisions involving the sameparticle affects the efficiency of the algorithm. In general, by

allowing only pairwise collisions DSMC imposes a high

lower bound on the accessible values of the kinematic vis-

cosity coefficient, which may be important if higher Rey-

nolds number flows are desired. Since it is a particle-based

scheme the multiparticle-collision model does not suffer

from numerical instabilities. The velocity distribution is

Maxwellian and the hydrodynamic equations are isotropic.

Because of these features it can serve as mesoscopic

model for solvent dynamics which can be coupled to a full

molecular dynamics treatment of solute degrees of freedom.

The examples of simulations of fluid flow around large ob-

jects presented in Sec. IV are relevant for studies of the

interactions among large colloidal particles in solution. The

method has also been used to study the coupling among

small ‘‘effective’’ monomer units in a polymer through sol-

vent dynamics.15 This is an example of the coupling of the

full molecular dynamics of a polymer chain to the mesoscale

dynamics of the solvent. Consequently one may study the

validity of existing polymer models and the influence of fluid

flow on polymer conformational dynamics. The collision

model is easily extended to treat more complex situations

such as the inclusion of interactions that give rise to phase

segregation or reactive hydrodynamic flows. To simulate this

effect the motion of a small number of complex solute mol-

ecules is coupled to the solvent through interaction potentials

that affect motion of the ideal particles during the streaming

step while preserving the integrals of motion and the phase

space volume. The model also may be used to probe the

dynamics on mesoscopic length scales where fluctuations

may play a role in determining the character of the phenom-

ena.

The emphasis in this paper was both on the formulation

of the multiparticle collision model and the development of

its theoretical underpinnings, namely, the existence of an H

theorem and the derivation of hydrodynamic equations. The

simulations have served to verify the theoretical predictions

and demonstrate that the model can be used to simulate fluid

flows for applications to the dynamics of complex fluids.

FIG. 4. Development of the boundary layer in flow past disk set in motion.

FIG. 3. Vortex shedding in a stationary flow past circular cylinder. The dark

line segments represent tangents to the velocity field at their median point.

The length of a segment is proportional to the magnitude of the velocity.



14.139.227.196 On: Fri, 10 Jan 2014 09:10:56

8/13/2019 1.478857 a very good paper


ACKNOWLEDGMENT

This work was supported in part by a grant from the

Natural Sciences and Engineering Research Council of

Canada.

APPENDIX: PROOF OF H-THEOREM

In this Appendix we show that H (t ) defined in Eq. 21

decreases on each evolution step. The proof of theH-theorem relies on the following convexity inequality:

s

As Bs ln Bs s

As Bs lns

As Bs ,

A1

where A is normalized by s A(s)1.

Introducing the quantity R (n) defined as

R nVn,V̆n1

ˆ

i1

n

viV ˆ Vv̆i,

A2

whose integral over V̆

(n)

is unity, we may rewrite Eq.

20

as

H t ,nN

e

n!

V nd Vnd Xn

i1

n

f vi ,xi , t

lni1

n

f vi ,xi , t d V̆n R nVn,V̆n. A3

In each term in the above sum we exchange the order of the

V̆(n) and V( n) integrations. Use of the convexity inequality

A1 leads to the following relation:

d Vn RnVn,V̆ni1

n

f vi ,xi , t lni1

n

f vi ,xi , t

f ̃ nV̆n,Xn,t ln f ̃ nV̆n,Xn,t , A4

where

f ̃ nV̆n,Xn,t d Vn RnVn,V̆ni1

n

f vi ,xi , t .

A5

Hence, we may write

H t ,nN

e

n!

V nd V̆nd Xn f ˜ nV̆n,Xn, t

ln f ˜ n

V̆n

,Xn

, t . A6If we define the quantities Z and f ˆ ( n) as

Z V

nd V̆nd Xn f ̃ nV̆n,Xn,t

n , A7

f ˆ in v̆i ,xi , t

1

Z V [ n1]

d v̆1d x1•••d v̆iˆ d xiˆ •••d v̆nd xn f ̃ n

V̆n,Xn,t , A8

where the hat over the integration variables in definition A8

indicates that these variables should be omitted in the inte-

gration, and let

Ai1

n

f ˆ in ,

B f ̃ nV̆n,Xn,t

Z i1

n

f ˆ in

,

in the inequality A1 we obtain

1

Z V n

d V̆nd Xn f ̃ nV̆n,Xn, t ln f ̃ n V̆n,Xn,t

Z i1

n

f ˆ in

V

nd V̆nd Xn

f ̃ nV̆n,Xn, t

Z ln

V n

d V̆nd Xn

f ̃ nV̆n ,Xn, t

Z 0. A9

The last equality follows from the fact that the argument of

the logarithm is unity in view of the definition of Z . From

this inequality we deduce that

V

nd V̆nd Xn f ̃ nV̆n,Xn,t ln f ̃ nV̆n,Xn,t

V

nd V̆nd Xn f ̃ nV̆n ,Xn, t ln Z

i1

n

f ˆ in. A10

Finally, using the identity,

i1

n

V

nd v̆ixi Z f ˆ

in v̆i ,xi , t ln Z 1/ n f ˆ

in v̆i ,xi , t

V n d V̆nd Xn f ˜ nV̆n ,Xn, t ln Z i1

n

f ˆ in v̆i ,xi , t ,

we establish the following relation from Eq. A6:

H t i, ,nN1

e n1

n! V

d v̆d x Z 1/ n f ˆ in

v̆,x, t ln Z 1/ n f ˆ in v̆,x,t

A d v̆d xC f ln C f

B d v̆d x f v̆,x, t 1ln f v̆,x,t 1 . A11

In the above expression the inequality A arises from apply-

ing inequality

A1

with A

given by the Poisson distribution A(n)(e n1 / (n1)! ) and equality B follows from

the invariance of the integral with respect to translations by

the streaming transformation. Consequently, the value of the

H functional at time t 1 does not exceed its value at time t .

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A. Malevanets and J. M. Yeomans in preparation.



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