Isotropic stochastic rotation dynamics

Sebastian Muhlbauer,∗ Severin Strobl, and Thorsten PoschelInstitute for Multiscale Simulation, Friedrich-Alexander-Universitat Erlangen-Nurnberg,

Nagelsbachstraße 49b, 91052 Erlangen, Germany(Dated: May 29, 2017)

Stochastic Rotation Dynamics (SRD) is a widely used method for the mesoscopic modeling ofcomplex fluids, such as colloidal suspensions, or multiphase flows. In this method, however, theunderlying Cartesian grid defining the coarse-grained interaction volumes induces anisotropy. Wepropose an isotropic, lattice-free variant of stochastic rotation dynamics, termed iSRD. Instead ofCartesian grid cells, we employ randomly distributed spheres. This eliminates the requirement of agrid shift, which is essential in standard SRD to maintain Galilean invariance. We derive analyticalexpressions for the viscosity and the diffusion coefficient in relation to the model parameters, whichshow excellent agreement with the results obtained in iSRD simulations. The proposed algorithm isparticularly suitable to model systems bound by walls of complex shape, where the domain cannotbe meshed uniformly, such that standard SRD would fail.

PACS numbers: 47.11.-j, 05.40.-a, 02.70.Ns

I. INTRODUCTION

During the last years particle-based fluid simulationmethods have been well established as an alternative tocontinuum methods. They have important advantages,especially in situations where the continuum assumptiondoes not hold, but also for complex fluids, such as col-loidal suspensions, or multiphase flows. With molecu-lar dynamics, for example, complex fluids can be sim-ulated on the molecular level including the actual mi-croscopic interaction laws [1]. In contrast, the particlesemployed in Direct Simulation Monte Carlo (DSMC) [2]do not represent physical particles, but rather probabilityquanta, composing the velocity distribution function. InDSMC, the particle trajectories are not computed deter-ministically, as in molecular dynamics. Instead, binarycollisions are performed to model the transport of mo-mentum, and eventually solve the nonlinear Boltzmannequation [3]. Thus, unlike computational fluid dynamics(CFD), based on the numerical solution of hydrodynamicequations, DSMC does not rely on relations between thehydrodynamic fields, and is also reliable in cases wherethe hydrodynamic description of the system is problem-atic, for instance in the presence of shocks [4]. Furtherimportant fields of application are flows at moderate tohigh Knudsen number, such as rarefied gases [5], flows inthe vicinity of boundaries [6], flows in microfluidic devices[7], where the mean free path does not fulfill the condi-tion of being much smaller than the system dimensions.In general, DSMC consumes significantly larger compu-tational resources than field-based CFD. However, therange of validity of the method is wider than for CDF,since DSMC relies exclusively on the validity of the Boltz-mann description. Moreover, there are also cases wherethis simulation method is computationally efficient, e.g.,

∗ Corresponding author.E-mail address: sebastian.muehlbauer@fau.de

for rarefied gas flows with a large mean free path, andrather few particle collisions, since the computationalcost is proportional to the particle collision rate.

For application to denser gases, however, Stochas-tic Rotation Dynamics (SRD), developed by Malevanetsand Kapral [8], is more efficient, compromising betweenDSMC and CFD: Instead of numerous binary collision,one so-called multi-particle collision is performed to ex-change momentum between all particles within certaincoarse-graining volumes, which usually are the cells of aCartesian grid spanning the simulation volume. The sizeas well as the shape of the coarse-graining volumes hassignificant influence on the transport coefficients. There-fore, a grid composed of cubic cells is used for the ma-jority of applications. In order to model complicated do-mains, an additional surface grid, describing the bound-aries, has to be embedded into the regular simulationgrid. It was shown that in its basic form, SRD does notpreserve Galilean invariance, which can be corrected bya random grid shift, that is, the simulation grid has tobe shifted randomly before each collision step [9]. Thegrid shift assures Galilean invariance, however, it doesnot correct the anisotropy caused by the underlying cu-bic grid [10]. Ihle et al. [10] conjectured that additionalrandom grid rotations would restore isotropy in SRD.

For the case of DSMC, which suffers from similar prob-lems, in order to achieve isotropy, Donev et al. [11]suggest a grid-free version, termed isotropic DSMC(iDSMC). While in standard DSMC only particles fromthe same grid cell are allowed to collide, in iDSMC ran-dom particle pairs are chosen to transfer momentum, ifthey are not further apart than a given distance, disre-garding their cell affiliation. This procedure leads to anisotropic interaction between the quasi-particles [11, 12].

Following a similar idea, we propose a grid-free, trulyLagrangian version of SRD which we call isotropicStochastic Rotation Dynamics (iSRD).

We will first provide a short review of SRD and the cor-responding relations for the transport coefficients. Sub-

2

sequently, we will describe our approach to eliminate theinfluence of the spatial discretization, and derive analyt-ical expressions for the viscosity and the diffusion coef-ficient in this case. Consequently, we compare these ex-pressions with measurements of the transport coefficientsobtained from simulations.

II. STOCHASTIC ROTATION DYNAMICS

SRD is a Lagrangian approach, where the fluid is repre-sented by particles. Each time step of the simulation, ∆t,comprises a streaming and a collision step. The stream-ing step propagates the particles according to their cur-rent velocity,

~ri(t+ ∆t) = ~ri(t) + ∆t~vi(t) , (1)

where ~ri and ~vi denote the position and velocity of par-ticle i, respectively. The collision step randomly rotatesthe fluctuating contribution to the particles’ velocitiesdue to the temperature of the fluid according to the col-lision rule

~vi(t+ ∆t) = ~u(t)|i + R · [~vi(t)− ~u(t)|i] , (2)

where ~u(t) is the macroscopic flow velocity field and~u(t)|i is the value of the field at the location of parti-cle i. In SRD, the field ~u is obtained by coarse grainingfrom the velocities of all particles of the system, using aCartesian grid spanning the simulation domain. Thus,the flow velocity corresponding to the ith particle is es-timated as the mean velocity of all particles residing in

the same grid cell, ~u(t)|i ≡ 〈~v 〉C

, and the particle’s ther-

mal velocity is[~vi(t)− 〈~v 〉C

]. In three-dimensional sim-

ulations, the rotation matrix, R, represents a rotationby a constant rotation angle, α, where the rotation axisis randomly chosen in each collision step for each gridcell [8, 13].

As an example and for later reference, in Fig. 1 wepresent the flow profile for plane Poiseuille flow with sim-ple bounce-back boundary conditions [14] at the wallsand periodic boundaries in the other two spatial direc-tions. If not specified otherwise, here and in the fol-lowing we use dimensionless parameters, that is, the cellwidth, a = 1, time step, ∆t = 0.1, thermal velocity,vth ≡

√k T/m = 1, and the rotation angle α = π/2.

The number of particles is chosen such that on averageM = 10 particles reside in each grid cell. The flow isdriven by an external acceleration of 5× 10−3. Temper-ature is kept constant by means of the cell-level thermo-stat which was developed by Hecht et al. [15, 16] based onRefs. [17, 18]. To obtain the flow profile shown in Fig. 1,we average the particle velocities within slices parallel tothe channel walls, considering two different slice widths.For the symbols (◦), the slice width is chosen to be equalto the cell width, a. These cell averaged velocities agreevery well with the fitted parabola drawn as a dashed

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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

position

velocity

FIG. 1. Plane Poiseuille flow simulated using SRD withoutgrid shift. The discretization of the channel, which is 20 cellswide, is shown by the vertical thin lines. The particle veloci-ties are averaged within slices oriented parallel to the channelwalls. The symbols (◦) show the average particle velocitywithin slices having width a. The dashed line is a parabolafitted to this averaged velocity profile in the interval [−5, 5],i.e., distant from the walls. The solid line represents the de-tailed flow profile, obtained by averaging within slices of widtha/20. In this example the Reynolds number is approximately58. In dimensional quantities, this corresponds to water atroom temperature (20◦C) flowing through a 2 mm wide chan-nel at 29 mm/s.

line. Considering, however, the solid line representing thefiner resolved flow profile, which is obtained by averagingwithin slices having a width of a/20, we perceive distinctsteps, connecting the plateaus within the simulation cells.This effect reveals a violation of the molecular chaos as-sumption: In case the mean free path is small comparedto the cell width, many particles remain in one grid cellfor several time steps, colliding with each other repeat-edly and, thus, correlating the velocities of the particlesin the same cell. One way to maintain molecular chaos,and to make SRD Galilean-invariant is the so-called gridshift [9, 13]: Before each collision step the simulationgrid is shifted by a random vector ~s = [sx, sy, sz], withsx, sy, sz ∈ [−a/2, a/2]. This procedure is sensible, onlyas long as the simulation domain is discretized using aregular cubic grid.

III. TRANSPORT COEFFICIENTS IN SRD

The viscosity of a fluid simulated with SRD is knownto have two contributions, which are denoted as the ki-

3

netic and the collisional part [19]. Kikuchi et al. [20] de-rived analytical expressions for these two contributions,assuming that a grid shift is performed in the SRD simu-lation. For their computations, the grid shift is essential,since their derivation of both the kinetic and the colli-sional contribution depends on the assumption of molec-ular chaos. Considering stationary Couette flow withthe flow profile ~u(y) = [γ y, 0, 0], where the shear rate,γ ≡ ∂ux(y)/∂y, is constant, Kikuchi et al. find for the ki-netic contribution to the viscosity for three-dimensionalflows:

νkinSRD =kB T

m∆t

(1

2+

f

1− f

), (3)

where kB , T and m denote the Boltzmann constant, thetemperature and the particle mass, respectively. Theterm

f(α, n) = 1−(n− 1

n

)2

5(2− cosα− cos 2α) (4)

describes the relative change in the correlation 〈vx vy〉due to the collision operator from Eq. (2) in case exactlyn particles are contained in the considered grid cell,

〈vx(t+ ∆t) vy(t+ ∆t)〉 = f(α, n) 〈vx(t) vy(t)〉 . (5)

However, the number of particles, n, contained in the col-lision cells is not constant but Poisson distributed. Av-eraging over n ∈ {1, 2, ...,∞} yields

f(α,M) = 1− M − 1 + e−M

M

2

5(2− cosα− cos 2α) (6)

and

〈vx(t+ ∆t) vy(t+ ∆t)〉 = f(α,M) 〈vx(t) vy(t)〉 , (7)

with M being the average number of particles per cell.An analogous procedure leads to an analytical expressionfor the diffusion coefficient, having no collisional contri-bution [13],

DSRD =kB T

m∆t

(1

2+

g

1− g

), (8)

where

g(α,M) = 1− M − 1 + e−M

M

2

3(1− cosα) . (9)

In order to obtain the collisional part of the viscosity,Kikuchi et al. [20] compute the momentum exchangealong the direction of the velocity gradient due to thecollision step for the considered Couette flow and obtain

νcollSRD =a2

18 ∆t

M − 1 + e−M

M(1− cosα) . (10)

The results for the kinetic part of the viscosity, Eq. (3)together with Eq. (6), as well as for the collisional con-tribution, Eq. (10), match the corresponding simulationresults [20].

IV. ISOTROPIC SRD

A. Description of the method

In standard SRD, even with grid shift, the underlyingCartesian simulation grid induces anisotropy [10]. Usingisotropic coarse graining volumes, this problem can beeliminated in a natural way, and without the necessity ofperforming random grid-rotations [10] to correct for theanisotropy introduced by the lattice. Instead of usingcubic lattice cells for coarse graining, we suggest to usespheres of constant size, which are randomly distributedover the simulation domain. Consequently, the localmean velocity needed for the collision step (see Eq. (2)) isnow evaluated by averaging the velocities of the particles

residing in the same sphere, S, that is, ~u(t)|i ≡ 〈~v 〉S

, and

the thermal velocity of the ith particle is[~vi − 〈~v 〉S

].

Note that this approach is grid-free. For computationalefficiency we use an auxiliary grid to sort the particlesinto the coarse graining spheres. This grid is not neces-sarily Cartesian but can be chosen irregular, such thatthe sum of all grid cells covers the simulation domain.Unlike for standard SRD, this grid does, however, notaffect the simulation results, as we will demonstrate inSec. IV F. In analogy to isotropic DSMC [11, 12] we referto this method as isotropic SRD (iSRD).

B. Definition of the coarse graining volumes

In standard SRD with grid shift, the coarse grainingvolumes are the cubic cells of a Cartesian grid with anumber of basic properties:

i. The coarse graining volumes are homogeneouslydistributed in physical space.

ii. The total number of coarse graining volumes isgiven by the ratio between domain volume andcoarse graining volume. It is, therefore, invariantin time.

iii. At each time, each particle resides in exactly onecoarse graining volume.

We will see that in iSRD these properties do not holdtrue strictly, but only in a statistical sense. It will beshown, however, that the obtained simulation data agreewell with analytical results.

In difference to standard SRD, in iSRD the locations ofthe spherical coarse graining volumes are determined ran-domly. The centers of the coarse graining spheres shall beuniformly distributed in the simulation domain, however,each unit of physical space contains the same number ofcoarse graining spheres only on average. Note, that therandomness in the sphere positions has the same effectas the grid shift in SRD and ensures Galilean invariance.

Sorting the SRD particles into the coarse grainingspheres is a computationally expensive process, as it has

4

to be repeated in each time step. Therefore, we suggestto employ an auxiliary grid. In Sec. IV F we will showthat this auxiliary grid, while significantly enhancing theefficiency of the simulation, does not affect the simulationresult.

The number of coarse graining spheres, kl, within cell,l, of the auxiliary mesh is a Poisson distributed randomnumber with probability density

Pλl(kl) =

λkllkl!

e−λl . (11)

In the simulation, we first determine the number of coarsegraining spheres in the lth cell according to the distribu-tion Pλl

, Eq. (11), and then choose the locations of thesphere centers randomly inside the cell. Since particlescan be located in more than one coarse graining volume,they can take part in more than one multi-particle col-lision per time step. In order to avoid any systematiceffect, overlapping coarse graining spheres should be pro-cessed in random order.

The distribution of coarse graining volumes shall behomogeneous over the domain (i.), that is, independentof the cell volume, V Cl . Therefore, for two arbitrary cellsl and l′,

λlλl′

=V ClV Cl′

, (12)

that is, the expectation values for the number of coarsegraining volumes in a cell relate as the cell volumes. To-gether with the normalization condition∑

l

V Cl =∑l

λlVS (13)

we obtain

λl =V ClV S

, (14)

where V S is the volume of a coarse graining sphere. Sum-ming over all cells and denoting the volume of the entiredomain by V , we obtain

λ ≡∑l

λl = V/V S (15)

and, thus, the expectation value for the total number ofcoarse graining spheres in a given volume, V , does notdepend on the auxiliary grid, that is, (ii.) is statisticallyfulfilled.

Let us finally consider property (iii.): The probabil-ity that a particle is located in a randomly placed coarsegraining sphere of volume V S is the ratio V S/V = 1/λ,where V is the total volume of the domain. Assume, Ncoarse graining spheres are placed randomly and indepen-dently in the domain. The probability, Aj , for a certainparticle to be simultaneously located in exactly j coarsegraining spheres then obeys a binomial distribution

Aj = B (j |N, 1/λ ) ≈ B (j |λ, 1/λ ) , (16)

where the expectation value for N is given in Eq. (15).Albeit the total number of coarse graining spheres is afluctuating quantity, for the majority of applications thedomain volume is much larger than the coarse grainingvolume and, therefore, λ� 1, implying that the relativefluctuations, ∆N/N ≈ 1/

√λ, are small. We can further

exploit the convergence of the binomial distribution to aPoisson distribution, leading to

Aj ≈e−1

j!. (17)

Thus, the number of coarse graining spheres, j, a givenparticle is contained in, obeys a Poisson distribution withexpectation value 1. Hence, while in standard SRD ineach time step each particle is located in exactly onecoarse-graining volume, in iSRD this is not the case, and(iii.) holds true only on average.

C. iSRD for the force-free case – diffusion

For the first test of the iSRD model, we consider purediffusion in a periodic domain, being the most simplecase. For this simulation and – if not specified otherwise– also for the following simulations we use the radius of

the spherical coarse graining volume R = [3/ (4π)]1/3 ≈

0.620, such that V S = 1. Thus, on average the simula-tion domain contains one coarse graining sphere per unitvolume. Each coarse graining sphere contains M = 10particles on average. Figure 2 shows the mean squareddisplacements of the particles. After a short initial tran-sient, the mean squared displacement increases linearlywith time. The dashed line shows a linear fit to the simu-lation data. This indicates that diffusive mass transport

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

time

mea

nsq

uare

dd

isp

lace

men

t

mean squared displacement

linear fit for the range [2, 10]

FIG. 2. Mean squared displacement measurement of a force-free system obtained by iSRD. The dashed line shows a linearfit to the simulation data, disregarding the initial transient.The auxiliary grid consists of 32 × 32 × 32 cubic cells.

is reproduced correctly in iSRD.

5

D. iSRD for stationary Couette and Poiseuille flow– momentum transport

To demonstrate the simulation of momentum trans-fer with iSRD, we simulate stationary Couette flow aswell as plane Poiseuille flow. In both cases, the walls aremodeled through bounce-back boundary conditions [14].Temperature is kept constant by means of the cell-levelthermostat described in Ref. [15, 16]. Figures 3 and 4show the velocity profiles obtained from the simulationtogether with analytical solutions for reference. The vis-cosity is not known a priori, and is, therefore, determinedfrom the curvature of the flow profiles in Fig. 4. Thechannel walls are modeled both with and without ghostparticles. Further details are discussed in Sec. IV E.

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0.0

0.1

0.2

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0.5

coarse graining sphere

position

vel

oci

ty

without ghost particles

with ghost particles

linear flow profile

FIG. 3. Couette flow simulated using iSRD with and withoutghost particles. The dashed line shows the analytical solutionfor the given boundary velocities. The size of a coarse grainingsphere is shown at the top left.

For both cases, the analytical solution given by theNavier-Stokes equations is recovered, that is, iSRD re-produces the momentum transport correctly, except forthe region close to the walls where we obtain a finite slip.In contrast to the curve shown in Fig. 1, the flow profilesobtained by iSRD do not reveal the kinks caused by thespatial discretization inherent to the interaction schemein SRD.

E. Reducing slip at the walls – ghost particles

The finite slip at the system walls modeled by bounce-back boundaries is a well known problem of virtuallyall particle-based, computational fluid dynamics meth-ods, see, e.g., Ref. [21]. This on meso- and macroscopicscales undesirable behavior can be corrected by intro-ducing ghost particles [14, 22]. Here, we apply the modeldeveloped in Refs. [23, 24] for smoothed-particle hydro-dynamics (SPH) simulations: In case a coarse graining

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coarse graining sphere

position

vel

oci

ty

without ghost particles

with ghost particles

parabolic flow profiles

FIG. 4. Plane Poiseuille flow simulated using iSRD with andwithout ghost particles. The dashed lines are parabolas fittedin the interval [−5, 5]. For the viscosity we obtain ν = 0.568with ghost particles and ν = 0.569 without. In this exam-ple the Reynolds number is approximately 15.5. In dimen-sional quantities, this corresponds to water at room tempera-ture (20◦C) flowing through a 2 mm wide channel at velocity7.8 mm/s.

sphere reaches out of the domain and into the wall, wemirror the particles from inside the domain at the in-tersected boundary. The velocity of a ghost particle iscomputed by inverting the corresponding particle’s ve-locity ~vi with respect to the wall velocity ~uwall, that is,

~v ghosti = 2 ~uwall−~vi. In order to make the interaction be-

tween fluid and wall symmetric, additional coarse grain-ing spheres are generated with centers located beyond thedomain boundary. This effectively leads to an extrapo-lation of the flow velocity beyond the domain boundary.

The blue lines in Figs. 3 and 4 show the flow profilesobtained by iSRD for Couette flow and plane Poiseuilleflow where ghost particles are employed. In both cases,we obtain excellent agreement between the simulationresults and the theoretical solution.

F. Independence of iSRD on the auxilliary grid

The results of iSRD do not noticeably depend on thechosen auxiliary grid. For demonstration, we computethe velocity profile due to Poiseuille flows in a cubic do-main with identical parameters, except for the spatial dis-cretization of the auxiliary grid, see Fig. 5a. Three casesare investigated: A Cartesian grid (see solid black linesin Fig. 5b), a non-uniform regular grid (dashed red lines),and a non-uniform tetrahedral discretization (blue dotted

6

lines). For both considered time step sizes, ∆t = 0.1 and∆t = 1, and for all grids, the velocity profiles coincideperfectly. Hence, the grid geometry does not affect thesimulation results. This is not surprising, since for thecase of iSRD the grid is only a data structure intended toaccelerate the simulation. In contrast, for standard SRDit is a fundamental feature of the method, and the simu-lation results are in direct relation to the parameters ofthe grid. The dependence of the simulation result on thegrid is the reason for problematic behavior of SRD in thevicinity of boundaries, in particular for flows in domainsof complex geometric shape where the domain cannot bemeshed uniformly. Since iSRD does not suffer from thisproblem, iSRD can be applied to systems where standardSRD would fail.

(a)

(b)

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0.45

0.50

coarse graining sphere

∆t = 0.1

∆t = 1.0

position

vel

oci

ty

Cartesian

non-uniform regular

non-uniform tetrahedral

FIG. 5. Plane Poiseuille flow simulated using iSRD, includingghost particles, for the three different auxiliary grids depictedin (a). The mesh on the left consists of cubic cells with con-stant width, as indicated by the black scale in (b). The redscale refers to the mesh in the center, comprising non-uniformhexahedra with square basis and varying extent in the direc-tion of the velocity gradient, i.e., normal to the walls. Themesh on the right is composed of non-uniform tetrahedralcells.

V. TRANSPORT COEFFICIENTS IN ISRD

For standard SRD, Kikuchi et al. [20] derive an ana-lytical expression for the kinetic contribution to the vis-cosity, νkinSRD, by considering the change of velocity cor-relations 〈vx vy〉 in stationary Couette flow due to theaction of the streaming and the collision operators. Theeffect of the streaming operator is the same for SRD andiSRD, while the collision operator acts differently: Whilein SRD at any time, each particle is located in exactlyone coarse-graining volume, this is not the case for iSRD.To derive an expression equivalent to Eq. (7) for the caseof iSRD, we consider the effect of the collision operatoron the velocity correlation 〈vx vy〉i, referring to particlei under the condition that the particle is contained inj ∈ {0, 1, 2, . . . } coarse graining spheres. The expec-tation value of the kinetic contribution to the viscosity,νkiniSRD, is then obtained as a weighted average of thesecontributions for all j.

If at a certain time step, particle i is not located inany coarse graining sphere, that is, j = 0, the collisionstep does not change its contribution to the velocity cor-relation, 〈vx vy〉i. If the particle is contained in exactlyone coarse graining sphere (j = 1) the velocity correla-tion changes according to Eq. (7). For j = 2, when theparticle is located in two coarse graining spheres, S1 andS2, its contribution to the velocity correlation is

〈vx(t+∆t)vy(t+∆t)〉i=f(α, nS1)f(α, nS2)〈vx(t)vy(t)〉i ,(18)

where nS1 and nS2 are the total numbers of particles inthese spheres. For the general case, that particle i iscontained in exactly j coarse graining spheres, we have

〈vx(t+ ∆t)vy(t+ ∆t)〉i = 〈vx(t)vy(t)〉ij∏

k=1

f (α, nSk)

≈ 〈vx(t)vy(t)〉i [f (α,M)]j,

(19)

where we employ Eq. (6) with the approximation thatthe numbers of particles contained in the coarse grain-ing volumes are independent. Using the probability, Aj ,given in Eq. (17), for finding particle i in j coarse grainingspheres, and averaging over all particles, we define

f(α,M) ≡∑j

Aj fj(α,M)

Eq. (17)≈ ef(α,M)−1 , (20)

and find an expression for the change of the velocity cor-relation function due to a collision step in iSRD,

〈vx(t+ ∆t) vy(t+ ∆t)〉 = f(α,M) 〈vx(t) vy(t)〉 , (21)

having the same functional form as Eq. (7). Eventu-ally, we obtain the kinetic contribution to the viscosityin iSRD,

νkiniSRD =kB T

m∆t

(1

2+

f

1− f

), (22)

7

which has the same functional form as νkinSRD, givenin Eq. (3). The diffusion coefficient can be derived anal-ogously to the kinetic part of the viscosity. We find

DiSRD =kBT

m∆t

(1

2+

g

1− g

), (23)

with

g(α,M) =∑j

Aj gj(α,M)

Eq. (17)≈ eg(α,M)−1 . (24)

For the collisional contribution to viscosity we fol-low Ref. [20], and examine the redistribution of momen-tum due to the collision operator for stationary Cou-ette flow with ~u(y) = [γ y, 0, 0] and constant shear rateγ = ∂ux(y)/∂y. We divide the coarse graining sphere of

~u

x

y∆y

y = 0

y = y0

y = d

1

2

FIG. 6. Redistribution of momentum within one coarse grain-ing sphere in stationary Couette flow. The coarse grainingsphere is divided into two subvolumes by a horizontal planeat y = y0.

diameter d into two subvolumes, 1 and 2 , separatedby the plane y = y0 with 0 ≤ y0 ≤ d, as shown in Fig. 6.The two subvolumes accommodate n(1) and n(2) parti-cles with n(1) + n(2) = n, traveling at average velocities

〈vx〉(1) and 〈vx〉(2) in x-direction. The average velocity in

the entire coarse graining sphere is 〈vx〉S . Therefore,

∆vSx ≡ 〈vx〉(1) − 〈vx〉(2) =

n(1) + n(2)

n(2)

(〈vx〉(1) − 〈vx〉S

).

(25)The average distance of pairs of particles picked fromdifferent subvolumes, denoted by ∆y, is the distance ofthe geometric centers of the two subvolumes,

∆y =3 d3

8(32 d− y0

) (y0 − 1

2 d) . (26)

The collision rule, Eq. (2), conserves momentum, there-fore, the transfer of momentum between the subvolumesis equal to the change of momentum in subvolume 1 dueto the collision step.

The shear stress, σxy, defined as the transport of mo-mentum per units of time and cross sectional area, is [20]

σxy =m

A∆t

[2

3n(1) (1− cosα)

(〈vx〉(1) − 〈vx〉S

)], (27)

where m is the particle mass and A denotes the charac-teristic area, the momentum transfer due to one coarsegraining sphere refers to. To obtain A we consider the

A?

V ? = A? d

d2

d2

x

y z

FIG. 7. The characteristic area, A, is the average area themomentum transport within one single coarse graining sphererefers to. The size of area A can be obtained considering theaverage number of coarse graining spheres, N?, in a referencevolume V ? ≡ A? d, which is continued periodically in x andz-directions.

shaded plane, A?, in Fig. 7. The average number ofspheres intersecting this plane is equal to the expecta-tion value of the number of spheres, N?, having theircenters located in the rectangular volume V ? ≡ A? d,

N? =V ?

V S=

6A?

πd2, (28)

thus,

A =A?

N?=

π

6 d2. (29)

Combining Eqs. (25) to (27) and (29), and the averagemass density in the coarse graining sphere,

ρ =nm

V S=

(n(1) + n(2))m

V S, (30)

we obtain the collisional contribution to viscosity,

νcolliSRD =σxyρ γ

=σxy ∆y

ρ∆vSx

=n(1)

(n− n(1)

)4n2 ∆t

d4 (1− cosα)(32 d− y0

) (y0 − 1

2 d) . (31)

The number of particles n(1) in subvolume 1 obeys abinomial distribution, B

(n(1)|n, p

), with probability

p =V (1)

V S=y20 (3d− 2y0)

d3. (32)

Using the property of the binomial distribution,∑n(1)

(n− n(1)

)B(n(1)|n, p

)= n p (n− 1) (1− p) ,

(33)we rewrite Eq. (31), and obtain

νcolliSRD =(n− 1) (1− cosα)

4n∆t

× 2d y20y0 + 1

2d

[1− y20 (3d− 2y0)

d3

]. (34)

8

Averaging over the position of the plane with respect tothe coarse graining sphere, y0, which is uniformly dis-tributed in [0, d], yields

νcolliSRD =(n− 1) (1− cosα)

4n∆t

2 d2

15(35)

and averaging over n, which is, in good approxima-tion, Poisson distributed, leads to the final expressionfor the collisional contribution to the viscosity coefficientin iSRD,

νcolliSRD =d2

30 ∆t

M − 1 + e−M

M(1− cosα) . (36)

Note that Eq. (36) is of the same mathematical structureas the corresponding expression for standard SRD, νcollSRD,given by Eq. (10).

VI. QUANTITATIVE VALIDATION OF ISRD

A. Benchmarking cases

To check the validity of iSRD we compare the simula-tion results with the derived theoretical expressions forthe diffusion coefficient and the kinematic viscosity givenby Eqs. (22), (23) and (36). We consider two physicalsystems, force-free fluid and plane Poiseuille flow. Theparameters, we vary, are the time step, ∆t, the rota-tion angle, α, and the average number of particles percoarse graining sphere, M . The other parameters of oursimulation method, iSRD, are kept constant. We treattime and position as dimensionless parameters. Thus,the viscosity and the diffusion coefficient are consideredin dimensionless form, as well.

B. Diffusion coefficient

The diffusion coefficient is computed from the meansquared displacement of the particles in a force-free pe-riodic simulation domain. The auxiliary mesh comprisesa total of 4096 cubical cells. The numerical result is av-eraged over several individual measurements for each pa-rameter set to reduce statistical errors. Two differenttime steps, ∆t = 0.1 and ∆t = 1, are considered. Fig. 8shows the diffusion coefficient in dependence on the ro-tation angle, α. The theoretical result given by Eq. (23)is indicated by the dashed line in Fig. 8. The simula-tion data are in excellent agreement with the theoreticalresults over the full range of the rotation angle, α.

The numerical simulation results for the diffusion co-efficient as a function of the average number of particlesper coarse graining sphere are shown in Fig. 9. For largenumber of particles, M , the theoretical result, Eq. (23),is well reproduced for both of the considered time steps,∆t = 0.1 and ∆t = 1, indicated by the good agree-ment of the numerical data (symbols in Fig. 9) with the

0 π/6 π/3 π/2 2/3π 5/6π π0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

rotation angle α

diff

usi

on

coeffi

cien

tD

∆t = 0.1

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

diff

usi

on

coeffi

cien

tD

∆t = 1

FIG. 8. Diffusion coefficient as a function of the rotationangle, α, for two values of the time step. The left verticalaxis applies to the data for ∆t = 0.1 and the right axis to∆t = 0.1, respectively. The dashed line shows the theoreticalresult, Eq. (23).

100 101 102 1030.15

0.16

0.17

0.18

0.19

0.20

0.21

0.22

0.23

0.24

particles per coarse graining sphere M

diff

usi

on

coeffi

cien

tD

∆t = 0.1

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

diff

usi

on

coeffi

cien

tD

∆t = 1

FIG. 9. Diffusion coefficient as a function of the average num-ber of particles per coarse graining sphere, M , for two valuesof the time step. The left vertical axis applies to the datafor ∆t = 0.1 and the right axis to ∆t = 1, respectively. Thedashed line shows the theoretical result, Eq. (23).

dashed line showing the diffusion coefficient as given byEq. (23). For smaller M , the simulation data deviatefrom the analytical results. The reason for this devi-ation is the assumption that the population number ofparticles within overlapping coarse graining spheres is in-dependent, which was exploited in the derivation of thetransport coefficients. This approximation is, however,less justified for small particle numbers. In contrast, forlarge M , the fluctuations in particle number have hardlyany noticeable effect, rendering the discrepancy negligi-ble. For ∆t = 1 the observed divergence is smaller thanfor ∆t = 0.1. This is due to the fact that for large ∆t,the mean free path is also larger, and spacial correlations

9

in particle number decay faster. As shown in Fig. 8, thisdeviation hardly depends on the rotation angle, α.

C. Kinematic viscosity

For plane Poiseuille flow, we determine the kinematicviscosity from the curvature of the flow profile in thesteady state. The simulation domain is covered by anauxiliary Cartesian grid comprising 8000 cells. The num-ber of simulated time steps, Nt, is chosen such thatNtM = 16× 106, resulting in smooth profiles and smallfluctuations of the viscosity. The mean free path

0 π/6 π/3 π/2 2/3π 5/6π π0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

rotation angle α

kin

emati

cvis

cosi

tyν

∆t = 0.1

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

kin

emati

cvis

cosi

tyν

∆t = 1

FIG. 10. Kinematic viscosity as a function of the rotationangle, α, for two values of the time step. The left verticalaxis applies to the data for ∆t = 0.1 and the right axis to∆t = 1, respectively. The dashed line shows the theoreticalresult, Eqs. (22) and (36).

and, thus, the transport coefficients depend sensitivelyon temperature. Therefore, we maintain a constant tem-perature by means of the cell-level thermostat describedin Refs. [15, 16]. Figure 10 shows the simulation resultsfor both ∆t = 0.1, where the collisional contribution toviscosity dominates, and ∆t = 1, where the kinetic con-tribution dominates. The simulation data for the vis-cosity as a function of the rotation angle show excellentagreement with the analytical results, Eqs. (22) and (36).For the small time step, ∆t = 0.1, we obtain minor de-viations, which are less distinct for ∆t = 1. Figure 11shows the simulation data for the viscosity as a functionof the average number of particles per coarse grainingvolume, M . We obtain minor, systematic deviations forboth values of ∆t, which we attribute to the thermostat,since such deviations are not apparent in the diffusioncoefficient, where no thermostat is used. For the consid-ered time steps, ∆t = 0.1 and ∆t = 1, the simulationresults for large M deviate from the theoretical results(Eqs. (22) and (36)) by approximately 0.3% and 0.5%,respectively. Similar to the diffusion coefficient discussedabove, we also perceive systematic deviations between

100 101 102 1030.45

0.50

0.55

0.60

particles per coarse graining sphere M

kin

emati

cvis

cosi

tyν

∆t = 0.1

0.9

1.0

1.1

1.2

kin

emati

cvis

cosi

tyν

∆t = 1

FIG. 11. Kinematic viscosity as a function of the averagenumber of particles per coarse graining sphere, M . The leftvertical axis applies to the data for ∆t = 0.1 and the rightaxis to ∆t = 1, respectively. The line shows the theoreticalresult, Eqs. (22) and (36).

the simulation data and the theoretical results at smallparticle number. For the viscosity, this error is less pro-nounced than in case of the diffusion coefficient, presum-ably because it is partially compensated by the influenceof the thermostat.

VII. CONCLUSION

We introduce isotropic SRD, which is a modificationof standard SRD. Instead of using Cartesian grid cellsas coarse-graining volumes, we generate coarse grainingspheres randomly within the simulation domain. This al-lows to maintain Galilean invariance without the need fora grid shift, which is restricted to regular grids. In con-trast to traditional stochastic rotation dynamics, iSRD isa truly grid-free, particle-based fluid simulation method.Only an auxiliary grid is needed in order to efficientlysort the particles into the coarse graining spheres. Whilethe structure of the auxiliary grid can affect the compu-tational cost, we demonstrate that it does not affect thesimulation results. The proposed algorithm is, therefore,particularly suitable to simulate the fluid flow throughdomains of complicated shape.

Moreover, we show that iSRD reproduces both massand momentum transport correctly. We also provide an-alytical expressions for the transport coefficients depend-ing on the simulation parameters. The measured val-ues for the transport coefficients accurately follow theseexpressions, especially for higher particle numbers percoarse graining volume. Further, the use of spheres ascoarse graining volumes prevents the existence of pre-ferred directions for momentum transport, yielding afully isotropic simulation method.

10

For both iSRD and SRD, the mean free path doesnot depend on the density, and therefore, the knowndependence of the transport coefficients in gases on thetemperature is not recovered. Introducing an additionaltemperature-dependent collision probability, as proposedby Gompper et al. [13], allows to control the time step∆t locally. Since the diffusion coefficient is proportionalto ∆t, this idea renders the diffusion coefficient freelyadjustable. Viscosity, however, cannot be adjusted simi-larly, because viscosity has one component proportionalto ∆t and another one proportional to 1/∆t. Since iniSRD the diameter of the coarse graining spheres can beadjusted locally, it can be used to recover the temper-

ature dependence of the viscosity. Therefore, iSRD cansimultaneously reproduce the temperature dependence ofboth the viscosity and diffusion coefficient.

ACKNOWLEDGMENTS

We thank Dan Serero for sharing his expertise in ki-netic theory and Kuang-Wu Lee for fruitful discussionsregarding stochastic rotation dynamics. We also thankthe German Research Foundation (DFG) for fundingthrough the Cluster of Excellence “Engineering of Ad-vanced Materials”, ZISC, and, FPS.

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