Direct simulation for a homogeneous gasHasan Karabulut

Citation: American Journal of Physics 75, 62 (2007); doi: 10.1119/1.2366735 View online: http://dx.doi.org/10.1119/1.2366735 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/75/1?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in An efficient direct simulation Monte Carlo method for low Mach number noncontinuum gas flows based on theBhatnagar–Gross–Krook model Phys. Fluids 21, 033103 (2009); 10.1063/1.3081562 Accuracy and efficiency of the sophisticated direct simulation Monte Carlo algorithm for simulating noncontinuumgas flows Phys. Fluids 21, 017103 (2009); 10.1063/1.3067865 A direct simulation Monte Carlo method for rarefied gas flows in the limit of small Mach number Phys. Fluids 17, 107107 (2005); 10.1063/1.2107807 Rarefaction effects on shear driven oscillatory gas flows: A direct simulation Monte Carlo study in the entireKnudsen regime Phys. Fluids 16, 317 (2004); 10.1063/1.1634563 Direct Simulation of a Flow Produced by a Plane Wall Oscillating in Its Normal Direction AIP Conf. Proc. 663, 202 (2003); 10.1063/1.1581550

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Direct simulation for a homogeneous gasHasan KarabulutRize Faculty of Arts and Sciences, Department of Physics, Karadeniz Technical University,53100 Rize, Turkey

�Received 13 April 2006; accepted 22 September 2006�

A probabilistic analysis of the direct simulation of a homogeneous gas is given. A hierarchy ofequations similar to the BBGKY hierarchy for the reduced probability densities is derived. Byinvoking the molecular chaos assumption, an equation similar to the Boltzmann equation for thesingle particle probability density and the corresponding H-theorem is derived. © 2007 AmericanAssociation of Physics Teachers.

�DOI: 10.1119/1.2366735�

I. INTRODUCTION

Direct simulation Monte Carlo method �DSMC� is a stan-dard method for solving the Boltzmann equation numeri-cally. In this method space is divided into cells of volume �Vand a large number of “particles” �N=103–106� represent thereal gas molecules. The evolution of the gas for a short time�t is calculated in two steps. In the first step all particles arepropagated for a time �t without collisions. In the secondstep some randomly chosen pairs of particles in the same cellare allowed to collide and change their velocities withoutchanging their positions. The number of pairs �n� chosen tomake collision attempts is given by n=RN2�t /2V, where Ris a parameter we specify, N is the number of particles in thecell, and V is the volume of the cell. We call n the number ofcollision attempts because not every pair chosen makes acollision. A pair makes a collision with a probability u�T /R,where u is their relative velocity and �T is the total crosssection. The results are not sensitive to the value of the pa-rameter R as long as R is big enough so that very few pairsviolate the condition u� /R�1, because the average numberof successful collision attempts,

n�u�T�

R=

RN2�t

2V

�u�T�R

=N2�u�T�

2V�t , �1�

is independent of R. Here �u�T� is the average of u�T overall possible pairs. Although taking R very big is acceptabletheoretically, for practical reasons R should not chosen betoo big either.

The original method is due mainly to Bird. A seminalpaper1 by Bird gave some heuristic arguments to justify itsuse. There are many good references on the subject. Refer-ence 2 has a good tutorial on the subject and the monographby Bird3 is a complete reference on the developments up to1994. Books on rarified gas dynamics devote many chaptersto the subject and Refs. 4 and 5 are useful references in thiscategory.

A variant of the method was derived by Nanbu6 startingfrom the Boltzmann equation. To represent the evolution ofthe real gas such methods should converge to the true solu-tion of the Boltzmann equation in the limit N→�, �V→0,and �t→0. Convergence proofs were given by Babovsky7

and Babovsky and Illner8 for Nanbu’s method and byWagner9 for Bird’s method.

For the evolution of the velocity distribution of a spatiallyuniform gas there is no need to divide physical space intocells and we can just work in velocity space. Although Bird

3

recommended dividing real space into cells for studying the

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spatially homogeneous gas, we will show that this division isunnecessary. If we consider velocity space only and colliderandom pairs, we should obtain the evolution of the velocitydistribution. The purpose of this paper is to study this sto-chastic process.

These efforts to solve the Boltzmann equation using sto-chastic methods were driven by scientific applications andthere was no motivation to use them as a pedagogical tool. Itis surprising that similar stochastic algorithms for the homo-geneous gas were conceived by people interested in usingthem as a pedagogical tool to demonstrate the evolution of agas to the Maxwell-Boltzmann distribution. The earliest ofsuch articles of which the author is aware is that of Novakand Bortz,10 who studied the evolution of a gas of two-dimensional disks. Their algorithm is based on taking ran-dom pairs and colliding them with a probability proportionalto u�. Eger and Kress11 modified this algorithm and Bonomoand Riggi12 applied the modification to hard disks. There arealso other papers13,14 that do not use DSMC-type stochasticprocesses to demonstrate the Maxwell-Boltzmann distribu-tion. Although the DSMC method was well known, thesepapers do not reference papers on DSMC. Apparently theidea of stochastic methods for the evolution of a homoge-neous gas was conceived for pedagogical applications inde-pendently.

As mentioned, the DSMC algorithm can be used to dem-onstrate the approach of a velocity distribution to theMaxwell-Boltzmann distribution. The algorithm also givesan estimate of how many collisions are required to reachequilibrium and how various parameters affect the evolutionof the system.

Although direct simulation is intuitively appealing, it isnot clear that direct simulation algorithms represent the evo-lution of a real gas. The convergence proofs we have citedare formal and difficult to read. In this paper we prove thatthe normalized single particle probability distribution satis-fies the Boltzmann equation for a homogeneous gas. Theproof is relatively easy and intuitively appealing and is fa-miliar to physicists from the BBGKY hierarchy.15

In Sec. II we consider the stochastic algorithm for a ho-mogeneous gas. We derive a hierarchy of equations for theprobability distribution of particles similar to the BBGKYhierarchy.15 We use the molecular chaos assumption to de-rive an equation similar to the Boltzmann equation for thesingle particle probability distribution f�v�. We derive anH-theorem for f�v� and prove convergence to equilibrium.

We also show how the equation for f�v� reduces to the Bolt-

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zmann equation for a particular choice of collision probabili-ties and derive Bird’s “time counter” and “no time counter”methods.

II. ANALYSIS OF THE DIRECT SIMULATIONALGORITHM FOR A HOMOGENEOUS GAS

Consider a homogeneous gas of N�1 molecules withoutinternal degrees of freedom. We randomly select pairs ofmolecules to collide. All possible pairs have an equal prob-ability of 2 / �N−1�N to be selected. Suppose the velocities ofthe pair are vA and vB. The conditional probability that afterthe collision they have the velocities vC and vD in the inter-vals d3vC and d3vD is T�vA ,vB ;vC ,vD� d3vCd3vC. �From nowon we will denote d3v as dv for simplicity.� We also assumethe symmetries

T�vA,vB;vC,vD� = T�vC,vD;vA,vB� , �2a�

T�vA,vB;vC,vD� = T�vB,vA;vD,vC� . �2b�

The total probability is unity and therefore

� T�vA,vB;vC,vD�dvCdvD =� T�vA,vB;vC,vD� dvAdvB = 1.

�3�

Every selected pair makes a collision, although, as we willshow, by defining T�vA ,vB ;vC ,vD� some of the collisions donot change velocities. After each collision new velocities ofthe molecules are replaced by the old ones, and we select anew pair for the next collision. There is very small probabil-ity of choosing the same pair. If that happens, we let themcollide again. We do not need to keep a record of pairs thathave made collisions already.

We define f �N��v1 ,v2 , . . . ,vN� as the probability density forthe molecules. Because the molecules are indistinguishable,we require that the f �N� be totally symmetric:

f �N��v1, . . . ,vi, . . . ,v j, . . . ,vN�

= f �N��v1, . . . ,v j, . . . ,vi, . . . ,vN� . �4�

We also define the reduced probability densities

f �M��v1,v2, . . . ,vM� =� f �N��v1,v2, . . . ,vN�

�dvM+1dvM+2 ¯ dvN. �5�

Because we will be dealing with pairs of particles, it is usefulto define

f i,j�M��vA,vB� = f �M��v1, . . . ,vi = vA, . . . ,v j = vB, . . . ,vM� .

�6�

That is, the velocities of the i , j pair are replaced by vA ,vB inthe f �M��v1 ,v2 , . . . ,vM� where i , j�M. We will also use thenotation f �M��v ;n� for f �M��v1 ,v2 , . . . ,vM� after the nth colli-sion.

The function f �N��v ;n� satisfies the equation

f �N��v;n + 1� =1

N�N − 1��i=1

N

�j�i

N � f i,j�N��vA,vB;n�

�T�vA,vB;vi,v j� dvAdvB. �7�

The meaning of Eq. �7� is clear. If i , j is the last pair of

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molecules that has collided, then the probability of havingvi ,v j pairs after the collision is the probability of havinginitial velocities vA ,vB �represented by f i,j

�N��vA ,vB�� multi-plied by the probability of ending with vi ,v j �represented byT�vA ,vB ;vi ,v j��. The sum over i , j and the factor 1 /N�N−1� in Eq. �7� represents the fact that all pairs are possiblewith probability 1 /N�N−1�.

If we integrate Eq. �7� over vM+1, vM+2 , . . ., vN, we obtain

f �M��v;n + 1� =�N − M��N − M − 1�

N�N − 1�f �M��v;n�

+2�N − M�N�N − 1� �

i=1

M � f i,M+1�M+1��vA,vB;n�

�T�vA,vB;vi,vM+1� dvAdvBdvM+1

+M�M − 1�N�N − 1� �

i=1

M

�j�i

M � f i,j�M��vA,vB;n�

�T�vA,vB;vi,v j�dvAdvB. �8�

The f �M��v ;n+1� depends on f �M+1��v ;n�; Eq. �8� representsa hierarchy of equations similar to the BBGKY hierarchy.15

The first equation in the hierarchy is

f �1��v;n + 1� = �1 − 2/N�f �1��v;n� +2

N� f �2��vA,vB;n�

�T�vA,vB;vC,v� dvAdvBdvC. �9�

If we make the assumption of molecular chaos,

f �2��vA,vB;n� = f �1��vA;n�f �1��vB;n� , �10�

we obtain a nonlinear equation for f �1��v ;n� similar to theBoltzmann equation. For large N this approximation is al-most exact as shown by the following argument. The veloci-ties v1 ,v2 can be correlated only if particles one and twohave collided with each recently. But this probability is oforder 1 /N, which implies that for large N, the velocity dis-tributions of any two particles are uncorrelated. Present per-sonal computers can handle N=105–106, so the assumptionis almost exact.

The key assumption in the argument for the validity of Eq.�10� is “recently.” Two particles might be correlated for ashort time, but after they have made a few collisions withother particles the correlations are expected to disappear.

Another simplification occurs for large N. The factor of2 /N in Eq. �9� is small, and thus we can consider �=2n /N tobe a continuous parameter, which we call the collision time.Then ��=2/N and �f �1��v ;n+1�− f �1��v ;n�� /�� can be writ-ten as �f �1��v ;�� /�� and Eq. �9� becomes

�f �1��v;����

= − f �1��v;�� +� f �1��vA;��f �1��vB;��

�T�vA,vB;vC,v� dvAdvBdvC. �11�

From now on we will suppress the superscript �1� and thecollision time � in f �1��v ;��. Equation �11� can be expressedas

�f�v���

= − f�v� +� f�vA�f�vB�T�vA,vB;vC,v� dvAdvBdvC.

�12�

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A. The H-theorem and approach to equilibrium

By using the relation

f�v� =� f�v�f�vC�T�vA,vB;vC,v� dvAdvBdvC, �13�

which follows from Eq. �3� and the normalization of f�vC�,we can write Eq. �12� as

�f�v���

=� �f�vA�f�vB�

− f�v�f�vC��T�vA,vB;vC,v� dvAdvBdvC. �14�

This form is similar to the Boltzmann equation.We can derive an H-theorem for this equation. We define

H��� as

H��� =� f�v�ln�f�v�� dv , �15�

and use Eqs. �2� and �14� to express dH /d� as

dH

d�= −

1

4� ��f�T�vA,vB;vC,v� dvAdvBdvCdv , �16�

where

��f� = �f�vA�f�vB� − f�v�f�vC���ln f�vA�f�vB�

− ln f�v�f�vC�� . �17�

The function ��f� can be shown to be always nonnegative.We argue that �x−y��ln x−ln y� is non-negative for all posi-tive x and y; ln x is an increasing function and thus x−yand ln x−ln y always have the same sign. Their product isalways either positive or zero and zero occurs for x=y.T�vA ,vB ;vC ,v� is intrinsically positive. Therefore the inte-grand is positive and dH /d� is negative.

Following the usual arguments of the H-theorem, the de-crease of H stops only when

ln f�vA� + ln f�vB� = ln f�v� + ln f�vC� �18�

is satisfied, which implies that ln f�v� is a collision invari-ant. If we choose T�vA ,vB ;vC ,v� such that the total mo-mentum and energy is conserved in each collision, thenln f�v� must be expressible as a linear combination ofthese collision invariants as

ln f�v� =m

2�v − v0�2 + const, �19�

where is the temperature in energy units �kB=1� and m isthe mass of a molecule. Here v0 is the velocity of the centerof mass of the system. Hence we have shown that the systemapproaches the Maxwell-Boltzmann distribution.

B. Structure of T„vA ,vB ;vCv…, and connection with theBoltzmann equation

We define new variables

vT = �vA + vB�/2, u = vA − vB, u = u , �20a�

vT� = �v + vC�/2, u� = vC − v, u� = u� , �20b�

where vT and vT� are the center of mass velocities before and

after the collision. The Jacobian of the transformation is

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unity and integrations can be written in terms of the newvariables. Momentum conservation is imposed onT�vA ,vB ;vC ,v� as

T�vA,vB;vC,v� = �3�vT − vT��G�u,u�� . �21�

The integral in Eq. �12� is then written as

I =� f�vA�f�vB�T�vA,vB;vC,v� dvAdvBdvC �22a�

=� fv +u� + u

2� fv −

u − u�

2�G�u,u�� dudu�. �22b�

The conditions in Eqs. �2� and �3� become for G�u ,u��

� G�u,u��du =� G�u,u��du� = 1, �23�

G�u,u�� = G�u�,u� . �24�

Energy conservation requires that u=u�. If we define unitvectors u=u /u and n=u� /u and the angle � between them as�cos �= u · n�, we can write G�u ,u�� as

G�u,u�� =��u� − u�

u2 g��,u� . �25�

The condition in Eq. �23� becomes, for g�� ,u�,

� g��,u� dn = 1, �26�

where g�� ,u�dn is the probability of scattering into the solidangle dn in the center of mass frame. Then the integral I inEq. �22a� becomes

I =� fv +u

2+ un/2� fv −

u

2+ un/2�g��,u� dudn .

�27�

If we write the f�v� term in Eq. �12� as

f�v� =� f�v�f�v − u�g��,u� dudn , �28�

which follows from Eq. �26� and the normalization of f�v�,we can write Eq. �12� as

�f�v���

=� �f�vA�f�vB� − f�v�f�v − u��g��,u� dudn , �29�

where

vA = v +u

2+ un/2, �30a�

vB = v −u

2+ un/2. �30b�

Equation �29� is almost in the form of the Boltzmann equa-tion.

The Boltzmann equation represents a dilute gas for whichthe collision probability is proportional to u�T�u�, where�T�u� is the total cross section. We consider a large enough

number R such that the ratio u�T�u� /R for a selected pair is

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almost always less than unity. For �T�u�=�0 the constantR /�0 can be chosen to be a few �say five� times the rmsvelocity. Then when a pair is selected, we take a randomnumber r and allow the collision to occur if r u�T�u� /R;we select another pair if r�u�T�u� /R. Although this proce-dure insures that the collision probability is proportional tou�T�u�, it appears to violate the condition in Eq. �26� that allthe selected pairs have a collision. To satisfy the condition inEq. �26� we select g�� ,u� as

g��,u� =u���,u�

R+ 1 −

u�T�u�R

���u − n� , �31�

where ��� ,u� is the differential cross section. The latter isrelated to the total cross section �T�u� by

�T�u� =� ���,u�dn = 2�� ���,u� sin��� d� . �32�

The second term in Eq. �31� transfers the initial velocities tothe final velocities with the probability 1−u�T�u� /R and thecollision becomes a null collision. The ��u− n� function re-quires u= n, which implies u�=u because u�=u from energyconservation and u�=u�n and u=uu. Hence, vA−vB=vC−v.We also have vA+vB=vC+v from center of mass velocityconservation. These two equations yield vC=vA and v=vBand the velocities have not changed. A normal collision oc-curs with probability u�T�u� /R. It is easy to verify thatg�� ,u� given in Eq. �31� satisfies the condition in Eq. �26�.

If we substitute g�� ,u� in Eq. �31� into Eq. �29�, we obtain

�f�v����/R�

=� �f�vA�f�vB� − f�v�f�v − u��u���,u� dudn ,

�33�

where vA and vB were given in Eq. �30�. Equation �33� isessentially the Boltzmann equation with the difference thatf�v� is the probability density in velocity space whereas theBoltzmann equation is written in terms of the probabilitydensity in both physical and velocity space. If the volume ofthe cell containing the molecules is V, then we can write Eq.�33� for F�v�= �N /V�f�v� as

�F�v��t

=� �F�vA�F�vB� − F�v�F�v − u��u���,u� dudn ,

�34�

where t=�V /RN=2nV /RN2 is interpreted as the physicaltime. Equation �34� is the Boltzmann equation for a homo-geneous gas.

III. DISCUSSION

Let us summarize the direct simulation Monte Carlo algo-rithm for solving the Boltzmann equation. We choose a suf-ficiently large R such that only a negligible fraction of theselected pairs �say less than one in a thousand� violate thecondition u�T�u� /R�1. Then we select pairs randomly andlet them collide with probability u�T�u� /R. The latter isachieved by generating a random number r and letting thecollision occur if r�u�T�u� /R. If a pair collides, then in thecenter of mass system the collision occurs within the solid

ˆ ˆ ˆ

angle dn with probability P���dn= ���� ,u� /�T�u��dn. Sup-

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pose that we put the z-axis along u and we need to determinen=u� /u, which is determined by the angles � and �. Todetermine � we need to generate a random value of � byconverting the random numbers produced by a uniform prob-ability distribution to random numbers in the interval �0,��according to the probability distribution P���. The � anglesin the interval �0,2�� are equally likely. In this way wedetermine the final velocities of the particles as u� and −u� inthe center of mass frame. By adding the center of mass ve-locity we find the final velocities in the lab frame. Afterstoring the final velocities of the particles, we choose anotherpair and repeat the same process. The physical time is t=2nV /N2R, where n is the number of pairs chosen to makeattempts for a collision. If the number of collisions in a giventime is required, we can count the successful attempts for acollision. In Ref. 3 this algorithm for keeping track of thetime is called the “no time counter method.”

The original method of Bird3 to keep track of the time wasthe time counter method. Consider a narrow interval ofu�T�u� values. For a large n there will be �n pairs withu�T�u� values in this interval. Of these, only �u�T�u� /R��nof them will make collisions corresponding to a time interval�2V /N2R��n. Thus the elapsed time per successful attempt is

�t =�2V/N2R��n

�u�T�u�/R��n=

2V

N2u�T�u�. �35�

In the time counter method we let every pair collide, increasetime by �t �t→ t+�t� after each collision, and keep selectingpairs and colliding them until we reach the desired time.Every collision will cause a different time increment depend-ing on the value of u�T�u�. One disadvantage of this methodis that if a collision with a low u�T�u� occurs, the time in-crement will be large. Such collisions can occur with pairshaving almost equal velocities. The time counter method wasdeclared “obsolete” in Ref. 3. It is useful to be aware of themethod because it was widely used.

If the purpose of the simulation is to demonstrate that thevelocity distribution approaches the Maxwell-Boltzmann dis-tribution, we could let all the selected pairs make a collisionand the velocity distribution will converge to the Maxwell-Boltzmann distribution. This simplification corresponds tou�T�u� /R=1 or �T�u�=R /u, where the total cross section isinversely proportional to the relative velocity. Also, if it isdesired to not discuss cross sections and the time trackingmethod, it is convenient to assume isotropic scattering in thecenter of mass frame. Then u� can be calculated by taking arandom unit vector n and multiplying it by u. These twosimplifications make the programming easier and an under-graduate student with some programming background canwrite a program demonstrating the Maxwell-Boltzmann dis-tribution.

Direct simulation methods are also applicable to radiativeprocesses and chemical reactions and the present formalismgeneralizes to all these cases in a more or less straightfor-ward fashion for homogeneous gases. Such generalizationscan be a useful teaching tool and a fertile field for studentprojects.

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3G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of GasFlows �Clarendon, Oxford, 1994�.

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Chap. 3.

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