Is the mean free path the mean of a distribution?Steve T. Paik Citation: American Journal of Physics 82, 602 (2014); doi: 10.1119/1.4869185 View online: http://dx.doi.org/10.1119/1.4869185 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/82/6?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Analytic determination of the mean free path of sequential reactions Am. J. Phys. 80, 316 (2012); 10.1119/1.3681015 Mean Free Paths, Ion Drift Velocities, and the Poisson Distribution Am. J. Phys. 39, 534 (1971); 10.1119/1.1986207 Remarks on the Mean Free Path Problem J. Acoust. Soc. Am. 36, 556 (1964); 10.1121/1.1919002 Reciprocal of the Mean Free Path J. Acoust. Soc. Am. 36, 551 (1964); 10.1121/1.1919000 On the Relation of Two Mean Free Paths Am. J. Phys. 5, 260 (1937); 10.1119/1.1991234

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Is the mean free path the mean of a distribution?

Steve T. Paika)

Physical Science Department, Santa Monica College, Santa Monica, California 90405

(Received 6 September 2013; accepted 7 March 2014)

We bring attention to the fact that Maxwell’s mean free path for a dilute hard-sphere gas in thermal

equilibrium, ðffiffiffi2p

rn�1, which is ordinarily obtained by multiplying the average speed by the

average time between collisions, is also the statistical mean of the distribution of free path lengths

in such a gas. VC 2014 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4869185]

I. INTRODUCTION

For a gas composed of rigid spheres (“molecules”), eachmolecule affects another’s motion only during collisions,ensuring that each molecule travels in a straight line at con-stant speed between collisions—this is a free path. The meanfree path, as the name suggests, should then be the averagelength of a great many free paths, made by either an ensem-ble of molecules or a single molecule followed for a longtime, it being a basic assumption of statistical mechanics thatthese two types of averages are equivalent. However, themean free path is ordinarily defined in textbooks as the ratioof the average speed to the average frequency of collisions.Are these two different ways of defining the mean free pathequivalent? In this article, we answer this question in the af-firmative for a hard-sphere gas and, along the way, discussaspects of classical kinetic theory that may be of interest inan upper-level undergraduate course on thermal physics.

Consider a gas of hard spherical particles of mass m andradius a that have attained thermal equilibrium and spatialuniformity. The mean free path quoted in textbooks isk ¼ ð

ffiffiffi2p

rn�1, where r is the total scattering cross section

and n is the average number density throughout the gas.1 Inthree dimensions, r¼p(2a)2. This expression for the meanfree path is valid only in the limit where the gas is dilute, butnot so dilute that the mean free path becomes comparable tothe size of the container. In this article, we will discuss amore general form of the mean free path: k ¼ ½

ffiffiffi2p

rnvðnÞ��1,

which can be applied even when the gas is not dilute. Here,v(n) can be regarded as a finite-density correction that startsfrom a value of 1 in the extreme dilute limit and increasesmonotonically with density.2 When the density of spheresapproaches the value obtained by packing the spheres asclosely as possible, v�1 vanishes so that the mean free pathis zero.3 The factor v has been referred to in the literature asEnskog’s v, due to its appearance in the Enskog kinetictheory of transport.

The expression with v¼ 1 can be attributed to Maxwell,who defined the mean free path kMaxwell as the ratio of themean speed hvi, evaluated in the equilibrium velocity distribu-tion f(v), to the average collision frequency per molecule x,

kMaxwell �hvix: (1)

The average collision frequency is the reciprocal of the aver-age time between collisions, so Eq. (1) is simply the productof the average molecular speed and the average time of flightbetween collisions. Working in scaled units where massesare measured in units of m and velocities are measured inunits of (kT/m)1=2, the velocity probability density is

f ðvÞ ¼ ð2pÞ�3=2expð�v2=2Þ. Here, T is the temperature and

k is Boltzmann’s constant. Angle brackets denote an instruc-tion to take an average in the equilibrium velocity distribu-tion: h� � �i �

Ð� � �f ðvÞ dv.

The average collision frequency x can be obtainedthrough a standard flux argument wherein one counts thetotal number of collisions xDt received by a randomly cho-sen target molecule during an arbitrary time interval Dt. Letus label the target molecule “1”; there is a probability f(v1)dv1 to find it with velocity centered around some v1. Alongany particular direction through 1 there will be an incomingstream of scatterers all moving, roughly speaking, in the gen-eral direction toward 1. For a subset of these scatterers withvelocities centered around v2, only those in a cylindrical vol-ume jv1 � v2jrDt can reach the target in the allotted time.These scatterers do not necessarily have collinear flightpaths; some will have oblique paths that might cause them toexit through the side wall of the imaginary cylinder prior tomaking contact with 1. Therefore, one imagines making Dtsufficiently small to ensure that all molecules in the cylinderdo collide with 1. The number of scatterers in this cylinder isobtained by multiplying the volume by a density, yieldingjv1 � v2jrDt n f ðv2Þ dv2. The condition of molecular chaos isassumed: in the encounter between two spheres their initialvelocities are uncorrelated prior to the collision. Thisassumption allows one to express the collision probability interms of a product of equilibrium velocity distributions.Integrating over all possible velocities for the target and thescatterers yields the total number of collisions and dividingby Dt gives the average rate.

If the volume occupied by the molecules themselves is anappreciable fraction of the container volume, then it is incor-rect to assume that the local density in the collision cylinderis n. The inadequacy of n is explained by the observationthat, given a sphere, it is impossible to find another sphere atradial separation less than a molecular diameter, whereas forslightly larger separation there tends to be an increased like-lihood of finding a sphere relative to the case where anequivalent volume is examined at random in the gas. Whilethere is no attractive force between the spheres, this surplusin density is produced because two nearby spheres tend toreceive collisions on all sides except those sides facing eachother, resulting in an external pressure that tends to keep themolecules from separating. Such an effect leads to an effec-tive attraction between the spheres.4 Since the collision fre-quency is proportional to the number density in theimmediate neighborhood of a target molecule, we mustreplace n by nv, where v> 1 represents the ratio of the localdensity of molecular centers at the point of contact to the av-erage density throughout the container. In a uniform gas vdepends only on the reduced density n(2a)3.

602 Am. J. Phys. 82 (6), June 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 602

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The above discussion implies that

x ¼ rnvð ð

dv1 dv2 jv1 � v2jf ðv1Þf ðv2Þ: (2)

A change of variables from (v1, v2) to (v1, vrel� v1� v2), theabsolute value of the Jacobian being unity, along with theevenness of the velocity distribution, allows us to write

x ¼ rnvð

dvrel vrel

ðdv1 f ðv1Þf ðvrel � v1Þ: (3)

Since the inner integral defines the equilibrium probabilitydensity for the relative velocity, the outer integral simplycomputes the mean relative speed in the gas. Therefore,

kMaxwell �hvix¼ hvihvreli0

1

rnv; (4)

where h� � �i0 denotes the average in the relative velocity dis-tribution.5 A straightforward calculation shows that the ratio

hvi=hvreli0 ¼ 1=ffiffiffi2p

.Although we computed the average collision frequency x

using a time average, there is a clever argument given byEinwohner and Alder that shows how to formulate this cal-culation in terms of an ensemble average.6 Their derivationis described in the Appendix. The meaning of v is very clearin this approach: v is the radial distribution function for pairsof spheres evaluated just outside their point of contact.

The average collision frequency x is also a statistical mean,in the equilibrium velocity distribution, of the collision rater(v) that describes the probability per unit time that a givenmolecule with speed v encounters another molecule. Weassume, by virtue of the low density of the gas, that this colli-sion rate is independent of the past history of the molecule. Inother words, the probability that a given molecule suffers acollision between an arbitrary time t and tþ dt is r(v) dt, so

x � hrðvÞi ¼ð

dv f ðvÞrðvÞ (5)

and this exposes Maxwell’s mean free path as the ratio oftwo averages,

kMaxwell ¼hvihrðvÞi : (6)

It is not obvious that this ratio is the first moment of theprobability distribution for free path lengths. Historically, al-ternative definitions for the mean free path have been sug-gested. Tait’s definition takes the distance that a particle of apreselected speed travels from a given instant in time untilits next collision and averages over the equilibrium distribu-tion of velocities,7

kTait �v

rðvÞ

� �: (7)

Another definition, which is possible from dimensional con-siderations, is to first average the inverse collision rate overall velocities and multiply by the mean speed,

kother � hvi1

rðvÞ

� �: (8)

The explicit formula for r(v) is known and it is detailed inthe classic texts of the subject.2,7 Although this formula isnot needed to understand the central claim in this article, inSec. IV we present a self-contained derivation of this rate forhard spheres in three dimensions that serves as an alternativeto the traditional approach of scattering theory. A particu-larly lucid discussion of the probability rate r(v) from the tra-ditional viewpoint can be found in Ref. 1. Using the explicitformula for the speed-dependent collision rate, one finds thatdefinitions (7) and (8) differ from Maxwell’s by about 4%(Ref. 8) (see Table I).

Our primary goal is to emphasize a point that has not beenstressed in the literature, that Eq. (6) can be regarded as themean of the probability distribution for lengths of free pathsin a hard-sphere gas. The problem of constructing such a dis-tribution was thoroughly analyzed by Visco, van Wijland,and Trizac,9,10 building upon a key earlier observation byLue.11 In addition to studying the distribution analytically,Visco et al. used computer simulations to study a hard-diskgas in two dimensions and obtained excellent agreementbetween their analytical and numerical results. We shallreview their construction and see how it leads, almost trivi-ally, to the claim about Maxwell’s mean free path.

The remainder of this article is organized as follows. InSec. II, we construct the probability distribution for the freepath time and in Sec. III, we generalize that procedure to findthe distribution for free path length. In Sec. IV, we obtain thecollision rate for hard spheres and in Sec. V we use that rateto discuss explicit results for the hard-sphere gas.

II. PROBABILITY DISTRIBUTION FOR THE TIME

OF A FREE PATH

The first order of business is to find the probability P(t) dtthat any given molecule, after surviving a time t since its lastcollision, suffers a collision in the interval t to tþ dt. To thisend, select a collision at random from the gas and focusattention on one of the molecules involved in that collision.Following Refs. 9 and 10, we call this the “tagged”molecule.

Prior to the collision, the tagged molecule has maintainedsome constant velocity v for a time t. The conditional proba-bility that a molecule with velocity v has survived for at leasttime t is (Ref. 12) pðtjvÞ ¼ exp½�rðvÞt�. Multiplying this con-ditional probability by r(v)dt one obtains the probability thatthe molecule will suffer a collision during an infinitesimaltime interval succeeding time t,

pðtjvÞ rðvÞdt: (9)

It remains to characterize how likely different velocities vare for the tagged molecule. Since the tagged molecule isobtained by first identifying a collision, the probability distri-bution must account for the likelihood of finding a velocityin the range v to vþ dv in any randomly selected collision.

Table I. Various mean free paths.

Due to k(rnv)

Maxwell 0.7071

Tait 0.6775

Other 0.7340

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This probability, fcoll (v) dv, can be found as follows. At anygiven moment there is a fraction f(v1) dv1 of the gas that hasvelocity in the range v1 to v1þ dv1, and of that a fractionr(v1) dt will undergo a collision in the next short time inter-val dt. Therefore, the relative likelihood of the tagged mole-cule having some velocity centered around v1 as opposed to,say, some v2, is given by the ratio

fcollðv1Þ dv1

fcollðv2Þ dv2

¼ rðv1Þ dt f ðv1Þ dv1

rðv2Þ dt f ðv2Þ dv2

: (10)

This implies that the tagged molecule’s velocity is obtainedfrom the probability distribution,

fcollðvÞ dv � x�1rðvÞf ðvÞ dv; (11)

where the factor x�1 is required by normalization and thedefinition given by Eq. (5). Lue refers to Eq. (11) as the“on-collision” velocity distribution. Molecules with large-r-than-typical speeds collide more often than those withsmaller-than-typical speeds, so in any collision one is morelikely to find molecules with larger velocities than suggestedby the nominal equilibrium velocity distribution. The distri-bution is thus a product of a Gaussian and a weight thatenhances the relative probability of finding faster particles.13

This skewing of the nominal distribution can be seen explic-itly; in Sec. V it is shown that for a hard-sphere gas theon-collision speed distribution has a most probable valueslightly greater than that for the Maxwell speed distribution.

Sampling velocities from the equilibrium velocity distri-bution would be inconsistent with the requirement that thetime t represents the entire time of free flight of the taggedmolecule, since that entails choosing a particle at random atany moment in its flight rather than some moment a veryshort time preceding an encounter with another sphere. Inthe former case, the time t would exclude the remaining timeleft on the particle’s linear path to its next collision.

The desired probability distribution for the time t of a freepath is obtained by multiplying the probability to find a mol-ecule with velocity centered on v that has just undergone acollision at time zero, with the probability that it will survivefor some time t without receiving a collision, with the proba-bility that it will suffer its next collision between time tand tþ dt, and finally summing over all possible velocities.This amounts to multiplying expressions (11) and (9) andintegrating over v,

PðtÞdt ¼ð

fcollðvÞ dv pðtjvÞ rðvÞ dt: (12)

Visco et al. point out that this distribution is consistent withthe sensible requirement that the mean time between subse-quent collisions is equal to the reciprocal of the mean colli-sion rate defined in Eq. (5); that is,

Ð10

tPðtÞ dt ¼ x�1.

III. PROBABILITY DISTRIBUTION FOR THE

LENGTH OF A FREE PATH

By a similar construction, we can obtain the probabilitydistribution P(‘) d‘ for the length ‘ of a free path. The quan-tity P(‘) d‘ is formed from a product of three terms: theprobability to find a molecule with velocity centered on v ina collision selected at random at time zero, the probabilitythat it will survive without collision for at least a time equal

to t¼ ‘/v, and the subsequent probability to suffer a collisionbetween t¼ ‘/v and t¼ (‘þ d‘)/v. One must then sum overall possible velocities so that

Pð‘Þ d‘ ¼ð

fcollðvÞ dv pðt ¼ ‘=vjvÞ rðvÞ d‘v

) Pð‘Þ ¼ 1

x

ðdv

rðvÞ2

vf ðvÞe�rðvÞ‘=v: (13)

The expectation value of ‘ in this distribution is easily eval-uated by exchanging the order of integration: the ‘ integralevaluates to [v/r(v)]2 and the remaining integral defines themean speed in the Gaussian distribution,ð1

0

‘Pð‘Þd‘ ¼ hvix; (14)

identical to Eq. (6).

IV. DERIVATION OF THE COLLISION RATE FOR

HARD SPHERES

The hard-sphere gas is a particularly clean system becausecollisions are instantaneous and it is unnecessary to considersimultaneous collisions of three or more spheres. We con-sider a gas that has attained steady state and is spatially uni-form, for which the H-theorem implies that the velocitydistribution f(v) is independent of t and r; that is, f(v) is notchanged by molecular collisions. Let our gas consist of Nmolecules in a box of volume L3 at temperature T. Imaginethat a given molecule, labeled 1, has traveled some distance‘ between collisions. This molecule has been moving withconstant velocity v1 so that the time spent traveling in astraight line is t ¼ ‘=jv1j. During this time, all N� 1 othermolecules have passed by without making contact. Usingreasoning based on excluded volume, we will calculate theprobability that no other molecule hits 1 during this time. Inthis framework, it is also natural to include the lowest-orderfinite-density correction in n�N/L3, which can be found byfollowing the procedure outlined in the classic text byChapman and Cowling.2

At the moment when two molecules collide, the distancebetween their centers is 2a. Around each molecule we candraw a concentric “associated sphere” with radius 2a. A col-lision occurs when the center of a molecule lies on the asso-ciated sphere of another molecule. Moreover, that center cannever lie within another molecule’s associated sphere. Theassociated sphere construction is helpful in two respects: (i)it shows that the inhabitable volume for the center of mole-cule 1 is L3 less the total volume taken up by the N� 1 otherassociated spheres,14 and (ii) two molecules with overlap-ping associated spheres shield their common area in such away that no other molecule’s center can lie on that area (seeFig. 1).

Since we can only be certain that molecule 1 has main-tained a constant velocity during time t, it makes sense touse the rest frame of this molecule to evaluate the probabilitythat other molecules do not collide with it. It will be conven-ient to imagine that the center of molecule 1 occupies a sin-gle point, say, the origin of our box. During time t, theassociated sphere of molecule 2 sweeps out a piecewise cy-lindrical volume equal to15 pð2aÞ2jv2 � v1jt. The probability

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that the origin is not in the swept-out volume of 2’s associ-ated sphere would appear to be

p2 ¼½L3 � ð4p=3Þð2aÞ3ðN � 1Þ� � rjv2 � v1jt

L3 � ð4p=3Þð2aÞ3ðN � 1Þ: (15)

However, due to the previously mentioned shielding effectthere is a reduced likelihood of molecule 2 colliding withmolecule 1 since, at any given moment, only a fraction ofthe surface area of molecule 2’s associated sphere is avail-able to make contact with molecule 1’s center. That fractionis explicitly computed in Ref. 2 and what follows is an expo-sition of their procedure. The probable number of molecularcenters within a range r to rþ dr of the center of any chosenmolecule is16 n4pr2dr. Note that for a spatially uniform gasthis number is independent of the positions of other mole-cules. Two overlapping associated spheres whose centers areseparated by a distance 2a< r< 4a obstruct, on either

sphere, a dome with lateral area 4pa(2a� r/2). Therefore,the probable area unavailable to a molecule is

ð4a

2a

n4pr2 dr � 4pað2a� r=2Þ ¼ 11

3p2ð2aÞ5n: (16)

The available fractional area of molecule 2’s associatedsphere that can collide with molecule 1’s center is thus

4pð2aÞ2 � ð11=3Þp2ð2aÞ5n

4pð2aÞ2¼ 1� 11

12pð2aÞ3n: (17)

Expression (17) must multiply the swept-out volume ofmolecule 2’s associated sphere. As a check, if expression(17) is zero then there is unit probability that the origin liesoutside the swept volume. The correct modification ofexpression (15) is therefore

p2 ¼L3 � ð4p=3Þð2aÞ3ðN � 1Þh i

� rjv2 � v1jt 1� ð11p=12Þð2aÞ3nh i

L3 � ð4p=3Þð2aÞ3ðN � 1Þ¼ 1� rnvjv2 � v1j‘=v1

N; (18)

where v1 ¼ jv1j and

v � 1� ð11p=12Þð2aÞ3n

1� ð4p=3Þð2aÞ3ðn� L�3Þ

¼ 1þ 5

12pð2aÞ3nþ Oðn2Þ þ OðL�3Þ: (19)

In a large-volume limit, the correction of O(L�3) is sublead-ing and may be neglected. Moreover, we have neglected cor-rections imposed by the boundary. We see that effect (i)tends to increase the probability of collisions whereas effect(ii) tends to decrease it. Higher-order corrections to v areknown.2

A similar argument can be made concerning the probabil-ity p3 that the swept volume of molecule 3’s associated

sphere does not contain the origin. We assume the probabil-ities p2, p3, etc. to be independent, essentially invoking themolecular chaos assumption. However, this does not pre-clude the N� 1 particles from interacting with each otherduring time t; their collisions will inevitably “kink” eachother’s cylinders. The probability that none of the N� 1associated spheres hits molecule 1’s center during this timeis given by the product of the individual probabilities,

pð‘jv1Þ ¼YNi¼2

pi ¼YNi¼2

1� rnvjvi � v1j=v1

N‘

� �: (20)

In this product, the values of vi are obtained from the equilib-rium velocity distribution. One way to estimate this productis to discretize the velocity distribution into a finite numberof bins and assign to each vi a particular value representativeof a bin. In the jth bin there will be Nf(vj) d3vj particles, andthus

pð‘jv1Þ ¼Y

bins j

�1� rnvjvj � v1j=v1

N‘

�Nf ðvjÞ d3vj

¼ expX

j

�rnv‘jvj � v1j

v1

f ðvjÞ d3vj

¼ exp �rnv‘v1

ðd3v f ðvÞjv� v1j

� �: (21)

In the second step, we took the limit of large N (and largevolume so as to keep the concentration fixed and small com-pared to 1) and in the third step the sum was generalized to acontinuous integral over all possible velocities. The collisionrate is found by comparing Eq. (21) with pðtjv1Þ¼ exp �rðv1Þ‘=v1½ � to obtain the well-known formula

Fig. 1. The “shielding” effect. If a third molecule is sufficiently close to mol-

ecule 2, then the surface area of molecule 2’s associated sphere (dashed)

that is accessible to molecule 1’s center is reduced by a dome (grayed). For

further discussion of this effect, see Ref. 4.

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rðv1Þ ¼ rnvð

d3v f ðvÞjv� v1j: (22)

As one might expect, the probability of collision is highwhen the cross section is large, the gas is dense, or if therelative speed is large.17 The integral can be calculated by

orienting the z-axis along v1 so that the magnitude of the rel-

ative velocity v� v1 can be computed from the law of

cosines asffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ v2

1 � 2vv1cos hp

, where h is the polar angle.

Performing the angular and radial integrations produces

rðv1Þ ¼ rnv

ffiffiffi2

p

r ð10

dv v2e�v2=2 � vþ v21=3v; v > v1

v1 þ v2=3v1; v < v1

(

¼ rnv

ffiffiffi2

p

rexp � v2

1

2

� �þ v1 þ v�1

1

erf

v1ffiffiffi2p� �" #

;

(23)

where erf(n) is the error function. The collision rate is finiteas v1! 0, monotonically increasing, and behaves asymptoti-cally as rðv1Þ=rnv � v1 þ Oð1=v1Þ. The fact that r(v1) scalesas v1 for large speed is completely expected, based on thefact that the majority of molecules have relatively lowspeeds compared to the very few molecules in the tail of theMaxwell distribution. For v1 much larger than the character-istic speed set by the temperature, it is as if all moleculesexcept for molecule 1 are frozen in place. Then it is easy tosee that the frequency of scattering events incurred by mole-cule 1 is directly proportional to its speed. Notice that theterm in Eq. (23) that dominates the asymptotic behavior doesindeed come from the v1> v domain of integration. More tothe point, if we approximate the relative speed jv� v1j byjust v1 in Eq. (22), then the integral evaluates to v1.

V. EXPLICIT RESULTS FOR THE HARD-SPHERE

GAS

The collision rate found in Eq. (23) can be averaged in theequilibrium velocity distribution to obtain the mean collisionrate

x ¼ hrðvÞi ¼ 4ffiffiffipp rnv: (24)

We can compare predictions for this rate with the numericalresults of molecular dynamics simulations of a hard-spheregas over a wide range of density. In Fig. 2, we plot Eq. (24)against data from Ref. 11. As expected, Eq. (24) deviatesfrom the actual collision frequency as the density of the gasincreases. The discrepancy is less than 10% for densitiesbelow 0.3(2a)�3.

The on-collision distribution in terms of speed,fcollðvÞ4pv2dv, is evidently independent of themolecular size and density of the gas. The mean speedis ð3þ pÞ=ð2

ffiffiffippÞ � 1:7325 and the most probable speed

is found numerically to be 1.5769. These may be con-trasted with the corresponding values

ffiffiffiffiffiffiffiffi8=p

p� 1:5958

andffiffiffi2p� 1:4142, respectively, for the Maxwell speed

distribution, which are lower. The on-collision speed dis-tribution results from multiplying the Maxwell speed dis-tribution by the monotonically increasing function r(v)/x,which skews the distribution toward larger v. These twodistributions are plotted in Fig. 3.

The free-path-length distribution given in Eq. (13), whenappropriately normalized, can be expressed in terms of ascaling function, that is,

kMaxwellPð‘=kMaxwellÞ ¼ Fð‘=kMaxwellÞ; (25)

where

FðxÞ � 1

p

ð10

dv

vwðvÞ2exp �v2 � 1ffiffiffiffiffiffi

2pp v�2wðvÞx

� �(26)

with

wðvÞ � ve�v2 þ ð2v2 þ 1Þðv

0

e�t2 dt: (27)

The scaling function F does not depend on any dimension-less parameters other than its argument, which is the ratio ofthe free path length and the mean free path. In particular, Fis independent of the reduced density n(2a)3. This striking

Fig. 2. Average collision frequency in the hard-sphere gas versus density.

The frequency is given in units of ðkT=mÞ1=2ð2aÞ�1and the density is meas-

ured in units of (2a)�3. The solid line is the predicted rate x from Eq. (24)

using v ¼ 1þ ð5p=12Þð2aÞ3n, and the dashed line is the predicted rate with

v¼ 1. The dots are the reciprocal of the average time between collisions

taken from Table I of Ref. 11. The density range shown is between 0, the

extreme dilute limit, and 1, which is somewhat close to the limit of close

packing of the spheres within the container. The fractional error between the

predicted and simulated collision frequency grows with increasing density.

The fractional error is less than 10% for nð2aÞ3 < 0:3.

Fig. 3. Different types of probability density functions for speeds. Speed is

measured in units of (kT/m)1=2 and each density function has the inverse

unit. The dashed curve represents the Maxwellian probability density

4pv2f ðvÞ for randomly picking a molecule with speed v from the gas. The

solid curve represents the probability density 4pv2fcollðvÞ for finding a mole-

cule with incoming speed v when one member of a colliding pair is exam-

ined at random. This figure appears in Ref. 11.

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density independence of the scaled free-path-length distribu-tion was noticed in early molecular dynamics simulations offluids.18

A plot of ln F is shown in Fig. 4 contrasted with aplot of �x in order to show the deviations from pure ex-ponential behavior. The special result F(x)¼ e�x wouldresult only if all molecules in the gas were stationaryexcept for the one molecule executing a free path. Thissituation would correspond to a collision rate rðvÞ / v forwhich the computation of Eq. (13) can be done explicitlyand yields a simple exponential. To evaluate thelong-distance behavior of F(x) given by Eq. (26) one canuse the method of steepest descent;19 it turns out thatFðxÞ � e�x=

ffiffi2p

. Further discussion of the distribution canbe found in Refs. 9 and 10.

In Fig. 5, we plot the scaling function F and dataobtained from Einwohner and Alder’s early molecular

dynamics simulation of the hard-sphere fluid at three dif-ferent densities. Even for very dense fluids the agreementis excellent.20

ACKNOWLEDGMENTS

The authors gratefully acknowledge helpful discussionswith Jacob Morris and correspondence from Fr�ed�eric vanWijland. The authors thank Laurence Yaffe and both ref-erees of this journal for their many comments and ques-tions that have significantly improved and clarified thiswork.

APPENDIX: THE MEANING OF v

In Appendix A of Ref. 6, Einwohner and Alder prove thatthe mean collision rate per molecule x(n) in a hard-spherefluid with number density n is xðnÞ ¼ xðn! 0ÞvðnÞ. Here,xðn! 0Þ ¼ ð4=

ffiffiffippÞrn

ffiffiffiffiffiffiffiffiffiffiffikT=m

pis the collision frequency

in the limit of infinite dilution and v(n) is directlyrelated to the probability that, when given a sphere,there will another one located at a center-to-center dis-tance of 2a. Here, we discuss the salient points of theirproof.

Suppose that the N spheres experience a total of M colli-sions during a time t0 that is much longer than the averagetime between collisions. The total collision rate can be writ-ten as a time average

Nx ¼ M

t0

¼ 1

t0

ðt0

0

dtXM

k¼1

dðt� tkÞ; (A1)

where {tk} are the times at which the collisions occur.Imagine that one knows the trajectories ri(t) of all N spheresparametrized by the time t. Then the sum of delta functionsin time can be reexpressed as a sum of delta functionsinvolving the trajectories, giving

XM

k¼1

dðt� tkÞ

¼XN

i¼2

Xi�1

j¼1

dðrijðtÞ2 � 4a2Þ hð�rij � vijÞj2rij � vijj:

(A2)

We use the notation rij� ri� rj and a similar one forthe relative velocity. The expression in terms of theparticles’ phase space coordinates is more complicatedbecause a collision requires that two conditions be met:that the sphere centers are one diameter apart and thatthe relative velocity of the spheres must make anobtuse angle with the separation vector, the vectorpointing from the center of sphere j to the center ofsphere i. The latter condition is necessary to pick outonly those events in phase space where spheres are justabout to collide and to avoid those events where spheresdepart or pass parallel without touching; it has been incorpo-rated using the unit step function h, which equals 1 forpositive argument and 0 otherwise. Note that a Jacobianfactor involving the relative velocity is required by thechange of variables. The equality of time and ensemble aver-ages is now invoked:

Fig. 4. Logarithm of the scaling function for the free-path-length density.

The solid curve is ln FðxÞ. In the text it is shown that the distribution

of free path lengths Pð‘Þd‘, when parametrized in units of the mean

free path as x¼ ‘/kMaxwell, is given by F(x)dx. For contrast the dashed

curve is a straight line with negative unit slope. The disagreement

between ln FðxÞ and �x shows that the free-path-length density is not a

simple decaying exponential. Its behavior simplifies somewhat for

asymptotically large x: the dominant term is exponential and of the form

ln FðxÞ � �x=ffiffiffi2p

, although this slope is only apparent for x�O(100)

and higher.9,10

Fig. 5. Scaling for the distribution of free path lengths. The solid

curve is F(x). In the text it is shown that the distribution of free path

lengths P(‘)d‘, when parametrized in units of the mean free path as

x¼ ‘/kMaxwell, is given by F(x)dx. The data are obtained from Table

VIII of Ref. 6. The numerical simulations were carried out at three

densities: nð2aÞ3 � 0:47 (triangles), 0.71 (crosses), and 0.88 (circles).

This figure appears in Ref. 18 but with a subtle difference.20 The

inset shows the difference between the theoretical curve and simulated

data.

607 Am. J. Phys., Vol. 82, No. 6, June 2014 Steve T. Paik 607

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1

t0

ðt0

0

OðtÞdt ¼

ðe�H=kTOðfriðtÞg; fviðtÞgÞ dr1 � � � drN dv1 � � � dvNð

e�H=kTdr1 � � � drN dv1 � � � dvN

; (A3)

where H ¼ ðm=2ÞPN

i¼1 v2i þ Uðr1;…; rNÞ is the classical energy of the system. The potential energy is a sum of two-body

terms that depends only on the separations between particles: U ¼P

i<j /ðjrijjÞ, where / is infinite if jrijj is strictly less than2a and zero otherwise. Since the Boltzmann factor factorizes into a product of kinetic and potential terms, the velocity inte-grals can be done exactly. The delta function then collapses two of the position integrals so that

x ¼ 4ffiffiffipp

ffiffiffiffiffiffikT

m

rpð2aÞ2 N � 1

V

V2

ðdr3 � � � drN e�Uðjr12j¼2a;jr13j;…Þ=kTðdr1 � � � drN e�Uðjr12j;jr13j;…Þ=kT|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

v

: (A4)

For a uniform fluid v is a function of the reduced densityn(2a)3. If we assume that the molecular chaos approximationholds even when the gas is dense, so that the density of mo-lecular centers in the immediate neighborhood of a moleculeis not correlated to the velocity of that molecule, then wewould also expect that the collision rate r(v) experienced byany molecule moving with speed v is directly proportional tothe same factor v derived above (see the discussion inRef. 2); in particular, v does not depend on v.

a)Electronic mail: paik_steve@smc.edu1See, for example, F. Reif, Fundamentals of Statistical and ThermalPhysics (Waveland Press, Long Grove, IL, 2008), pp. 463–471.

2S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases (Cambridge U. P., New York, 1970), pp. 58–96, 297–300.

3The densest packing of spheres results in a periodic array of spheres that

occupy approximately 74% of the total available volume. This corresponds

to a number density of nð2aÞ3 ¼ffiffiffi2p

.4P. Cutchis, H. van Beijeren, J. R. Dorfman, and E. A. Mason, “Enskog and

van der Waals play hockey,” Am. J. Phys. 45, 970–977 (1977). These

authors focus on a hard-disk fluid in two dimensions but mention the gen-

eralizations to hard spheres in three dimensions.5Explicitly, h� � �i0 �

Ð� � �gðvrelÞ dvrel, where gðvrelÞ ¼ ð4pÞ�3=2

expð�v2rel=4Þ.

6T. Einwohner and B. J. Alder, “Molecular dynamics. VI. Free-path distri-

butions and collision rates for hard-sphere and square-well molecules,”

J. Chem. Phys. 49, 1458–1473 (1968).7J. H. Jeans, The Dynamical Theory of Gases (Cambridge U. P., London,

1904), pp. 229–241.8In passing we remark that if the collision rate were independent of speed

then all three definitions of the mean free path would give identical values.

For instance, in Morton Masius, “On the Relation of Two Mean Free Paths,”

Am. J. Phys. 5, 260–262 (1937), it is explicitly demonstrated how Tait’s defi-

nition yields the same value as Maxwell’s definition of the mean free path.9P. Visco, F. van Wijland, and E. Trizac, “Non-Poissonian statistics in a

low-density fluid,” J. Phys. Chem. B 112, 5412–5415 (2008).10P. Visco, F. van Wijland, and E. Trizac, “Collisional statistics of the hard-

sphere gas,” Phys. Rev. E 77, 041117 (2008).11L. Lue, “Collision statistics, thermodynamics, and transport coefficients of

hard hyperspheres in three, four, and five dimensions,” J. Chem. Phys.

122, 044513 (2005).12This is a proper probability, not a density. Note that pð0jvÞ ¼ 1. This for-

mula can be obtained by imagining that the finite time t is discretized into

N equal intervals dt. Then the probability to not undergo a collision in Nsteps is pðtjvÞ ¼ ½1� rðvÞdt�N ¼ ½1� rðvÞt=N�N . In the limit that N! 1,

pðtjvÞ becomes exponential.

13A similar reasoning is used to obtain the molecular speed distribution of a

hot gas effusing through a small hole in an oven. Recall that the Maxwell

speed distribution is weighted by an additional factor of the speed,

reflecting the fact that faster molecules bombard the hole with greater fre-

quency than slower molecules. That relative frequency is the ratio of their

speeds.14This is an approximation based on the observation that the number of over-

lapping associated spheres in a dilute gas is small compared to the total

number of spheres. In order to correct for the possibility of overlap we

note that the probable number of molecules close enough to any given

molecule for their associated spheres to overlap is proportional to n.

Therefore, the probable excluded volume on any molecule scales as n, and

there are order N molecules. Thus, the overlap correction to the naive

inhabitable volume estimate appears at O(n2) in the factor v.15It is unnecessary to include contributions to the volume from the hemi-

spherical endcaps. This is accounted for when we reduce the amount of

inhabitable volume of molecule 1 due to the volumes of the other

molecules.16Actually, the correct expression is ngðrÞ4pr2 dr, where g(r) is the radial

distribution function for the hard-sphere gas. Expanding in the density,

g(r)¼ 1þO(n), so the lowest order term, 1, is sufficient to calculate v to

O(n) accuracy.17In Ref. 1, a remark given on p. 470 summarizes a rigorous traditional deri-

vation of the collision rate r(v1) that is more general than the one given in

this article. Reif’s expression involves an additional integration over all

solid angle of a differential scattering cross section that can be computed

for an arbitrary interaction potential between molecules.18B. J. Alder and T. Einwohner, “Free-path distribution for hard spheres,”

J. Chem. Phys. 43, 3399–3400 (1965).19For very large x the main contribution to the integral comes from an envi-

ronment of v wherein the function �ð2pÞ�1=2v�2wðvÞ, which multiplies

the factor x in the exponent of the integrand, goes through a maximum;

this happens as v !1. Therefore, the dominant asymptotic behavior of

the integral is of the form expðbxÞ, where b ¼ limv!1 �ð2pÞ�1=2v�2wðvÞ¼ �1=

ffiffiffi2p

.20In Ref. 6, the authors actually formulate the scaled free-path-length distri-

bution without the additional factor of r(v)/x, because they incorrectly

assumed that the probability to find a molecule with a given speed in a col-

lision was given by the Maxwell speed distribution. In our notation they

found FincorrectðxÞ ¼ ð2ffiffiffi2p

=pÞÐ1

0dvwðvÞexp½�v2 � ð1=

ffiffiffiffiffiffi2ppÞv�2wðvÞx�.

Somewhat fortuitously, the error incurred by using the wrong distribution

makes a very small adjustment to the shape of the scaling function. The

absolute value of the difference between F and Fincorrect is no greater than

about 0.07 and this occurs for small values of the argument. This discrep-

ancy is noticeable in Fig. 1 of Ref. 18. Note that the correct scaling func-

tion predicts F(0)� 1.0921, which matches the data better at very small

free path length than is reported in that letter.

608 Am. J. Phys., Vol. 82, No. 6, June 2014 Steve T. Paik 608

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