We bring attention to the fact that Maxwell's mean free path for a dilute hard-sphere gas in thermal equilibrium, , 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.
REFERENCES
1.
See, for example,
F.
Reif
, Fundamentals of Statistical and Thermal Physics
(Waveland Press
, Long Grove, IL
, 2008
), pp. 463
–471
.2.
S.
Chapman
and T. G.
Cowling
, The Mathematical Theory of Non-Uniform Gases
(Cambridge U. P.
, New York
, 1970
), pp. 58
–96
, 297–300.3.
The 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 .
4.
P.
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 generalizations to hard spheres in three dimensions.5.
Explicitly, , where .
6.
T.
Einwohner
and B. J.
Alder
, “Molecular dynamics. VI. Free-path distributions and collision rates for hard-sphere and square-well molecules
,” J. Chem. Phys.
49
, 1458
–1473
(1968
).7.
J. H.
Jeans
, The Dynamical Theory of Gases
(Cambridge U. P.
, London
, 1904
), pp. 229
–241
.8.
In 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 definition yields the same value as Maxwell's definition of the mean free path.9.
P.
Visco
, F.
van Wijland
, and E.
Trizac
, “Non-Poissonian statistics in a low-density fluid
,” J. Phys. Chem. B
112
, 5412
–5415
(2008
).10.
P.
Visco
, F.
van Wijland
, and E.
Trizac
, “Collisional statistics of the hard-sphere gas
,” Phys. Rev. E
77
, 041117
(2008
).11.
L.
Lue
, “Collision statistics, thermodynamics, and transport coefficients of hard hyperspheres in three, four, and five dimensions
,” J. Chem. Phys.
122
, 044513
(2005
).12.
This is a proper probability, not a density. Note that . This formula 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 N steps is . In the limit that N → ∞, becomes exponential.
13.
A 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 frequency than slower molecules. That relative frequency is the ratio of their speeds.
14.
This is an approximation based on the observation that the number of overlapping 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 χ.
15.
It is unnecessary to include contributions to the volume from the hemispherical endcaps. This is accounted for when we reduce the amount of inhabitable volume of molecule 1 due to the volumes of the other molecules.
16.
Actually, the correct expression is , where g(r) is the radial distribution function for the hard-sphere gas. Expanding in the density, g(r) = 1 + O(n), so the lowest order term, 1, is sufficient to calculate χ to O(n) accuracy.
17.
In Ref. 1, a remark given on p. 470 summarizes a rigorous traditional derivation 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.
18.
B. J.
Alder
and T.
Einwohner
, “Free-path distribution for hard spheres
,” J. Chem. Phys.
43
, 3399
–3400
(1965
).19.
For very large x the main contribution to the integral comes from an environment of v wherein the function , which multiplies the factor x in the exponent of the integrand, goes through a maximum; this happens as v →∞. Therefore, the dominant asymptotic behavior of the integral is of the form , where .
20.
In Ref. 6, the authors actually formulate the scaled free-path-length distribution without the additional factor of r(v)/ω, because they incorrectly assumed that the probability to find a molecule with a given speed in a collision was given by the Maxwell speed distribution. In our notation they found . 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 discrepancy is noticeable in Fig. 1 of Ref. 18. Note that the correct scaling function predicts F(0) ≈ 1.0921, which matches the data better at very small free path length than is reported in that letter.
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