Content uploaded by Robert Shour
Author content
All content in this area was uploaded by Robert Shour on Oct 13, 2018
Content may be subject to copyright.
Clausius’s 1860 article on mean path2
lengths3
that appears to anticipate dark energy4
Robert Shour5
13th October, 20186
In 1860 Clausius wrote a short article, now obscure and not easy to find,7
that appears to predict dark energy. This article includes a PDF of his 18608
article so that readers can judge Clausius’s article for themselves.9
Keywords dark energy, dimension, the 4/3 law, scaling.10
1 Introduction11
First version: October 29, 2017; second: March 16, 2018; third: October 2018, sec. 1.412
4/3 for 3/4.13
1.1 Clausius’s 1860 article14
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science in15
Vol. XIX, Fourth Series covering the months of January through June 1860 at page 43416
published a letter from Professor Clausius On the Dynamical Theory of Gases (Clausius,17
1860). The primary purpose of this article is to reproduce it after the conclusion of this18
introduction.19
One reason for making it more easily accessible is that it is not well known and not20
easy to locate. It is not included in the selected readings on Kinetic Theory compiled by21
Stephen Brush (Brush, 1948). Likely it can only be found in the Philosophical Magazine22
itself, and finding the right volume is not easy. On the Internet Archive, for example,23
searching The Philosophical Magazine 1860 at October 2017 gives Vol XX, July to24
December 1860, not the January to June volume containing Clausius’s article.25
Another reason is that Clausius gives a proof that is one way of demonstrating the26
4/3 law, a law that is universal and scale-invariant and connected to metabolic scaling,27
the fractal envelope of Brownian motion and to dark energy, though those connections28
do not seem to have been generally noticed.29
Yet another reason is that the version of the 4/3 law that Clausius derives mathe-30
matically involves lengths: a four dimensional system of nparticles in motion in three31
1
dimensional space has a mean path length that is 3/4 of the distance between the cor-32
responding equal sub-volumes of the same space absent motion. Or, in other words,33
distances between centers of the sub-volumes of the counterpart static three dimensional34
system are 4/3 as long as the mean path length for the four dimensional system which35
includes one dimensional motion.36
This last point makes Clausius’s 1860 letter of some especial significance in 201737
because in that version the lengths are 4/3 longer in the three dimensional system38
compared to the four dimensional system, an implicit prediction of the dark energy39
discovered in 1998.40
This article reproduces in addition to Clausius’s 1860 article excerpts from the first41
article in which Clausius mentioned the 3/4 relationship , Maxwell’s different calculation,42
and an editorial note from Maxwell’s collected works commenting on Clausius’s 3/443
calculation.44
1.2 About Clausius45
Rudolf Clausius was born 1822 and died in 1888. He was the youngest of eighteen46
children (Cropper, 2001, p. 104). Not much biographical information exists but more47
than is in Cropper’s Great Physicists can be found on the World Wide Web on the48
MacTutor History of Mathematics Archive (O’Connor and Robertson, 2017). One can49
readily share Cropper’s regret (p. 105) of the absence of biographical information; “we50
should mourn the loss”.51
Clausius laid foundations for much of the field of thermodynamics. His contributions52
to the study of thermodynamics are immense. He devised the concept of mean path53
length, derived a mathematical characterization of entropy and in 1865 coined the word54
entropy (Clausius, 1867, p. 357).55
Clausius’s foundational texts on thermodynamics written in the 1860s and 1870s bear56
a striking resemblance to modern texts, likely because texts on thermodynamics have57
been modeled on Clausius’s great works (Clausius, 1867, 1879).58
1.3 Clausius’s 3/459
In 1858, Buijs-Ballot in an article in the Annalen (Buijs-Ballot, 1858) objected that the60
idea of molecular motion in straight lines was refuted by, for example, the way tobacco61
smoke remains for “so long extended in immoveable layers”.62
Clausius responded to this point. In 1858 Poggendorff’s Annalen published an article63
by Clausius on mean path lengths, Ueber die mittlere L¨ange der Wege (Clausius, 1858).64
A translation into English appeared in 1859 in the Philosophical Magazine (Clausius,65
1859). That English translation is included in Brush’s anthology of works on kinetic66
theory (Brush, 1948, Vol. 1). Clausius pointed out that molecules travel in straight67
lines that are short when they are interrupted by a collision.68
He sets up a problem to solve (p. 84-85 in the Philosophical Magazine, p. 139 in Vol.69
1 of Brush):70
2
If, now, in a given space, we imagine a great number of molecules moving71
irregularly about amongst one another, and if we select one of them to watch,72
such a one would ever and anon impinge upon one of the other molecules,73
and bound off from it. We have now, therefore, to solve the question as to74
how great is the mean length of the path between two such impacts; or more75
exactly expressed, how far on an average can the molecule move, before its76
centre of gravity comes into the sphere of action of anther molecule.77
We will not discuss this question, however, immediately in the form just78
given: we will propose instead a somewhat simpler one, which is related to79
the other in such a manner that the solution of the one may be derived from80
that of the other.81
Clausius then remarks:82
The mean lengths of path for the two cases (1) where the remaining83
molecules move with the same velocity as the one watched, and (2) where84
they are at rest, bear the proportion to one another of 3
4to 1.85
Clausius thus proposed to solve the more general problem of what is the mean path86
length among moving molecules by generalizing his result from the case where only87
one of the molecules is moving and all the other molecules are still. This is a common88
mathematical technique: solve the general case by eliminating unnecessary or extraneous89
assumptions that preserve the fundamentals of the problem. Once he finds the mean90
path length of one moving particle relative to all the other particles which are held91
stationary relative to the moving particle he will, he infers, have the general answer.92
That this is his understanding of the relationship between the specific case and the93
general case is confirmed in the 1860 article in which he provides the mathematical94
details. He there writes (Clausius, 1860, p. 485) about his 3/4 comment in the 185995
Philosophical Magazine article:96
In my solution I confined myself to the case where the molecule µmoves,97
and the others m,ml, . . . remain at rest; but at the same time I asserted that98
in the case where the latter molecules also move with the same velocity as µ,99
the number of collisions increases in the ratio of 1 : 4
3. I did not prove this100
assertion, because for the object I then had in view it was not necessary. . . .101
What appeared in his 1859 article in English was:102
in our consideration of the mean path, it is not the question to determine103
exactly its numerical value, but merely to obtain an approximate notion of its104
magnitude; and hence the exact knowledge of this relation is not necessary.105
Unfortunately, generalizing the 3/4 result from the case where only molecule is moving106
to the case where all are moving in invalid. In describing the two cases he sets up two107
3
distinct reference frames; the mathematics in one of the cases — one molecule moving108
amid unmoving molecules — does not generalize to the other — all molecules moving.109
The case of one moving molecule with all other molecules still is not a specialization110
of the general case of all moving molecules. Instead it is a means to measure the ratio of111
the average distance in a given volume between nstationary nodes relative to the mean112
path length of nmoving molecules. Note that for nequal sub-volumes within the given113
volume the average and actual distances between nodes would be the same.114
By way of analogy, imagine a pin-ball game set up with nbumpers so small their115
size does not materially affect calculations of path length. With a large number of116
trials measuring the distance a ball travels before hitting a bumper one can estimate117
the average distance between bumpers. Clausius’s set-up, of one molecule moving and118
the rest stationary, is analogous to the pin-ball game set-up. One molecule moving119
among stationary molecules reveals the average distance between stationary molecules120
within the given volume. The moving molecule is an experimental prop to reveal average121
distances; it is not a representative molecule from a sea of moving molecules.122
When Clausius holds all molecules in the given volume still and allows only one to123
move he is indirectly measuring the average distance between nimmoveable centers. He124
is not measuring the representative distance between any two moving molecules in a125
system where all molecules are moving.126
Hence in his 1858 mathematics that appears in 1859 in the Philosophical Magazine,127
he does in fact calculate the ratio of the average distance between a system of nmoving128
molecules relative to the average distance between the nunmoving centers of nsub-129
volumes in the given volume. If we treat radial motion as having one dimension, in effect130
Clausius is comparing the mean path length of nmoving molecules for a 4 dimensional131
system to the mean distances between centers for the 3 dimensional nsub-volumes.132
There are two reference frames.133
Clausius has (but for the flawed generalization), typically, reasoned brilliantly. His134
inference that if we vary the assumption and put all molecules into motion the same135
3/4 relationship applies, is false. This last inference Maxwell shows, indirectly, is wrong136
because a different mathematical approach to the more general problem leads to √2 not137
4/3.138
1.4 Maxwell’s √2139
Maxwell in 1860 in the Philosophical Magazine (Maxwell, 1860) at p. 27 gives the140
problem: To find the probability of a particle reaching a given distance before striking141
any other. In the course of providing an answer he notes that for the speed vof a particle142
relative to another particle for moving particles, if two particles have the same speed,143
v1=v2then v=√2v1. Maxwell notes that M. Clausius obtains 4
3instead of √2.144
In effect, Maxwell works in one reference frame only, with all moving molecules. If145
we assume that all the molecules move independently of all other molecules, then on146
average the molecules move orthogonally to each other. If they all move at the same147
average speed v1=v2then the speed of a moving molecule relative to some other moving148
molecule colliding with it can be determined as the hypotenuse of a right-angled isosceles149
4
triangle, hence √2v1.150
In effect, Clausius and Maxwell have solved different problems and accordingly their151
answers do not coincide. Clausius solved the problem of the ratio of mean path length152
for moving particles to average distances between centers of counterpart sub-volumes.153
Maxwell solved the problem of the average relative distance between two moving particles154
in a sea of moving particles.155
The editor of Maxwell’s collected works (Maxwell, 1890), W. D. Niven, includes a foot-156
note at p. 387 volume 1 on the 4/3 result obtained by Clausius compared to Maxwell’s157 √2. The purpose of the footnote is to show where Clausius went wrong in his mathe-158
matics. The footnote must be right; Clausius’s mathematics giving a 3/4 ratio does not159
apply to a system with all moving molecules. It only applies to the two reference frame160
problem that Clausius proposed as a simplification.161
I first read Clausius’s 1859 mean path lengths article in July 2011 and until October162
2017 I did not notice that the problem was in the generalization inference he made in163
the 1859 article that preceded his observation.164
1.5 Ironies165
Clausius’s remark “in our consideration of the mean path, it is not the question to166
determine exactly its numerical value” is one of several ironies in this account.167
The exact numerical value 3/4 is equivalent to finding that including motion as a168
fourth dimension, lengths in a 3 dimensional system are 4/3 as long as those in a 4169
dimensional system.170
One irony is that Clausius discounts the numerical significance of 3/4 (the 4/3 result171
expressed inversely) that appears to predict dark energy first observed 138 years later.172
Another irony is that Clausius himself makes an error not in his mathematics but in173
inferring that his particular case of one moving molecule among stationary molecules174
generalizes to the case of all molecules moving. Clausius’ inappropriate generalization175
resulted in his 1860 paper being discounted as erroneous in its entirety, which it is not.176
1.6 Clausius’s version of the 4/3 law177
Since Clausius’s attack on the problem of a single moving molecule involves no measure-178
ment of distance, his ratio result is scale invariant.179
Suppose now that energy is proportional to length traveled (or average energy is180
proportional to average length traveled). Then we can equivalently say for counterpart181
4 and 3 dimensional reference frames with the four dimensional reference including linear182
motion:183
A 3 dimensional system has 4/3 as much energy per dimension as the counterpart184
4 dimensional system.185
A 4 dimensional system has 4/3 the degrees of freedom of a 3 dimensional system186
per event or unit time.187
5
Lengths in a 3 dimensional system are 4/3 of the length of those in the 4 dimen-188
sional system (which is what Clausius shows using trigonometry and calculus).189
The same scale invariant result incorporating the important 4/3 appears in an inter-190
mediate step of the proof of Stefan’s Law about black body radiation by Boltzmann,191
which can be found in various texts such as (Planck, 1914, p. 61), (Allen and Maxwell,192
1948, p. 742-743), (Longair, 2003, p. 297). This is likely the most important instance of193
the 4/3 law because on average undirected energy in nature is distributed isotropically194
as in black body radiation and this scaling or exponent relationship is scale invariant. In195
other words, isotropic energy distribution is universal at least on average and therefore196
Clausius’s 3/4 observation (and its 4/3 inversion) is therefore also universal.197
A circulatory system is a 4 dimensional system consisting of one dimensional flow198
through 3 dimensional pipes supplying the 3 dimensional tissues of an organism. Since199
energy distribution capacity increases by a power of 4 relative to an increase in mass200
by a power of 3, an organism’s metabolic rate increases by a 3/4 power relative to mass201
to maintain a constant intracellular temperature for constant temperature animals, a202
result known as Kleiber’s Law (Kleiber, 1932, 1947, 1961; Dodds et al., 2001).203
Other examples of the 4/3 law include, in 1845, Waterston’s 4/3 vis viva in connection204
with a gravitating plane (Waterston, 1892), Richardson’s 4/3 scaling of wind eddies205
(Richardson, 1926) proved by Kolmogorov in 1941 (Kolmogorov, 1991b,a), the 4/3 fractal206
envelope of Brownian motion (Lawler et al., 2001) proved in 2001, and possibly a 4/3207
power in connection with cellular transmission (Jafar and Shamai, 2007). The 4/3208
also appear in networks since the measured mean path length for information networks209
(the transmission network has one dimensional information flow set in a 3 dimensional210
environment) is close to 4/3 times the value of the natural logarithm which accords with211
measurement (Watts and Strogatz, 1998). The natural logarithm is the scale factor, in212
steps, for an isotropic homogeneously scaled network.213
1.7 On the significance of Clausius’s 3/4 mean path length214
The discovery in 1998 of what is called dark energy is a reason to resurrect interest in215
Clausius’s 1860 article. Both the faintness of the observed type IA supernovae (SN)216
compared to what is expected and measurements of the energy density of dark energy217
(so called) connect to Clausius’s mathematics that shows lengths are 4/3 longer in a 3218
dimensional space relative to a 4 dimensional one.219
Astronomers found that “high-redshift SNe Ia are ≈0.25 mag fainter than expected220
in our universe with its presumed mass density but without a cosmological constant (or221
which is not accelerating)” (Riess, 2000, p. 1287); also (Perlmutter, 2003). “Both teams222
found that distant SNe are ≈0.25 mag dimmer than they would be in a decelerating223
Universe” (Frieman et al., 2008, p. 12). These observations correspond to the observed224
supernovae being 4/3 farther away than expected, as one would predict from Clausius’s225
1860 paper.226
Then there is energy density. The lengths ratio in Clausius’s 1860 paper implies that227
energy density in a 4 dimensional system relative to a 3 dimensional system gives:228
6
E/(1)3
E/(4/3)3=43
13≈0.7033
0.2967.(1)
The 0.2967 in equation (1) is the cosmological term Ωm, the proportion of energy in the229
universe attributed to the energy density of matter. Astronomers around 2000 observed230
that some supernovae imply Ωm= 0.30 (Riess et al., 2000; Perlmutter, 2003, p. 65),231
very close to the 0.2967 that the 4/3 law predicts. More recent measurements are even232
closer, 0.2965, (Betoule et al., 2014), which is about a half per cent different than the233
predicted value.234
Might we say that Clausius is the grandfather of the dark energy concept? Or at least,235
of a universal scale-invariant 4/3 law?236
1.8 The fourth dimension problem237
In the preceding is an analysis of Clausius’s 1860 work and its connection to a universal238
scale-invariant 4/3 law as well as to dark energy. But this is far from an end point239
because of a deep problem that sits at the heart of the 4/3 theory.240
The problem arises this way. One way of characterizing the 4/3 law is as the equal-241
ization of energy distribution among degrees of freedom in counterpart 4 dimensional242
and 3 dimensional spaces. But the fourth dimension corresponds to motion in a 3243
dimensional space or to time if the motion is constant; this connects to Minkowski’s244
space-time characterization of special relativity (Minkowski, 1918; Saha and Bose, 1920;245
Minkowski, 2012). But how can we say that energy (which produces motion) itself is246
distributed along a fourth dimension that corresponds indirectly to linear motion? That247
seems to be a circular characterization. Or, from the perspective of time as a fourth248
dimension, how can energy be considered equally distributed among 3 dimensions each249
of which stands for a length and one that stands for time? This is still a problem even250
though Minkowski suggested in his space-time lecture that all four dimensions had equal251
standing as dimensions.252
I don’t know the answer to that question. I am not sure that the question is properly253
framed or phrased. One might suspect the problem arises because there is a more254
fundamental way to characterize motion or time that physics has not found. Perhaps255
motion and time are perceptions or even illusions resulting from the expanding universe,256
perhaps motion is relative to the expanding background? There are also the problems257
of explaining the connections between the 4/3 law and general relativity and quantum258
mechanics.259
Despite the serious problems that the 4/3 law and its connection to dark energy260
present, still Clausius’s 1860 paper deserves serious reconsideration.261
1.9 Following the Bibliography are:262
1. P. 84 and 85 in the Philosophical Magazine from Clausius’s 1859 article that men-263
tions 3/4.264
2. P. 27 and 28 of Maxwell’s 1860 article, Prop. X., with √2.265
7
3. Clausius’s 1860 article (pages 434-436) deriving the 3/4 ratio.266
4. Footnote at p. 387, Vol. 1, Maxwell’s collected works.267
Bibliography268
Allen, H. S. and Maxwell, R. S. (1948). A Text-book of Heat. Macmillan and Co.269
Betoule, M. et al. (2014). Improved cosmological constraints from a joint analysis of the270
SDSS-II and SNLS supernova samples. A&A, 568(A22).271
Brush, S. G., editor (1948). Kinetic Theory — An Anthology of Classic Papers with272
Historical Commentary. Pergamon Press.273
Buijs-Ballot, C. H. D. (1858). Ueber die Art von Bewegung, welche wir W¨arme und274
Elektricit¨at nennen. Annalen der Physik, 179(2):240–259.275
Clausius, R. (1858). Ueber die mittlere L¨ange der Wege, welche bei der Molecularbewe-276
gung gasf¨ormiger K¨orper von den einzelnen Molec¨ulen zur¨uckgelegt werden; nebst eini-277
gen anderen Bemerkungen ¨uber die mechanische W¨armetheorie. Ann. Phys, 181:239–278
258.279
Clausius, R. (1859). On the Mean Length of the Paths described by the separate280
Molecules of Gaseous Bodies on the occurrence of Molecular Motion: together with281
some other Remarks upon the Mechanical Theory of Heat.(Translation by Guthrie, P.282
of Annalen, No. 10, 1858). Philos. Mag., 17. Fourth Series(112):81–91.283
Clausius, R. (1860). On the dynamical theory of gases. Philos. Mag., 19, Fourth284
Series:434–436.285
Clausius, R. (1867). The Mechanical Theory of Heat with the applications to the steam286
-engine and to the Physical Properties of Bodies (translated by Tyndall, J.). John van287
Voorst.288
Clausius, R. (1879). The Mechanical Theory of Heat (translated by Browne, Walter R.).289
Macmillan.290
Cropper, W. H. (2001). Great Physicists. Oxford Univ. Press.291
Dodds, P. S., Rothman, D. H., and Weitz, J. S. (2001). Re-examination of the “3/4-law”292
of metabolism. J. Theor. Biol., 209:9–27.293
Frieman, J. A., Turner, M. S., and Huterer, D. (2008). Dark energy and the accelerating294
universe. arXiv:0803.0982v1.295
Jafar, S. A. and Shamai, S. (2007). Degrees of freedom region for the MIMO X channel.296
arXiv:cs/0607099v3.297
8
Kleiber, M. (1932). Body size and metabolism. Hilgardia, 6:315.298
Kleiber, M. (1947). Body size and metabolic rate. Physiol Rev, 27(4):511–541.299
Kleiber, M. (1961). The Fire of Life: an introduction to animal Bioenergetics. John300
Wiley & Sons.301
Kolmogorov, A. N. (1991a). Dissipation of energy in the locally isotropic turbulence302
(translation by V. Levin of 1941 article in Russian). Proc R Soc A, 434(1890):15–17.303
Kolmogorov, A. N. (1991b). The local structure of turbulence in incompressible viscous304
fluid for very large Reynolds numbers (translation of 1941 article in Russian by V.305
Levin). Proc. R. Soc. Lond. A, 434:9–13.306
Lawler, G. F., Schramm, O., and Werner, W. (2001). The dimension of the planar307
Brownian frontier is 4/3. Math Res Lett, 8:401–411.308
Longair, M. S. (2003). Theoretical Concepts in Physics, Second edition. Cambridge309
University Press.310
Maxwell, J. C. (1860). Illustrations of the dynamical theory of gases. – Part I. on the311
motions and collisions of perfectly elastic spheres. Philos. Mag., 19, Fourth Series:19–312
32.313
Maxwell, J. C. (1890). The Scientific Papers of James Clerk Maxwell. Cambridge314
University Press.315
Minkowski, H. (1918). Time and space (translation of 1908 lecture from German by E.316
H. Carus). The Monist, XXVIII:288–302.317
Minkowski, H. (2012). Space and Time - Minkowski’s Papers on Relativity (Translated318
by Fritz Lewertoff and Vesselin Petkov). Minkowski Institute Press, Montreal, Quebec,319
Canada.320
O’Connor, J. J. and Robertson, E. F. (2017). Rudolf Julius Emmanuel Clausius. Mac-321
Tutor History of Mathematics Archive.322
Perlmutter, S. (2003). Supernovae, dark energy, and the accelerating universe. Phys.323
Today, April:53–60.324
Planck, M. (1914). The Theory of Heat Radiation (Translator Masius, M.) — Second325
Edition. P. Blackiston’s Son & Co.326
Richardson, L. F. (1926). Atmospheric diffusion shown on a distance-neighbour graph.327
Proc R Soc A, 110(756):709 – 737.328
Riess, A. G. (2000). The case for an accelerating universe from supernovae. Publ. Astron.329
Soc. Pac., 112(776):1284.330
9
Riess, A. G., Schmidt, B. P., et al. (2000). Tests of the accelerating universe with331
near-infrared observations of a high-redshift type Ia supernova. Ap.J., 536:62–67.332
Saha, M. N. and Bose, S. N. (1920). The Principle of Relativity – Original Papers by A.333
Einstein and H. Minkowski translated into English. University of Calcutta.334
Waterston, J. J. (1892). On the physics of media that are composed of free and perfectly335
elastic molecules in a state of motion. Philos T Roy Soc A, 183:1–79.336
Watts, D. J. and Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks.337
Nature, 393:440.338
10
P. 84 and 85 in the Philosophical Magazine from Clausius's
1859 article
with Remarks upon the Mechanical Tlleory
of
Heat. 85
more simply assume that
a11
molecules
move
at the same rate;
and
for
this
case
we
obtain the following result
:-The
meall
lengths
of
patll
for
the two cases
(1)
whe1'e
the remaining molecules
move with
tIle
same velocity
as
tlte
one
watched, and (2) where
they are at rest,
beal'
the proportion
to
one
another
of
i
to
1.
It
would not be difficult to prove thc correctness of this rela-
tion:
it
is, however, unnecessary for us
to
devote our time to
it;
for, in our consideration of the mean path,
it
is
not the question
~o
determine exactly its numerical value, but merely to obtain
an approximate notion of its magnitude; and hence the exact
knowledge of this relation is not necessary,
It
is even suffi-
cient for our purpose
if
we
may
assume
as
certain that the mean
path among moving molecules cannot
be
greater than among
stationary ones; this
will
certainly be
at
once admitted. Under
this hypothesis,
we
will confine the discussion of the question
to that case where the molecule watched alone moves, while all
the otllers remain at rest.
'I
Moreover, without affecting the question in anything,
we
may
suppose a mere moving point in place of the moving molecule;
for
it
is in fact only the
~entre
of gravity of the molecule
which has to be considered.
(5.) Suppose, then, there is a space containing a great num-
ber of molecules, and that these are not regularly arranged, the
only condition being that the density is the same throughout,
i.
e.
in equal parts of the
space
there are the same numbers of mole-
cules. The determination of the density may
be
performed con-
veniently for our investigation
by
knowing
how
far apart
two
neighbouring molecules would
be
separated from one another
if
the moleeules
were
arranged cubically, that is,
so
arranged
that the
whole
space might
be
supposed divided into a number
of equal very small cubic spaces, in
whose
corners the centres
of the molecules
were
situated. We shall denote this distance,
that is, the side of one of these little cubes,
by
A,
and shall call
it
the mean distance
of
the neighbouring molecules.
If,
now,
a point
moves
through this space in a straight line,
let us suppose the space
to
be
divided into llarallel layers pet'-
pendicular to the motion of the point, and let us detet'mine how
great is tlte probability that tlie point
wilt
pass freely
tlU"ougl~
a
layer
of
the tltickness x without encounle1'ing the sphere
of
action
0/
a molecule.
Let us first take a layer of the thickness 1, and let us denote
by the fraction of unity a the probability of the point passing
through this layer without meeting with any sphere of action:'
then the corresponding probability
for
a thickness 2
is
a9.
; for if
such a layer be supposed divided into
two
layers of the thickness
1, the probability of the points passing free thl'ough the first
layer, and thereby arriving
at
the
second, must
be
multiplied by
Digitized
by
GOdgle
P. 27 and 28, Maxwell's 1860 article
28 Prof. Maxwell on the Motions and Collisions
Putting N=lwhen x=Q, we find e'" for the probability of a
particle not striking another before it reaches adistance x.
The mean distance travelled by each particle before striking is
-=/. The probability of aparticle reaching adistance =nl
without being struck is e~". (See apaper by M. Clausius, Phi-
losophical Magazine, February 1859.)
If all the particles are at rest but one, then the value of a. is
where sis the distance between the centres at collision, and N
is the number of particles in unit of volume. If vbe the velo-
city of the moving particle relatively to the rest, then the num-
ber of collisions in unit of time will be
and if u, be the actual velocity, then the number will be v^x
;
therefore
^\
where i\ is the actual velocity of the striking particle, and vits
velocity relatively to those it strikes. If v^ be the actual velocity
of the other particles, then v= \^v^ +v^^. If Vy =v^ then
v= a/2vi, and _
«= v/27rs2N.
Note.—M. Clausius makes «=|7rs^N.
Prop. XI. In amixture of particles of two different kinds, to
find the mean path of each particle.
Let there be N, of the first, and N2 of the second in unit of
volume. Let Sj be the distance of centres for acollision between
two particles of the first set, s^ for the second set, and s' for col-
lision between one of each kind. Let v^ and v^ be the coefficients
of velocity, M, Mg the mass of each particle.
The probability of aparticle Mj not being struck till after
reaching adistance x^ by another particle of the same kind is
The probability of not being struck by aparticle of the other
kind in the same distance is
Therefore the probabihty of not being struck by any particle
before reaching adistance xis
Clausius's 1860 article (pages 434-436)
Footnote at p. 387, Vol. 1, Maxwell's collected works
