Physikalische Zeitschrift. 27th edition, 1926. Pages 95–101.
On Einstein’s gas theory
∗
E. Schrödinger
1
Basic principle
The essential point of the new gas theory recently worked out by Einstein1 is
surely that a new statistics, the so-called Bose statistics2 , should be applied
to the motion of the gas molecules. To consider the new statistics as primary,
i.e. something which cannot be explained further, goes against the natural
instinct3 . It seems that it rather conceals a certain interdependence or interaction between the gas molecules, which in this form, however, is rather
difficult to analyse.
We might expect to obtain a deeper insight into the true nature of the
new theory if we succeed in keeping the logically well-founded experimentally
tested statistical methods in tact, while changing the assumptions in a place
where this does not require intellectual sacrifice. For this, the following
simple idea is useful: Einstein’s gas theory is obtained if one applies to
the gas molecules the same statistics which, when applied to the “atoms of
∗
Translation by T. C. Dorlas
1
A. Einstein, Berl. Ber. 1924, p. 261; 1925, p. 3.
Bose, Zeitschr. f. Phys. 26, 178, 1924.
3
See A. Landé, Zeitschr. f. Phys. 33, 571, 1925. However, I cannot agree completely
with this paper.
2
1
light”, leads to Planck’s radiation law. But Planck’s radiation law can also
be gotten by applying the “natural” statistics to the ”eather resonators”,
i.e. the degrees of freedom of the radiation (field)4 . The atoms of light
then feature only as the energy levels of the resonators. The transition from
the natural to the Bose statistics can always be achieved by exchanging the
roles of the concepts “manifold of energy states” and “manifold of carriers
of these states”. One therefore has to view the gas simply in a way similar
to the black body radiation picture which does not agree with the extreme
light quantum view. The natural statistics, i.e. the usual Planck method
of partition functions, will then lead to Einstein’s gas theory. This means
nothing other than to be serious about the De Broglie5 -Einstein6 wave theory
of moving particles according to which these are nothing but a kind of “foam
crest” on a fundamental radiation wave.
The elaboration of this idea seems to me to be of sufficient interest to
explain it here further.
2
The gas as a system of oscillators. Determination of the free energy.
We start from the usual assumption that each of the n molecules of a
monatomic ideal gas in a volume V can take a discrete set of precisely determined states, where its energy has the values
²1 , ²2 , . . . ²s . . . .
(1)
At any moment in time, arbitrarily many molecules can have the same state.
But I am only making these standard assumptions for the sake of clarity.
In fact I will assign a degree of freedom to the entire system such that the
4
J. H. Jeans, Phil. Mag. 10, 91, 1905; P. Debye, Ann. d. Phys. 33, 1427, 1910. Cf.
the first chapter of Planck, “Wärmestrahlung”. See also M. von Laue, Ann. d. Phys. (4)
44, 1197, 1914.
5
L. de Broglie, thesis, Paris (Ed. Masson & Cie.), 1924. Same title, Ann. de physique
(10) , 3, 22, 1925.
6
Einstein, l.c. §8.
2
s-th degree of freedom has (“oscillates with”) the energy ns ²s when, in the
usual representation, ns molecules are in the state (with energy) ²s . The s-th
degree of freedom thus corresponds to that of a one-dimensional harmonic
oscillator; indeed, it has the possible energy values
0, ²s , 2²s , . . . ns ²s . . .
(2)
The whole system is thus, for the purpose of the calculation, an ensemble
of linear oscillators, just like a solid, or better like a volume of radiation
since the number of oscillators is not finite. The spectrum of the “principal
oscillations” is given by (1); we can keep it quite arbitrary for the moment.
We treat this system according to Planck’s method of a sum over states.
The general term of this sum is
1
e− kT (n1 ²1 +n2 ²2 +···+ns ²s +... ) .
(3)
If we were dealing with a solid or with radiation then the numbers ns would
not be restricted by any condition, and the sum (3) would split in the usual
way into a product of elementary sums which can each be summed as geometric series:
∞
YX
Y
ns ²s
1
e− kT =
(4)
²s .
− kT
1
−
e
s ns =0
s
The product would have finitely many factors in the case of a solid, and
infinitely many in the case of radiation. The special character of our system
is that the ns have to satisfy the condition
X
ns = n.
(5)
It seems to me that the most valuable conclusion of the present work is the
fact that this condition, which is so trivial and obvious in the usual gas
theory, is the crucial distinguishing feature of a gas according to Einstein’s
theory.
We should therefore retain in the partition sum only such terms (3) for
which (5) holds. In other words, we should select from the infinite product
(4) those members which are homogeneous of degree n in the infinitely many
variables
²s
(6)
xs = e− kT ; s = 1, 2, 3, . . .
3
This exercise can be solved by the beautiful residue theorem of function theory7 In the right-hand side of (4) we write zxs instead of xs , we multiply
the whole expression by z −n−1 , and we look for the residue of the thus obtained function of the complex variable z at the point z = 0. This residue is
obviously 2πi times the partition sum Z that we want to compute:
I
∞
Y
1
1
Z=
dz z −n−1
.
(7)
2πi z=0
1
−
zx
s
s=1
The question of the convergence of the product shall be ignored for the
moment. In the following we assume that the sequence ²s is nonnegative,
increasing and even increasing to infinity. The integrand is then regular in
the interior of the circle |z| < x11 , except at z = 0. It has, other than the pole
at z = 0 a sequence of poles at
zs =
1
1
≥ ;
xs
x1
s = 1, 2, 3, . . .
(8)
on the real axis, accumulating at infinity, where there is an essential singularity. To evaluate (7), we employ the method of steepest descents.8 The
integrand has an extraordinarily sharp minimum on the real axis between 0
and x11 , say at z = r, owing to the extremely large value of n. This same
function value is a sharp maximum on the circle |z| = r, namely, because
the derivatives of an analytic function are independent of the direction and
the second differential takes very large negative values for imaginary increments, and very large positive values for real increments. One then moves
the contour of integration to this circle and can restrict the integration to a
small neighbourhood of z = r, as can easily be shown.9 The position r of
the minimum is obtained by logarithmic differentiation:
∞
n + 1 X xs
−
+
= 0.
r
1 − rxs
s=1
7
(9)
The elegant method used here was introduced into statistics by C. G. Darwin and R.
H. Fowler, Phil. Mag. 44, 450, 1922 (see also p. 823 and 45, 1, 1923).
8
B. Riemann, Ges. Math. Werke u. Wissenschaftliche Nachlaß, p. 424 (2nd Ed.
Leipzig, Teubner 1892). P. Debye, Math. Ann. 67, 5354, 1909. See also Darwin and
Fowler, l.c. page 462.
9
See the previous note.
4
Now we write z = rei ϕ on the circle, and expand the logarithm in the
integrand as a power series in ϕ. Using (9), we find
Ã
!
∞
Y
1
log z −n−1
1 − zxs
!
à s=1 ∞
∞
Y
1 2 X
xs
1
−n−1
= log r
− ϕr
+ ....
(10)
2
1
−
rx
2
(1
−
rx
)
s
s
0
s=1
Inserting into (7), and writing dz = i dϕ, we can, to very good approximation,
which we will not attempt to estimate here, neglect terms in ϕ3 etc. and
extend the integration from −∞ to +∞. This yields
Ã
Z=
2πr
∞
X
s=1
xs
(1 − rxs )2
!− 12
r
−n−1
∞
Y
1
.
1
−
rx
s
s=1
(11)
For Planck’s Ψ-function, i.e. the negative free energy divided by the temperature (Ψ = k log Z) we get
(∞
X
Ψ = −k
log(1 − rxs ) + (n + 1) log r
s=1
Ã
!)
∞
X
1
xs
+ log 2πr
.
2
(1 − rxs )2
s=1
(12)
We recall that the meaning of xs is as in (6). The number r is determined
by (9), i.e.
∞
X
xs
n+1=r
.
(90 )
1
−
rx
s
s=1
Comparing the results (6), (12) and (9’) with Einstein’s results (Berl. Ber.
1924, pp. 263 f., Eqns. (81 ), (92 ), (6a), (13) and (14)), we easily observe
that there is complete agreement, provided that the last member in curly
brackets in our Eqn. (12) can be neglected. Our quantity r is called e−A
by Einstein10 The fact that we obtain everywhere n + 1 instead of n is of
course irrelevant. Moreover, our formulas are basically more general since
10
Our r is also Einstein’s degeneracy parameter λ. For the moment, we do not have the
s
condition r < 1, but only r < x1s . Our quantity rxs is called e−α by Einstein.
5
the “frequency spectrum” (1), or in other words, the energy levels of a single
molecule, have not been specified.
Concerning the neglect of the last member of (12), this can in general
be justified by the fact that the corresponding sum is, though larger, not of
higher order than that in (9’). The said term will thus be of order log(n + 1)
and is in general small compared with the first two. We will return to this
question later.
3
Determination of the frequency spectrum.
It would be illogical to be satisfied here with Einstein’s determination of
the molecular energy levels (1) by subdivision of the molecular phase space
into quanta. Indeed, for us, (1) is the frequency spectrum of the volume
V of gas. We will follow closely L. de Broglie in calculating this spectrum
using the representation of a molecule with rest mass m and velocity v = βc
as a “signal”, one could say a “crest of foam”, of a system of waves with
frequencies in the neighbourhood of
mc2
p
ν=
h 1 − β2
(13)
and phase velocity u satisfying a dispersion law which by the above equation,
is given by
c
c2
u= =
(14)
β
v
(v plays the role of the signal velocity, as was shown by de Broglie, and can
be easily checked). The problem is to count the number of oscillation modes
for a wave in a volume V satisfying this dispersion law. If s is the number
of modes with wavelengths between ∞ and λ then, as is well known,
4πV −3
λ ,
(15)
3
if we consider the oscillating quantity (space function) as a scalar – only then
do we get a sensible result. Using
s=
λ−1 =
mcβ
ν
= p
u
h 1 − β2
6
(16)
we get
s=
4πV
m3 c3 β 3
.
3 h3 (1 − β 2 )3/2
(17)
From this one obtains easily the quantum of energy of the s-th eigen mode:
r
mc2
J 2 s3/2
²s = hνs = p
= mc2 1 + 2 2
mc
1 − β2
Ã
µ
¶− 13 !
4πV
J =h
.
(18)
3
J is a certain quantum of momentum, depending only on the volume, but
not on the nature of the gas. In the case of the gas theory, where only very
small values of β are relevant, it is more convenient and more conventional,
if less logically consistent, to replace ²s by the (classical) kinetic energy. One
can then write approximately
Ã
!
1
²s = mc2 p
−1
1 − β2
!
Ãr
2 s2/3
J
1+ 2 2 −1
= mc2
mc
µ
¶− 23
2
J 2 2/3
h2 4πV
=
s3 ,
(19)
s =
2m
2m
3
which of course agrees exactly with Einstein’s equation (8), p. 263, l.c.
On the other hand, if β is close to 1, then (18) yields approximately
1
hνs = Jcs 3 or
4πV νs3
s =
,
3 c3
(20)
which is, up to the infamous polarisation factor 2, the well-known formula
for the degrees of freedom of black-body radiation.
The different exponents of s (s2/3 in (19), s1/3 in (20)) is the second characteristic difference between black-body radiation and a gas. The increase
of the exponent to 2/3 in the region of long wavelengths is a consequence of
the dispersion. The dispersion of the phase velocity is by (13) and (14) very
7
significant for long waves, but vanishes for short waves when β approaches 1.
This behaviour is just the opposite of that for the so-called elastic branches of
the spectrum of a solid11 with different kinds of atoms. The correspondence
is even perfect in the region of long waves, where in both cases the frequency
is almost independent of the wavelength, and hence the phase velocity is directly proportional to the wavelength. Whether this correspondence has any
further significance has yet to be investigated.
Here we must emphasise an important difference between our result (19)
for the energy levels and Einstein’s hypothesis. In (19) we cannot allow the
value s = 0, whereas Einstein does. Namely, for a stationary molecule, β = 0,
the wavelength (16) becomes infinite whereas the wavelength of the longest
oscillation mode in the volume V is of order V 1/3 . In any case, (19) is a
poor approximation and the same is well-known to be the case for (15). It is
known that the spectral law of the lowest frequencies depends on the shape of
the volume. Phenomena for which the derivation has to take account of the
(energy) distribution (19) for small values of s, should therefore be treated
with caution. In general, one can only say that such phenomena should,
according to the here presented theory, depend on the shape of the volume.
I am convinced that the nonexistence of the rest state does not cause
any problems in Einstein’s gas theory. However, the phenomenon of “condensation” described by Einstein will of course disappear. The degeneracy
parameter r, called e−A by Einstein, can exceed unity up to the value
2
4πV − 3
h2
1
(
)
2mkT
3
=e
.
x1
(21)
I am not going into any further details, but would like to express the opinion
that the inclusion of the rest state is not in agreement with the foundations
of the theory (for the above reasons: because this corresponds to an infinite
wavelength).
We have to give serious consideration to the fact that the frequency spectrum (18) or (19), as well as the dispersion law (13), (14), contains the
11
See for example the excellent account by G. heckmann in the “Ergebnissen der exakten
Naturwissenschaften” 4, 118, 1925.
8
molecular mass, so that it is not a property of the volume alone, but depends of the character (though not the quantity!) of its contents. It is not
correct to think that the nature (i.e. the mass) of the particular molecule
helps to determine the frequency ν of the waves, but that the phase velocity
u depends in a universal way on ν. This is not the case. If we denote by
mc2
ν0 =
h
(22)
the rest frequency of the molecule, then it follows from (13) and (14) that
the following explicit dispersion law holds
cν
u= p
.
ν 2 − ν02
(23)
This dispersion law itself also depends on the rest frequency; it appears a
bit like a resonance frequency. The universal radiation of which the particles
are supposed to be the “signals”, or perhaps the singularities, is therefore
something essentially more complicated than the radiation of Maxwell’s theory. Indeed, this is so, not just because it has dispersion, but in particular
because the dispersion law for the phase velocity of a group of waves also
depends on what type of singularity is created by this group of waves. This
reminds one on the one hand of “waves with finite amplitude”, and on the
other hand it could be related to the fact that de Broglie’s phase waves of a
particle moving in a straight line are plane waves in the sense that surfaces of
constant phase are planes, but certainly not in the sense that the oscillation
of all points in such a plane is the same.
4
Calculation of mean values and fluctuations.
For this the method explained in §2 is particularly suited as is known from
the above mentioned works of Darwin and Fowler. The e-power (3) is a measure for the probability to find the system with the occupation numbers, or
quantum numbers, n1 , n2 , . . . ns . . . , provided of course these number combinations are allowed according to (5). To find the average value of a quantity
f (n1 , n2 , . . . ns . . . )
9
we have to multiply by the e-power (5) and work with the product in the
same way as with the e-power alone in the case of the partition sum Z. The
result, divided by Z, yields the desired f . It is especially easy to calculate
the average value of a product of the ns this way. The multiplication by (3)
can be interpreted as the differential operation
−kT
∂
,
∂²s
or upon introduction of the variables (6),
xs
∂
.
∂xs
This equivalence remains valid for an arbitrary number of iterations, either
with the same or with different s. This operation can obviously be commuted
with the integration (7), so that, e.g.,
ns =
xs ∂Z
∂ log Z
1 ∂Ψ
=
=
.
Z ∂xs
∂ log xs
k ∂ log xs
And for an arbitrary product,
µ
¶µ
¶µ
¶
∂
∂
∂
Ψ
ns nt nu · · · =
... .
∂ log xs
∂ log xt
∂ log xu
k
(24)
(25)
Here Ψ is of course the function (12). The indices s, t, u . . . can coincide at
arbitrary positions - however, this must be known before doing the differentiations as the result obviously depends on it.
Doing the differentiations in (24) exactly for the function (12) would
lead to very complicated expressions for two reasons. First of all, the last
logarithm in (12), which in fact is only a small correction, would lead to
contributions which are hard to comprehend. Secondly, the quantity r is
itself a function of xs by (9’), which has to be taken into account when
differentiating with respect to x³s . But ´
the latter is also of little consequence:
1
the variation of a single pole z = xs changes the position of the saddle
point value only in an imperceptible way. I think that these details do not
merit closer investigation at the moment. If we ignore them then only the
10
first member of (12) yields a (significant) contribution to the mean values,
i.e. one gets
µ
¶µ
¶
∂
∂
ns nt nu . . . = −
∂ log xs
∂ log xt
µ
¶
∞
X
∂
···
log(1 − rxs ).
(26)
∂ log xu
s=1
Thus we have, for example,
rxs
ns =
1 − rxs
rxs (1 + rxs )
r 2 xs xt
n2s =
n
n
=
,
s t
(1 − rxs )2
(1 − rxs )(1 − rxt )
rx
s
n2s − (ns )2 =
= ns (ns + 1)
(1 − rxs )2
n2s − (ns )2
1
=1+ .
2
(ns )
ns
The expression for ns agrees with equation (11), p. 263 of Einstein l.c. The
expression for the relative fluctuation of the square is a special case of l.c.,
p. 9, formula (34a), which was derived in a completely different way (by
considering auxiliary semipermeable walls). Our expression is a special case
in the sense that it concerns the fluctuation of the number of molecules in a
single “cell”, so that instead of the reciprocal of the number of cells, we get
unity.
But also the general expression for the fluctuation of the total number of
molecules in a collection of cells of arbitrary size is contained in the above
formulas, and even in somewhat greater generality than the above, as the
collection of cells does not need to consist of neighbouring cells, but can be
chosen in any wild or arbitrary way. If we denote such an aggregate of cells
by Σs , then the above formulas yield
!2
Ã
!2 Ã
X
X
X
=
ns
ns (ns + 1).
ns −
s
Hnece
s
s
P
P
P
2
( s ns )2 − ( s ns )2
1
s (ns )
=
.
+P
P 2 2
2
(ns )
( s ns )
s ns
11
(27)
If the cells are sufficiently “close”, so that xs and ns are approximately equal
inside the cell complex, then the first quotient on the right-hand side is the
reciprocal number of cells.
We will further use the results (25) to estimate, in hindsight, the last
logarithm term of (12) which we neglected. The expression for n2s − (ns )2
shows that
Ã
!
∞
X
1
xs
log 2πr
=
2
(1 − rxs )2
s=1
"∞
#
X
1
1
=
log 2π + log
ns (ns + 1)
2
2
s=1
"
#
∞
X
1
1
2
=
log 2π = log n +
(ns )
2
2
s=1
< log(n + 1) +
1
log 2π.
2
(28)
This term is therefore completely negligible with respect to the previous term
(n+1) log r in (12), as long as the degeneracy parameter r does not approach
1. The latter is only the case at rather strong degeneracy. In that case a more
accurate approximation is needed which might show that the additional term
is no longer completely irrelevant. The above estimate is in any case only a
plausibility argument - in a proper proof the results obtained by neglecting
the term to be estimated cannot be used.
5
About the possibility of representing molecules or quanta of light by plane waves.
At first sight, it is somewhat uncomfortable to view the phase wave of a
particle as a plane wave, as in the discussions by de Broglie, l.c. Obviously,
one can constitute a “signal” which is almost entirely concentrated on a thin
“cut” of the entire space, by superposition of a large number of plane waves
with common normal direction and frequencies in a small interval. On the
other hand, one might be unsure whether and how it would be possible to
12
restrict the signal to a small part of space in all three directions. This can
be achieved, according to Debye12 and v. Laue13 by varying not just the
frequency but also the direction of the plane wave over a small solid angle
dω, and by integrating a continuum of infinitesimal wave functions in these
frequency and solid angle intervals.
In any case, it is not possible to achieve, in this “model of a light quantum”
by classical waves, which mixes many wavelengths in all directions, that the
waves remain together permanently. It rather disperses to an expanding
volume after it traverses a focal point.
If one could avoid this conclusion by a quantum modification of the classical laws for waves, this would open the way to a resolution of the light
quantum dilemma.
6
Summary.
The true meaning of Einstein’s gas theory is the fact that the gas must be
treated as a system of linear oscillators, just like a volume of radiation or
a solid. However, while there are infinitely many modes of oscillation in a
volume of radiation without any restriction on the quantum numbers, and
for a solid there are only finitely many, again without restriction, the gas
has infinitely many modes of oscillation, but with the restriction that for
a “materially confined” gas, the sum of the quantum numbers is constant,
as it corresponds, in the usual manner of speaking, to the total number of
molecules.
The frequency spectrum of the volume of gas is obtained according to
de Broglie by quantising the standing phase waves which are possible in the
volume V by the method due to Jeans and Debye. This leads to a somewhat
different frequency spectrum (νs proportional to s2/3 rather than s1/3 ) due
to the dispersion of the phase waves.
12
13
P. Debye, Ann. d. Phys. 30, 755, 1909.
M. v. Laue, Ann. d. Phys. 44, 1202, 1914 (§2 of the above cited work).
13
Whenever, in a particular case, the experimental facts require that Bose
statistics should be applied to a certain class of objects (which is obviously
far from having been established in the case of a gas), then, in my opinion,
one should conclude that this class of objects is not one of individuals, but
consists of states of energetic excitations. The Bose statistics is then simply
an intermediate stage and can be replaced by “natural statistics” applied to
a different class of objects.
(Submitted 15 December 1925.)
14