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ENTROPY, STRAlN AND THE
FERMl-DIRAC STATISTICS

THESIS TUE THE DEGREE OF M. A.
Guy Berry

3931

 

MICHIGAN STATE LIBRARIES

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To Doctor William Scribner Kimball
without whose aid and suggestions
this paper could not have been

corflpletedo

EHTROPY, STRAIN AND THE
FSRnI—DIRAC STATISTICS

A Thesis

submitted to the Faculty of

MICHIGAN STATE COLLEGE
of
AGRICULTURE AHD APPLIED SCIEKCE

In partial fulfillment of the
Requirements for the Degree of

Master of Arts
by
Guy Berry

1931

93909

ENTROPY, STRAIN AND THE FEHLI-DIRAC STATISTICS
I Introduction

Boltzmann's famous relation between the entropy S of

a gas and W the probability of its state is given by*

u) S:kLmW+C.

here k is Boltzmann's constant and C is a constant depending
upon the zero from Which entropy is to be measured. The
probability W of the gaseous state is calculated rigorously
in statistical mechanics by counting the ”complexions" or
possible arrangements of molecules in phase Space. The pro-
duct of the number of complexions each raised to a suitable
power measures the probability, being a physical quantity
called (for the sake of definiteness) the weight. This
weight represents the W of equation (1) and forms the basis
of entropy calculations in statistical mechanics.

There has recently been deveIOpedMI a "geometrical
weight methoc" based on a physical concept called the "range"
or volume occupied per molecule in phase space. The geo-
metrical weight method has been extended*** to inelude the
new statiStical mechanics of Einstein, Bose, Fermi, and Dirac.
Problems usually handled by the classical statistiCal mechanics
all seem within the scepe of this new method. One important
advantage is that Stirling's approximations and associated
‘ E. Bloch; The Kinetic Theor of Gases, p. 69.

** W. S. Kimball?jifit?3p_-EEHZPEBbEBTETty, Journal of Physical
Chemistry, vol. 6% (19297 p. 1058, hereafter referred to
as, Kimball, Entrepy and Probability.

***s. Chandrasekhar; On the Probabilitr_Meth0Q in the New

Statistics, PhIToSOphical Magazine, vol._§'319305 p. 621,
hereafter referred to as Chandrasekhar.

difficulties* do not enter at all into this direct geometrical
treatment. (Since factorials never enter). Another advantage
is the geometrical picture of weight“I which the method affords.
This contrasts favorably with counting complexions to find the
measure of thermodynamic probability.

The most noteworthy achievement of this geometrical weight
method is the new eXplanation, which it suggests, of the second
law of thermodynamics. By using this method it has been shown***
that the equilibrium equations which (by the Lagrange method
of conditional maxima and minima) determine a state of maximum
entropy and probability are equilibrium equations between stress
and strain. Entrepy then appears as k times the total strain Y.
(2) B : k log W‘+ C = kY.

The second law of thermodynamics is accounted for as being due

to the Operation of these internal forces or stresses in velocity
or momentum as well as ordinary Space. The equilibrium state

of maximum entr0py is really the state of maximum strain brought
about by these forces.

Accordingly the known equilibrium**** equations that lead

to the Maxwell-Boltzmann distribution law may be put in the form

(5) _Wf£dm:‘)‘%“£0/u¢=W.Q/ifl
7.

L

* J. Rice; Statistical mechanics, p. 282.

*‘ Kimball; Entropy and Probability, sec. 8, Fig. 3.

*** W. S. Kim a1 ; Entropy, Elastic Strain, the Second Law of
Thermod namics; etc., Journal of Physical Chemistry,_—
vol. 35 (19315 p. 611, sec. 5, hereafter referred to as
Kimball; Elastic Strain.

**‘*R. C. Tolman; Statistical Mechanics, Chap. 4.

 

wherein the Qi is an undetermined constant and the 7( is the
Maxwell-Boltzmann distribution function. The 773' in the
right member are the ranges* or internals between successive
molecules in velocity or momentum Space, and the zafr are

the X velocity components for each molecule.

(4) 7;. : 7,1. -11. :- 4.

When )\ is given its value -2%} in terms of the absolute
temperature T, determined from the distribution function in
the usual way**, equations (3) take the form of equilibrium

between stress and strain*** acting in velocity or momentum

 

space,
(5) I4 7—: m: : “M M, 9/24, z w: 21,- (7/14. '
7', ”id 7".

In this thesis we extend the geometrical weight method
to include real gases in which the volume is restricted
according to Van der Waal's equation and the energy of molecules
obeys the Pauli exclusion principle****. We find that the
isothermal bulk modulus of gases obeying these restrictions
is the same as that of the classical perfect gas provided
molecular attractions are neglected. This holds true both
for momentum space and ordinary Space. Also the state of
’ Kimball; Entgopy and Probability, sec. 8.
** L. B. Loeb; Kinetic Theory gjpgaggg, Chap. 4.

**’ Kimball; Elastic Strain.
****J. Rice; Statistical MeCHanics, p. 277.

maximum weight or entropy for a given internal energy, again
correSpondS to a state of equilibrium between stress and strain
acting in momentum Space as well as ordinary space just as was
the case for the gas law. The entrOpy again appears as k times
the total strain in phase space. The velocity distribution
function which these equations represent is the same Fermi-Dirac
distribution as usually derived by the statistical method, and
which Chandrasekhar has shown* can be derived by the geometri-
cal weight method.

Perhaps the most important result of the present inves-
tigation is that it affords a step toward a mechanical ex-
planation of the Pauli exclusion principle. We show that
forces acting in momentum Space exclude particles from occupied
cells with the same vigor as they are excluded from the oc-
cupied positions in ordinary Space as represented by Van der-
Waal's constant b. Thus Pauli's exclusion principle is ex-
plained in exactly the same way as the second law of thermo-
dynamics. Each is the result of the objections** which groups
of particles have toward compression in phase space. The
corrections introduced by the Pauli exclusion principle show
the degree of objection which particles feel when compression
tends to put additional ones in an occupied phase volume. The
mechanical vigor of the objections is the same in momentum
space as in ordinary space as measured by the gas law and
Van der Waal's equation.

* Chandrasekhar.
*' Kimball; Elastic Strain, p. 620.

II The Bulk Modulus and the Stress-Strain

Relations For Real Gases In Ordinary Space

From the gas law we calculate the isothermal bulk modulus

by differentiation at constant temperature.

(5) 7’: *— 5%? :z‘w,427

V

Here p is the pressure, V the volume, and n is the number of
molecules per 0.0. Multiplying both numbers of equation (5)
by n gives the modulus for stress and strain in momentum space.

.4, , 1

”I 7’

L’—

7.

It is interesting to note by comparing (6) and (7) that the
modulus of elasticity for change in volume is the pressure, p,
the same as for change in momentum Space.

For real gases obeying van der Waal's equation,
a I ' _ ‘ :
{WW/V W ”7

the isothermal bulk modulus calculated as above takes a

modified form.

(8) f'I/PIVJI’jp’

If we consider only repulsive forces due to impact the po-
tential energy term jgiidue to molecular attractions drops
out and Van der Waal‘s equation simplifies to

p (V—b): RT.

The simplified equation gives

__'_ 0/79 , M
<9) ’9“ .17 22-77:)
V-/L F B

for the bulk modulus instead of expression (8). In the right
member of equation (9) we let V:- ’1'}: and—N4» : 734 for sake of
later comparisons (see (14)). The expression (9) shows that,
if molecular attractions are neglected and account taken only
of molecular size, the modulus of elasticity p is the same as
for gases obeying the gas law. Hence the incorporation of
what might be called "the Van der Waal exclusion principle"

has no effect on the modulus of elasticity.

III Entropy Strain, and Weight for Real

Cases in Ordinary Space.

If we use the gas law and restrict attention to iso-
thermal changee, the first law of thermodynamics gives the
relation between entrOpy change and the work term

pdv RHV

/c2 I ._
.5- : L a. :: '—-——-——/ __
(10) d 7 T V

The right member shows the relation between entropy and

strain.

If we consider real gases represented by Van der Waal's
equation and again restrict attention to isothermal changes,
the work term will be modified;

_ fl] ._ 0’17/V , [IV
(11) d5” 792 " 60* 27'1/ 'f ”RV—A.

The entropy is equal to the sum of the strains Y multiplied

by Boltzmann's constant k.

Elll/ ... I ‘ ..__/t :—
(12) g .-: R z 012 _ amw 1/ r C /rY

It has been shown* for a perfect gas, in which molecules
are mere points, how the range in ordinary Space, 3% , the
volume occupied per molecule, is related to the weight which
measures the thermodynamic probability for one out of N
particles. The N is the number of molecules in a gram molecule,

the amount considered.

For real gases, following the idea of Van der Waal's equation
and his constant b, we see that the range instead of being %% ,
will be restricted by the effective volume attributed to each

molecule by whatever method is used in calculating b.

* Kimball; Entropy and Probability, p. 1567.

Thus
_4_ ”4.4.2.4.;

is the range or volume which each particle has to itself
being the previous range reduced by ,4,, the effective
volume displaced by its physical size. In the right member
we let F, a constant distribution function, represent the
number of particles per unit volume in ordinary Space just
as f7 represents the number per unit volume in momentum
space. Also‘8==:f: is evidently the number of particles
that could be packed into a unit volume in case there is

no free motion. Hence the weight per molecule is

._ _— N I- E.
(15) w.- ,w= lain—WA, : Vvé —. 7w 3,,

For N molecules the weight is

_ , w . w. Jv
(ls) W— IV-vt/ 1%) (#7,?)

which when substituted in Boltzmann's equation gives
#5 - Wt 9/1

(1’) “ V’/

and

(18) ='k log W + C : Nk log (V—b)-+ C = kY.

The expression (1?) Checks equation (11) and the entrOpy
strain equation (2) shown* to hold for gases obeying the

gas law. It will be observed that b and B represent the
fact that only one particle at a time can occupy a volume
diSplaced by its physical dimensions. We might refer to
this as the Van der Waal exclusion principle by analogy with
or perhaps a part of the Pauli exclusion principle which

excludes extra particles from an occupied call in phase space.

IV Entropy Strain, and Weight for Real Gases

In Velocity and Momentum Space.

As in the classical statistics the number of particles
in a unit volume of velocity or momentum Space is also repre-

sented by a distribution function

(19) gig = flim/jw/ : 47; : ,-—.
o/ay

wherein (in : dudvdw is a volume element in the space, and

u, v, and w are the velocity components. The volume occupied
by a particle at this place in velocity space is given by the
reciprocal of this number as indicated. Where the gas law
applies this volume equals the range (vz-z7q ) that leads to
the classical distribution law**. It has to be multiplied by
N to give the weight per molecule, and the product for N
molecules gives the total weight W being related to entropy

* Kimball; Elastic Strain.
*’ Kimball; EntrOpy and Probability.

 

10

and the haxwell-Boltzmann law in the familiar way*.
For real gases we restrict the range given by (19)

according to the Pauli exclusion principle. Thus

/ —-”Ln/4;:‘V‘wC~ :,_/...
(20) Yi:—)—;}~al—f A t ’ jc‘

is the free empty range left to a particle at that place in

velocity or momentum Space after deducting C; the minimum

range it can occupy. Here A 2 if is evidently the mrnber of
such minimum ranges or cells in a unit volume of momentum
space, being the usual symbol** used to represent the Pauli
exclusion principle. equation (20) is seen to be identical
with (14) except that it applies to velocity or momentum Space
instead of ordinary Space, and suggests possible physical
relationship between the Pauli exclusion principle and the

Van der Waal exclusion principle".

A causal relationsnip between these two has already been
considered*** and then dismissed**** by shrenfest. The present
treatment, shows that both are due to figmg cause, i.e. modus
operandi of elastic forces in a restricted phase space. This
of course merely takes us another step forward in explanation
towards the ultimate "let cause".

‘ Kimball; Entropy gpd_Probability.

** Chandrasekhar. .

‘** P. shrenfest; Relation bgtween the Reciprocal Impenetra-
bility 2£_Hatter and Pauli's mxclusion Principlg,
Nature, volt—TISHTTSZT) p. 196.

****P. Ehrenfest; Relation between the Reciprocal Impenetra-

bility‘g£_Matter and Pauli's gxglusion Principle,
Nature, vol. 119 (1927) p. 602.

11

Thus we see that a, r: 2,“ is like [,2 Zé = % , the latter
being the volume in ordinary space that a particle must
occupy to the exclusion of other particles by virtue of its
very physical existence.

Having obtained the expression for range we multiply
this by N to give the weight per molecule which measures the
probability that any one of N molecules may occupy this

particular range, thus
( ) /Vn j? l/ A j!

and for N molecules the weight is

 

I a— f’. I fp‘ I \l ”N
. _ V' , ____ N / '" Irrp-é. ‘ - l- f: ____n _,.
(22) W“ N (7177.017. ”‘73) ' N 7i)/Z_é/( / ofljz V?”

where the g's are the reciprocals of the ranges. Sub-
stitution in Boltzmann's equation as before gives the entropy.

fry/tr: /
f ’ I N N f’l’ .- évi I
(23) 5 1': /f/€/ej W-f—C :- ”Luff/V +7§ ("inf 71-)+C .. A“ 7‘, ._ /1’ Y

Equations (21) and (22) have been given by Chandrasekhar’.
They are here shown in terms of the free range (20) left to
a particle after the volume a, E Al which it diSplacee in
momentum space is deducted according to Pauli's exclusion
principle. It is significant that the weight is still N”

times the product of the free ranges reserved to the particles

* Chandrasekhar.

12

in momentum space, and the entropy is the sum of the corres-
ponding strains exactly as it was in the case of a gas obey-
ing the gas law.

Although the argument of this section refers to velocity
or momentum space it may be generalized to include phase space
(action Space) and leads to formulas of the same form as (21)

(22) and (23).

v The Equilibrium Equations rcr Stress and

Strain in the Fermi-Dirac Statistics.

To find the maximum value of W subject to a constant
energy, E, we use the method of Lagrange. The function ¢
is formed by adding E times an undetermined constant ;\ , to
w; 40: was . Take N partial derivatives of the now N inde-

pendent rd: , and set each partial derivative equal to zero.

'1

<24) at... calla: ..__ .,
Bu; 3% fiuz E M/jJ;.L7 J‘+A"H¢ ’0

But in view of (20)

(as) w 1:». A (a; r — w: + A tad

7c. 4]" '
wherein the partial derivatives are taken with respect to
the LX velocity component of each molecule. Equation (25)

shows that there is a constant ratio, namely,

n H _ 97) 14: Fly, 2' ¢}‘7\_‘(’/: [/14] “:7 _. Mg!
(20) — A — 7” .14 lit/p25».
1’7” ”n f A .'

13

between energy change per molecule (between successive
molecules) and the corresponding strain just as was shown

to be the case for the Maxwell—Boltzmann distribution. This
ratio is of course the elastic modulus per molecule. To find
its value we substitute in (26) the value of the X. determined

by the integrals

We“) , For} P ._.
(27) 3/jilujdfi : / 2] hymn/12:1;
0 J fl
wherein
, /
( 28) fit! J I m- —._.’»
(76"1w; it

is determined from (25) and the f; is the energy associated
with each molecule. The determination of C from these integrals
presents difficulties which do not concern equation (26). The
.A however which appears in the eXponential term of equation
(28) is required* by thermodynamics to have the value -Z%% .

It follows that ,ffig) given by (28) will have the same form

as the Fermi-Dirac distribution law. This formula of the new
statistics is thus derived by the geometrical weight method

as already shown by Chandrasekhar**. Substituting-jfi%;for.)

in equation (26) gives the elastic modulus in terms of mean

values of u and r:

 

(29) A/ w —- 213:4 0/.“ :: -— Wm, M 0&2:
’ " 17. mi ,. 1,) .
”r 7/ A

* J. Rice; Statistical Mechanics, p. 274.
*‘Chandrasekhar.

l4

Multiplying both members of equation (29) by n gives

(30) :m/ff”: WMMP/M_ 4777101141;
% 172— "M " g 9/7; // _:——_£-
7 7 A

These formulas are like (6) and (7) and snow that in the
case of the Fermi—Dirac statistics as well as the classical
law, the elastic modulus per molecule is kT. For a unit
volume it is equal to the pressure (nkT : p). Likewise the
'elastic modulus appearing in equation (30) is the same as
that in (9) which refers to real gases in ordinary space.
Now (9) represents real gases in which the Van der Waal
constant b restricts the volume and no intermolecular
attractions are considered. Thus there appears a striking
parallelism (as represented by (29) and (30) on the one hand
and (9) on the other) between the behavior of molecules in
velocity Space and ordinary Space. In both types of space
molecules "object" to being crowded together with the same
vigor (same modulus of elasticity) whether their range is
curtailed by the Pauli exclusion principle or the Van der Waal
exclusion principle. Furthermore this resistance to com-
pression is the same in each case as for the simple theory

in Which the exclusion principles are neglected*.
VI Intermolecular Attractive Forces.

On the other hand the elastic modulus given by equation

(7) is not the same as in the other cases, since here

‘ Kimball; Elastic Strain, p. 620.

15

intermolecular attractions are considered. It is recognized‘
in connection with the new statistics that the Fermi-Dirac
distribution law, with the negative correction term (":§f)
appearing in (21) and (22), correSponds experimentally to the
case of an extra repulsion between particles, and is well
verified by a gas of electrons or protons. Where attractive
forces exist as with neutral molecules, the Bose-Einstein
statistics ought to hold. If we were to introduce attractive
forces into equilbrium equations like (24) and (25), it seems
likely that they would correSpond to the new statistical
expressions for weight like (21) and (22) except with a
positive correction term (1nfif). Such changes** give the
Bose-Einstein statistics which can probably be fitted into
the stress~strain,naad entrOpy theory. The present treat-

ment, however, considers only the repulsions correSponding

to the Pauli exclusion principle.
VII Experimental Verification.

‘The most important verification of the stress strain
relations and the allied entrepy strain equations (2), (18),
and (23) is that they afford a natural and obvious explana-
tion*** of the second law of thermodynamics. The "ergodic"
hypothesis, on the other hand, has provén quite unsatisfactory
to most physicists****.

‘ P. Ehrenfest and J. R. Oppenheimer; Note on the Statistics
2;: Nuclei, Physical Reviews, trot-753719.?) p. 333.
*‘ Chandrasekhar.

it. Kimball; Elastic Strain, sec. 7.
‘***R. C. Tolman; Statistical Mechanics, p. 39.

16

The present viewpoint illuminates the heretofore unex-
plained Pauli exclusion principle. Not only is there a striking
analogy between the equilibrium equation (9) for ordinary space
and (29) and (30) for momentum space, but the modulus of elas-
ticity is the same for the two cases, indicating the Operation
of the same mechanical causes in the two realms. Even the

exclusion of occupied regions in the two cases introduces the

;C
A

for the Pauli exclusion principle) in an exactly analogous

correction terms (% =3? for Van der Waal's equation, and

way. This shows that their mechanical effect in these force
equations is of the same type.

The operation of the Pauli exclusion principle is ex-
plained in exactly the same way as the second law of thermo-
dynamics. Each of these is mechanically accounted for by the
operation of forces acting in momentum Space. They are a
measure of the force which excludes particles from occupied
cells in momentum space. The correction term given by the
Pauli exclusion principle expresses the degree of the objection
to forces tending to put particles in occupied phase positions.

Furthermore it is worthy of mention that the fundamental
relations of the Fermi-Dirac statistics are obtained without
the usual technical rules for counting complexions and the
associated difficulties. It is significant that the two
points which the statistical methOd does not explain, namely,
the second law of thermodynamics and the Pauli exclusion
principle, are explained by the present mechanical stress

strain theory.

J.

L.

Bibliography

Bloch; Tpg_Kinetic neory p£_ga§§§

S. Kimball; Entrony and Probability, Journal of Physical
Chemistry, vol. 33 (1929) p. 1558.

S. Kimball; Entropy, Elastic Strain, the Second Law 9;
Thermodynamics, etc., Journal of Physical Chemistry,

vol. 35 (1931) p. 611.

. Chandrasekhar; 9g3the ggpbabilipy_hethodiig the New

Statistics, Philos0phical Magazine, vol. 9 (1930) p. 621.

Rice; Statistical Mechanics.

. C. Tolman; Statistical Mechanics.

B. Loeb; Kinetic Theory pf Gases.
Ehrenfest; Relation between the Reciprocal Impenetrabiligg
p§_hatter and Pauli's Exclusion Principle, Nature, vol. 119

 

Ehrenfest; Relation between the Reciprocal Impenetrability

9£_Matter and Pauli's Exclusiqp Principle, g ggrrection,

 

Nature, vol. 119 (1927) p. 602.
Ehrenfest and J. R. Oppenheimer; Note ppgthe Statistics 2;

Nuclei, Physical Reviews, vol. 33 (1931) p. 333.

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