Before reading this essay, go to your kitchen and find a bottle of Italian salad dressing. Get one that’s been sitting still for a while at a fixed temperature—that is, one in thermal equilibrium. You’ll find an oil-rich layer at the top of the bottle and a vinegar-rich layer at the bottom (see Fig. 1). But think for a moment before spreading it over a delicious salad and eating up. That bottle’s in thermal equilibrium, so it’s in a state of maximum entropy. Doesn’t entropy mean “disorder”? No one would call a stack of 50 pennies and 50 dimes disordered if all the dimes were on the top and all the pennies at the bottom. So why is this salad dressing at thermal equilibrium segregated like an ordered stack of coins?

1.
By “uniform” I mean “uniform on a macroscopic scale.” All phases of all materials are of course granular on the atomic scale.
2.
Todd P.
Silverstein
, “
The real reason why oil and water don’t mix
,”
J. Chem. Educ.
75
,
116
118
(
1998
).
3.
Thomas A.
Moore
and
Daniel V.
Schroeder
, “
A different approach to introducing statistical mechanics
,”
Am. J. Phys.
65
,
26
36
(
Jan.
1997
).
4.
D. F.
Styer
, “
Insight into entropy
,”
Am. J. Phys.
68
,
1090
1096
(
Dec.
2000
).
5.
“Wheeler’s first moral principle. … Guess the answer to every puzzle. Courage: no one else needs to know what the guess is. Therefore make it quickly, by instinct. A right guess reinforces this instinct. A wrong guess brings the refreshment of surprise.”
Edwin F.
Taylor
and
John Archibald
Wheeler
,
Spacetime Physics
, 1st ed. (
W. H. Freeman and Company
,
San Francisco
,
1963
) p.
60
; 2nd ed. (1992), p. 20.
6.
This conclusion cannot be avoided by invoking “the law of large numbers” or “the thermodynamic limit.” For if we had a million times more squares scattered on a million times more sites, then the configurations in Fig. 2 would be absolutely typical for a window containing one-millionth of the total sites.
7.
A more extreme example comes from family size. In the 2010 census, the average American family had 3.14 members—π to three significant digits—despite the fact that no family whatsoever had 3.14 members. See Table II in
Daphne
Lofquist
,
Terry
Lugaila
,
Martin
O’Connell
, and
Sarah
Feliz
, “
Households and Families: 2010
,”
U.S. Census Bureau
, issued April
2012
, at http://www.census.gov/prod/cen2010/briefs/c2010br-14.pdf.
8.
See, for example,
Ihsan
Barin
,
Thermochemical Data of Pure Substances
(
VCH Publishers
,
Weinheim, Germany
,
1995
).
9.
Richard P.
Taylor
,
Adam P.
Micolich
, and
David
Jonas
, “
Fractal analysis of Pollock’s drip paintings
,”
Nat.
399
,
422
(
June
3,
1999
).
10.
Parker
Tyler
, “
Jackson Pollock: The infinite labyrinth
,”
Magazine of Art
43
(
3
),
92
93
(
March
1950
).
11.
Ray Nelson
Faulkner
,
Art Today: An Introduction to the Fine and Functional Arts
(
Holt, Rinehart and Winston
,
New York
,
1956
), p.
433
.
12.
Stephen G.
Brush
,
The Kind of Motion We Call Heat
(
North-Holland
,
Amsterdam
,
1976
).
13.
Given the vagueness of the word “disorder,” we must ask the exact meaning of the phrase “a gas at equilibrium is disordered.” First consider an ideal gas subject to no external forces: An atom is equally likely to occupy any position within its container, and pairs of atoms are uncorrelated. The velocity of the atom is equally likely to point in any direction. But the speed of the atom is very much not “equally likely to have any value”: indeed, the probability of the atom having speed between v and v + dv is given by the Maxwell-Boltzmann distribution $2π(mkBT)3v2e−mv2/2kBTdv.$ In contrast, for a real gas subject to no external forces, an atom is equally likely to occupy any position within its container, but pairs of atoms are correlated. (See for example David Chandler, Introduction to Modern Statistical Mechanics [Oxford University Press, New York, 1987], Chap. 7.) The velocity remains equally likely to point in any direction, but the speed probabilities do not follow the Maxwell-Boltzmann distribution (which is derived assuming that the only energy is kinetic).
14.
Sadi
Carnot
,
Réflexions sur la Puissance Motrice du Feu
(
Chez Bachelier, Libraire
,
Paris
,
1824
).
15.
Rudolf
Clausius
, “Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie,” read at the Philosophical Society of Zürich on April 24, 1865. (Translated by John Tyndall as “On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat” in
The Mechanical Theory of Heat, with its Applications to the Steam-Engine and to the Physical Properties of Bodies
[
John Van Voorst
,
London
,
1867
], pp.
327
374
, at p. 357.)
16.
Ludwig
Boltzmann
, “
Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen
,”
Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Vienna
,
66
,
275
370
(
1872
).
(Translated by
Stephen G. Brush
as
“Further Studies on the Thermal Equilibrium of Gas Molecules” in
The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary
[
Imperial College Press
,
London
,
2003
], pp.
262
349
; excerpts quoted here appear on pp. 262, 263, and 291.)
17.
Perhaps because he was at the time engaged in a priority quarrel with Clausius. See
Giovanni
Gallavotti
, “
Ergodicity: A historical perspective
,”
Eur. Phys. J. H
41
,
181
259
(
2016
).
18.
Hermann
von Helmholtz
, “
Die Thermodynamik chemischer Vorgänge
,”
Sitzungsberichten der Akademie der Wissenschaften zu Berlin
,
1
,
23
(
Feb.
2,
1882
).
(Translated as “The Thermodynamics of Chemical Processes.” See
Ralph
Baierlein
and
Clayton
Gearhart
, “
The disorder metaphor
,”
Am. J. Phys.
71
,
103
[
Feb.
2003
].
See also the translation by
E. F. J.
Love
in
Physical Memoirs Selected and Translated from Foreign Sources
[
Taylor and Francis
,
London
,
1888
], pp.
43
62
, at p. 56.)
19.
Satyendra
Ray
, “
The solid-gas, – A criticism of the definition of entropy as disorderliness
,”
Proc. Indian Sci. Congress
21
,
166
167
(
1934
).
20.
Karl K.
Darrow
, “
The concept of entropy
,”
Am. J. Phys.
12
,
183
196
(
Aug.
1944
).
Jeffrey A.
Phillips
, “
The macro and micro of it is that entropy is the spread of energy
,”
Phys. Teach.
54
,
344
347
(
Sept.
2016
).
See the appendix at TPT Online, http://dx.doi.org/10.1119/1.5126822 under the Supplemental tab for 16 related critiques with publication dates between 1944 and 2016.
21.
R. H.
Fowler
and
E. A.
Guggenheim
,
Statistical Thermodynamics
(
Cambridge University Press
,
Cambridge, UK
,
1939
) does not mention disorder at all, but remarks that the related concept of absolute entropy “has caused much confusion and been of very little assistance” (p.
192
).
22.
Henry
Adams
,
The Education of Henry Adams
(
Houghton Mifflin Company
,
Boston
,
1918
).
23.
John
Adams
, “
Statement for the Defense at the Boston Massacre Trial
” (
Dec.
3-4, 1770), see http://founders.archives.gov/documents/Adams/05-03-02-0001-0004-0016.
24.
Reference 22 at p. 451.
25.
Henry
Adams
, “The Rule of Phase Applied to History,” written Jan. 1, 1909, published on pp. 264–311 of
The Degradation of the Democratic Dogma
(
Macmillan
,
New York
,
1919
), excerpts quoted here appear on pp.
302
, 305, and
308
.
26.
Henry
Adams
, “A Letter to American Teachers of History,” written Feb. 16, 1910, published on pp. 134–263 of
The Degradation of the Democratic Dogma
(
Macmillan
,
New York
,
1919
), excerpts quoted here appear on pp.
209
and
251
.
27.
Gilbert L.
Wedekind
,
Spiritual Entropy
(
Xulon Press
,
Fairfax, VA
,
2003
), p.
68
.
28.
Using the rusting of iron to illustrate entropy arguments is particularly common and particularly specious. The rust reaction is 4 Fe + 3 O2 → 2 Fe2O3. Anyone can look up the entropy values of iron, oxygen, and iron oxide at room temperature and atmospheric pressure. (See Ref. 8, pp. 675, 1239, and 702.) The entropy of the reactants is 724.6 J/K, the entropy of the products is 174.8 J/K. The rust reaction results in an entropy decrease! Of course, this doesn’t mean that when a car rusts, the entropy of the universe decreases: although the entropy of the iron plus oxygen decreases, the entropy of the surroundings increases by even more. But the same observation holds for the reverse reaction: in a blast furnace, oxygen is stripped from iron oxide to produce elemental iron. In this process, too, the entropy of the universe increases.
29.
Abigail
Van Buren
, “
Dear Abby: Improving ‘Rotten Apples
,’”
Los Angeles Times
(
Aug.
31,
1970
), p.
C5
.
30.
This statement was particularly damaging to me, because I read it the summer after I was introduced to the concept of entropy in my high school chemistry class. My understanding of entropy was weak at the time, and after reading this statement by S.A.S. I felt that I would never grasp it. How could morality be related to the entries in a table of entropy values? How could morality be measured in units of joules/kelvin? Little did I realize, at age 15 years, that the weakness lay in S.A.S., not in me.
31.
Alessandra
Stanley
, “
Red Update Alert: Poseidon Goes Belly Up Again
,”
New York Times
(
Nov.
18,
2005
).
32.
Editorial Board
, “
No Justice, No Police
,”
New York Times
(
Jan.
6,
2015
).

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