## Primer on Entropy - Part II B LO19987

AM de Lange (amdelange@gold.up.ac.za)
Tue, 24 Nov 1998 12:01:01 +0200

Dear Organlearners,

PART II -- THE GENESIS OF ENTROPY (Continue)

The Twin
---------
That doyen of experimentalists, probably the greatest experimentalist
ever to have lived, Michael Faraday, wrote in 1951 in his "Some
thoughts on the Conservation of Forces" (the word "energy" was still
not in use) that not only is the conversion of the "live forces"
(forms of energy) important, but at least as important is the "force"
(the concept "entropy production" was not yet known) which drives
these conversion of "forces". He also wrote that heat was peculiar
from all other "conservation forces" (forms of energy) in that not all
of it could be converted into the rest. It is strange how this thought
made not the least impression on his fellow scientists. [It noteworthy
that Max Planck, who began to think along the same lines fifty years
later, felt the same isolation, even though thinking along these lines
helped him to discover the quantum effect. People were just not ready
to try and explain why heat had such a unique role.]

Clausius. By 1850 he already succeeded to redescribe Carnot's findings
by making use of the Law of Conservation of Energy. He showed that the
efficiency of a steam engine is given by the ratio of the work
delivered by the engine divided by the heat which enters the engine.
Thus he managed to formulate the efficiency in terms of the ratio of
absolute temperatures T, namely (Tin-Tout)/Tin where Tin is the higher
"absolute temperature" T of the steam going IN and Tout is the lower
"absolute temperature" T of the steam going OUT of the engine after it
has done work.

Carnot cycle. He derived, using the Carnot cycle, among others the
equation Qin/Tin = Qout/Tout. (Whereas T is the symbol for absolute
temperature, Q is the symbol for heat.) In this equation Qin is the
major heat entering the engine at the higher temperature Tin while
Qout is the minor heat leaving the engine at lower Tout. He kept on
hammering this equation with questions, trying to understand it. It
worried him that he had to work with such a highly idealised cycle
called the "reversible" Carnot cycle to able to derive the equation
Qin/Tin = Qout/Tout. In a reversible Carnot cycle the engine has to
work so slowly that it appears as if nothing is happening in the
engine -- only being, no becoming. This is so contrary to real steam
engines. For almost fifteen years, week after weak, year after year,
he struggled with this problem while his fellow scientists enjoyed the
life of their "newly found higher law" -- feeding it, letting it grow.
What does the quantity Q/T (heat Q divided by the absolute temperature
T) in the equation Qin/Tin = Qout/Tout really means?

Then one day, it is not clear when, he had the dazzling insight that
"heat" and "heat content" is not the same thing! The "heat content" is
the "energy" within a body which gives rise to its temperature T. In
other words, the "heat content" is the "thermal energy" of a body. But
heat itself is "thermal energy in flow", flowing from a higher to a
lower temperature. Only when there is a temperature difference, will
the being-like "thermal energy" manifest itself as the becoming-like
"thermal energy in flow" (heat). He immediately realised that Q/T
describe a systemic quantity which he called entropy S (Greek:
"en"=in, "trepo"=turn, transform). (By calling it a systemic quantity,
we mean that the system has the property S just as it has other
properties like energy E, temperature T, pressure P volume V, etc.).
The important thing to understand was that Clausius realised for the
first time ever that

a CHANGE in the entropy S of a system can be
CALCULATED in terms of a "reversible heat flow"
divided by the absolute temperature.

He most probably did the following calculation which helped the twin
to get born! This calculation is very easy, even though you may never
have done it before. We are going to do a "mind experiment". Let us
picture a long metal bar with ends labelled by "a" and "b"

(a)=======================(b)

Assume that end "a" is at a lower temperature ("intensity of fire")
than end "b". The recommended symbol for "absolute temperature" is T.
We will symbolise the "absolute temperature" at end "a" by Ta and at
end "b" by Tb. The "absolute temperature" is a temperature of which
the zero of its scale begins at the lowest possible temperature in
nature. The unit of the temperature scale "celsius" is symbolised by C
The temperature scale "celsius" will not qualify because its zero
begins at the melting point of ice. Much lower temperatures are
possible. But the temperature scale "kelvin" will do. Its unit is
symbolised by K. The unit K is exactly equal to the unit C. But the
zero of the temperatures scale "kelvin" begins at the lowest possible
temperature ("absolute zero"). This lowest temperature in degrees
Celsius is -273C.

Assume end "a" is at an absolute temperature Ta=300K (colder) and end
"b" is at absolute temperature Tb=400K (hotter). Representing this
information on the bar will lead to:

(a)=======================(b)
300K 400K

The freezing point of water is 0C in "celsius" and 273K in "kelvin".
Since the units 1C and 1K are equal, the temperature 300K corresponds
to 27C which is roughly room temperature. Likewise 400K is about 27C
hotter than 100C, the boiling point of water.

Whereas temperature measures the "intensity of fire", heat measures
the amount of energy released by the fire. Therefor heat is measured
in the unit of energy which has the name "joule" and the symbol J.
(Before Joule, people measured heat in terms of the unit "calory".
Joule showed that 1calory = 4.18joule.) Because "b" is at a higher
temperature than "a", heat will flow from "b" to "a". Assume that
1200J heat will flow from "b" to "a". Representing this information on

(a)=<=========<==========<=(b)
300K 400K
+1200J -1200J

The arrows shows the direction in which the heat will flow. The
+(plus) sign before 1200J at end "a" and the - (minus) before 1200J at
end "b" are very important. The + sign at "a" says that heat ENTERS
"a" and thus INCREASES its energy. The - sign at "b" says that heat
LEAVES "b" and thus DECREASES its energy. Exactly as much energy
(-1200J) leaves "b" as the energy (+1200J) which enters "a". Thus,
should we add these two quantities up as follows:
+1200J -1200J = 0J
the result "zero joule" tells us that the total energy is constant
(conserved). In other words, no energy has been created or destroyed.
This is the Law of Energy Conservation (LEC).

Now let us do very simple calculations, using Clausius brilliant
insight that the CHANGE in the entropy S of a system can be CALCULATED
in terms of a "reversible heat flow" divided by the absolute
temperature. Thus:
entropy change at "a"
= (+1200J/300K)
= +4J/K
Note how we divide the magnitude 1200 of the heat by the magnitude 300
of the temperature to get the magnitude 4 of the change in entropy.
Also note how we "divide" the unit J ("joule") of heat by the unit K
("kelvin") of temperature to get the unit J/K ("joule per kelvin") of
entropy. Note how the + sign carries over. The + sign says that the
change of entropy at "a" is an "increase". Similarly, we calculate
entropy change at "b"
= (-1200J/400K)
= -3J/K
Let us add these two changes together:
The TOTAL entropy change for WHOLE bar
= entropy change at "a" + entropy change at "b"
= (+4J/K) + (-3J/K)
= (+4-3)J/K
= +1J/K

Observe it again!
The TOTAL entropy change for WHOLE bar = +1J/K
The + sign says that the total entropy has increased!! This sort of
thing can happen anywhere in the universe because anything in the
universe has a temperature.

The twin has been born! Clausius probably jotted down the following
words just to repeat it in his famous paper of 1865:

The energy of the universe is constant
The entropy of the universe increases.

(Note that we have not yet included the important words "towards a
maximum". We will get to that later.)

What strange thoughts must have went through the mind of Rudolf Julius
Emmanuel Clausius? Maybe he thought: "Rudolph, get back to the nest of
constancy and conservation. You are not an eagle which can soar into
the sky to view new worlds. You are a chicken which has to be fed by
the grown ups. Testing your wings against the wind blowing past the
edge on the rock face was but youthful insurgence." But then he would
also have thought: "Rudolph, nothing is wrong with your calculations.
They are so easy that everybody can check them. Yet they somehow
remind you of the intense fire burning in you, spreading the your
thoughts like heat in all directions. Do not return to the nest, but
soar further to explore the new worlds you are seeing."

It is a pity that his actual thoughts have not been recorded anywhere.
Maybe he did try to publish it somewhere, but his peers thought it too
trivial to qualify for publication. Who knows? Whereas many thinkers
participated together as a Learning Organisation in the emergence of
the Law of Energy Conservation (LEC), only one thinker as a Learning
Individual experienced the emergence of the Law of Entropy Production
(LEP). Whereas many shared joy and curiosity as adjoints of the
emergence of the LEC, only one experienced these adjoints in the
emergence of the LEP. Whereas the LEC quickly grew into a gigantic
complexity, the stunning simplicity of the LE P prevented further
growth (complexification). Why? The "twin syndrome" had lifted its

Clausius was a soaring eagle. But let me explain the "twin syndrome"
by telling about a bird of a different kind.

Twin syndrome
-------------------
Birds like eagles and vultures can fly all day, but a raptor bird like
the Turkey Buzzard (Southern Ground Hornbill) prefers to walk. The TB
is found in Southern Africa. Except in the game parks, the TB is becom
ing vary scarce because of its habits. It is as big as a stork, but
with a much shorter neck and legs. It is covered with black feathers,
making its appearance stern like that of a clergyman. Part of its neck
is bare, almost like a turkey, showing a blood red tissue as if it is
continuously at war, ready to kill with its long, pointed bill. The TB
lives in families containing up to five breeding pairs. During the
breading season in summer a TB family will jealously guard their
territory which may be up to 100 square kilometres. A TB makes a deep
droning sound "dru-dru-drudru" which can carry as far as 5 kilometres.
Should a human hear that sound and imitate it, members of the family
will soon come along, flying in short bursts and running in between,
ready to chase away the intruder by all means.

A TB pair breeds once every two or three years, laying two eggs in a
hollow tree. The male closes the hollow with a clay wall, except for a
small opening through which the female (and later the offspring) will
be fed by the male and possible other members of the family. When the
first egg hatch, the chick eats ferociously. Thus when the second egg
hatches, the first chick is strong enough to kill the second chick. It
immediately begins with that task, succeeding in a couple of days. The
female and the surviving chick stay for months in the hollow until the
chick has matured near the end of the season. Then the male breaks the
clay closure down so that they can come out. Thus at least two broods
of TB (involving approximately 5 years) are necessary to keep their
numbers constant.

I will not describe the twin syndrome by the history of humankind, for
example Esau and Jacob. I will rather use the description above as
metaphor to analyse the genesis of the concept entropy.

The eggs of the brood was laid by Copernicus in 1543. In 1852
(Thomson) the first chick (Law of Conservation of Energy) hatched. In
1865 (Clausius) the second chick (Law of Entropy Production) hatched.
Thirteen years separating the two chicks. It may seem to be a long
time interval, but in the history of the brood (309 years) it is not
that long an interval. Fortunately, the interval was too short for the
first chick to become strong enough to kill the second chick. But a
desperate struggle for survival ensued. This is what happened.

The "problem" with the Law of Entropy Production (LEP) is that it is
fundamentally different to the Law of Energy Conservation (LEC) and
all other conservation laws. Whereas each of them conserve a quantity
(by keeping it constant), this law does the opposite. Thus it cannot
fit into them and they cannot fit into it. The first way to consider
LEC and LEP is as "dialectical duals", as conflicting opposites. This
leads to, among many things, the "twin syndrome". For example, LEC
(the books must balance) acts very much like an accountant whereas LEP
(money has to be made) acts very much like an economist -- and these
two are often at each other's throats! The second way to consider them
is as "complementary duals" like verbs and nouns are. The one is
needed to understand the other. For example, in the calculation we
employed LEC (remember the + - signs?) to come to LEP (only the +
sign).

Soon after LEP was formulated, two Scots (note!), namely William
Thomson and James Clerk Maxwell and the German Ludwig Boltzmann took
it upon them to fit the LEP into the LEC (the accountancy law). But
first they had to explain what temperature is. Thus they developed in
a couple of years one of the most remarkable theories of hard core
science, namely the Kinetic Molecular Theory (KMT). The KMT rests on
seven hypotheses:
1 Gas-like matter consists of particles (atoms or
molecules) far way from each other.
2 These particles are like hard points without an
inner structure.
3 These particles are in continuous, random motion so
that they collide only occasionally with each other or
the walls of the container.
4 When these particles collide, the collisions are perfectly
elastic.
5 The total translational kinetic energy of all the particles
is equal to the thermal energy of the gas.
6 The thermal energy of the gas is proportional to its
absolute temperature.
7 The average kinetic energy of a molecule is equal to the
total kinetic energy divided by the number of the molecules.

With these hypotheses (originally called postulates by the KMT
thinkers) they concluded many things, among others that:
1 The pressure of the gas is proportional to the total kinetic
energy of all the particles.
2 The absolute temperature of the gas is proportional to the
average of the velocity square of a single particle.
3 The velocities of the molecules followed a Gaussian (bell
shaped) distribution (function f(v), Maxwell-1867).
Thus they were able to derive all the gas laws (Boyle, Charles,
Guy-Lussac).

But the KMT thinkers introduced by these HYPOTHESIS some fixed mental
models which are deadly to the understanding of "entropy" and "entropy
production". To pinpoint these mental models, I have reorganised the
"postulates" (since their formulation a century ago) in the hypotheses
above.
1 The KMT thinkers reduced by hypothesis 1 matter into
particles which seldom come into contact with each other.
But what about liquids and solids in which the particles are
continuously in contact with each other and hence
influencing one another?
2 They reduced by hypothesis 2 matter into point-like particles
without an inner chemical structure. They knew nothing of the
hundreds of thousands of different shapes of molecules
brought about by the organisation of the atoms inside them.
3 They reduced by hypothesis 3 all behaviour into chaotic
(disordered, random) linear motion between the particles.
They knew nothing about the behaviour of electrons and
nuclei within the well ordered structure of the molecule.
(The electron and nucleus was not even discovered then!)
The electrons and nuclei TOGETHER perform fantastic
nonlinear motions such as rotations, vibrations, pulsations
and jumps.
4 They reduced by hypothesis 4 all contacts into elastic,
unproductive collisions so that the law of conservation
of kinetic energy could hold and hence temperature could
be proportional to total kinetic energy. But what about
inelastic, productive collisions which result into new kinds
of molecules? In other words, they forced chemistry out
of the models of physics, making physics the study of
conservative systems. But a chemist knows that when
molecules react with each other, the thermal energy
(and hence total kinetic energy) changes drastically.

Thus Boltzmann (1872) set out to give account of the LEP in terms of
the LEC. To do this, he had to make an additional "postulate", based
on the outlook which the "postulates" of KMT provided. He postulated
that the entropy S of a gas is proportional to the logarithm of its
probability P for DISORDER, namely

S = k log P

To calculate P, he made use of Maxwell's distribution function f( )
for molecular velocities. But soon he encountered a serious problem.
What happens with the distribution of the velocities the moment when
molecules collide? Well, he envisaged a new kind of disorder emerging
from Maxwell's disorder. Thus he split Maxwell's distribution function
f( ) into two terms, f(regular) for the old DISORDER involving motion
and f(collision) for the new SUPER DISORDER. He went on and on,
eventually defining his famous H quantity as

H = int.f.log f.dv

where "int" is the integral of mathematical calculus. When he began to
investigate how H changes through the course of time, he found that H
decreases. But this was exactly opposite to the entropy S which
increases! How could that be? He tried time and again to solve this
anomaly, but he could not succeed. Eventually he became very
disillusioned with his own work.

However, others (like Planck, Einstein and Schroedinger) did not give
up. They all tried their hands on this dilemma, but with no success.
Yet nobody questioned his notion of a "new super disorder" caused by
collisions. Why not? By hypothesis 4 they closed (postulated) their
world of physics into a conservative system, thus excluding
non-conservative systems like chemistry and biology. Since none of
them was a chemist and very few chemists studied physics so deep,
nobody had the experience to question the "super disorder" introduced
by the term f(collisions). Meanwhile, in 1926, Erwin Schroedinger
became the father of Quantum Mechanics by deriving his famous wave
equation. But he kept on trying to solve Boltzmann's dilemma. In 1929,
then already famous, he wrote: "His [Boltzmann's] line of thought may
be called my first love in science. No other has ever thus enraptured
me or will ever do so again."

It is the same Schroedinger who in 1944 ("What is life") claimed that
life and the Law (LEP) which predicts increasing disorder cannot go
together. Thus he introduced his notion of "negentropy" (negative
entropy) to explain how life evolves into more order. Living organisms
escape with negentropy the LEP which (as he believed according to his
mental model) drives the universe into more disorder. Maybe
Schroedinger was thinking of the H quantity (which never was even
named) when he announced his notion of "negentropy". But he never even
indicated how this "negentropy" should be measured or calculated from
measurements. Compare it with the original definition of entropy by
Clausius which tells clearly how to do it, namely

the CHANGE in the entropy S of a system can be
CALCULATED in terms of a "reversible heat flow"
divided by the absolute temperature.

Clausius never wrote anything about chaos and disorder. The Law of
Energy Conservation (LEC) and the Law of Entropy Production (LEP) were
for him two twins complementing each other. Chaos was the brain child
of Thomson, Maxwell, Boltzmann and others who tried to "explain" LEP
in terms of conservative physics. This branch of physics became known
as Statistical Mechanics. Later arrivals to physics like Planck,
Einstein, Heisenberg and Schroedinger tried to solve Boltzmann's
contributions to physics, but Boltzmann's dilemma in Statistical
Mechanics remained a mystery. Yet none of them ever expected
Boltzmann's interpretation of entropy as a measure of chaos to be the
culprit. None of them ever expected Boltzmann's split of Maxwell's
( ) function into a regular disorder (caused by moving molecules) and
a super disorder (caused by colliding molecules) to be the very point
where the elder twin lost the fight over the younger twin.

It is not that they were completely unaware of some serious "error"
having been made somewhere along the line. Einstein was the first to
raise his voice after Max Born gave the probability interpretation to
the wave function of quantum mechanics. He wrote to Born: "You believe
in the God who plays dice, and I in complete law and order in a world
which objectively exists, and which I, in a wildly speculative way, am
trying to capture. I firmly believe, but hope that someone will
discover a more realistic way, or rather more tangible basis than it
has been my lot do." Note how he and Born formed "dialectical duals"
(like order "versus" chaos) rather than "complementary duals" (like
order "and" chaos). Also Heisenberg became convinced late in life,
after having studied in depth high energy conversions which happen
when cosmic rays collide with matter, that somewhere in their thinking
of physical systems a serious error has been made, one which he
desperately tried to pinpoint. Pauli had similar misgivings. Even
Schroedinger's attempt with "What is life", despite his fixed mental
model concerning disorder, must be seen in this light of trying to
discover the error.

Like the Turkey Buzzard, many fine people and great thinkers were also
victims of the "twin syndrome". We cannot change the behaviour
(including the "twin syndrome") of these beautiful birds. But can
each of us change ourselves? Can we heal ourselves from the "twin
syndrome"? Can we perceive beyond the "dialectical dual" a possible
"complementary dual"? Let us see.

Best wishes

```--

At de Lange <amdelange@gold.up.ac.za>
Snailmail:    A M de Lange
Gold Fields Computer Centre
Faculty of Science - University of Pretoria
Pretoria 0001 - Rep of South Africa

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