Primer on Entropy - Part III A LO20018

AM de Lange (
Tue, 24 Nov 1998 12:30:19 +0200

Replying to LO19979 --

Dear Organlearners,


Why higher laws?
I wrote that most extraordinary of the period 1835-55 was the great
excitement among those who struggled to articulate the Law of Energy
Conservation (LEC). Most obvious to them was the holistic implications of
this law. Although they did not invented or used the term "information
explosion" they were already tacitly aware of it. The LEC was most welcome
because it provided them with a web to keep physical and chemical
information together.

But there was also another reason for their excitement. Since so many
different forms of energy became known to people, all suggesting new
sources of energy, they began to invent devices to create energy -- to
become rich through such devices. The stream of such proposed devices to
patent offices, even up to today, began to pick up. But none of these
devices ever worked in practice so that no patent could be issued. These
unsuccessful applications are nothing else than attempts to falsify the
LEC. Thus the LEC became subjected to falsification more than any other
law -- and it stood the test.

There was much less excitement for the Law of Entropy Production. Nobody
realised the holistic importance that temperature was a property of all
bodies, inanimate or living. Yet the LEP became tested in patent offices
even more than the LEC. When one form of energy is converted into another,
the conversion is not perfect because some of the former energy gets
converted into heat rather than the second form of energy. It is called
the "degradation of energy". All devices proposed to circumvent the
degradation of energy, failed to do so. Only one man became very excited
that the LEP stood the test of falsification, namely sir Arthur Eddington.
He writes: "If somebody points out to you that your pet theory of the
universe is in disagreement with Maxwell's equations -- then so much the
worse for Maxwell's equations. ... But if your theory is found to be
against the second law of thermodynamics I can give you no hope, there is
nothing for it but to collapse in deepest humiliation."

By now you should have become aware that I am leading towards something.
Two qualities concerning both LEC and LEP caused scientists to become very
excited: "holism" and "falsification". These two qualities have as much to
do with the abstract world of mind as with the material world of brain.
These two qualities connect the physical world with the spiritual world.

But there is also a third quality which I must point out to you which
nobody else seems to have pointed out. Physicists and chemists can measure
tens of different quantities, using different measuring instruments. In
all the laws (except LEC and LEP) of physics and chemistry, these laws
relate measurable quantities to each other. In none of these laws (except
LEC and LEP) is there a quantity which could not be measured. In other
words, falsification of any of these laws merely required sufficient
measurements until some measurements were found which falsified the law.
With this process it was discovered that some laws were continent
(general, never an exception) like Newton's law of gravitation while
others laws were contingent like Ohm's law or Boyle's law. By simply
measuring two or more quantities, law upon law was discovered.
Measurement, accuracy and precision became the call. The success of this
call began to mesmerise people all over the entire academical spectrum,
including the humanities.

However, the two quantities energy and entropy cannot be measured directly
like mass, volume, current, force, power, etc. There is no measuring
instrument available to do so up to this day. These quantities can only be
CALCULATED in terms of other measured quantities. Furthermore,
mathematicians have been able to show that all numerical calculations
depend on the LAWS OF MIND. Thus these two quantities do no have any
direct counterpart outside the human mind. For example, I can show you
for the temperature 27C (which is a mental construct) a thermometer (not
part of me) reading 27 on its scale. I can show you for the electrical
current 5.8 amp (which is also mental construct) an ammeter (which is also
not part of me) with its needle pointing at 5.8. But I cannot point to any
technological device outside me which registers an energy of 1200J or an
entropy of +4J/K. I have to point at least two different technological
devices (needed for the measuring) as well as a "reckoner" (my brain or
any material innovation such as a slide ruler, a calculator or even a
computer). In other words, without a "reckoner" there is no sense in even
speaking of energy and entropy.

I cannot stress this third property enough which makes LEC and LEP higher
laws. Even being higher laws, LEC and LEP are in a certain sense still
very primitive because they depend on the mental activity "calculation".
There are many other kinds of mental activities possible making use of
numbers, some of them of a much higher order than calculations. Using
actual measurements and these higher activities can lead to powerful

Temperature and chaos/order.
>From the Kinetic Molecular Theory, even with its limitations as I have
pointed out, we know that for gases their absolute temperature T is
proportional to the "average of the velocity squared" of an individual
gas molecule. Now what does this "average of the velocity squared"
means? It means that should we measure the velocity of a molecule on
many occasions, it will have many different values. This is so because
after every collision with another molecule, its velocity changes.
Should we square each of these values and then determine the average,
the value will be proportional to the temperature.

But there is also another way we could understand the relation between
temperature and velocity by making use of Maxwell's concept of a
distribution function f( ). We make mentally a snapshot at a particular
point in time of all the molecules in the gas. Somehow our snapshot (a
frozen picture or being) has to show the velocity (becoming) of each
molecule. Let us now count the molecules having a velocity with a certain
magnitude (without direction) and plot the result on a graph. You will
have to do it yourself on a piece of paper next to you. Please take the
time and do it. Name the horizontal axis by "magnitude of velocity" and
the vertical axis by "number of molecules having certain velocity". You
can make a rough sketch of the graph as follows. Try to sketch a "bell
shaped" curve. The one end of the bell begins in the origin. First it
increases faster and faster until half-way up the bell. From there it
increases slower and slower so that at the top of the bell it cannot
increase any more. For the other half of the bell, let the graph decrease
faster and faster until halfway whereafter it decreases slower and slower.
By the time it reaches the horizontal axis, it cannot decrease anymore.
The bell should not be completely symmetrical -- it should be a little bit
fatter on the outward side, the side away from the origin.

The graph tells us that there are very few molecules having a velocity
close to zero. As we move away from the zero velocity, more and more
molecules have a higher velocity. The top of the bell indicates that a
maximum number of molecules have a velocity indicated by the middle of the
bell. As we move still further away the number of molecules having an even
higher velocity decreases. At the tail end very few molecules have a very
high velocity. This tail end had no special meaning for physicists like
Thomson, Maxwell and Boltzmann. But for the chemists of today, it is of
extreme importance. It is exactly these molecules with a very high
velocity and thus kinetic energy which can "penetrate" into another
molecule, thus changing its internal ORDER (molecular structure). In other
words, it is exactly these molecules which through their collisions do not
create a "super disorder" upon the existing "old disorder", but a new
"super order". This "super disorder" is nothing else than chemistry. This
is how new chemical compounds with new molecular structures are created.

By the way, the surface area of the bell tells us how much molecules there
are in total for the whole volume of gas. Explanation? Picture by your own
drawings the graph as made up by many bars, each with equal thickness. The
length of any bar gives the number of molecules. But since the thickness
of the bars are constant, so will the surface area of each bar also
indicate the number of molecules. Adding all the surface areas of all the
bars together gives the total number of molecules.

How can we understand absolute temperature? Let us think of the average
velocity of all the molecules. Where is this average velocity situated on
the graph? If you expect it to be along the top (middle) of the bell, you
have the general idea. (Actually, it is slightly displaced to the tail
side, but the reason is not important now.) Let us now try to picture what
happens when the temperature is increased. The bell becomes much fatter,
but also flatter (shorter) because its total surface area (number of
molecules) must stay the same. (It is as if you take the tail of the bell
into your fingers and then pull it further away from the origin. Like a
piece of elastic rubber it has to stretch horizontally and shrink
vertically! Do you get the idea?) Try to draw a second bell exactly to
this "elasticity" specifications, but for the higher temperature. (The
second bell must have the same surface area as the first one.) Now compare
the tops of the two bells. By raising the temperature, the top has moved
outwards (and lower). In other words, the average velocity of the
molecules has increased (and less molecules have this average velocity).

But there is something else which you also have to try and grasp. This is
very, very important. By raising the TEMPERATURE, the "flatter and fatter"
the bell becomes. It tells us that there is now a GREATER DIVERSITY
(statistians call it variance) in the velocities of the molecules. NB NB
The "flatter" (shorter) means that the number of molecules having a
certain velocity has become less. NB NB The "fatter" (broader) means that
the variation in velocity have increased -- there are more different
velocities to be considered. Think this "elasticity" over and over again
until you have grasp it. Draw the bell on a flat piece of rubber and
stretch it over and over again until you get the picture in your mind. Do
not go on without grasping the picture -- it will be futile. It is exactly
this "diversity in motion" which Boltzmann named as disorder and which
others later renamed to chaos. Please note that by increasing the
TEMPERATURE, the DISORDER (diversity in motion) increases. Please note
that nothing yet has been said of entropy! In other words, we have not yet
made any connection between entropy and disorder.

Reversible world: factious or factual
Einstein was a man with very strong convictions in certain matters.
With respect to reversible or irreversible systems, he strongly
believed that irreversibility is an illusion. Let us now again take
under consideration Clausius' definition for a change in entropy:
the CHANGE in the entropy S of a system can be
CALCULATED in terms of a "reversible heat flow"
divided by the absolute temperature

It clearly mentions a REVERSIBLE heat flow. Now what exactly is a
reversible heat flow?

Let us think of a system "b" with a absolute temperature Tb, say 400K
("kelvin"). Let us think of another system "bb"with a temperature Tbb very
close to 400K, say 399.999999K. If it is not close enough, add another
couple of nines. Bring system "b"and "bb"into thermal contact with each
other. Heat will flow very slowly from "b"with temperature Tb to "bb" with
slightly lower temperature Tbb. This slowness was already noticed by
Newton through his cooling law. The closer Tbb becomes to Tb the more
reversible the exchange of heat becomes. When Tb and Tbb becomes exactly
equal, no NETT heat will flow from "b"to "bb".

But let us create the following "thought experiment", whether it is an
illusion or not. Some molecules of "b"with higher kinetic energy will
collide against molecules of "bb"with lower kinetic energy. Thus the
molecules of"b"part a very slight amount (quantum amounts) some of their
thermal kinetic energy to the molecules of "bb". Thus a minute amount of
heat flow from "b"to "bb". But simultaneously, the same thing happens
reversibly to "bb". It parts a slight amount of its thermal kinetic energy
to "b". The nett result is that no heat flows from "b"to "bb"or vice
versa. But let us assume that we have all the time in the world and now
add up all the minute portions of heat flowing from "b"to "bb". We will
have to add billions and billions of such minute portions to get -1200J of
heat. (The minus sign says that "b"looses energy by the heat flow going
out of it.). Simultaneously, we do the same thing for "bb", but for the
heat flowing out of it back into "b". Thus "bb"will lose -1200J of heat
which is the gain +1200J at "b". In total, after such a long time has
elapsed and billions of additions have been made, "b" loses -1200J of heat
to "bb"and gains +1200J of heat from "bb". This is what the
"reversible"flow of heat between "b"and "bb" is about.

Let us now think about entropy, applying Clausius' definition. System "b"
loses an amount of entropy equal to -1200J/400K = -3J/K through the heat
flow to "bb", but gains an amount of entropy equal to +1200J/400K = +3J/K
through the heat flow from "bb". When we add these two entropy changes
together, the result (+3J/K)+(-3J/K) = 0J/K. In other words, no entropy
has been produced!! This is what characterises conservative, reversible
processes -- no entropy is produced. But we must also remember that NO
DIFFERENCE in temperature exists and that everything happens VERY SLOWLY.
Since "b" gains as much heat (+1200J) as it loses heat (-1200J), the
temperature of "b"stays the same. But since the temperature stays the
same, the diversity in motion (molecular velocities) stays the same. Thus,
if temperature indicates disorder and because the temperature stays the
same, the disorder stays the same for reversible changes.

This is the world of the reversible physics of conservative systems -- no
entropy is produced, no differences in temperature are accommodated and
everything happens infinitely slowly. Perhaps this world of reversibility
is reality or perhaps it is an illusion. Let you be the judge.

Let us allow for a slight irreversibility into the system. Say Tbb=390K,
slightly lower than Tb=400K. Thus nett heat will flow from "b"into "bb".
Because system "b"loses some thermal kinetic energy, its temperature will
lower slightly. As a result there will be slightly less diversity in
motion in "b". The bell curve will become slightly taller and thinner.
Thus there will be slightly less disorder in "b". In other words, there
will be a change in disorder the moment we move from reversible conditions
(no differences, long times) to irreversible conditions (differences
become important, short time intervals become important). Perhaps this
world of irreversibility is reality or perhaps it is an illusion. Let you
be the judge.

Irreversible heat flow

Let us consider again the metal bar with two ends "a"and "b" at
temperatures Ta=300K and Tb=400K. Allow a quantity heat of 1200J to flow
from "b"to "a". The setup is:

300K 400K
+1200J -1200J
+4J/K -3J/K

We calculated the
entropy change at "b"
= -1200J/400K
= -3J/K

The flow of heat is irreversible because it will always flow from "b"
(higher temperature) to "a" (lower temperature). It will never flow form
"a"to "b". In other words, it seems as if the flow of heat has a PURPOSE
and that it will never fail that purpose. This is what irreversibility
means. Yet we have used Clausius' definition for reversible flow of heat.
Did we not make a blunder with irreversible consequences here? No. To see
why, we will have to perform again a "thought experiment" in order to
bring the idealised reversible world into this real irreversible world.

Divide the bar in eleven equal sections so that the two outer sections
form ends "a"and "b". Say that the temperature is everywhere in a section
the same, but that it jumps with 10K from section to section. Thus in the
entire section "b"the temperature is 400K, in the entire section just
before "b" it is 390K, the section before that it is 380K, ...., so that
it is 310K in the section before "a" and 300K in the section "a".

300K 310K 390K 400K

-1200J of heat will flow reversible (infinitesimally slowly with no
temperature difference) along section "b"until it reaches the section just
before "b". The entropy change is -1200J/400K= -3J/K along "b" until it
leaves "b". Then it flows over the border into the next (second) section.
If the next section had the same temperature 400K as the end section "b",
the entropy of the second would have increased by +1200/400K= +3J/K. (The
+ sign says heat enters the second section). But the temperature of the
second section is 390K so that when the heat enters the second section,
the entropy change is now +1200/390K=+3.077J/K. This "jump"is irreversible
(fast with a temperature difference). Thus the irreversible jump from temp
400K to 390K causes entropy of (3.077J/K - 3J/K) = +0.077J/K to be

The heat will flow along the second section reversible (infinitesimally
slowly with no temperature difference) until it reaches its end, ready to
"jump" into the third section with temperature 380K. When the heat leaves
the second section, its entropy decreases by -3.077J/K. But when the heat
enters the third section, its entropy increases by +1200J/380K = 3.158J/K.
Thus again the irreversible (fast with temperature difference) jump of the
heat causes entropy of (3.158J/K - 3.077J/K) = +0.081J/K to be produced.
This process go on and on until the heat leaves the second last section
with an entropy change of -1200J/310K= -3.871J/K. It enters the last
section "a" with an entropy change of +1200J/300K = +4J/K. Thus the
entropy production over the last irreversible jump is (4J/K - 3.871J/K) =
+0.129J/K. If we now add up all the entropy produced at all the
irreversible jumps, the total production will be close to, but not exactly
to +1J/K.

If the jumps appear to be too irreversible for your liking, divide the bar
not in 11 sections, but in say 1 000 001 sections. Do a million such
calculations. For example, the jump from "b"to the next section will
produce (1200J/399.9999K - 1200J/400K) = +0.000 000 75J/K. Add a million
such tiny jumps together. The total entropy production will now be very
close to (+1200J/300K -1200J/400K) = +1J/K. If it still appears to be too
irreversible, divide it in trillions of sections. The total entropy
produced is now almost exactly +1J/K.

What have we learnt here? By BEGINNING with an IRREVERSIBLE world (rather
than a reversible one) and then AFTERWARDS modelling reversible sections
over it, our picture still stays irreversible. This is the real world.
But should we begin with an reversible world (like in the former section
"Reversible world: factious or factual") and then afterwards try to model
irreversible sections over it, our whole strategy collapse. This what
Eddington meant by "our theory ... to collapse in deepest humiliation."

Irreversibility, chaos and order.
It is now high time to get this chaos/order conflict behind us. But to
do it, we will have to use all our powers of Systems Thinking as never

Let us consider our experimental bar once again.

300K 400K
+1200J -1200J
+4J/K -3J/K

The irreversible flow of 1200J heat from 400K to 300K produces entropy
of +1J/K. The entropy decrease at "b" is -1200J/400K=-3J/K and the
entropy increase at "a"is +1200J/300K= +4J/K

But what happens really at "a"? Because the heat +1200J ENTERS "a", its
TEMPERATURE increases slightly. Thus there is an increase in diversity of
motion, i.e an INCREASE in what Boltzmann called DISORDER. It is an easy
trap NOT to think of the increase in TEMPERATURE, but to think of the
increase +4J/K in ENTROPY. So what happens? We begin to interpret an
increase in entropy (rather than an increase in temperature) with an
increase in disorder. We begin to think like Boltzmann and all who
followed in his footsteps.

Even worse, let us now see with such a mental model what happens at "b".
Because the heat -1200J LEAVES "b", its TEMPERATURE decreases slightly.
Thus there is an decrease in diversity of motion, i.e an DECREASE in what
Boltzmann called DISORDER. But the entropy has also decreased by -3J/K. So
we keep on interpreting entropy in terms disorder, i.e an decrease in
entropy leads to a decrease in disorder.

However, let us now question the mental model. What is a decrease in
disorder? Is it not an increase in order? Yes, definitely. When the heat
-1200J leaves "b", its temperature decreases. The bell shape of the
distribution of the molecular velocities becomes thinner and taller,
moving somewhat towards the origin. This is an increase in order!

But let us keep the bigger picture in mind. Let us think about both "a"and
"b". At "a" the temperature increases slightly so that there is and
INCREASE IN CHAOS while at "b"the temperature decreases slightly so that
there is an INCREASE IN ORDER. In other words, we have an increase in BOTH
chaos (at "a") AND order (at b").

The last step in our description is NOT to think of the entropy increase
+3J/K at "a" or the entropy decrease -4J/K at "b". It is because these two
changes are indistinguishable from there reversible counterparts as has
been argued earlier (where we have cut the bar in eleven sections). What
we have to think about, is the total entropy change, the irreversibility
along the entire bar, the entropy production of +1J/K. It is this entropy
production for the WHOLE which causes increasing chaos at the one part
(end "a") of the whole AND which causes increasing order at another part
(end "b") of the whole. Boltzmann and his followers looked only at a
system "a" which receives heat. But the Law of Energy Conservation says
the heat has to come from somewhere. Thus by virtue of LEC we hae to
include system "b" -- the "other side".

Entropy production has caused both increasing chaos and increasing order
at different parts in the whole. Entropy production has caused a change in
the organisation (chaos and order) of the whole. If we do not think
holistically, then we will never see this "and" in the picture. But could
Boltzmann and his followers think holistically? Holism as an essence of
evolution has only been formulated 50 years after the work of Boltzmann
(1872) by Jan Smuts (1926) in his key work "Evolution and Holism"! And
even then Jan Smuts was discouraged by his closest friends to publish his
book because people were not ready to understand holism.

Best wishes


At de Lange <> 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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