## Primer on Entropy - Part III B LO20048

AM de Lange (amdelange@gold.up.ac.za)
Tue, 24 Nov 1998 12:54:07 +0200

Dear Organlearners,

PART III -- THE GENESIS OF ENTROPY PRODUCTION (Continue)

Which twin comes first?
-----------------------------
We have established in the previous section that irreversibility (or
the production of entropy) is responsible for increases in both chaos
AND order. But the question now is: Do chaos and order play equal
roles in the flow of heat and thus change in organisation?

Let us consider our experimental bar once again.
(a)=<=========<==========<=(b)
300K 400K
+1200J -1200J
+4J/K -3J/K

1200J heat flows from a higher temperature 400K to a lower temperature
300K.

But observe how our minds think. Heat, originally at "b", reacts to
something at "b" as the CAUSE. Then it flows along the bar. Finally it
arrives at "a" where it has some EFFECT. Maybe you perceive the higher
temperature at "b" as the cause and the lower temperature at "a" as the
effect. Maybe you perceive, with the help of the last few sections, the
increase in order at "a" as the cause and the increase in disorder at "b"
as the effect. Maybe you perceive neither cause nor effect, namely that
increase in order at "b" and increase in disorder at "a" contributes
equally to the flow of heat. Which of these possibilities are the best one
to adhere to -- or is there a possibility that we have not yet thought of?

Our problem with the experimental bar is that it is such a simple set-up.
There are no other processes changing the organisation (chaos and order).
Thus there are no other processes to help us to decide which comes first
-- chaos or order. Specifically, there are no bifurcations (forkings)
which can play the role of distinguishing marks to decide which comes
first -- chaos or order. The only process which we have, is the flow of
heat. Thus we must try our best to answer the question in terms of the
heat flow alone.

I refer you again to the situation in which we have partioned the bar into
eleven sections, each section having a constant temperature. We had from
"b" to "a" the temperatures 400K, 390K, 380K, .... 310K, 300K. When the
heat of 1200J flows over the border between two sections, we calculated
the following jumps in irreversible entropy production, going form "b" to
"a": +0.071, +0.081, ..., +0.129. In other words, near "b" (higher temp)
the partial entropy production is less than +0.1J/K while near "a" (lower
temp) it is more than +0.1J/K. It means that the partial entropy
production increases while going from "b" (higher temp, increasing order)
to "a" (lower temp, increasing chaos). In other words, the "entropy
production" increases as we go from "b"where order increases to "a" where
chaos increases.

If we are now mentally capable to perceive "entropy production" as the
primordial cause for the flow of heat, then bigger partial entropy
production at "a" (increasing chaos) is more of a cause than the smaller
partial entropy production at "b" (increasing order). Thus the increasing
chaos at "a" which pulls the heat towards it has a slight advantage over
the increasing order at "b" which pushes the heat away from it.
Consequently, when there is a clear distinction possible between chaos and
order in complex systems, we can expect chaos to happen before order. In
other words, when entropy gets produced, first chaos and then order.

Is it not remarkable that even the history of the concept of entropy
followed the same course. For almost 100 years, since the days of Ludwig
Boltzmann, everybody believed that entropy has merely to do with chaos.
Only during the seventies Ilya Prigogine began to uncover the role which
entropy plays also in order.

Do you still remember what troubled Faraday and later Planck so much
about the conversion of energy from one form XXX to another form YYY?
Yes, some how heat (another form of energy) was always involved. In
other words, we do not get a pure conversion
XXX => YYY
where XXX + YYY = constant
XXX => YYY + HEAT
where XXX + YYY + HEAT = constant

Now we can begin to see why. Since molecular chaos (highest partial
entropy production) has to be manifested first before any other order can
also be manifested, the degradation of any form of energy during its
conversion into heat is a witness to this first manifestation. Prigogine
refers to this witnessing as the dissipativeness of systems. In other
words, irreversible self-organising systems have to be dissipative
systems. Living systems are irreversible self-organising systems. Thus
living systems have to be dissipative systems. They have to degrade some
of their energy conversions into heat. Were the Greeks really so foolish
to see fire as one of the four elements of life?

Minimal entropy production
----------------------------------
We may easily think that with the wordings
LEC: The energy of the universe is constant
LEP: The entropy of the universe increases
we have formulated the Law of Energy Conservation and the Law of
Entropy Production sufficiently. However, it is not the case.

Let us first consider LEC. When people like Mayer, Joule, Grove, Faraday
and Thomson struggled to articulate the LEC, they emphasised the constancy
of the total energy as the essential feature of the LEC. But they also
stressed that as its main clause, we should never forget that this law
makes only sense when one form of energy is converted into another. In
other words, if the amount of one form of energy decreases (no constancy),
then the amount of at least one other form of energy must increase (no
constancy) by precisely such an amount that the total sum (whole) of all
the forms of energy remains constant.

To summarise. The formulation
LEC: The energy of the universe is constant
is actually senseless. We must add the clause
LEC: The [total] energy of the universe is constant
[while forms of energy are converted to one
another.]

Now what about LEP? What is essential feature? The total entropy has to
increase. But what is its main clause, that which we need to know in order
to understand its essential feature? Clausius himself supplied the clause:

LEP: The [total] entropy of the universe increases
[towards a maximum]

How did Clausius come to this insight? Well, let us consider our
experimental bar once again.

(a)=<=========<==========<=(b)
300K 400K
+1200J -1200J
+4J/K -3J/K

The total entropy production is +1J/K

After a while, because of the heat flowing from "b" to "a", the
temperature of "b" will become lower, say 390K, while that of "a" will
become higher, say 310K. Thus our experimental bar has become.

(a)=<=========<==========<=(b)
310K 390K
+1200J -1200J
+3.871J/K -3.077J/K

The total entropy production now is +0.794J/K

Compare this last result +0.794J/K with the former result +1.000J/K. The
entropy production decreases gradually. Eventually there will be no
entropy production when the maximum has been reached. The two temperatures
will be the same so that a reversible state has been reached. This is also
the equilibrium state.

Irreversible against irreversible
-------------------------------------

This "minimal entropy production" is a great worrying factor to people,
apart from the "chaos only" factor. We have seen that LEP does not predict
only an increase in chaos. But does the LEP predict a reversible death,
the great equalising of all differences at equilibrium? No. It is here
where complexity enters the picture. Our metal bar in which only a flow of
heat takes place, is too simple. It contains only one form of energy
(thermal energy) which increases at one place and decreases in another
place without any oscillating feedback loop. The feedback loop requires
that heat must flow in the opposite direction, i.e from low to high
temperatures. As soon as we can build a feedback loop in the system, we
will observe how at the one place the difference in temperature oscillate
between increase to decrease while at the other place it oscillates
exactly oppositely, namely decrease to increase.

However, does this flow from lower to higher temperature not defy LEP?
Not at all. Consider again our experimental bar for this seemingly
impossible case. Note that the arrows points in the opposite direction.

(a)=>=========>==========>=(b)
300K 400K
-1200J +1200J
-4J/K +3J/K

Note that not only the arrows have reversed, but also the signs of the
heat. The total entropy production is -4J/K +3J/K = -1J/K. The negative
sign indicates a decrease. Nothing will happen if the total entropy has to
decrease. Heat will not flow on its own from a lower temperature 300K to a
higher temperature 400K. If you find such a place any where in the world,
buy it. It is worth at least a trillion dollars, more than Bill Gates
could ever muster.

Does it mean that heat will never do this? No. What about a fridge
(cooler box). Heat flows from the colder inside to the hotter outside.
Aha, finally a case where LEP fails. No. We are not thinking
holistically enough. Every fridge has an electrical motor which pumps
refrigerating gas through a closed loop. The fridge use much more
electrical energy than the heat energy which it "pumps" to the
outside. What becomes of this electrical energy? It is converted into
heat! So, in order to let 1200J of heat to flow from 300K at "a" to
400K at "b", we will need roughly 2400J electrical energy. When it is
converted into heat by the "pumping action", this heat is liberated at
400K. Thus an additional entropy is produced at "b" namely,
+2400J/400K = +6J/K. In other words, the total entropy production
is -4J/K +3J/K +6J/K = +5J/K. Again it is an increase which make the
working of the fridge possible. Think holistically -- do not fragment
any information from the picture.

into heat energy than the actual heat we wanted to let flow in the
non-spontaneous direction. Also note that the electrical energy was not
converted directly into heat like in an ordinary resistance heater. No,
the electrical energy was converted first into work (the pumping) and then
into heat by the properties of the refrigerating gas.

Work and irreversibility.
-----------------------------
We can have many forms of energy which can be converted partially into
work. If we add up every part of each form of energy which can do
work, the total of these parts is known as the "free energy" F. In
other words, not all of the "total energy" E (sum of all forms of
energy) can do work, but only its part called the "free energy" F. J
Willard Gibbs has shown us that the LEP can be translated into an
exciting new manner as follows:

the entropy of the universe increases

becomes

change in free energy F < work W

The sign "<" means smaller. It says that the left hand side, namely
"change in free energy F" has to be smaller than the right hand side,
namely "work W". Only when in an irreversible process this expression
becomes true, will that irreversible process happen.

Here is an simple example of what this latter expression tells us. A
little boy Jack at the bottom of a hill may have a free energy
(gravitation potential energy) of 20 000J and at the top of the hill 80
000J. The displacement of Jack from the bottom to the top of the hill is
said to be non-spontaneous if we have to displace Jack and not Jack
himself. The non-spontaneous means that we have to do work on Jack to get
him to the top of the hill.

Thus to displace Jack from the bottom to the top of the hill his free
energy must change from 20 000J to 80 000J. The change in free energy is
given by 80 000J - 20 000J = +60 000J. Let us assume that we try to
displace Jack with as little work as possible. What about +10 000J work?
Substitute these values in the expression

change in free energy F < work W
+60 000J < +10 000J

But this is false -- the 60 000 is not smaller than 10 000. Thus Jack
will not become displaced to the top of the hill, although he might
become displaced a little bit upwards. We will have to do at least 60
000J of work, say 70 000J, to displace Jack. Thus

+60 000J < +70 000J

become a true expression. The extra work of 10 000J is wasted as heat by
trying to take Jack quickly up the hill.

Obviously, what goes for us, go for Jack also. If Jack has to go up the
hill himself, he must overcome this non-spontaneous displacement by doing
at least +60 000J of work. Any work less and he would not get to the top
of the hill.

On the other hand, Jack might want to go down from the top of the hill to
the bottom of the hill. This is a spontaneous process. Even a rolling a
mindless ball with no legs will do it. The free energy F of Jack will now
decrease with -60 000J. Let us say that Jack drives in a little cart down
hill so that he does not have to work. Substitute these values in the
expression

change in free energy F < work W
-60 000J < 0J

The expression is true (a negative number is smaller than zero).. Jack
will be able to free downhill in his little cart.

Jack may even do some work while going downhill. Assume a rope is fixed to
his cart, pulled around a pulley at the top of the hill, then taken down
hill and fixed to another little cart in which Jill sits. Assume the
weight of Jill is 80% of Jack's weight. The change in free energy for her
would be 80% of 60 000J, i.e 48 000J. To get Jill up the hill will need at
least 48 000J work. See what now happens when Jack goes downhill, pulling
Jill uphill by letting his change in free energy do the work. Substitute
these values in the expression

change in free energy F < work W
-60 000J < -48 000J

The expression is true. Jack will be able to "work" downhill in his little
cart (clever Jack, letting Jill think that he did the work.)

However, should her weight be 120% of that of Jack, the values in the
expression will give

change in free energy F < work W
-60 000J < -72 000J

The expression is not true any more. Jack cannot get more work from his
change in free energy than the maximum allowed by it, namely -60 000J.
(Maybe clever little Jack will find a new explanation to fool Jill.)

Free energy, total energy and entropy
---------------------------------------------
What is the relationship between the free_energy F, the total_energy E
and the entropy S of a system? Between 1876-78, 4 years after
Boltzmann first major publication, J Willard Gibbs published a long
paper (323 pages in several instalments) in an almost obscure
Connecticut journal. In that paper he did something truly
remarkable -- he opened thermodynamics (LEC and LEP) up to the world
of chemistry. Whereas Boltzmann's work lead to frustration upon
frustration, Gibbs work paved the way for modern chemistry. Among
other things, he argued with brilliant insight that F, E and S are
related by the extremely simple equation

F = E -TS

(Actually, his equation was G = E + PV - TS. Although it is a little more
complex, he made things even easier for chemists in practice. It is sad
that few Americans know about Gibbs -- in a rating about all Americans
ever, he will easily get my vote.)

Now let us see what we can learn from this equation. We know that the
total_energy E is the sum of all forms of energy. We know that F is the
sum of each part of every form of energy which could be converted into
work. In other words, the free_energy F is that part of the total_energy E
which could be converted into work. Now what is work itself? Work is a
flow of energy with which the FUTURE organisation of a system can be
changed. With work we can build a wall (increasing order) or build and
blast a bomb (increasing chaos). It is up to us how we want to change the
future of any organisation -- to change the organisation of the future
--only increasing chaos or also increasing order.

If this is the case, then what does (E - F) mean? The (E - F) is that part
of the total_energy E which cannot be used to change the organisation of
the future since we have removed the free_energy F from it. In other
words, (E - F) is that part of the total_energy E needed to sustain the
PRESENT organisation (chaos and order). Divide (E - F) by the absolute
temperature T. The result of that division is that (E - F)/T is not any
more a quantity in the dimension of energy, but a quantity with unit
[J/K]. Yes, it is nothing else than

S = (E - F)/T

In other words, the entropy S of a system is a measure (indication) of the
PRESENT organisation of the system, an organisation which involves both
chaos and order. By changing the entropy S of the system, the organisation
of the present system will be changed into some different organisation of
the future system.

How we change the organisation of the system, is up to us. But one thing
we will never get away with -- we will have to change it by changing the
entropy S of the system. And how will we do that? By knowing more about
"entropy production" in general -- by learning what Ilya Prigogine had
discovered about "entropy production", even if it merely concerns the
physical world. What we must understand, is that the "entropy production"
can be achieved by the free energy F and only F alone! Hence the "free
energy" F is critically important in understanding the change of entropy
by "entropy production".

The quantity (E - F), or the "total_energy E minus the free_energy F" is
not only important to Thermodynamics. It is also one of the two basic
ingredients of Quantum Mechanics (QM). The other basic ingredient of QM is
that position and momentum are related to each other through the
quantisation of energy (e = hf) and relativity theory (E = mc^2) which
leads to the DeBroglie relation. In QM the F is replaced by some specific
potential energy like the Coulomb potential of the nucleus. The difference
between Thermodynamics and Quantum Mechanics is that the former (T)
involves billions times billions of molecules while the latter (QM)
involves only one molecule. Whether billions times billions of molecules
or only one molecules, the factor (E - F) is crucial! This factor will
remain to be crucial even when we enter the realm of mind.

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

Learning-org -- Hosted by Rick Karash <rkarash@karash.com>
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