You guys are too smart for me. I’m trying to understand what you are talking about, though.
From: Martin Taylor mmt-csg@MMTAYLOR.NET
Sent: Tue, Aug 18, 2009 4:55 pm
Subject: Re: Entropy & Reorganization
[Martin Taylor 2009.08.18.14.41]
and Bill and others.
one or all of you
comment on the relationship bet
ween entropy and reorganisation?
between entropy and limits, and statistical entropy or/and
(if one dares).
I imagine I’m the one who is expected to answer this, but it’s a
question about which I have never thought. It’s certainly an
interesting question and answering it might be worthwhile in developing
ideas about reorganization. But, as you might be aware, my mind is
largely elsewhere than CSGnet and PCT theory at the moment. Maybe next
On the other hand, I will point out (as I do every few years) that the
essence of control is the export of entropy from the controlled
variable to the environment in the face of environmental interactions
that tend to increase the entropy of the controlled variable. In other
words, every control system is a refrigerator.
Thinking about it from
that point of view, and thinking of reorganization as control, with the
perceptual hierarchy as its controlled variable, one is led to expect
that the essence of reorganization is reduction of the entropy of the
perceptual control hierarchy – in other words the imposition of
organization and order such as conflict-reducing modularity.
That’s very vague, but that probably would be the direction I would
start in thinking about addressing your question seriously.
As for the relation between statistical and informational entropy, I
can repost a message first posted in about 1993 and in edited form in
1996. In the light of recent threads, I draw attention to the last
couple of paragraphs (but you have to have read at least some of the
earlier material in order to understand them).
Control, Perception, and Entropy–a tutorial
The following takes off from an article in Physics Today, Sept 1993,
pp32-38 “Boltzmann’s entropy and time’s arrow” by Joel L. Lebowitz.
Boltzmann’s view of entropy is one of three classical views, the
others being those of Clausius and of Gibbs. Clausius entropy applies
only to macroscopic systems in equilibrium, and Gibbs entropy to a
statistical ensemble of microstates corresponding to a specific
macrostate, whereas Boltzmann entropy applies to any describable
system. Boltzmann entropy agrees with the other two when they ar
applicable, and is therefore a more general concept of entropy than
the others, which I will ignore in the rest of this posting.
I will give an introduction to the Boltzmann view of entropy as I
understand it from the Lebowitz article, and then suggest how this
implies that a control system is a cooling system, and why the energy
received from or supplied to the environment by the perceiving side of
the control system must be small compared to the energy involved in
the disturbance and the output side of the control system. I shall
also argue that there is a natural relationship between Boltzmann
entropy and Shannon information, beyond the formal similarity of the
equations that describe them.
Where does the concept of “entropy” apply?**
Entropy is a measure of disorder or lack of structure. Many people
understand that entropy tends to increase over time, but are not aware
that this is true only in a closed system. In an open system that can
import energy from and export energy to its environment, entropy can
increase or decrease, and structure can be maintained indefinitely. An
increase in entropy means that the system becomes less ordered or
structured. Any system has a maximum entropy, beyond which it cannot
increase. In common language, in a system at maximum entropy,
everything is random. Wherever there is structure, entropy is less
than it might be for that system.
Entropy is DEFINED over a DELIMITED system, not a closed one. A closed
system is one in which the elements do not interact with anything
outside the system, whereas a delimited system consists of a well
defined set of elements, which may interact in any way with the world
outside the system provided they do not lose their identity. A
delimited system may be open or closed.
It’s all perception, so the interests of the observer do matter, in
agreement with the comment in a message that induced me to write the
original posting from which this is derived: “To me, order/disorder is
a purely subjective concept.” It is, and moreover, it depends on the
perceptual functions used to observe the system. This is why I
commend the Lebowitz paper as a way of seeing how physical entropy
might relate to communicative information.
Entropy is a measure that has a value at an instant in time, and it
can be defined over limited parts of structures that are intimately
intertwined with other parts of the same structure, such as the
electrons in a metal, or the radiation field in a gas. If there are
two equal-sized defined systems of the same degree of disorder, the
entropy of the two lumped together is twice the entropy of one.
Entropy can decrease only if the system exports some to the world
outside, increasing the ordering of the system at the expense of
disordering the world outside. This can be done only if there is a
non-equilibrium energy flow through the open system under
consideration, from one part of the world outside to a different part
of the world outside. A living control system is just such an open
system, getting its energy ultimately from the sun and depositing its
entropy in the form of less organized waste products. The business of
a living control system is to create and maintain structure–to keep
entropy less than its maximum within the living body–and it can do
this for as long as it can maintain an energy flow through itself.
A living system maintains its entropy at a more or less constant
level, lower than would be the case for its components if it were not
alive. When a living organism dies, its components decay and are
scattered around the world, raising their entropy to a level consistent
with that of the world as a whole.
A describable system: 2 balls in a box
A describable system, which any physical systems is, can be described
in terms of a phase space. The dimensions of the phase space are
those variables on which the value makes a difference to the system’s
aspects of interest.
One way of looking at a system constructed of simple elastically
interacting balls (an idealized gas, in other words) is to record the
location and velocity of each ball in space, a 6-dimensional vector.
This 6-vector can be represented by=2
0a point in a 6-D space, which is
the phase space for that ball. For a gas of N balls, the phase space
has 6N dimensions.
If the ball constitutes a “closed system” in the sense of no energy
transfer, the velocity components of this vector will not change as
time goes on, at least until the ball bounces off a wall of the closed
container, if one exists. If there are walls, any such bounces do not
cause the ball to gain or lose energy. All that happens is that the
velocity vectors reverse their sign with respect to the orientation of
the box wall, and the point moves to a new point in the velocity
subspace at the same distance from the origin. The ball’s energy is
determined by its location in the 3-D velocity subspace–in fact by
the ball’s radial distance from the origin in that subspace, which
does not change in the bounce, by definition of “closed system”.
In a gravitational field, or if the ball were magnetic and the box
permeated by a non-uniform magnetic field, the energy would also
depend on the location. For a given energy, the radius of the sphere
in the velocity subspace would depend on the location of the phase
point in the 3-D position subspace. The ball would slow or speed up
depending on where in the box it happened to be. But we will ignore
such complications for the time being.
Trivial, so far.
Now let’s add another ball makin
g a 2-molecule ideal gas. The phase
space now has 12 dimensions. Ignoring position-dependent fields, the
total energy in the system is still represented by the radial distance
of the point from the origin in the (now 6-D) velocity subspace.
What happens when the balls meet? They bounce off each other, losing
no energy overall. Each of the six velocity components changes,
though. The system moves to a new point in the 12-D subspace. When
the balls bounce off each other, the system as a whole is still
closed, and no energy is gained or lost in the collision. The
position of the point in the 6-D velocity subspace is changed to a new
position, at the same distance from the origin. This new position will
not change until the balls bounce off each other or the wall again.
In the bounce, with high likelihood, vector components much larger
than average will be reduced, and components much smaller than average
will be enhanced, so the location of the point in phase space is most
likely to be found not very close to the axes of the velocity
subspace. More typically, the individual velocity vector lengths will
be distributed around some intermediate value.
Can there be order and disorder in such a 2-ball system? It depends
on how you choose to look at the system. There might be something
special about a state in which one ball was stationary at in the&nb
centre of the box, all the energy being concentrated in the other
ball. In the phase space, such a situation would be represented by a
point that lies in a particular 3-D subspace of the whole 6-D space.
There might be something special about a state in which the two balls
travelled together as a pair, in which case the three velocity vectors
for one ball would be defined by those of the other ball. Measure one,
and you know them both. Or there might be something special about a
state in which the balls exactly mirrored each other’s motion (as they
would if they were equal gravitational masses in outer space). The
particular subspace is defined by the observer, not by the momentary
behaviour of the balls. Each particular tightly defined subspace is
untypical, in Boltzmann’s sense, and in an everyday psychological
Even though any precise position for the system’s phase point is as
probable as any other position in the phase space, almost all of the
equiprobable points do not lie in or very close to any predefined
position. It is much like a bridge hand in which the deal gives each
player 13 cards of one suit. That hand is no less probable than any
other, but we see it as untypical because we have previously labelled
the cards so that we perceive certain relations to exist among them.
An observer is likely to ponder the possibility that the dealer might
It is more probable that a bridge hand will be “typical” in having
each player receive cards of at least three suits. There are 24
different hands in which the four players all hold 13 cards of one
suit, but many thousands of hands in which each player has cards of at
least three suits. The hand is “typical” because it is a member of the
large class rather than of the small class. For the two-ball system,
it is most probable (typical) that its phase point will be in a region
of the phase space that represents both balls as moving. A state in
which one ball is nearly stopped and the other carries all the energy
can happen, but it is not typical.
Another aspect of typicality in the ideal gas (which refers to the
problem of time-asymmetry) is that if the phase point defined by (L,
V) is typical, then the phase point defined by (L, -V) is also typical
(L and V are the location and velocity vectors defining the position
of the point in the location and velocity subspaces). A typical
situation does not become less typical if all of the balls bounce off
the wall. The same applies if any subset of the V vector components
has a sign reversal.
Replacing the phase point (L, V) by (L, -V) is to reverse time. A
brief snapshot of the gas described by the two different phase points
would show no characteristic difference between them. The differe
is in the detail of which ball or molecule is going in which
direction, but this ordinarily does not matter. What does matter is
that if the gas is in a small atypical region of the phase space,
collisions between the balls are more likely to move the phase point
into a typical region than the reverse. It is more likely that when
bridge hands are shuffled, the deal after a pure- suit deal will be
distributed at least three suits in each hand than that the reverse
Similarly, if the gas is in an atypical region of the phase space at
time t, it will most probably be in a more typical region at time
t+delta t, and have come from a less typical region at time t-delta t.
If all the velocity vectors were replaced by their inverses, the gas
would revert to its less typical prior state. This is most improbable,
even in the two-ball “gas.” Although the states (L, V) as observed,
and its mirror image (L, -V) are of identical typicality, the two
microstates are not equally likely to occur in practice. One changes
over time from less typical to more typical states, and the other
becomes less and less typical over time. However, a casual observer
shown a brief snapshot of the two could not tell the difference
More complex: 2 balls in a swarm of others
Notice that none of the above description requires that there be a box
confining the balls.&n
bsp; If there is no box, then there will be only one
bounce of the two balls against each other, or none, and the situation
is less interesting.
Let’s remove the box, but add a lot more balls that individually do
not interest us. Call them “nondescript” because we will not describe
them within the phase space. They may, however, interact with the two
balls that interest us. We will still look only at the original two
balls, and the position of their phase point in their 12-D phase
space. Now the balls can encounter each other OR any of the
nondescript other balls. The phase space description of the two balls
is unchanged, but the behaviour of the phase point is different.
If the two balls bounce off one another, the point moves to another
position at the same distance from the origin in the velocity
subspace. But if they bounce off a nondescript ball, the radius of
the shell in which they live may change. If the nondescript ball was
moving very fast, the total velocity of the interesting ball will
probably be increased in the collision. The phase-point of the 2-ball
system will move further from the origin in the 6-D velocity
subspace. The open 2-ball system will have gained energy.
Conversely, if the nondescript ball happened to be moving very slowly,
the interesting ball is likely to lose speed and the nondescript ball
to gain i
Temperature is proportional to the energy in a system. In our 2-ball
system, the total energy is equal to the sum of the energies
associated with each of the velocity vectors, or 0.5sum(vi^2) where
vi is the velocity on axis i and the balls are assumed to be of unit
mass. The temperature, T, of the system is not affected by how many
balls are in the gas, and so we can write T = kmean(0.5*vi^2).
Another way of saying this is that the energy per degree of freedom in
the system is kT/2. The proportionality constant k is known as
In typical regions of the phase space, the velocity vectors are
distributed more or less evenly around some intermediate value, very
few being either very large or very small. When one of the interesting
balls collides with a ball from the environment, and it gains energy
from the collision, this increases the temperature of the system of
interesting balls, and decreases the temperature of the system of
environmental balls. When an interesting ball collides with a slower
environmental ball, conversely, the temperature of the interesting
system is lowered and that of the environment raised.
The radius of the sphere centred on the zero-velocity origin of the
velocity subspace is sqrt(sum(vi^2)), which means that the temperature
of the system is proportional to the square of the length of the
ity component of the phase vector, divided by the dimensionality
of that part of the phase space.
Typical and atypical description states: the measurement of entropy**
Now let’s add a lot of interesting balls, bringing the set to size N.
The phase space for this set now has 6N dimensions. Again, we don’t
care whether there are other uninteresting balls with which they
interact. The phase space has typical and untypical regions, defined
by the observer (exactly as any CEV is defined by the perceptual
functions of the observer). One thing to note is as before: if a
point is in a typical region, it will almost never move into an
untypical region as a consequence of reversing some or all of its
components in the velocity subspace.
To see that the typicality of a region of the phase space depends on
the observer, imagine an observer looking at the phase space plot
through the N-dimensional equivalent of a scrambled fibre-optic pipe.
In a fibre-optic pipe, a large number of glass fibres are placed
together so that at each end of the pipe their ends form a single
plane onto which a pattern can be focused. If the alignment of the
fibres were the same at both ends, a pattern focused onto one end of
the pipe would be seen glowing on the surface at the other end of the
pipe. But if the fibres are scrambled, what is seen at the other end
is a random hodge-podge of light and dark. There is, however, a very
specific configuration of what looks like random dots that, when
entered at the front end, emerges as a straight line at the back end.
Patterns of random dots are “typical” of dot patterns in general, in
that almost all such patterns of dots look alike; but a straight line
pattern of dots is untypical, there being only a relatively few ways
that the dots can be so arranged. A random procedure for locating the
dots could produce a straight line, but it would be most unlikely to
do so. A “random” pattern at one end of the pipe that produces a
straight line at the other is a most untypical random pattern, and the
observer that saw the line would so assess it.
Returning to the system of balls, the observer’s definition of what
constitutes a particular “atypical” structure is like the scrambling
of the fibres in the fibre-optic pipe. The pipe may be straight
through, and the observer may define a “natural” atypical
configuration, such as that half the balls have zero velocity and are
arranged in a repeated regular pattern of locations (a crystal), while
the rest of the balls have high velocity and are not specifically
related on location. Or the observer may define some arbitrary pattern
that another observer would see as quite typical.
Each point in the phase space defines a possible “microstate” of the
space. Any volume of the phase space
defines a “macrostate.” Any
configuration with a phase point that the observer saw as conforming
to the arbitrary pattern would belong to the macrostate for the
pattern, and any point that did not so conform would belong to the
larger “typical” macrostate of the phase space.
Boltzmann’s entropy is concerned with the volume of the phase space
that might be considered typical of the present state of the system.
For example, in the 2-ball case in which one ball was motionless or
nearly so, it would presumably not matter which of the balls was
stopped and which was moving, so the region of phase space typical of
the “one-stopped” state would at least include two thin slices near
the two 3-D subspaces of the 6-D velocity space which represent one of
the balls as stopped. In the Physics Today article, the “one-stopped”
state would be an example of a “macrostate,” whereas the specific
phase point associated with one such condition would be a
To be accurate, it should be noted that macrostates do not normally
have discrete boundaries. It would be more proper to treat a
microstate not as being “inside” or “outside” a specified macrostate,
but as having a specified fuzzy set membership in the macrostate. The
“volume” of the macrostate would then be the integral over the phase
space of the membership function of the microstates. But we will
continue to treat this integral as20if it were the volume of a
well-defined region of the phase space.
The value of Boltzmann’s entropy is proportional to the log of the
size of the subspace typical of the current state. It does not depend
on whether the system is open or closed (whether there are
“nondescript” balls, in the ideal gas example). It does depend on the
system being delimited, (e.g., in the ideal gas example, knowing which
balls are being described). If the phase space is continuous, there
may not be a natural scale of measurement for the volume of the
typicality space, but if it is discrete, the natural unit is the
volume of one discrete cell. However since the value of the entropy is
a logarithmic function of the size of the typicality region, changing
the scale unit only adds or subtracts a fixed quantity to all
entropies measured using the unit. It makes no difference when we
consider changes in the entropy of a defined system, as is usually the
If the system is closed, the region of phase space available to it is
limited to a shell of constant radius in the velocity subspace. That
shell has a volume, the log of which which represents the maximum
entropy possible for the system (in a high-dimensional system,
nontypical subspaces have a vanishingly small total contribution to
If the system is open, the shell to which its phase point is confined
pand or contract, depending on whether energy is on balance
transferred into or out of the system. As the shell radius changes,
so does the maximum entropy possible for the system, and if (as is
highly probable) the system is in a state of near-maximum entropy, its
entropy will increase as it gains energy (gets hotter).
Notice that nowhere in the definition of the Boltzmann entropy does
the construct “probability” appear; it appears only consequentially.
At least in the ideal gas, any point in phase space is as likely as
any other to be occupied. Hence the probability of a point being in
any particular macrostate is proportional to the size of the region
typical of that macrostate relative to the total phase space volume.
In systems with attractor dynamics, this statement does not hold, and
probability and entropy go their separate ways. A low-entropy
condition is highly probable in a system with attractor dynamics–or
in the environment of a control system.
As Lebowitz’s article points out, Boltzmann’s entropy is numerically
consistent with other definitions of entropy if the situation is
appropriate for the application of the other definitions, such as
Gibbs, which depends on probabilities, or Clausius, which applies to
Time’s arrow, as described in the Physics Today article, depends on
the fact that it is easier to get out of a tiny volume of the phase
into a larger volume than to find a way from an arbitrary point
in the large volume back into the tiny volume. Taking that to the
extreme, for a specific example, suppose that there were a macrostate
for which the typical region consisted of just one microstate (L, V).
Then, if after some time the system had evolved to a microstate (L’,
V’) in a larger macrostate and all the V vector components were
reversed exactly, the system would eventually evolve back to the
original microstate. But if any of the reversals were inexact, or if
some of the components were left alone, the system would not return to
the original microstate. The incomplete reversal would leave the
system in the same evolved macrostate, but in a microstate outside the
original small volume. Entropy (almost) always increases in a closed
Entropy does not always increase in an open system. I, being outside
the open system of concern, can select the balls of interest and
deliberately place them in any state I desire, in location and
velocity. I can “control” them, at least insofar as I can perceive
their exact locations and velocities. I can have them move all in
parallel at the same velocity (a flow regime), or stand on top of one
another, or whatever. These structures have microstates with
relatively small typicality regions. I, the observer-manipulator, have
reduced the entropy of the system of i
nterest, putting the phase point
in a non-typical subspace of the descriptive space I have defined by
my perceptual functions. I can keep the structure in such a
low-entropy state, provided I can continue to observe and correct its
departures from this non-typical region under the influence of the
“nondescript” parts of the universe. I can control my perception of
I argue that Boltzmann entropy can be applied to any descriptive
space, and in particular it can become Shannon entropy or uncertainty
under appropriate conditions. Why is this so? Consider my control of
the balls of the open system; I define a state in which I wish them to
be. I can force them into this state only as accurately as I can
perceive them. The region “typical” of my perception consists of all
those microstates I cannot distinguish. The less uncertain I am about
the location and velocity of all the balls, the smaller this
typicality region and the lower the entropy of the ball system as I