logic> hierarchies> thermodynamics
This page is about the thermodynamics of organic systems.
Mechanicalism
assumes closed and static worlds. Therefore – dichotomously
– organicism must start out with the opposite assumption.
Which leads us to an “unboxed” model of
thermodynamics.
We will have a hierarchically-organised system that
has a middle ground exhibiting the signature property of scale
symmetry. Or in thermodynamic
terms, a system that is energetically open and smoothly expanding. In
organicism, in contradictory
fashion, it is inflation or growth that comes for free. It is a
fundamental attribute of a "world". By contrast with the usual
mechanical view, it is stillness or
conservation that would cost extra.
Of course, it is not usual to talk about thermodynamics as being part
of “logic”. But what is thermodynamics? It is about
the behaviour of large scale
systems made up of many small scale or atomistic parts. If you get
enough bits of stuff in interaction, then eventually they will form a
system that settles down towards some average or ambient state. The
whole they create will
develop a typical internal statistical picture.
And logic is about predictable regularity - the way things have to
happen. The laws of thermodynamics allow us to say that given starting
conditions A, you can be
absolutely certain you will arrive sooner or later at final
conditions B. So even if thermodynamic outcomes appear to be based on
randomness and averaging - a broader statistical view - they
are still logical and we ought to include them as
part of any general
causal model of reality.
Thermodynamics is only not part of classical models of logic because
classical models are mechanical and so desire to dispense with the
systems-level
view. The mechanical approach says systems behaviour is not intrinsic,
but subsequent. It emerges itself as a result rather than being a
necessary part of the causal process. But again, here we are taking a
holistic stance in which both systems and their parts are fundamental
and so thermodynamics does become an intrinsic part of causal modelling
– its middle ground as I have said.
And the theme of the discussion here is that there are two quite
different thermodynamic pictures that will seem fundamental depending
on whether you employ mechanical or organic logic.
Briefly - deep breath - the mechanical view of any thermodynamic system
is essentially boxed. That is static and closed. This produces a
certain characteristic statistics. The disorder, entropy or randomness
of the system will seem Gaussian. It will have a single (smallest)
scale. But the organic view is un-boxed. It is
the view of a system that is open and dynamic or freely expanding. And
now its statistics, its description of the
“disordered” middle ground, will be fractal or
scale-free.
This organic approach to thermodynamics obviously connects to the
extended models of entropy proposed by Renyi, Tsallis and others. Where
classical Boltzmann entropy has a single (smallest) scale of
interactions, these new entropies have fractal scale –
correlations over all possible scales. But let’s not rush
into the technicalities. For the moment I just want to paint a
broad-brush philosophical picture of the issues.
Boltzmann's gas in a box
We can begin by thinking about the standard mechanical image of a
thermodynamic system – a bunch of gas particles trapped in a
box. This is the classical Maxwell-Boltzmann story of an ideal gas or
the canonical ensemble.
The raw ingredients are some freely moving particles. There are no
complicated interactions between the particles as you might have with
water molecules, like dragging electrostatic bonds. The particles are
Newtonian billiard balls travelling inertially from one elastic
collision to the next. It is the simplest kind of system possible
– or the one that maximises the causal property of locality
(the L in RAMML).
The particles are then trapped inside a rigid container. There is a box
that sets fixed or static global limits. There is no need to call on
some emergent global effect like surface tension to organise the limits
of the system. Following Boltzmann, we can simply take this container
to exist.
And note something critical here. There is space (and time) inside the
box. So the interior of the box is a void – an a-causal
backdrop that neither contributes nor hampers the action taking place
within it. And the walls likewise do nothing – except for one
thing. They reflect any straying particles back into the heart of the
system. The walls of the box serve to keep its contents all together in
the one space (and all at the same time).
This is of course pure atomism. Except instead of the void being
infinite in its extent, someone has taken the trouble to enclose a
portion of it. And if you are thinking the existence of the box now
sounds rather artificial, well it is.
Anyway, let’s talk about order and disorder in such a world.
The idea of order and disorder – tidiness and messiness - is
central to thermodynamics. On the one hand we have organisation,
information, crispness. On the other (dichotomously!) we have chaos,
noise, chance, randomness.
The first law of thermodynamics is the conservation principle. In any
closed system, the contents are going to be preserved. Change is in a
sense illusory because there is only ever a process of rearrangement or
mixing.
This of course assumes that the “stuff” of the
system, its substance, has existence (an inertial quality) rather than
merely persistence (a contextual quality). Unlike say a pattern of
ripples or whorls in water, a bunch of gas particles can be granted
this kind of substantial presence. They won’t be slipping
into and out of existence of their own accord. And if the ingredients
inside a box are stable, then obviously the total amount is going to be
conserved. The gas particles can bang about as much as they want, but
like angry bees buzzing in a jar, they can never escape.
The second law of thermodynamics then states that order must always
slide down the slippery slope towards disorder. Given freely behaving
particles, ones that do bang about blindly, then the insides of a
closed box must become more and more messy until eventually it reaches
some equilibrium limit of messiness. With time, the contents of the box
will become scrambled so that any further changes result in no apparent
change. And once scrambled, the contents will have no way of
unscrambling themselves.
The second law is often likened to a teenager’s bedroom. The
room may start spic and span. But lock a teenager in it for a weekend
and it will follow an inexorable path towards maximum entropy or
disorder. All order will be degraded by random acts until it reaches a
state of disorder where the further shuffling around of books, clothes
and CDs can do nothing to deepen the prevailing sense of chaos. And
because books, clothes and CDs are helpless to put themselves back in
drawers or cupboards, there is no way back up the slope to the initial
state of order.
The first and second laws of thermodynamics are of course
“paradoxically” dichotomous in many ways. So while
the first law is time-symmetric – the world remains
essentially the same, unchanged in its overall contents, as it passes
through time – the second law is time-asymmetric. It has an
arrow of time in that the world looks different at the end than it does
at the start. Once order (or its synonym, energy) is spent, it cannot
be recovered.
clumped~spread-out order
Now back to the gas particles trapped in a box. A point often missed in
discussions of thermodynamics is that there are two possible extremes
of order in this situation, two ways of being perfectly neat and
regular.
First we can imagine all the gas particles starting clumped together in
one corner of the box. This unlikely situation might be created by an
experimenter using a syringe to inject a little squirt of gas. Anyway
we can see how the gas would soon spread out to fill the box. The
initial state of order would be disordered, the position of the gas
particles all messed up by their random wanderings.
According to Newtonian mechanics, it is possible that through the
chaotic activity of the particles, they might one day all happen to
bounce back to the same corner and regroup. The original state of high
order might be recreated. But we also know that this would be a
vanishingly unlikely occurrence. It would be like water suddenly
deciding to flow up hill.
Each possible configuration of the particles in the box is termed a
microstate. And there are countless gazillions of microstates that have
much the same look – that old messy spread-outness. Only a
few would resemble the initial clumping. So if each microstate occurs
at random, then the chances of the clumped look reappearing becomes a
very long shot, while a messy looking microstate is pretty much
guaranteed.
It is simple statistics. Microstates are effectively pulled out the
hat. There are gazillions of messy microstates holding a ticket and
only a very few tidy ones available for selection.
Now the image of all the particles starting out clumped in one corner
is the conventional way of representing an initial state of order in
such a system. But there is - surprise, surprise - a second asymmetric
and dichotomous alternative. Instead of being all clumped in one
corner, the gas particles might instead find themselves starting out
exactly evenly spaced, as if painstakingly arranged by some fiendish
experimenter on a lattice-like grid.
So we have thermodynamic order dichotomised into spread~clumped. And
released from either of these two extreme of orderliness, gas particles
would immediately start to wander off, becoming messy and disordered.
But while we have two possible states of order, there is only one
resulting state of disorder. There is only one stable outcome, one
equilibrium balance, that can form the emergent middle ground of a
hierarchy. What is its statistical profile?
Well we can see that the messiness of an ideal gas at equilibrium looks
more spread out than clumped. As a state, the disorder of the particles
in the box lies quite close to a perfectly regular or lattice-like
arrangement. There is a randomness, but it is rather compressed. Indeed
it seems squashed to a single scale – a smallest possible
scale.
normal distributions
Randomness with a single scale is called a “normal”
or Gaussian distribution. It yields the famous bell curve and also the
s-curve of a cumulative distribution functions - two ways of graphing
essentially the same thing.
The bell curve has a single mean, a central average. The probability of
an event or occurrence then tails away rapidly towards the extremes.
There is an asymptotic approach to a limit (and where have we heard
that one before?).
If we measure a group of people for height, IQ or many other
characteristics, we will get (roughly) this distribution. A lot of
people in the middle between five foot and six foot. Then a very few
people at four foot and seven foot. Maybe a few in a billion at three
and eight foot. And even here, syndromes such as
dwarfism or pituitary gland tumours usually come into the explanation.
We can get an intuitive sense of how Gaussian distributions are
created. It is like the system in question - like the genome
that builds the typical human - is aiming at a target and then there is
a
bit of error in the results. Point a gun at a bulls-eye and there will
be a grouping with a degree of scattering. Most shots will be near the
target and then a few may miss by quite a bit. The randomness is due to
a process which has a target outcome but which is constrained rather
than controlled. The results will always be close but never exact/
Another version of this kind of directed randomness is the Poisson
distribution. This
the Gaussian or normal distribution, but skewed or compressed to one
side. Typically we see this pattern emerge with a time-based event like
radioactive
decay.
Now radioactive decay is in fact very mysterious being a quantum
process. But as it will be useful later, let’s describe how
it is an example of a Poisson distribution. Take a lump of
plutonium-241, an isotope with a half-life of 14.4 years. This
half-life figure means that in 14.4
years, half of the atoms will have decayed. After 28.8 years a further
half the
remaining atoms – or a quarter of the original –
will also have decayed. And so on.
We can see that the tail of this
Poisson distribution thus stretches out towards infinity. Or at least
some very large numbers. The halving of the remaining half can keep
going for quite a time, which is why trace residues of radioactive
materials can linger so long.
It works the other way too, which is why the first half of the Poisson
curve is squashed up. In half the half-life, or 7.2 years, a quarter of
the atoms will have already decayed. In half of that time, or 3.6
years, an eighth will have already decayed. So while the
halving gets stretched out in one direction, in the other it gets
compressed. The Poisson curve is crisply bounded
to one side (the moment, for example, when we first start to count
decay events), yet essentially unbounded to the other.
Where this all gets quantum-curious is that individually every
plutonium atom acts as if it is independent and memoryless. The chances
of some atom decaying at any particular moment is unaffected by how
long it has been sitting about. It is not like a time-bomb with a
fizzing fuse. The fact that it did not go off a moment ago, or the
moment before that, has no impact on its likelihood of going off right
now. It might or it might not in quite random fashion. Yet
collectively, globally, we see a statistical pattern emerge. Half the
atoms in the lump are likely to go off after 14.4 years.
One way of making sense of this paradox is that each atom of a certain
size has a break-up threshold – a point at which nuclear
forces are no longer strong enough to bind it together. The atom
trembles the whole time due to quantum uncertainty, and at some point a
fluctuation is large enough to carry it over the threshold. But still
the fluctuation itself seems causeless.
So the skew is about leaping a threshold or making a phase
transition. The decay is in a sense held back because smaller
fluctuations are ignored and then critical size fluctuations are
dramatised as "an event". The size of the particular threshold for
the system is consequently what sets the mean - the half-life time.
A Poisson distribution is a special case because it overlays a
yes/no binary decision about the occurence of an event over a deeper
level of randomness - a spread of fluctuations which may or may not
bust a particular threshold. And does this deeper level of statistics
fit some profile? Well if it is not Gaussian, it must be scale-free.
But anyway, for the moment, let’s continue building up a
sense of what a Gaussian
or normal random distribution is all about.
Another classic example is coin flipping. With a fair toss, the chances
of heads or tails is 50/50 with every go. The process is memoryless
– each toss is independent of any historical trends
– and so the odds always remain 50/50. Yet it is also equally
certain that enough coin tosses will produce excursions from the mean.
You will get a surprising run of heads or tails. Indeed, to be properly
random, such runs must occur with predictable regularity. We could
indeed say that something like a run of seven heads has a half-life of
250 throws. That is on half the trials that length, it will happen at
least once.
Coin-tosses, roulette wheels, radioactive decay and other games of
chance have no memory. So let’s consider quickly the
difference of a random process with a memory, such as brownian motion
or a random walk.
Imagine tracking a
jittering particle that with every step can jump in
one of four directions. It can go left, right, forwards or backwards.
Well a quarter of the time it will go back the way it just came,
erasing its last step. And at other times it may repeat its last jump,
even making runs of jump in the same direction like a sequence of
heads.
So the particle is behaving randomly. Yet now this system (also called
a Levy flight) has a memory because each new step takes off from
wherever the particle last landed. It is like a drunk staggering around
an infinitely large field. The actions are random but carry a
history of the past with them
When we look at the statistics of such a random walk, it is fractal.
Scale-free. Patchy instead of smooth. The particle can spend
some time wandering one small corner of the space before lurching away.
It is all about fits and starts as the particle appears to be battling
two opposed tendencies, one to cluster, the other to flee. And
because this dichotomous impulse is freely expressed over
spatiotemporal scale, the resulting wanderings become fractal or
scale-free.
disorder fills the middle ground
Now let’s return to the statistical picture created by gas
particles allowed to mess about in a sealed box. As we said, there are
two possible initial states of order but only one disorderly outcome
for an ideal gas trapped in a container. This disorder has a single
scale.
And it is a scale centred around the system’s lower or local
bound of orderliness. The disorder looks a long way from clumped order
but ends up looking very close to smooth order.
Why is this so? The particles released from either kind of order
– clumped or grid-like – will start to bang about
like angry bees. They will go off in any which direction and randomise
their positions.
They will also randomise their velocities. It would not matter if we
released all the particles with a uniform speed or with a wide variety
of speeds. Through their elastic collisions they would bump about in
ways that had the effect of creating a homogenous distribution of
speeds.
If the initial state was a mix of very hot and very cold
particles, then each would rub off against the other in a way that
moved both towards some joint average temperature.
Equally, if all the particles were set
off with exactly the
same temperature, the same kinetic energy, then the randomness
of their collisions would give some particles
a little bit of an extra kick, others left slightly slowed down. From
one single value, there would grow a Gaussian spread of values. So
from either pole of greater order, the population of particles
would slide its way to the same equilibrium balance of
disorder.
And the box walls would play a crucial part in this. Remember
that the gas particles are flying about freely. Without a box
to contain them, they would simply wander away from each
other. But the box walls reflect every straying particle immediately
back into the
fray. The particles are forced to keep mixing and keep
averaging.
In fact we can see that
it is the global boundary – the walls
of the box – that imposes the actual scale of the disorder.
The number of particles in the box is constant. None escapes. And the
void inside the box is also held at a constant size. So these two
factors combine to set the scale of the lower boundary of
order.
Lower bound order is defined as an exactly grid-like
distribution of paticles. For some particular boxed system,
this grid spacing is then simply the quantity
of empty space divided by the number of available particles.
It spells out how the box should be filled for the lower boundary to
become completely smooth.
But of course the box walls can only constrain the
wanderings of the particles towards this desired smoothness. The
particles have enough remaining freedom, enough unconstrained variety,
to express a randomness around the lower bound mean value. The
particles can be tamed almost to a grid-like smoothness,
but not quite. There must remain a smallest scale Gaussian jitter.
So disorder fills the middle ground of this system,
the space
between its two extremes of order - the global box and the local grid.
But this disorder, being rigidly boxed, does get compressed to
a
single scale - and the smallest scale. Again what seems a pretty simple
situation - a bunch of free particles trapped inside an empty box -
harbours quite a metaphysical complexity. It certainly no longer seems
like the ontologically most simple possible situation. Surely
we can do better?
now lift the lid of the box
Thermodynamics offers a statistical description of a system. It is how
things pan out over the long run. And the organic argument is that a
system is always a hierarchy. There will be the triadic structure of an
orderly upper limit, an orderly lower limit, and then a middle ground
of disorderly-ness where things freely mix, completely expressing their
local freedoms within the prevailing global context of constraints. In
short, between the levels of the container and the particle we find
chaos.
Or at least this is the mechanical model of an organic hierarchy. The
classical Maxwell-Boltzmann system of an ideal gas trapped in a closed
box illustrates the basic logic of our holistic story. Yet being
mechanical, it has to assume a few key things.
It
assumes for example that the particles can fly free while they are
travelling through a void. So axiomatically, any
meaningful intermediate
levels of scale are banished from the start. The system collapses
towards its lower
boundary of order – the Gaussian jitter of particles about a
grid – because all larger scales of existence in the box have
no memory. A vacuum is seen as an a-causal nothingness, an
empty realm which has spatiotemporal scale yet is not
capable of being marked by the events taking place within it.
So in Boltzmann’s box, causality is not just dichotomised
– that is, separated and then allowed to mix over all scales.
It is broken apart or dualised. You have the local freedoms of the
particles making one scale of causal action. Then a long stretch of
nothingness and quite suddenly at a certain distance, or spatiotemporal
scale, the particles run into a solid reflective barrier. The box walls
eventually overwhelm the free flight of the particles, forcing them
to homogenise towards a smallest grain of disorder.
Now how are we going to change this mental image around to create
something more like a properly organic model of an organic hierarchy?
Well we can start by simply taking away the walls of the box.
What happens if we begin with an orderly clump of particles released
into an infinite void, a place without walls to confine them? Or
equivalently, if they are released into a box that is expanding at
precisely the same rate as the particles can spread (much as the
spacetime of our universe is said to expand at the rate of its material
contents)?
Well now we will get a different statistical picture of their
distribution. Instead of being constrained to smoothness, they will
move out and start to become fractally patchy. Like the
drunkard’s walk, we will switch from single scale Gaussian
statistics to fractal, scale-free, long-tail and power-law statistics.
Or what some have called Mandelbrotian statistics.
The signature of this kind of statistics is that it enjoys scale
symmetry. Every scale is present yet none is privileged. You have small
scale randomness, large scale randomness, and all the scales
in-between. It is symmetrical in that moving up or down the scale of
observation leaves the world looking the same. A fractal world is flat
in a direction – an axis or dimension - that we rarely notice
even can exist.
And recognising that the middle of our triadic hierarchy
has scale symmetry neatly ties
up our story of organic logic. We said an organic system is
formed by the dichotomous separation towards asymmetric limits of
scale. Dichotomisation literally opens up a space. But then as
this gap yawns,
it is also getting back-filled by a mixing – a disordering
– of the aysmmetric order being created. And this act of
disordering results in an internal axis of symmetry.
So we end up with a dichotomy of asymmetry and symmetry. Remember that
vaguenesss, the starting point for our causal story, is itself best
modelled as an unbroken symmetry. Then vagueness gets broken -
dichotomised towards the local and the global - to create
an asymmetric realm of scale. Completing this tale of organic
development we finally have a disordering of what has been divided. And
as this takes place across all possible scales, we again return to the
"unmarked flatness" of a symmetry. Except instead of being a vague
symmetry, now it is a crisply developed symmetry.
So as it should, the causal picture all ties back together.
But before returning to organic hierarchies, we need to do a little
more mathematics. To have an exact description of the disordered middle
grounds of organic hiearchies, we ought to consider various examples
of fractal
systems.