NonEquilibrium Thermodynamics

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Non-Equilibrium Thermodynamics
And Complex Systems Theory
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Galileo
1564-1642
The simplest phenomena studied by science can
thus be interpreted as the key to understanding
nature as a whole the complexity of the later is
only apparent . . .
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Isaac Newton
1642-1727
. . . the separation of the world into the
profane and sacred, into the world that is
subject to chance and degradation, and a sacred
one that is meaningful . . .
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Henri Bergson
1859-1941
The basic characteristics of trajectories are
lawfulness, determinism, and reversibility.
Everything is given.
5
Pierre-Simon Laplace
1749-1827
For the science of Laplace . . . a description
is objective to the extent to which the observer
is excluded and the description itself is made
from a point lying de jure outside the world,
that is, from the divine viewpoint to which the
human soul, created as it was in Gods image, had
access at the beginning.
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Rudolf Julius Emmanuel Clausius
1796-1832
For an isolated system, equilibrium appears as
an attractor of non-equilibrium states.
Entropy always increases.
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Albert Einstein
1879 -1955
Supreme purity, clarity, and certainty at the
cost of completeness.
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William Thompson
1824 - 1907
Thomson formulated his new principle the
existence in nature of a universal tendency
toward the degradation of mechanical energy.
Note the word "universal," which has obvious
cosmological connotations.
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William Thompson
1824 - 1907
Effect-producing differences in nature
progressively diminishes. The world uses up its
differences as it goes from one conversion to
another and tends toward a final state of thermal
equilibrium, "heat death." In accordance with
Fourier's law, in the end there will be no longer
any differences in temperature to produce a
mechanical effect.
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William Thompson
1824 - 1907
Or, from a recent textbook the unpalatable
truth appears to be that the inexorable
disintegration of the universe as we know it
seems assured, the organization which sustains
all ordered activity, from men to galaxies, is
slowly but inevitably running down, and may even
be overtaken by total gravitational collapse into
oblivion."
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The Problem of Problems
How can something become more complex and diverse
with time, when the trend is for everything to
decay, wear down, wear out?
Laws of Thermodynamics
1st Law You cant win. Energy can be changed
from one form to another, but it cannot be
created or destroyed it is conserved.
2nd Law You cant break even. In all energy
exchanges, if no energy enters or leaves the
system, the potential energy of the state will
always be less than that of the initial state."
Or, entropy always increases.
3rd Law You cant get out of the game. Absolute
zero cannot be attained by any procedure in a
finite number of steps. Absolute zero can be
approached arbitrarily closely, but it can never
be reached.
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The Problem of Problems
Is about the 2nd Law You cant break even.
Which is about . . .
Entropy
Thermodynamic Entropy - "For a closed system, the
quantitative measure of the amount of thermal
energy not available to do work."

• The higher the entropy the more uniform heat is
  distributed.
• Entropy in a closed system can never decrease.
• It's a negative kind of quantity, the opposite
  of available energy.

Logical Entropy - A measure of the disorder in a
closed system."

• The higher the disorder the higher the entropy.
• Entropy in a closed system can never decrease.
• Without someone to fix it a broken glass never
  mends.

The trouble is . . . .
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The Problem of Problems
Is about the 2nd Law You cant break even.
Which is about . . .
Entropy
We observe in the world all around us that order
does increase, things do become more complex with
time.

• Cars and airplanes have obviously become very
  complex but we can attribute that to teleology
  design and purpose.

• But, what about cities or economies? who can
  be shown to be in charge of their increasing
  complexity?

• And then, there is the most difficult problem
  the origin of life.

All this leads to . . . .
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Denis Diderot
1713 - 1784
Look at this egg with it you can overthrow all
the schools of theology and all the churches in
the world. What is this egg? An insensitive
mass before the germ is put into it . . . How
does this mass evolve into a new organization,
into sensitivity, into life? Through heat. What
will generate heat in it? Motion. What will the
successive effects of motion be?
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Denis Diderot
1713 - 1784
Just listen to your own arguments and you will
feel how pitiful they are.
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Illya Prigogine
1907 - 2003
It is not only living nature that is profoundly
alien to the models of thermodynamic equilibrium.
In the world that we are familiar with,
equilibrium is a rare and precarious state.
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The Three Phases of Science
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Order Out of Chaos
Ilya Prigogine in collaboration with other
members of the "Brussels School showed that
physical and chemical systems far from
thermodynamical equilibrium tend to self-organize
by exporting entropy and thus to form dissipative
structures. Both his philosophical musings about
the new world view implied by self-organization
and irreversible change, and his scientific work
on bifurcations and order through fluctuations
remain classics, cited in the most diverse
contexts.
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What is Required for a System to Evolve?
The significant problems we face today cannot be
solved with the same level of thinking we were at
when we created them. Albert Einstein
One of the perverse laws of the universe is that
we least understand those phenomena that have the
greatest bearing on our lives and future.
Eugene Linden, 2002
The Future in
Plain Sight
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The Pendulum as a Limit Cycle
Can the Behavior of a Pendulum Evolve???
Aside from slowing down does the pendulums
behavior change, spontaneously, on its own?
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The Pendulum as a Limit Cycle
The pendulum is an important system because
throughout history it has been the emblem of
classical mechanics and the epitome of clockwork
regularity.
In the 19th century the great French
mathematician Laplace said If we were to know
with precision the positions and speeds of all
the particles in the universe then we could
predict the future with certainty.
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Classical Science and Evolutionary Systems Science
Classical Science
Evolutionary Systems

• Reductionist

• Holistic and Emergent

To understand a system we must take it apart and
understand the pieces.
The whole is more than the sum of the parts the
whole must therefore be studied in its own right.
2. Linear
2. Non-Linear
A system where the effects are proportional to
the causes.
Of or relating to a system of equations whose
effects are not proportional to their causes -
sensitive dependence. Such a set of equations can
be chaotic.
3. Clear Boundaries
3. Diffuse Boundaries
The best system is self-contained, isolated,
complete in itself so it can be understood
without distractions or outside influence.
Organic. Coevolving. Because complex systems
are open they must exchange energy and
information with the surroundingsthe
environment. Therefore, boundaries are not
distinct.
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Classical Science and Evolutionary Systems Science
Albert-Laszlo Barabasi, 2002, Linked How
Everything Is Connected to Everything Else and
What It Means
Have you ever seen a child take apart a favorite
toy? Did you then see the little one cry after
realizing he could not put all the pieces back
together again? Well, here is a secret that never
makes the headlines We have taken apart the
universe and have no idea how to put it back
together. After spending trillions of research
dollars to disassemble nature in the last
century, we are just now acknowledging that we
have no clue how to continueexcept to take it
apart further. Reductionism was the driving
force behind much of the twentieth century's
scientific research.
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Classical Science and Evolutionary Systems Science
Classical Science
Evolutionary Systems
4. Discrete Solutions
4. Qualitative
Numeric solutionsanswers. There is a right
answer, and the more precision we have the
better. The devil is in the details.
The interactions among the agents are beyond
logical analysis, but the outcome of the
interactions produce patterns easily observed and
compared. The eye is quicker than the mind.
5. Centralized Control
5. Interconnected Systems
Straight forward cause and effect relationships,
or clear chains of command.
Every component affects every other component in
a complex network of positive and negative
feedback. There is no centralized control.
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Classical Science and Complex Systems Science
OK, OK. But, really, just how common are these
complex systems?
It has been said that if the universe is an
elephant, then linear theory (classical science)
can only be used to describe the last molecule in
the tail of the elephant and chaos (read complex
systems) theory must be used to understand the
rest. Or, in other words, almost all interesting
real-world systems are described by non-linear
(complex) systems.
http//en.wikipedia.org/wiki/Chaos_theory
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The Pendulum as a Chaotic System
Yet, Henir Poincare in a 1903 essay titled
Science and Method said this
If we knew exactly the laws of nature and
the situation of the universe at the initial
moment, we could predict exactly the situation of
that same universe at a succeeding moment. but
even if it were the case that the natural laws
had no longer any secret for us, we could still
only know the initial situation approximately.
Henri Poincaré (1854-1912)
If that enabled us to predict the succeeding
situation with the same approximation, that is
all we require, and we should say that the
phenomenon had been predicted, that it is
governed by laws. But it is not always so it may
happen that small differences in the initial
conditions produce very great ones in the final
phenomena. A small error in the former will
produce an enormous error in the latter.
Prediction becomes impossible, and we have the
fortuitous phenomenon.
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Attractors
1. To cause to draw near or adhere by physical
force Magnetic poles are attracted to their
opposites.
2. To arouse or compel the interest, admiration,
or attention of We were attracted by the display
of lights.
3. A set of physical properties toward which a
system tends to evolve, regardless of the
starting conditions of the system.
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Real Space and Phase Space
Limit Cycle Attractor
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Real Space and Phase Space
Point Attractor
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Universality
Properties of Complex Evolutionary Systems
Strange Attractors with Examples
A Variety of Attractors
Run Lorenz
Lorenz Applet
Run Xnext
Run Galaxy
Strange Attractor
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Universality
Properties of Complex Evolutionary Systems
Strange Attractors - Turbulence
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Limit Cycle Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Strange Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Limit Cycle Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Strange Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Limit Cycle Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Strange Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Limit Cycle Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Strange Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Limit Cycle Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Strange Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
Limit Cycle Attractor
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Properties of Complex Evolutionary Systems
The Pendulum as a Strange Attractor
http//hubble.physik.uni-konstanz.de/jkrueger/phsi
362/
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Observe that the pendulum bifurcation diagram in
the last slide shows the same period doubling
pattern as the Xnext bifurcation diagram below.
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Chaos non-technical, dictionary definition
1 a state of extreme confusion and disorder
2 any confused, disorderly mass
3 the formless and disordered state of matter
before the creation of the cosmos
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Definitions of Deterministic Chaos
1 the quantitative study of unstable aperiodic
behavior in deterministic non-linear dynamical
systems
2 Xnext rX (1-X)
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Henri Bergson
1859-1941
The basic characteristics of trajectories are
lawfulness, determinism, and reversibility.
Everything is given.
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Determinism and Predictability
Determinism is the philosophical proposition that
every event, including human cognition, decision
and action, is causally determined by an unbroken
chain of prior occurrences. It holds that no
random, spontaneous, mysterious, or miraculous
events occur.
Since you can write down equations and solve them
in order to predict the second event based on the
occurrence of the first event, the predictability
becomes the key issue. Another word for such
predictability is determinism the first event
determines the occurrence of the second.
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Determinism and Predictability
Determinism is the philosophical proposition that
every event, including human cognition, decision
and action, is causally determined by an unbroken
chain of prior occurrences. It holds that no
random, spontaneous, mysterious, or miraculous
events occur.
A determinisitic system is one whose behavior is
predictable Deterministic predictable Predictabl
e deterministic
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Marquis Pierre Simon de Laplace We may regard
the present state of the universe as the effect
of its past and the cause of its future. An
intellect which at any given moment knew all of
the forces that animate nature and the mutual
positions of the beings that compose it, if this
intellect were vast enough to submit the data to
analysis, could condense into a single formula
the movement of the greatest bodies of the
universe and that of the lightest atom for such
an intellect nothing could be uncertain and the
future just like the past would be present before
its eyes.
http//plato.stanford.edu/entries/determinism-caus
al/
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Modeling an Evolutionary System
Xnext A Model of Deterministic Chaos
(A.k.a. the Logistic or Verhulst Equation)
(1-X)
X next rX
X population size - expressed as a fraction
between 0 (extinction) and 1 (greatest
conceivable population).
X next is what happens at the next iteration or
calculation of the equation. Or, it is the next
generation.
r rate of growth - that can be set higher or
lower. It is the positive feedback. It is the
tuning knob
(1-X) acts like a regulator (the negative
feedback), it keeps the growth within bounds
since as X rises, 1-X falls.
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Modeling an Evolutionary System
Xnext and Deterministic Chaos
Iteration X Value 0 0.0200000 1
0.0529200 2 0.1353226 3
0.3159280 4 0.5835173 5
0.6561671 6 0.6091519 7
0.6428318 8 0.6199175 9
0.6361734 10 0.6249333 11
0.6328575 12 0.6273420 13
0.6312168 14 0.6285118 15
0.6304087 16 0.6290826 17
0.6300117 18 0.6293618 44
0.6296296 45 0.6296296 46
0.6296296 47 0.6296296 48
0.6296296 49 0.6296296 50
0.6296296
X next rX (1-X)
.65
.64
.62
.62
.61
.60
.58
.35
X .02 and r 2.7 X next rX (1-X) X next
(2.7) (.02) (1-.02 .98) X next .0529
.13
.05
.02
Run X-next
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r 2.7
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r 2.9
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r 3.0
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r 3.1
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r 3.3
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r 3.4
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r 3.5
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r 3.6
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r 3.7
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r 3.8
The harder a system is pushed the more unstable
it becomes
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r 3.9
The harder a system is pushed the more
unpredictable its behavior becomes
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r 4.0
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Until it vibrates so hard it flies apart and goes
extinct (becomes a closed system)
r 4.1
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And now, for a little slight of hand
Time Series Diagram
Bifurcation Diagram
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Modeling an Evolutionary System
Bifurcation Diagram
Population Size
Why are some systems stable, reliable,
predictable, and others are not.
2nd Bifurcation
1st Bifurcation
Refrigerators Computers Cars Airplanes
Weather Stock Market Human Behavior Outcome of
sport events
The harder you push a system, the higher the r
value, the more unstable it becomes.
r Values Rate of Growth
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"In my graduate school courses, we were always
taught that large nonlinear systems were
monsters, practically impossible to solve.
Exciting but unpredictable
Complex Systems (you are not taught)
Classical Science (what you are taught)
Arrggghhh
Safe . . .
But boring
72
Our lives are based on what is reasonable and
common sense Truth is apt to be neither. From a
well-worn path we step into a fog wherein lies
precipices, quagmires, and a howling widerness,
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Modeling an Evolutionary System
Bifurcation Diagram
Population Size
Chaos Theory
Complexity Theory
r Values Rate of Growth
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Evolution Via Self Organization
Ilya Prigogine 1917 -
Awarded the Nobel Prize in chemistry in 1977 for
his contributions to nonequilibrium
thermodynamics, particularly the theory of
dissipative structures.
Ilya Prigogine in collaboration with other
members of the "Brussels School showed that
physical and chemical systems far from
thermodynamical equilibrium tend to self-organize
by exporting entropy and thus to form dissipative
structures. Both his philosophical musings about
the new world view implied by self-organization
and irreversible change, and his scientific work
on bifurcations and order through fluctuations
remain classics, cited in the most diverse
contexts.
87
How Does Self Organization Occur ?
When the second law of thermodynamics says
everything should be running down to decay and
disorder
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Evolution Via Self Organization
Self Organized Criticality
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Evolution Via Self Organization
Autocatalytic Networks
Stuart Kauffman
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Evolution Via Self Organization
Cellular Automata
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Cellular Automata and Self Organization
Cellular Automata (CA) are simply grids of cells,
where the individual cells change states
according to a set of rules. The CA may be one
dimensional, or linear, like a string of cells in
a row (below), or two dimensional, like a
checkerboard
Local Rules/Global Behavior
Sample Local Rules
Survival Rules number of surrounding cells
necessary to make it to the next generation.
Birth Rules number of surrounding cells
necessary for a dead cell to come alive the next
generation.
Life3000
LifeWin
MericksCelebration
http//math.hws.edu/xJava/CA/
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Cellular Automata and Self Organization
Cellular Automata (CA) are simply grids of cells,
where the individual cells change states
according to a set of rules. The CA may be one
dimensional, or linear, like a string of cells in
a row (below), or two dimensional, like a
checkerboard
Local Rules/Global Behavior
Optimal Local Rule Set
Survival Rules 2/3 a live cell survives to the
next generation if at least 2 but no more than
three of the surrounding 8 cells are alive. Less
than 2 and it dies of loneliness more than 3 and
it dies of over crowding.-
Birth Rules 3/3 a dead cells comes alive the
next generation if 3, any 3, of the surrounding 8
cells are also alive.
Life3000
Spiral Waves Applet
MericksCelebration
Applet - MJCell
http//sjsu.rudyrucker.com/nksapplets.htm
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Sensitive Dependence in Cellular Automata
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Hysteresis and Oscillating Chemical Reactions
Boris P.Belousov (1893-1970)
Temporal Oscillations
Spatial Oscillations.
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Classical Systems Chemical Reactions
A B C D
Reaction Diffusion Systems
The BZ (Belousov-Zhabotinsky) Reaction
Applet 1
Applet 3
Applet 2
Ilya
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Common Reaction Mechanisms of all known Chemical
Oscillators
1. While the oscillations occur, the chemical
mixture is far from equilibrium (i.e. r is high),
and an energy releasing reaction occurs whose
energy drives the oscillating "sideshow."
2. The energy-releasing reaction can follow at
least two different pathways, and the reaction
periodically switches from one pathway to another.
3. One of these pathways produces a certain
intermediate, while another pathway consumes it,
and the concentration of this intermediate
functions as a "trigger" for the switching from
one pathway to the other.
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http//www-users.med.cornell.edu/dchristi/antispi
ral/antispiral.html
Illya Reaction Diffusion
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Reaction Diffusion
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Reaction Diffusion
101
Reaction Diffusion
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http//www-users.med.cornell.edu/dchristi/antispi
ral/antispiral.html
Complex Chemical Systems
Reaction-Diffusion and Activator-Inhibitor Systems
Moves fast
Reaction
Moving Waves
Moves slow
Inhibition
http//delfin.klte.hu/gasparv/menuh.html
Inhibitor
Stationary Waves/Spots
Activator
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Activaor-Inhibitor Systems in Biology
From patterns in animal hides, butterfly wings,
and shells, to the distribution of organisms in a
ecosystems, activator-inhibitor systems provide
explanatory mechanisms.
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Reaction-Diffusion/Activator-Inhibitor in
Geology Agate Structure
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Reaction-Diffusion/Activator-Inhibitor in
Geology Liesegang
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Reaction-Diffusion/Activator-Inhibitor in
Geology Liesegang
http//web.uct.ac.za/depts/geolsci/dlr/laingsburg/
Dscn0160.jpg
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Reaction-Diffusion/Activator-Inhibitor in
Geology Liesegang
http//www.geo.ucalgary.ca/macrae/t_origins/carbb
ones/leisegang.jpeg
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Reaction-Diffusion is the most likely cause of
the structure In the arms of a spiral galaxy
http//en.wikipedia.org/wiki/ImageGalaxy.m74.arp.
750pix.jpg
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Reaction Diffusion Systems
Fibonacci Spiral Phylotaxis in plants
http//www.drjax.co.uk/14420html20pages/139.html
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Reaction Diffusion Systems
Fibonacci Spiral
http//www.mathcurve.com/courbes2d/logarithmic/spi
raledor.shtml
http//www.taos-telecommunity.org/epow/EPOW-Archiv
e/archive_2003/EPOW-031117.htm
http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/phi2DGeomTrig.html
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Complex Evolutionary Systems
One of the practical difficulties with studying
and understanding oscillating Earth systems is
the scale of observation humans have.
We are way too small and live for way to short a
time for human memory, even human history, to
have kept track and remembered them.
For example, how an individual experiences a
hurricane.
http//delfin.klte.hu/gasparv/menuh.html
http//people.musc.edu/alievr/rubin.html
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Bistable Behavior in Recent Climate
The North Atlantic Oscillation
Recent Glacial/Interglacial Cycles
http//sinus.unibe.ch/klimet/wanner/nao.html
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Bistable Behavior in Recent Climate
The North Atlantic Oscillation
Recent Glacial/Interglacial Cycles
El Nino Southern Oscillation
http//sinus.unibe.ch/klimet/wanner/nao.html
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Bistable Behavior in Recent Climate
The North Atlantic Oscillation
Recent Glacial/Interglacial Cycles
El Nino Southern Oscillation
http//www.isse.ucar.edu/signal/15/articles.html
http//sinus.unibe.ch/klimet/wanner/nao.html
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Bistable Behavior in Recent Climate
The Southern Oscillation
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Oscillating Systems
Which leads to the more pertinent question -
when a system has more than one path available to
it, and periodically switches between them, when
and how does it choose between them?
Here we see the ability of a system to exist in
different states under the same conditions (i.e.
same r value).
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Bistable Behavior in Recent Climate
The Southern Oscillation
http//www.pmel.noaa.gov/tao/elnino/faq.html
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Bistable Behavior in Recent Climate
The Southern Oscillation
http//www.pmel.noaa.gov/tao/elnino/faq.html
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Bistable Behavior in Recent Climate
http//www.usgcrp.gov/usgcrp/images/glob_jan-dec_p
g.gif
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Bistable Behavior in Recent Climate
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Bistable Behavior in Recent Climate
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Bistable Behavior in Recent Climate
A Geological Example
Recent Glacial/Interglacial Cycles
20,000 Year Record
400,000 Year Record
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Sir Arthur Eddington
1882 - 1944
Not only is the universe stranger than we
imagine,
it is stranger than we can imagine.
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There is a whole new world out there . . .
Just waiting for you. . .
Go find it !