1.1 Deterministic and Random Aspects of Macroscopic Order

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Introduction 1.1 Deterministic and Random Aspects
of Macroscopic Order Science uncovering hidden
regularities in nature Belief in fundamental
regularities in spite of variability and
complexity
Once understood enlighten and simplify
everything. Johannes Kepler "The diversity of
the phenomena of Nature is so great, and the
treasures hidden in the heavens so rich,
precisely in order that the human mind shall
never be lacking in fresh nourishment" Order lt-gt
deterministic conception of the universe. Laws
lt-gt make exactly predictable events which at
first sight were undeterminable,
( eclipses or the path followed by comets in the
sky) Newton's success gt determinism of the laws
of nature basis scientific methodology.
postulate that behind any phenomenon as
mysterious as could be, purely
materialistic causes are acting which sooner or
later will be identified.
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• determinism -gt nature is precisely organized, i.
  e.,
• no place for such "imperfections"
• such as chance or
  randomness.
• discovery of the precision and beauty of
  celestial mechanics
• all doubts were brushed away, even for the most
  complex systems,
• that random processes play a significant role
  in the natural world
• further strengthening in the early progress of
  the natural sciences
• Dalton's law of definite proportions for the
  combinations of chemical species
• Boyle-Mariotte's law for the expansion of gases
• The idea that this holds in all fields dominated
  the development of science
• The achievements of the natural sciences were
  viewed as a universal model
• In the case of medicine this conviction is
  expressed in the response of Laplace to somebody
  who was astonished that he had proposed to admit
  medical doctors to the Academy of Sciences, since
  medicine at that time was not considered as a
  science.
• - It is, Laplace said simply, in order that
  they be among scientists.

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Some forty years later, medicine had clearly
advanced in the direction indicated by Laplace
when Claude Bernard noted that, "II y a un
determinisme absolu dans tout phenomene vital
des lors il y a une science biologique". With
a few decades of delay, the same ideas appear in
the social sciences. For Durkheim 1.3
determinism and science were one and the same
thing the idea that societies are submitted to
natural laws and constitute "un regne naturel" is
equivalent to admitting that they art'governed by
"the principle of determinism" which at that time
was so firmly established in the so-called exact
sciences. Yet it is now clear that the complete
rejection of randomness prevailing in
the nineteenth century was unwarranted.
Classical' determinism experienced a strong blow
when the atomic and subatomic properties of
matter revealed irreducible in-determinacies and
required the formulation of a new mechanics
quantum mechanics. But independently even of
these microscopic findings, the strength of
deterministic positions has also been eroded for
other reasons related to problems posed by the
properties of matter at the very macroscopic
level.
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.. In macroscopic physics the oldest and still
actual of these problems concerns the meaning of
entropy and of irreversibility which entropy
purports to describe. Taking a step which
shocked some of his most illustrious
contemporaries, Boltzmann proposed an
interpretation of entropy which supplemented the
laws of dynamics with stochastic considerations.
The latter have generally been regarded as
added approximations made unavoidable by the
practical impossibility to treat exactly the
dynamics of a manybody problem. This point of
view is likely to be revised. Indeed the idea
gains ground that irreversibility is already
rooted in dynamics and is not an illusion due to
approximations. It has been shown that in the
evolution of classical dynamical systems an
intrinsic randomness coexists with the perfectly
deterministic laws of dynamics 1.4- 7 (for a
review see 1.8 -10). More precisely, whereas
the motion of individual particles follows
trajectories which are deterministic in the
fullest sense of the word, the motion of regions
of phase space, i. e., of bundles of
trajectories, acquires stochastic features.
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irreversibility lt-gt the role of stochastic
elements lt-gt the constructive role of
irreversible processes in the formation of
large-scale supramolecular organisations
commonly called dissipative structures.
Striking dichotomy the behavior at
macroscopic level lt-gt behavior at the
microscopic level. space-time coherence of
chemical dissipative structures,
laser beams or
Benard rolls
possible can such a long-range macroscopic
order spontaneously appear and maintain itself
in spite of molecular chaos and internal
fluctuations? The same dichotomy processes
of self-organization taking place in biology.
Metabolic processes lt-gt chemical molecular
mechanisms Yet extraordinarily precise.
Production with an astonishing degree of
dependability of protein
molecules whose probability of occurrence by pure
chance is 0
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As another facet of the constructive role of
irreversible processes and of the dichotomy
between order and randomness which is involved
let us consider for a moment the mechanism of
biological evolution. Biological species and
even prebiotic macromolecular compounds are
self-organizing systems. They are in a perpetual
state of becoming which depends in an essential
manner on events of chance. At random and
independently of the direction of evolution, a
large pool of hereditary genetic variations
develops. This pool is the indispensable raw
material for evolution. In it, evolution finds
the favorable variations whose frequency in the
population it subsequently amplifies and
stabilizes via the precise well-defined rules of
heredity transmission. Thus the distinguishing
characteristic of evolution theory, which clearly
had no analog in the physical sciences at the
time when evolution theory was formulated, is
that it gives an unusually important role to
random events. Mutations are the random
triggers of progress. However their effects are
even more far reaching and decisive than that
these events of chance may decide at random
between different possible roads of evolution.
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• In recent years the mechanisms of
  self-organization in the physical sciences have
  be-
• come much better understood and a new
  appreciation of the role of chance in
• natural phenomena has emerged gt turn in ideas to
  which these advances led.
• They suggest that the macroscopic world is far
  less deterministic,
• e., predictable in the classical sense, than we
  ever thought.
• In fact, completely new aspects of randomness
  have come to light
• reappraisal of the role and importance of
• random phenomena in nature.

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First, mechanisms of self-organization ?
strongly dissipative systems (rather than
conservative, equilibrium-type systems where the
behavior of a dissipative system can easily be
predicted it has a unique attractor the
thermodynamic branch. Far from thermodynamic
equilibrium on the contrary, the same system may
possess a complex network of bifurcations. The
importance of elements of chance,such as internal
fluctuations then inevitably increases. Their
influence becomes crucial in the choices which
the system makes in the course of its evolu- tion
between the numerous basins of attraction, or
dissipative structures, to which bifurcations
give rise 1.14, 15. When an external
parameter is changing, somewhat as in biological
evolution, different scenarios can unfold some
attractors will be visited, others will not,
depending only on the random fluctuations which
occur at each instant of time. Remarkably, this
sensitivity to fluctuations already appears in
the simplest self-organizing hydrodynamical
systems. It is known, for example, that a Benard
system whose parameters are controlled with the
best possible experimental accuracy nevertheless
in two identical experiments evolve unpredictably
according to different scenarios 1.16.
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• The second blow to conventional ideas
• - a shock in the physical and biological
  sciences
• facility with which the scenarios govering the
  evolution of deterministic macroscopic systems,
  e. g., systems described by ordinary differential
  equations, generate
• irregular aperiodic solutions called chaotic or
  turbulent.
• deviate drastically from the scenario proposed by
  Landau
• to explain hydrodynamical turbulence
• the excitation of an infinite frequency
  modes in a continuous system.
• alternative scenario, (Ruelle and Takens) only
  three frequencies.
• "noisy behaviorlt-gt a strange attractor
• (appears after only three successive Hopf
  bifurcations).
• strange attractor gt sensitive dependence on
  initial conditions
• nearby trajectories separate exponentially in
  time gt turbulent behavior,
• can occur already in low-dimensional systems
  three first-order differential equations.

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• two other scenarios
• - the intermittent transition to turbulence
• - the period doubling scenario
• control parameter of a dissipative system -
  changed in a continuous systematic way
• chaos is not necessarily the "ultimate
• chaotic domains may alternate with temporally
  organized regimes
• Belousov-Zhabotinsky reaction hydrodynamical and
  chemical turbulence
• The investigation of self-organization in
  nonequilibrium systems which are
• coupled to fluctuating environments has brought
  forth the
• This course third direction to reappraise the
  role of randomness
• Break a naive intuitive belief
• the influence of environmental fluctuations
  (rapid random variations) is trivial
• (i) rapid noise is averaged out and thus a
  macroscopic system
• essentially adjusts its state to the average
  environmental conditions
• (ii) there will be a spreading or smearing
  out of the system's state around that average
  state due to the stochastic variability of the
  surroundings.

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Surprisingly, however, more often than not the
behavior of a nonlinear system in a noisy
environment does not conform to the common
intuitive expectations. Systematic theoretical
and experimental studies have demonstrated that
in general the behavior is stupendously different
from the aforementioned simple picture. In a
large class of natural phenomena environmental
randomness can, despite its apparently
disorganizing influence, induce a much richer
variety of behaviors than that possible under
corresponding deterministic conditions.
Astonishingly, an increase in environmental
variability can lead to a structuring of
nonlinear systems which has no deterministic
analog. Perhaps even more remarkably, these
transition phenomena display features similar to
equilibrium phase transitions and to transition
phenomena encountered in nonequilibrium systems
under deterministic external constraints as, for
instance, the Benard instability and the laser
transition. The notion of phase transition was
extended to the latter about a decade ago, since
certain properties which characterize equilibrium
phase transition are also found in these
phenomena.
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Point of the course it is possible to go even
one step further and to extend the concept of
phase transition to the new class of
nonequilibrium transition phenomena which are
induced by environmental randomness. Thus called
them noise-induced nonequilibrium phase
transitions or, for short, noise-induced
transitions. This choice of name is intended to
express that this new class of transition
phenomena is close kin to the classical
equilibrium phase transitions and to the more
recent class of nonequilibrium phase transitions.
However, it is not meant to imply, and it
should not be expected, that noise-induced
transitions display exactly the same features as
equilibrium transitions. Deterministic
nonequilibrium conditions already lead to a
richer transition behavior with such new
possibilities as the transition to sustained
periodic behavior known as limit cycle. More
importantly, for the new class of transition
phenomena, one cannot overlook the fact that the
new states, to which noise-induced transitions
give rise, carry a permanent mark of their
turbulent birth. They.are creatures of noise and
as such at first sight foreign to our deeply
ingrained deterministic conceptions of order.
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Outside the physical sciences also, though still
somewhat confused, the idea is gaining ground
that random factors which do not enter into
consideration in the usual Darwinian theory play
an important role in evolution. Writing about
the possible causes of the Permian extinction
which some 225 million years ago wiped out more
than 80 percent of all species living at that
time, Gould Ref. 1.57, p. 349 concludes in a
manner which sounds like a response to Johannes
Kepler that, "Perhaps randomness is not merely an
adequate description for complex cases that we
cannot specify. Perhaps the world really works
this way, aI)d many happenings are uncaused in
any conventional sense of the word. Perhaps our
gut feeling that it cannot be so reflects only
our hopes and prejudices, our desperate striving
to make sense of a complex and confusing world,
and not the ways of nature."
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In fact, the phenomenon of transitions induced by
external noise belongs to a whole stream of ideas
which really upsets our classical conceptions
concerning the relation between determinate and
random behavior. These ideas constitute a
refutation of our gut feeling about the role of
fluctuations. Though for noise-induced
transitions the situation is not as neat as it is
for classical equilibrium and nonequilibrium
phase transitions, it is far from unpredictable
and lawless. The notions arm concepts,
developed for classical transition phenomena and
essen- tially rooted in a deterministic
conception of nature, can be extended and adapted
to deal with situations where noise plays an
important role. A theoretical investigation is
thus made possible. More important even, the
situation is ac- cessible to experimental
investigation. Transitions induced by external
noise are an observable physical phenomenon as
documented by various experiments in
physico-chemical systems. Noise-induced
transitions are thus more than a mere theoretical
figment and their existence has profound
consequences for our under- standing of
self-organization in macroscopic systems. As
stated above, they force us to reappraise the
role of randomness in natural phenomena.
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The organization is as follows To place the
class of noise-induced transition phenomena in
its proper context, in the next sections we shall
briefly discuss disorder-order transitions under
deterministic environmental conditions and the
effect of internal fluctuations on them. Then
we shall present the phenomenon of noise-induced
transitions and have a first go at the question
of modeling macroscopic systems subjected to a
fluctuating environment.
To make this course self-contained we shall
present in a concise, but we hope nevertheless
clear way the mathematical tools needed for an
unambiguous treatment of non- linear systems
driven by external noise. This will be followed
by a precise and operational definition of
noise-induced transitions. Their properties will
be investigated in detail for Gaussian
white-noise. Experiments, in which
noise-induced transitions have been observed,
will be described.
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... 1.2 From Crystals to Dissipative
Structures Order-disorder transitions
discussed in Physics under
deterministic / constant in time / or time
periodic conditions. These conditions have been
universally adopted in the development of
thermodynamics and statistical mechanics. They
correspond to a simplified approach which in all
logic had to be pursued first and which fitted a
general inclination to play down the importance
of environmental randomness. Randomness being
synonymous with imperfection, it had to be
eliminated by all means in experiments in
theories it could only be a source of unnecessary
com- plications obscuring the fundamental beauty
of the processes of self-organization of matter.
Therefore deterministic stationary environmental
conditions have so far always been considered as
self-evident 'in macroscopic physics. We recall
in this section some important results which have
been established within this frame- work. Later
on this will permit us to situate more easily the
novelties which are brought in by noise-induced
transitions. '
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