Notes on Horsthemke and Lefever

1
Notes on Horsthemke and Lefever July , 2003
Sorin Solomon
2
1.2.1 Macroscopic Description of
Self-Organization - Constant
Environment macroscopic systems - described
by evolution equations
2.1
X(r, t) vectors whose components Xi are state
variables e.g. temperature, electrical
potential, velocity field f l (X(r,t))
functional relations expressing local evolution.
-may contain partial derivatives and
be non-linear, l control parameters e.g.
kinetic constants, diffusion coefficients,
fixed concentrations of some compounds, etc.).
3
Constraints imposed by the environment on
1.1
- Boundary Conditions - Dirichlet fix
the value of X(r, t) on the surface -
Neumann fix the values of the fluxes
on the surface S (n vector normal
to the surface). - Values of l Fixed
environment gt Time independent solution
(reference, banal, un-self-organized solution)
1.2
Fluctuations in the environment -gt
self-organization
4
Stability of the reference under to small
perturbations.
1.3
The time evolution of the perturbation x (r, t)
is then given by the solution of the system of
equations
1.4
obtained by linearization of (1.1). The elements
Aij are time independent, since we have
linearized around a time-independent reference
state, and thus (1.4) admits solutions of the form
1.5
5
The xk must satisfy the boundary conditions
imposed on the system but may have lower symmetry
properties than the reference state. In fact they
are simply the eigenvectors of the eigenv,alue
problem (k refers to the wave numbers possible)
1.6
The values of Re w k determine the rate at
which the perturbations of the state variables
evolve. Typically the life-time of disturbances
in the system is of the order of
1.7
gt t macro the macroscopic time scale of the
evolution of the system
6
The onset of a transition can then be found
simply by studying the behavior of Re w k as a
function of -the values of the control parameters
l and of the boundary conditions imposed on the
system. To be specific, let us assume that I we
explore the properties of the system by
manipulating a single control parameter l. At
that point l lc at which at least one Rewk
changes -from negative to positive, the lifetime
of fluctuations tends in first approximation to
become infinite. In other words there is a
slowing down of the rate of relaxation of the
fluctuations.
7
The value lc is called a point of bifurcation a
point at which one or several new solutions of
the equations (1.1) coalesce with the reference
state X considered. By new solution we, refer
here to the stable or unstable asymptotic regimes
which the system may approach respectively for t
-gt or t -gt . It is customary to
associate with a new solution of (1.1) a quantity
m, or order parameter, which vanishes at lc and
which measures the deviation from the reference
state, e. g., the difference between the
concentration of a compound on the new branch of
solutions and the value of its concentration on
the reference state, the amplitude of a spatial
or temporal oscillatory mode, etc.
8
Plotting these quantities as a function of l
yields a bifurcation diagram. An example for a
second-order phase transition is bifurcation
diagram is
9

• Below lc there exists a unique asymptotic
  solution
• - stable and
• highest symmetry compatible with the imposed
  constraints
• At lc this solution becomes unstable and
• new branches of solution of lower symmetry
  bifurcate.
• E.g.
• - classical hydrodynamical instabilities
  described by Benard or Taylor.
• onset of a coherent spatial pattern of
  convection cells in an initially
• unstructured fluid phase when the temperature
  gradient or the angular velocity
• gradient imposed across the systems passes.
  through a threshold value.
• - onset of temporal and/or spatial oscillations
  in chemical and enzymatic reaction systems
• Belousov-Zhabotinsky reaction and
• -the reaction of the glycolytic enzyme
  phosphofructokinase.

10
Typically in second-order phase transitions, the
order parameter grows for l gt lc like
1.8
and the relaxation time of fluctuations behaves
in the vicinity of lc like, compare with (1.7),
11
In addition to the behavior sketched in Fig. 1.1,
one finds in all branches of the physical and
biological sciences an overabundance of
discontinuous transition phenomena similar to
first-order phase transitions. They are
characterized by the existence of a branch of
solutions which bifurcates subcritically and is
part of an hysteresis loop as represented in Fig.
1.2.
Fig. 1.2. First-order phase transition. At lc a
new branch of solutions bifurcates. For lc lt l
lt l c two locally stable states coexist the
reference state (curve a) and a new branch of
(curve b)
12
When the external parameter l increases
continuously from zero, the state variables X may
jump at l l c from the lower branch of steady
states (a) to the upper branch (b). If l is then
decreased, the down jump from (b) to (a) takes
place at a different value lc This is the
simplest form of first-order transition possible
in spatially homogeneous systems a cusp
catastrophe for the homogeneous steady-state
solutions of (1.2) when plotted in terms of
appropriate control parameters ll, l2 these
states lie on a surface which typically exhibits
a fold (Fig. 1.3). The coordinates (Alc, A2c, Xc)
where the fold has its origin constitute the
critical point of the system.
13
Fig. 1.3. Cusp catastrophe
14
It would be a task out of proportion to the scope
of this book to try to present here a complete
panorama of what is known of the bifurcation and
self-organization phenomena observed even in the
simplest natural or laboratory systems,
especially since in the last ten years giant
advances have been accomplished in many widely
diverse fields. However, there are two aspects
of the organization of macroscopic systems in a
constant environment which are of fundamental
importance and which we need to recall
explicitly. Keeping these aspects in mind permits
us to situate more exactly the framework in which
noise induced transitions have to be discussed
the/question of the thermodynamic interpretation
of bifurcations, with Sect. 1.2.2 as a brief
summary of the influence of internal fluctuations
on bifurcation diagrams.
15
So far we have introduced the mechanisms of
self-organization and bifurcation without
considering the dependence of these phenomena on
the strength of the external constraints imposed
by the environment. We also mentioned the
simplest examples of bifurcation taking place
when a "banal" reference state becomes unstable.
However, since equations (1.1) are in general
highly nonlinear, their exploration in parameter
space usually reveals a whole network of further
instabilities. This network is responsible for
the complex behaviors and the multiplicity of
scenarios mentioned in Sect. 1.1. The richness
of dynamical behaviors in a macroscopic system is
specific to the domain far from thermodynamic
equilibrium. Conversely, in the parameter space
there exists a domain close to thermodynamic
equilibrium where the nonlinearities present in
(1.1) cease to playa role whatever system is
investigated. The dynamical properties of any
macroscopic system then become fairly simple and
can be apprehended in a model-independent
fashion. We should like to recall these
thermodynamic results here because they
demonstrate the clear-cut distinction which
exists between two types of order in a constant
environment.
16
First let us recall the situation at
thermodynamic equilibrium. If a system is
in contact with a constant environment which
furthermore is at equilibrium, i.e., which
imposes no constraints in the form of fluxes of
energy or matter on the system, then only the
class of coherent organization known as
equilibrium structures can appear. The standard
example is the crystal. The formation of
these structures obeys a universal mechanism
hich, at least qualitatively, is
well understood. This mechanism immediately
follows from the second law of thermodynamics. At
constant temperature and volume, it amounts to
looking for the type of molecular organization
that minimizes the system's free energy F E-
TS, where E is energy, T temperature and S
entropy. Self-organization is then the result of
competition between the energy and entropy. At
low temperatures, the system adopts the lowest
energetic configuration even if its entropy is
small as in the case of the crystal.
17
During the last fifty years physicists have
striven to understand how far the basic
simplicity and beauty of the lws which govern
self-organization under con- - stant equilibrium
conditions can be transposed to self-organization
phenomena taking place in sj'stems which are
subjected to a constant environmental stress and
which therefore cannot approach a state of
thermodynamic equilibrium. The motivations in
this direction are very strong, since clearly
many of the most organized systems of nature like
biological systems are submitted to
an environment far from thermodynamic
equilibrium. The natural way to try to extend
the ideas ex-plaining the formation of
equilibrium structures to nonequilibrium
situations is to look for the conditions
under which the dynamical' properties of
macroscopic systems can be expressed in terms of
a potential funcpon which plays the role of the
free energy. A first answer to this question was
qound jn the development of the linear
thermodynamic theory of irreversible processes.
18
(No Transcript)