Cristbal Lpez

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From microscopic dynamics to macroscopic
evolution equations (and viceversa)
Cristóbal López IFISC, Palma de Mallorca,
Spain http//ifisc.uib-csic.es clopez_at_ifisc.uib-cs
ic.es
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Outline

• First part From micro to macro.
• Introduction.
• Two simple Individual Based Models and their
  continuum description.
• Methods to derive continuum descriptions in
  terms of concentration or density fields.

Second part low dimensional systems from
macroscopic descriptions and data. Karhunen-Loeve
(KL)or Proper Orthogonal Decomposition (POD)
approach. - Brief introduction to KL. - A
dynamical system model for observed coherent
structures (vortices) in ocean satellite data.
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INTRODUCTION
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BIOLOGICAL OR CHEMICAL SYSTEMS
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The discrete nature of organisms or chemical
molecules is missed in general when a continuum
approach (reaction-diffusion) is used to model
processes in Nature. This is specially important
in situations close to extinctions, and other
critical situations.
However, continuum descriptions (in terms of
concentration or density fields) have many
advantages stability analysis and pattern
formation.
Therefore, there is the need to formulate
Individual Based Models (IBMs), and then
deriving continuum equations of these
microscopic particle systems that still remain
discreteness effects.
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TWO SIMPLE INDIVIDUAL BASED MODELS AND THEIR
CONTINUUM DESCRIPTION.
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FIRST EXAMPLE
One of the simplest IBM Brownian bug
model. Birth-death model with non-conserved total
number of particles
Young, Roberts and Stuhne, Reproductive pair
correlations and the clustering of organisms,
Nature 412, 328 (2001).
- N particles perform independent Brownian
(random) motions in the continuum 2d physical
space. - In addition, they undergo a branching
process They reproduce, giving rise to a new
bug close to the parent, with probability l (per
unit of time), or die with probability b .
The physical phenomenon that minute particles,
immersed in a fluid, move about randomly.
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LETS WRITE DOWN A MEAN-FIELD LIKE CONTINUUM
EQUATION
Modeling in terms of continuous concentration
field
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l gt b
If lgtb explosion If lltb extinction
If lb, simple diffusion
Total number of particles
l b
l lt b
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At the critical point (lb), fluctuations are
strong and lead to clustering
NOT SIMPLE DIFFUSION
Very simple mechanism Reproductive correlations
Newborns are close to parents. This is missed in
a continuous deterministic description in which
birth is homogeneous
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Making the continuum limit PROPERLY
Demographic noise
Fluctuations play a very important role and a
proper continuum limit must be performed.
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SECOND EXAMPLE
Nonlocal density-dependent. Conserved total
number of particles .

• N particles with positions (xi(t), yi(t)) in the
  2d continuum physical space.
• At every time step the positions of all the
  particles are update synchoronously as follows

N R(i) means the number of neighbors at distance
smaller than R from bug i
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Lets write down the continuum version
(mean-field)
ri(xi, yi)
Take the limit
Ito-Langevin
Fokker-Planck
Probability density or expected density
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Discrete particle model
Depending on the value of p
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That is

• Fluctuations (noise) may have an important role.
• The noise term is not trivial. Usually is a
  function of the density itself (multiplicative
  noise).
• In order to reproduce spatial structures
  mean-field like descriptions are better when the
  total number of particles is conserved.
• We have looked at pattern formation, but there
  are other features not properly reproduced. E.g.
  in birth-death models typically there are
  transitions extinction-survival where the values
  of the parameters are not captured.
• STATISTICAL PHYSICS HAVE DEVELOPED DIFFERENT
  METHODS TO OBTAIN THE RIGHT MACROSCOPIC
  EQUATIONS. IN FACT THIS IS A CENTRAL TOPIC IN
  STATISTICAL PHYSICS.

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MODELS WITH PARTICLES APPEARING AND DISAPPEARING
(NON CONSERVED NUMBER)
Second quantization or Fock space or
anhilation-creation operators or Doi-Peliti
techniques

• Put the particles in a lattice (of L sites), and
  consider the number of particles at each site
  (N1, N2,, NL).
• Write the Master Equation for the time evolution
  of the probability of these numbers.
• Represent the Master Equation in terms of a
  (quantum mechanical like) Hamiltonian constructed
  with creation and annihilation operators.
• Find the action associated to that Hamiltonian.
  Go again off-lattice by performing the continuum
  limit.
• Approximate the action by keeping only quadratic
  terms, so that a Langevin equation for an
  auxiliary density-like field can be extracted
  from it.

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WHAT IS A MASTER EQUATION?
It is a first order differential equation
describing the time evolution of the probability
of having a given configuration of discrete
states.
If P(N1, N2, ) probability of having N1
particles in the first node, etc
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MODELS WITH CONSERVED NUMBER OF PARTICLES
A system of N interacting Brownian dynamics
Gaussian White noise
Interaction potential
DENSITY
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SECOND PART OF THE TALK
How to obtain low-dimensional systems from
macroscopic descriptions and data. The
Karhunen-Loeve (KL) or Proper Orthogonal
Decomposition (POD) approach.
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SOME WORDS ON KL
Original aim To identify in an objective way
coherent structures in a turbulent flow (or in a
sequence of configurations of a complex evolving
field). What it really does Finds an optimal
Euclidian space containing most of the data.
Finds the most persistent modes of fluctuation
around the mean.
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KL or POD provides an orthonormal basis for a
modal decomposition in a functional space.
Therefore, if U(x,t) is a temporal series of
spatial patterns (spatiotemporal data series).
Temporal average
Amplitude functions
Empirical orthogonal eigenfunctions
They are the eigenfunctions of the covariance
matrix C of the data
Eigenvalue
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WHY THIS PARTICULAR BASIS?
It separates a given data set into orthogonal
spatial and temporal modes which most efficiently
describe the variability of the data set.
Therefore, can be understood as a
spatial pattern contained in the data set with
its own dynamics (coherent structure). The
stronger its eigenvalue the more its relevance
in the data set. The ai(t) provides the temporal
evolution of the corresponding coherent structure.
The decomposition is optimum in the sense that if
we order the eigenvalues by decreasing
size we
may recover the signal with just a few
eigenfunctions
Where and Resid has no physical
relevance.
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SATELLITE DATA OF SEA SURFACE TEMPERATURE
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Physical meaning
Temporal modes
Spatial modes
Seasonal variability
Two vortices
Almería-Orán front
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• Interesting property
• The minimum error in reconstructing an image
  sequence
• via linear combinations of a basis set is
  obtained when the
• basis is the EOF basis.

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Algerian Current Altimetry
lk
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The data filtered to the coherent structure
represented by eigenfunctions 3 and 4
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Eddies
Baroclinic instability
Two-layer quasigeostrophic model
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i3,4
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We can make bifurcation analysis, study
periodicities and etc with the simple dynamical
system.
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CONCLUSIONS AND PERSPECTIVES

• We have experience with mathematical/physical
  tools that allow to describe, with macroscopic or
  collective variables, systems of interacting
  individuals.
• We have experience with mathematical/physical
  tools to obtain low-dimensional dynamical systems
  from data of complex spatio-temporal fields.

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EXTRAS
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Continuum description
Master equation in a lattice
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Using the Fock space representation
Bosonic conmutation rules
Defining the many-particle state
.
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One can obtain a Schrodinger-like equation which
defines a Hamiltonian
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3. Going to the Fock space representation
Defining the many-particle state
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• Interesting properties
• The minimum error in reconstructing an image
  sequence
• via linear combinations of a basis set is
  obtained when the
• basis is the EOF basis.

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• 0. Extract the temporal mean of the image
  ensemble
• Y(x,t)Image(x,t) - ltImage(x,t)gtt
• Calculate correlation matrix
• Solve the eigenvalue problem
• And the reconstructed images are

Imagine we have a sequence of data (film)
Image(x,t)
Eigenvalues
p is the number of relevant eigenfunctions
k1,,p Empirical Orthogonal Eigenfunctions
(EOFs)
Temporal amplitudes
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Bifurcation analysis in the eddy viscosity
2 stable fixed points and 4 unstable
Hopf bifurcation. Limit cycle with aprox. 6
months period
Limit cycle persists and no new bifurcation occurs