Cristóbal López

IMEDEA, Palma de Mallorca, Spain

http://www.imedea.uib.es/PhysDept

clopez@imedea.uib.es

From microscopic dynamics to macroscopic evolution equations (and viceversa)

OutlineFirst 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.

INTRODUCTION

as CONTINUOUS FIELDS

BIOLOGICAL OR CHEMICAL SYSTEMS

or as DISCRETE INDIVIDUALS

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.

TWO SIMPLE INDIVIDUAL BASED MODELS AND THEIR CONTINUUM

DESCRIPTION.

One of the simplest IBM: Brownian bug model.

Birth-death model with non-conserved total number of particles

- 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 (per unit of time), or die with

probability .

Young, Roberts and Stuhne, Reproductive pair correlations and the clustering of organisms, Nature 412, 328 (2001).

*The physical phenomenon that minute particles, immersed in a fluid, move about randomly.

FIRST EXAMPLE

C→2C autocatalisis, or reproduction

C→0 death

Modeling in terms of continuous concentration field:

CDCt

C 2)(

LET’S WRITE DOWN A MEAN-FIELD LIKE CONTINUUM EQUATION

Total number of particles

If : explosionIf : extinction

If simple diffusion

CDt

C 2

At the critical point (), fluctuations are strong and lead

to clustering

Very simple mechanism: Reproductive correlations: Newborns are close to parents. This is missed in a continuous deterministic

description in which birth is homogeneous

NOT SIMPLE DIFFUSION

),(),(),(),( 2 txtxCtxCD

t

txC

)'()'(2)','(),( ttxxtxtx

Making the continuum limit PROPERLY

Fluctuations play a very important role and a proper continuum limit must be performed.

Demographic noise

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:

)()/)((2)()(

)()/)((2)()(

tNiNtDtytty

tNiNtDtxttx

yi

pRii

xi

pRii

N R(i) means the number of neighbors at distance smaller than R from bug i

Let’s write down the continuum version (mean-field)

Take the limit 0t

iiip

Ri dWttrdWNiNDtdr )),(()/)((2)(

)),(),((2

1),( 2 txtxt

tx

Ito-Langevin

Fokker-Planck

Probability density or expected density

Rrx

RR trdrxNiN||

),()()(

)(trx i

ri=(xi, yi)

p

Rrx

NtrdrtxDt

tx)/),()(,(

),(

||

2

Discrete particle model

Depending on the value of p

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.

Second quantization or Fock space or anhilation-creation operators or Doi-Peliti techniques

1. Put the particles in a lattice (of sites), and consider the number of particles at each site (N1, N2,…, N).

2. Write the Master Equation for the time evolution of the probability of these numbers.

3. Represent the Master Equation in terms of a (quantum mechanical like) Hamiltonian constructed with creation and annihilation operators.

4. Find the action associated to that Hamiltonian. Go again off-lattice by performing the continuum limit.

5. Approximate the action by keeping only quadratic terms, so that a Langevin equation for an auxiliary density-like field can be extracted from it.

MODELS WITH PARTICLES APPEARING AND DISAPPEARING (NON CONSERVED NUMBER)

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…

}'{ }'{

)()'()'()'()},({

N N

NPNNWNPNNWdt

tNdP

MODELS WITH CONSERVED NUMBER OF PARTICLES

A system of N interacting Brownian dynamics

N

kiki

i tDtxtxVdt

tdx

1

)(2)()()(

Interaction potentialGaussian White noise

N

ii xtxtx

1

))((),(

)),(()(),(),(),( 2 txyxVtydytxD

t

tx

DENSITY

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.

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.

SOME WORDS ON KL

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).

p

iii tatUtU

1

),(),( xxx

Temporal average

Empirical orthogonal eigenfunctionsAmplitude functions

They are the eigenfunctions of the covariance matrix C

of the data

),(),(),( tUtUC xxxx

)()(),( xxxxx

kkkC

Eigenvalue

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.

)(xk

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

k0...21 N

sidtatUtUN

iii Re)()(),(),(

1

xxx

Where and Resid has no physical relevance.pN

SATELLITE DATA OF SEA SURFACE TEMPERATURE

Physical meaningSpatial modes Temporal modes

Seasonal variability

Two vortices

Almería-Orán front

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.

Algerian Current Altimetry

k

)(ta3

)(ta4

),()(),()( yxtayxta 4433

The data filtered to the coherent structure represented by eigenfunctions 3 and 4

Baroclinic instability

Two-layer quasigeostrophic model

Eddies

),,(),,( tyxhf

gtyx

01

i=3,4

)),()(ˆ),()(ˆ),,((),,( yxtayxtatyxhf

gtyx 4433

01

)),()(),()((),,( yxtbyxtbf

gVxUytyx 4433

02

),()(ˆ),()(ˆ yxtayxta 4433

We can make bifurcation analysis, study periodicities and etc with the

simple dynamical system.

43 ˆ,ˆ aa

200

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.

-Study of spatial patterns for bacteria dynamics. Role of multiplicative noise term?

- Macroscopic descriptions of social systems with a particular topology of the interaction network?

- Patterns of behavior and Coherent structures in data. How can the KL help?

Relation with PATRES

EXTRAS

Continuum description

Master equation in a lattice

),..,..()1(

),..,..()1(),..1,..()1(

),..1,..()1(),...,(

1

11

11

NNNPNNW

NNNPNNWNNNPNNW

NNNPNNWdt

NNdP

iii

iiiiii

iiii

iii

iii

NiNNW

NiNNW

)()(

)()(

1

1

Using the Fock space representation

00

1

1

11

11

i

ijji

iii

iiii

a

aa

NNNNNNa

NNNNNNNa

,

,..,,..,,..,,..,

,..,,..,,..,,..,

Defining the many-particle state

NN i

Ni

iaNNP,..,

)(),...,(1 1

1 0.

Bosonic conmutation rules

One can obtain a Schrodinger-like equation which defines a Hamiltonian

)(0

)(0

2

/1

/1

)(

)(

iRjjjsi

iRjjjsi

iiiiiiiiii

aaN

aaN

aaaaaaaH

Hdt

td

3. Going to the Fock space representation

00

,

,..,1,..,,..,,..,

,..,1,..,,..,,..,

11

11

i

ijji

iii

iiii

a

aa

NNNNNNa

NNNNNNNa

Defining the many-particle state

NN i

Ni

iaNNP,..,

)(),...,(1 1

1 0

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.

')'()( kkkttt kaka

)()(),(1

xxxx

k

N

kkk ΨΨC

kllk ΨΨ )()( xxx

tt

p

kk

N

kk

tt YY

2

11

2 ˆ

0. Extract the temporal mean of the image ensemble: Y(x,t)=Image(x,t) - <Image(x,t)>t

1. Calculate correlation matrix:

2. Solve the eigenvalue problem:

3. And the reconstructed images are

ttYtYC ),(),(),( x'xxx

)()(),( xxxxx

kkkC

p

kkt katY

1

)()(),(ˆ xx

)(xk)(kat

k=1,…,p: Empirical Orthogonal Eigenfunctions (EOFs)

: Temporal amplitudes

Imagine we have a sequence of data (film): Image(x,t)

p is the number of relevant eigenfunctions

Eigenvalues

Bifurcation analysis in the eddy viscosity

212 2 stable fixed points and 4 unstable

212 Hopf bifurcation. Limit cycle with aprox. 6 months period

212 Limit cycle persists and no new bifurcation occurs

43 aa ˆ,ˆ

200),()(ˆ),()(ˆ yxtayxta 4433