Van Kampen Expansion: Its exploitation in some social and

economical problems

Horacio S. Wio Instituto de Fisica de Cantabria, UC-CSIC, Santander, SPAIN

(A) Electronic address: wio@ifca.unican.es URL: http://www.ifca.unican.es/~wio/

IN COLLABORATION WITH:

J.R. Iglesias (UFRGS, Brazil)

I. Szendro (Dresde)

M.S. de la Lama (IFCA-UC, Spain)

• Van Kampen's expansion approach in an opinion formation

model, M.S. de la Lama, I.G. Szendro, J.R. Iglesias and

H.S. Wio, Eur. Phys. J. B 51 435-442 (2006); and

ERRATUM, Eur. Phys. J. B 58 221 (2007).

Sketch of the talk:

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents; ► Macroscopic and Fokker-Planck Equation for Fluctuations;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents; ► Macroscopic and Fokker-Planck Equation for Fluctuations; ► Some results;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents; ► Macroscopic and Fokker-Planck Equation for Fluctuations; ► Some results; ► Inclusion of “Fanatics”;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents; ► Macroscopic and Fokker-Planck Equation for Fluctuations; ► Some results; ► Inclusion of “Fanatics”; ► Other Cases;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents; ► Macroscopic and Fokker-Planck Equation for Fluctuations; ► Some results; ► Inclusion of “Fanatics”; ► Other Cases; ► Failure and Extension of the -expansion;

Sketch of the talk:

► Introduction: brief description of van Kampen’s -expansion; ► The Model: Inclusion of Undecided Agents; ► Macroscopic and Fokker-Planck Equation for Fluctuations; ► Some results; ► Inclusion of “Fanatics”; ► Other Cases; ► Failure and Extension of the -expansion; ► Conclusions.

INTRODUCTION

Van Kampen Expansion of the Master Equation: Aim: avoid the inconsistencies that arise when obtaining, through a naïve approach, the macroscopic equation for a system described by a Master Equation. Exploiting a perturbative-like approach, the dominant order gives the macroscopic eq., while the following one yields a Fokker-Planck equation for the fluctuations.

INTRODUCTION

Van Kampen Expansion of the Master Equation: Aim: avoid the inconsistencies that arise when obtaining, through a naïve approach, the macroscopic equation for a system described by a Master Equation. Exploiting a perturbative-like approach, the dominant order gives the macroscopic eq., while the following one yields a Fokker-Planck equation for the fluctuations.

INTRODUCTION

Van Kampen Expansion of the Master Equation: Aim: avoid the inconsistencies that arise when obtaining, through a naïve approach, the macroscopic equation for a system described by a Master Equation. Exploiting a perturbative-like approach, the dominant order gives the macroscopic eq., while the following one yields a Fokker-Planck equation for the fluctuations.

The approach requires to identify , a system’s parameter, so large that allows to make expansions in its inverse. The original variable is transformed according to

INTRODUCTION

Van Kampen Expansion of the Master Equation: Aim: avoid the inconsistencies that arise when obtaining, through a naïve approach, the macroscopic equation for a system described by a Master Equation. Exploiting a perturbative-like approach, the dominant order gives the macroscopic eq., while the following one yields a Fokker-Planck equation for the fluctuations.

The approach requires to identify , a system’s parameter, so large that allows to make expansions in its inverse. The original variable is transformed according to

= macroscopic + fluctuations

INTRODUCTION

Van Kampen Expansion of the Master Equation: Using a step operator , defined through

the master eq. could be (formally) written as

INTRODUCTION

Van Kampen Expansion of the Master Equation: Using a step operator , defined through

the master eq. could be (formally) written as

INTRODUCTION

Van Kampen Expansion of the Master Equation: Using a step operator , defined through

the master eq. could be (formally) written as

Assuming is very large, and jumps are small, we can expand

INTRODUCTION

Van Kampen Expansion of the Master Equation: Changing variables in the pdf

we have that the lhs of the master equation changes to

INTRODUCTION

Van Kampen Expansion of the Master Equation: Changing variables in the pdf

we have that the lhs of the master equation changes to

INTRODUCTION

Van Kampen Expansion of the Master Equation: Changing variables in the pdf

we have that the lhs of the master equation changes to

Replacing everything into de master equation we obtain a complicated equation with terms of different order in

INTRODUCTION

Van Kampen Expansion of the Master Equation: Collecting terms of the same order in we obtain for the different contributions:

up to

that corresponds to the macroscopicmacroscopic equation.

INTRODUCTION

Van Kampen Expansion of the Master Equation: Collecting terms of the same order in we obtain for the different contributions:

up to

that corresponds to the macroscopicmacroscopic equation. Stability condition

INTRODUCTION

Van Kampen Expansion of the Master Equation: Collecting terms of the same order in we obtain for the different contributions:

up to

that corresponds to the macroscopicmacroscopic equation. Stability condition

The following order, , gives a “linear” Fokker-Planck eq.

describing the behavior of fluctuations around the macroscopic one.

INTRODUCTION

Van Kampen Expansion of the Master Equation: As the FPE is linear, we only need to calculate the mean value

the dispersion

The general solution will have the Gaussian form

INTRODUCTION

The previous Fokker-Planck eq. has a related Langevin eq. :

INTRODUCTION

Van Kampen Expansion of the Master Equation: Meaning:

The Model: Undecided Agents

The original model consist of only two groups, say A and B, with some rules that allows agents or members of one group, to convince the agents or members of the other. Here we consider that agents of group A don’t interact directly to agents of group B, but we include an intermediate group I, formed by “undecided” agents that mediates the interaction between A and B (Redner et al.) Members of groups A and B, could convince the members of group I. Also, we also assume that there is the possibility of an spontaneous change of opinion from group A to I or from B to I and vice versa. This implies some form of “social temperature”.

The Model: Undecided Agents

The different process we are going to consider are:

Convincing rules:

Spontaneous changes

The Model: Undecided Agents

The Master Equation looks as

The Model: Undecided Agents

According to the method, the original variables and transforms into

As indicated before, we should introduce the new variables into the Master Equation.

Macroscopic and Fokker-Planck Equation

Collecting terms corresponding to the different orders in we obtain, for the macroscopic equations (order )

Asymptotically, this set of eqs. has only one solution (or attractor)

Macroscopic and Fokker-Planck Equation

The following order ( ) give us a Fokker-Planck equation for the pdf of fluctuations

and

Macroscopic and Fokker-Planck Equation

Using the Fokker-Planck equation we can obtain information about the dynamics of fluctuations. We define mean values and correlations as

Macroscopic and Fokker-Planck Equation

Using the Fokker-Planck equation we can obtain information about the dynamics of fluctuations. We define mean values and correlations as

As the FPE is linear (Ornstein-Uhlenbeck-like) this is all the information we need to completely define the pdf

Macroscopic and Fokker-Planck Equation

We use as our reference state the symmetric case

Macroscopic and Fokker-Planck Equation

Similar eqs. for the mean values of fluctuations, while for the correlations

Some results:

Some results considering the symmetric case as well as some

departures from it

Some results:

Some results:

Some results:

Some results:

Some results:

The approach also allows to obtain information about the relaxation time around the stationary state. For the symmetric case we have

Inclusion of “Fanatics”

Inclusion of “fanatics” (or inflexible agents) transform our variables according to

and

Inclusion of “Fanatics”

Inclusion of “fanatics” (or inflexible agents) transform our variables according to

and

Without details, for the macoscopic equations we obtain

Inclusion of “Fanatics”

Other Cases

Another possibilities we are exploring regards some financial aspects related with the “herding effect”, and the “stylized facts” in finance, and also with lenguage competition. In particular we are analyzing a model discussed by Alfano & Milakovic (2007), Lux (2006), Pietronero et al. (2008). Such a model can be mapped into our scheme, if some kind of intermediate agents (in addition to bullish & bearish, fundamentalists & chartists, buyers & sellers, etc) is included. The point here is to reinterpret the results in terms, or the lenguage, adequate to the new context. Oscillatory behaviour in a single realization (McKane & Newman, Risau-Guzman & Abramson, etc).

Failure and extension of the -expansion

The inclusion of the intermediate group avoids a problem that could occur within the van Kampen’s approach: the case when the macroscopic contribution is multivalued. Without such an intermediate group it could happen that

there is more than one solution for the macroscopic equation, a fact associated to the breaking of the stability condition, that occurs when

Failure and extension of the -expansion

The inclusion of the intermediate group avoids a problem that could occur within the van Kampen’s approach: the case when the macroscopic contribution is multivalued. Without such an intermediate group it could happen that

there is more than one solution for the macroscopic equation, a fact associated to the breaking of the stability condition, that occurs when

Failure and extension of the -expansion

The inclusion of the intermediate group avoids a problem that could occur within the van Kampen’s approach: the case when the macroscopic contribution is multivalued. Without such an intermediate group it could happen that there is more than one solution for the macroscopic equation, a fact associated to the breaking of the stability condition, that occurs when

Problem: if the deterministic eq. is zero, as well as several of its derivatives. However, it is possible to overcome this drawback: after a transient, the highest order (macroscopic) doesn’t exists. The following order, indicating a time scale slower than before by a factor , and leads us to a Fokker- Planck eq. for the pdf of fluctuations.

Failure and extension of the -expansion

The possibility of spatial extension could be also included: dividing the system into cells, and defining a space (or cell) dependent pdf as

Failure and extension of the -expansion

The possibility of spatial extension could be also included: dividing the system into cells, and defining a space (or cell) dependent pdf as

The analysis of these and other related situations is possible, and is currently under way.

Conclusions

Conclusions

The -expansion is a versatile and powerful method of analysis.

Conclusions

The -expansion is a versatile and powerful method of analysis. It offers the possibility of obtainig, in a clear, unambiguous, and controlable way, not only the macroscopic eq. but also the dynamics of fluctuations.

Conclusions

The -expansion is a versatile and powerful method of analysis. It offers the possibility of obtainig, in a clear, unambiguous, and controlable way, not only the macroscopic eq. but also the dynamics of fluctuations. It also offers the possibility of studying (analitically) several of the models that have been recently discussed in the literature, gaining insight into the model dynamics, and complementing numerical simulations.

Conclusions

The -expansion is a versatile and powerful method of analysis. It offers the possibility of obtainig, in a clear, unambiguous, and controlable way, not only the macroscopic eq. but also the dynamics of fluctuations. It also offers the possibility of studying (analitically) several of the models that have been recently discussed in the literature, gaining insight into the model dynamics, and complementing numerical simulations. It is a method worth to be exploited in the research area of socio- and econophysics.

Thanks for your attention!