The equilibrium and nonequilibrium distribution of money Juan C. Ferrero Centro Laser de Ciencias...

The equilibrium and nonequilibrium

distribution of money

Juan C. Ferrero

Centro Laser de Ciencias Moleculares and INFIQCUniversidad Nacional de Córdoba, Córdoba

Argentina

Science → Prediction (Control)

Events Time Rate Consequences

Nature → Spontaneity → Endless approach to (irreversibility) equilibrium

(continuous evolution)

One approach to the problem is to learn through model calculations of known systems

ith money level of agent A

External input and output

Interaction transfer into i Interaction transfer out of i

(Pi1 n1 + Pi2 n2 + Pi3 n3 +…) ( P1i ni + P2i ni + P3i ni+…)

dni/dt = Pijnj - ni

Integration requires a model for Pij

Pij=N exp[-(Mi-Mj)/<M>d]

-10 -8 -6 -4 -2 0 2 4 6 8 100,0

0,2

0,4

0,6

0,8

1,0

Pro

ba

blity

M

0 100 200 3000

1

2

3

4

5

Pro

ba

bilty

de

nsity, %

Money, a.u.

0 100 200 300

An arbitrary, far from equilibrium distribution evolves to the BG population through near

Gaussian distributions

ith money level of agent A

Interaction transfer with A and B into Ai and Bi

Interaction transfer with A and B out of Ai and Bi

ith money level of agent B

AA(Pi1 n1 + Pi2 n2 + Pi3 n3 +…) +AB(Pi1

n1 + Pi2 n2 + Pi3 n3

BA( P1i ni + P2i ni + P3i ni+…) +BB( P1i

ni + P2i ni + P3i ni+…)

kiA

kiB

BiBA

AiAB

j

Ai

AAjiAA

j

Ai

BAjiBA

j

Aj

ABijAB

j

AAAijAA

Ai nknknPnPnPnPdtdn

j /

AiAB

BiBA

j

Bi

BBjiBB

j

Bi

BAjiBA

j

Bj

ABijAB

j

BBBijBB

Bi nknknPnPnPnPdtdn

j /

0 100 200 300 4000,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8P

op

ula

tio

n

Money

0 100 200 300 400-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

Pop

ulat

ion

Money

0 100 200 300 400 500

0,0

0,1

0,2

0,3

0,4

0,5

0,6P

op

ula

tion

Money

P(M) = N M(-1)exp(-x/)

0 100 200 300

1

10

100

A

B

B

Pa

ram

ete

rs G

am

ma

fu

nctio

n

Time

A

P(x) = N x(-1)exp(-x/)

• The initial BG population evolves to two different BG distributions through BG-like intermediate

distributions with different values of

1- Near Gaussian distributions

2- Multiple BG distributions with different values of

This provides two criteria for deviation from equilibrium:

0 10000,00

0,05

0,10

0,15

0,20

0,25

0,30

Tsallis N 0.00015 ±0.00046g 1.41594 ±0.66423B 0.00395 ±0.00327q 0.84609 ±0.32219

gamma N 0.00004 ±0.00011a 1.73002 ±0.49468b 162.45518±43.48357

Po

pu

laito

n

Money

Oct 92

200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00021 ±0.00217g 1.59288 ±2.86693B 0.01153 ±0.03983q 1.26 ±0

Gamma N 0.00063 ±0.00154a 1.19673 ±0.50889b 238.392 ±89.23977

Po

pu

latio

n

Money

Oct 94

200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00055 ±0.00003g 1.285 ±0B 0.00643 ±0q 1.15733 ±0

Gamma

N 0.0022 ±0.00269a 0.92627 ±0.25525b 291.26611 ±67.43559

Po

pu

latio

n

Money

Oct 97

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00056 ±0.00202g 1.28504 ±0.90502B 0.00643 ±0.00839q 1.1573 ±0.1986

gamma

N 0.0024 ±0.00297a 0.91412 ±0.25896b 291.92228±69.54917

Po

pu

latio

n

Money

Oct 98

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00009 ±0.00077g 1.79221 ±2.23108B 0.01275 ±0.03169q 1.21091 ±0.10661

Gamma

N 0.0012 ±0.0024a 1.07548 ±0.42203b 240.67984 ±82.5054

Po

pu

latio

n

Money

Oct 99

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30

Tsallis

N 0.00003 ±0.00037g 2.14814 ±3.83315B 0.0186 ±0.06854q 1.21 ±0

gamma

N 0.00007 ±0.00015a 1.74438 ±0.50266b 141.28245±37.8206

Po

pu

latio

n

Money

May 01

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00017 ±0.00092g 1.73744 ±1.54406B 0.01597 ±0.02735q 1.23915 ±0

Gamma

N 0.00067 ±0.00072a 1.25035 ±0.23333b 177.60977 ±29.48252

Po

pu

latio

n

Money

Oct 01

200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00105 ±0.00792g 1.35196 ±2.39257B 0.01555 ±0.053q 1.31 ±0

Gamma N 0.0031 ±0.00498a 0.93552 ±0.36537b 199.23211 ±66.46085

Po

pu

latio

n

Money

May 02

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30

Po

pu

latio

n

Money

May 02

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 0.00068 ±0.00734g 1.55782 ±3.73005B 0.02154 ±0.10659q 1.29 ±0

Gamma

N 0.00109 ±0.00317a 1.24716 ±0.67377b 131.67864 ±60.09182

Po

pu

latio

n

Money

Oct 02

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30

Po

pu

latio

n

Money

Oct 02

0 200 400 600 800 10000,0

0,2

0,4

0,6

Po

pu

latio

n

Money

May 03

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30

Po

pu

latio

n

Money

Oct 03

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis N 8.4778E-6 ±0.00008g 2.43731 ±2.49103B 0.01762 ±0.03605q 1.12533 ±0.10513

Gamma N 0.00004 ±0.00012a 1.90312 ±0.59478b 114.39555 ±32.47463

Money

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30

Po

pu

latio

n

Money

May 04

0 200 400 600 800 10000,00

0,05

0,10

0,15

0,20

0,25

0,30Tsallis

N 3.8136E-8 ±2.5551E-6g 4.38782 ±24.2868B 0.07691 ±1.01538q 1.15491 ±0.45587

Gamma

N 0.00107 ±0.00007a 1.17191 ±0b 182 ±0

Po

pu

latio

n

Money

• Before the crisis: A single Gamma function (bimodality was always present).

• As the crisis developed, the low and medium region of the data could only be fit to Gaussian functions. Distortion reached its maximum in May 2003 and returned to a more normal shape in 2004.

• A Gaussian shape in the distribution is expected, according to model calculations, for the evolution of a system far from equilibrium.

Conclusions:

• In the low and medium range, money follows BG distribution• This implies that a more egalitarian society (world) is obtained

increasing the degeneracy (). • The opposite holds if increases.• The tail of the distribution shows fractal behaviour (Pareto

power law) • The Tsallis function fits the whole range and should be

considered (Richmond and Sabatelli(2003), Anazawa et al (2003))

• The distributions can be mono o polymodal, in equilibrium or not

• BG distribution does not implies equilibrium (Shuler et al, 1964)• In the approach to equilibrium, the coldest partner wins (lower

)• Criteria for non equilibrium: 1) BG distribution with time

dependent 2) Gaussian shape

Predicting behaviours:

Thermodinamical formulation for mono and multicomponent systems

Model simulations of countries in crisis, like Argentina (time dependence)