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)