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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)