the equilibrium and nonequilibrium distribution of money juan c. ferrero
DESCRIPTION
The equilibrium and nonequilibrium distribution of money Juan C. Ferrero Centro Laser de Ciencias Moleculares and INFIQC Universidad Nacional de Córdoba, Córdoba. Argentina. Science → Prediction (Control) Events Time Rate Consequences - PowerPoint PPT PresentationTRANSCRIPT
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)