Transcript Diapositiva 1 - Saha Institute of Nuclear Physics

The equilibrium and
nonequilibrium
distribution of money
Juan C. Ferrero
Centro Laser de Ciencias Moleculares and INFIQC
Universidad Nacional de Córdoba, Córdoba
Argentina
Science
Events
Time
→
Prediction (Control)
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
External input and
output
ith money level of
agent A
w (Pi1 n1 + Pi2 n2 + Pi3 n3 +…)
Interaction transfer into i
w ( P1i ni + P2i ni + P3i ni+…)
Interaction transfer out
of i
dni/dt = wSPijnj - wni
Integration requires a model for
Pij
Pij=N exp[-(Mi-Mj)/<DM>d]
1,0
Probablity
0,8
0,6
0,4
0,2
0,0
-10
-8
-6
-4
-2
0
DM
2
4
6
8
10
5
Probabilty density, %
4
3
2
1
0
0
100
200
300
0
100
200
300
Money, a.u.
An arbitrary, far from equilibrium distribution
evolves to the BG population through near
Gaussian distributions
ith money level of
agent A
kiB
ith money level of
agent B
kiA
wAA(Pi1 n1 + Pi2 n2 + Pi3 n3 +…) +
wAB(Pi1 n1 + Pi2 n2 + Pi3 n3
Interaction transfer with A
and B into Ai and Bi
wBA( P1i ni + P2i ni + P3i ni+…) + wBB( P1i
ni + P2i ni + P3i ni+…)
Interaction transfer with
A and B out of Ai and Bi
dniA / dt  wAA  PijAAn Aj  wAB  PijABn Aj  wBA  PjiBAniA  wAA  PjiAAniA  k ABniA  kBAniB
j
j
j
j
dniB / dt  wBB  PijBBn Bj  w AB  PijABn Bj  wBA  PjiBAniB  wBB  PjiBBniB  kBAniB  k ABniA
j
j
j
j
0,8
0,7
0,6
Population
0,5
0,4
0,3
0,2
0,1
0,0
0
100
200
Money
300
400
0,7
0,6
0,5
Population
0,4
0,3
0,2
0,1
0,0
-0,1
0
100
200
Money
300
400
0,6
P(M) = N M(a-1)exp(-x/b)
0,5
Population
0,4
0,3
0,2
0,1
0,0
0
100
200
300
Money
400
500
100
Parameters Gamma function
bB
bA
10
aB
aA
1
0
100
200
Time
P(x) = N x(a-1)exp(-x/b)
300
• The initial BG population evolves
to two different BG distributions
through BG-like intermediate
distributions with different values
of b
This provides two criteria for
deviation from equilibrium:
1- Near Gaussian distributions
2- Multiple BG distributions with
different values of b
0,30
Tsallis
Oct 92
0,25
N
g
B
q
0.00015
1.41594
0.00395
0.84609
±0.00046
±0.66423
±0.00327
±0.32219
0,20
Populaiton
gamma
N
a
b
0,15
0.00004
±0.00011
1.73002
±0.49468
162.45518±43.48357
0,10
0,05
0,00
0
1000
Money
0,30
Tsallis
Oct 94
N
g
B
q
0,25
Population
0,20
0.00021
1.59288
0.01153
1.26
±0
±0.00217
±2.86693
±0.03983
Gamma
N
a
b
0,15
0.00063
1.19673
238.392
±0.00154
±0.50889
±89.23977
0,10
0,05
0,00
200
400
600
Money
800
1000
0,30
Tsallis
0,25
N
g
B
q
0,20
Gamma
Population
Oct 97
N
a
b
0,15
0.00055
1.285 ±0
0.00643
1.15733
±0.00003
±0
±0
0.0022 ±0.00269
0.92627
±0.25525
291.26611
±67.43559
0,10
0,05
0,00
200
400
600
Money
800
1000
0,30
Tsallis
Oct 98
N
g
B
q
0,25
gamma
0,20
Population
0.00056
±0.00202
1.28504
±0.90502
0.00643
±0.00839
1.1573 ±0.1986
N
a
b
0,15
0.0024 ±0.00297
0.91412
±0.25896
291.92228±69.54917
0,10
0,05
0,00
0
200
400
600
Money
800
1000
0,30
Tsallis
Population
Oct 99
0,25
N
g
B
q
0.00009
1.79221
0.01275
1.21091
0,20
Gamma
0,15
N
a
b
±0.00077
±2.23108
±0.03169
±0.10661
0.0012 ±0.0024
1.07548
±0.42203
240.67984
±82.5054
0,10
0,05
0,00
0
200
400
600
Money
800
1000
0,30
May 01
Tsallis
0,25
N
g
B
q
0.00003
±0.00037
2.14814
±3.83315
0.0186 ±0.06854
1.21
±0
0,20
Population
gamma
N
a
b
0,15
0.00007
±0.00015
1.74438
±0.50266
141.28245±37.8206
0,10
0,05
0,00
0
200
400
600
Money
800
1000
0,30
Tsallis
Oct 01
N
g
B
q
Population
0,25
0.00017
1.73744
0.01597
1.23915
0,20
Gamma
0,15
N
a
b
±0.00092
±1.54406
±0.02735
±0
0.00067
±0.00072
1.25035
±0.23333
177.60977 ±29.48252
0,10
0,05
0,00
0
200
400
600
Money
800
1000
0,30
Tsallis
N
g
B
q
May 02
0,25
0.00105
1.35196
0.01555
1.31
±0
±0.00792
±2.39257
±0.053
Gamma
Population
0,20
N
a
b
0.0031 ±0.00498
0.93552
±0.36537
199.23211
±66.46085
0,15
0,10
0,05
0,00
0
200
400
600
Money
800
1000
0,30
Tsallis
Population
Oct 02
0,25
N
g
B
q
0.00068
1.55782
0.02154
1.29
±0
0,20
Gamma
0,15
N
a
b
0.00109
1.24716
131.67864
±0.00734
±3.73005
±0.10659
±0.00317
±0.67377
±60.09182
0,10
0,05
0,00
0
200
400
600
Money
800
1000
0,6
May 03
Population
0,4
0,2
0,0
0
200
400
600
Money
800
1000
0,30
Tsallis
Population
Oct 03
0,25
N
g
B
q
8.4778E-6
2.43731
0.01762
1.12533
0,20
Gamma
0,15
N
a
b
±0.00008
±2.49103
±0.03605
±0.10513
0.00004
1.90312
114.39555
±0.00012
±0.59478
±32.47463
0,10
0,05
0,00
0
200
400
600
Money
800
1000
Population
0,30
May 04
Tsallis
0,25
N
g
B
q
3.8136E-8
4.38782
0.07691
1.15491
0,20
Gamma
0,15
N
a
b
0.00107
1.17191
182
±0
±2.5551E-6
±24.2868
±1.01538
±0.45587
±0.00007
±0
0,10
0,05
0,00
0
200
400
600
Money
800
1000
• 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 (a).
• The opposite holds if b 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
b)
• Criteria for non equilibrium: 1) BG distribution with time
dependent b
2) Gaussian shape
Predicting behaviours:
Thermodinamical formulation for mono and
multicomponent systems
Model simulations of countries in crisis, like Argentina
(time dependence)