The equilibrium and nonequilibrium distribution of money Juan C. Ferrero

2. The equilibrium and nonequilibrium distribution of money Juan C. Ferrero Centro Laser de Ciencias Moleculares and INFIQC Universidad Nacional de Córdoba, Córdoba Argentina

3. 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

4. External input and output ith money level of agent A w (Pi1 n1 + Pi2 n2 + Pi3 n3 +…) w ( P1i ni + P2i ni + P3i ni+…) Interaction transfer into i Interaction transfer out of i

5. dni/dt = wSPijnj - wniIntegration requires a model forPij

6. Pij=N exp[-(Mi-Mj)/<DM>d]

7. An arbitrary, far from equilibrium distribution evolves to the BG population through near Gaussian distributions

8. kiB ith money level of agent A ith money level of agent B kiA wAA(Pi1 n1 + Pi2 n2 + Pi3 n3 +…) + wAB(Pi1 n1 + Pi2 n2 + Pi3 n3 wBA( P1i ni + P2i ni + P3i ni+…) + wBB( P1i ni + P2i ni + P3i ni+…) Interaction transfer with A and B into Ai and Bi Interaction transfer with A and B out of AiandBi

12. P(M) = N M(a-1)exp(-x/b)

13. P(x) = N x(a-1)exp(-x/b)

14. The initial BG population evolves to two different BG distributions through BG-like intermediate distributions with different values of b

15. This provides two criteria for deviation from equilibrium: 1- Near Gaussian distributions 2- Multiple BG distributions with different values of b

28. 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.

29. 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

30. Predicting behaviours: Thermodinamical formulation for mono and multicomponent systems Model simulations of countries in crisis, like Argentina (time dependence)