Lagrange Principle and Boltzmann Distribution of Wealth

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1 Jürgen Mimkes, Physics Department, Paderborn University, Germany Geoffrey M. Willis, S. Elmsall, W. Yorkshire, UK Coworkers: Mario Hillebrand, Christian Denk, Thorsten Fründ, Stefan Kallerhoff

2 Jürgen Mimkes, Physics Department, Paderborn University, Germany Geoffrey M. Willis, S. Elmsall, W. Yorkshire, UK Introduction Lagrange Principle in Many Particle Systems Interpretation of the Lagrange Principle Application of the Lagrange Principle Coworkers: Mario Hillebrand, Christian Denk, Thorsten Fründ, Stefan Kallerhoff

3 Introduction

4 Lagrange principle = statistics with constraints = Gibbs free energy

5 Lagrange principle = statistics with constraints = Gibbs free energy

6 Surface of brass (Cu Zn) Binary system of copper and zinc

7 Surface of brass (Cu Zn) Binary system of copper and zinc Map of Bosnia 1991 Binary system of Serbs and Non Serbs (Bosnians, Croats)

8 Simulation

9 Lagrange Principle in Many Particle Systems

10

11 Mathematics of stochastic systems Normal distribution P(x) normal distribution,15,1, x P(l, r) = N! / (N l! N r! 2 N ) max! P(x) = (1/ (2π) σ) e - x² / 2 σ ² ) x = ½ (N r - N l ) σ = 1 / N

12 Probability P( N ; N ) = l r N! N! N l r 1! 2 N P = 31,25 % most probable distribution

13 Probability P( N ; N ) = l r N! N! N l r 1! 2 N P = 1,56 % most improbable distribution

14 Probability with constraints P =? P = 1 %? most probable distribution

15 Probability with constraints P =? defect distribution

16 Probability with constraints (Lagrange Principle) L = E + T ln P maximum! L : Lagrange function P : Probability E : constraint: ( ), fine T : Lagrange parameter Joseph de Lagrange ( )

17 Probability with constraints L = E + T ln P maximum! Equilibrium: L = N r E + T ln (N! / N l! N r!) = max x r = N r / N ( = 1 / 6 ) L = -N{x r E + T(x l ln x l +x r ln x r) } = max L / x r = -N T { ln x r E/T} = T { ln x r E/T} = (Boltzmann) x r = A exp{ - E / T}

18 Probability with constraints x r = A exp (- E / T) Probability,5,4,3,2, Energy E/T Boltzmann distribution of defects

19 Probability with constraints x r = A exp (- E / T) Probability Traffic offences,5,4,3,2, Energy E/T Number of offenders Number of points Boltzmann distribution of defects Distribution of traffic violations German Traffic Department 2

20 Application of the Lagrange Principle

21 Natural science L = E (order) + T ln P (disorder) maximum! T small : order (solid) T large : disorder (liquid)

22 Society L = E (order) + T ln P (disorder) maximum! T small : Order (improbable) T large : disorder (probable)

23 Literature, information: L = E (order) + T ln P / P E (disorder) maximum! T small : order (correct) T large: chaos (incorrect)

24 Distribution of energy: x = A exp (- E / T) Probability,5,4,3,2, Energy E/T Distribution of energy in gases: Number of atoms per energy class

25 Distribution of prices: x = A exp (- E / T) Automobile market D 1998 number cars in Mill. units 8, 7, 6, 5, 4, 3, 2, produced cars Boltzmann used cars 1,, average price per car in 1 DM Distribution of prices in free markets: Number of cars per tax class

26 Distribution of Wealth: x = A exp (- E / T) Property distribution in D 1993 Number of households in Mill. 2 1,5 1,5 5 1 property tax class in kdm Distribution of Wealth in free societies: Number of households per property class

27 German automobile market 1998

28 German automobile market 1998 x s = A exp (- P / T)? Produced cars 1998 Automobile market D 1998 price units in kdm in ,45 Mill. number cars in Mill. units 8, 7, 6, 5, 4, 3, 2, produced cars Boltzmann Boltzmann distribution? 1,, average price per car in 1 DM

29 German automobile market 1998 x s = A exp (- P / T)? Produced cars 1998 Automobile market D 1998 price units in kdm in ,45 Mill. Flensburg: new 3,75 Mill. used 7,45 Mill total 11,2 Mill. number cars in Mill. units 8, 7, 6, 5, 4, 3, 2, 1,, produced cars Boltzmann used cars average price per car in 1 DM

30 Distribution of wealth in Germany 1993

31 Lorenz distribution of wealth in Germany 1993 (DIW) amout of wealth y Wealth distribution in D ,8,6,4,2,2,4,6,8 1 number of households x Property distribution in Germany 1993 (DIW estimated) Total property Number of households Mean property per household Property class x in kdm Number N of Hh in % 992 Bill. DM 35,6 Mill. 278 kdm / Hh Property K of Hh in % = 1,5 <1 44,5 9, ,7 17, ,3 28, ,3 16,8 >1 2,7 28 y Boltzmann = x + (1-x) ln (1-x)

32 Distribution of wealth in Germany 1993 (DIW) N(k): number of Hh in class k K(k): amount of capital in class k a y( a) = K( k) dk / K( k) dk a x( a) = N ( k) dk / N ( k) dk N(k) = A exp( - k / T) Boltzmann distribution y(x) = x + (1 x) ln (1 x)

33 Boltzmann distribution of wealth for Germany 1993 (DIW) N(k) = A exp( - k / T) number of Hh in class k K(k) = A k exp( - k / T) amount of capital in class k N = N(k) dk = A T = 36 Mill. Hh K = k N(k) dk = A T 2 = 1 Trill.DEM T = K / N = 278. DEM / Hh

34 Comparing Boltzmann to DIW wealth data for Germany 1993 Property distribution in D 1993 Property distribution in D 1993 Number of households in Mill. 2 1,5 1,5 5 1 property tax class in kdm property in Bill. DM property tax class in kdm N (k) = (N /T) exp( - k / T) N (k) d k = N = 35,6 Mill. Hh K (k) = k (N /T) exp ( - k / T ) K (k) d k = N T = K = 9,9 Trill DM

35 Comparing Boltzmann to DIW wealth data for Germany 1993 Property in D 1993 (DIW and Boltzmann) Property in D 1993 (DIW and Boltzmann) households N(x) in % DIW data Boltzmann property tax class x Capital K(x) in % DIW data Boltzmann property tax class x N (k) = (N /T) exp( - k / T) N (k) d k = N = 35,6 Mill. Hh K (k) = x (N /T) exp ( - k / T ) K (k) d k = N T = K = 9,9 Trill DM

36 Comparing Wealth to Income Property distribution Income Distribution Number of households in Mill. 2 1,5 1,5 5 1 property tax class in kdm number of households 1,4 1,2 1,8,6,4, wage class Distribution of wealth Distribution of income

37 195 Data Boltzmann N (Number of households in 1) K (Total income) 65,3 Mrd DM 74,6 Mrd DM G. Göseke, Verteilung und Schichtung der Einkommen privater Haushalte in der BRD Anzahl Haushalte pro Einkommensklasse BRD 195 N=1525, E=65,3 Mrd DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM BRD 195 N=1525, E=65,3 Mrd DM Anzahl Haushalte pro Einkommensklasse in Boltzmann Daten Klasse: monatliches Einkommen in DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM Boltzmann Daten Klasse: monatliches Einkommen in DM Number of households per income class Income per income class

38 1955 Data Boltzmann N (Number of households in 1) K (Total income) 114,2 Mrd DM 124 Mrd DM G. Göseke, Verteilung und Schichtung der Einkommen privater Haushalte in der BRD Anzahl Haushalte pro Einkommensklasse in 1 Anzahl Haushalte pro Einkommensklasse BRD 1955 N=1623, E=114,2 Mrd. DM Boltzmann Daten Klasse: monatliches Einkommen in DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM BRD 1955 N=1623, E=114,2 Mrd. DM Boltzmann Daten Klasse: monatliches Einkommen in DM Number of households per income class Income per income class

39 196 Data Boltzmann N (Number of households in 1) K (Total income) 19 Mrd DM 198 Mrd DM G. Göseke, Verteilung und Schichtung der Einkommen privater Haushalte in der BRD Anzahl Haushalte pro Einkommensklasse BRD 196 N=1895, E=19,3 Mrd. DM Haushaltseinkommen pro Einkommenklasse in BRD 196: N=18.95 Haushalte, E=19,3 Mrd. DM Anzahl der Haushalte pro Einkommensklasse in Boltzmann Daten Einkommensklasse in DM / Monat Haushaltseinkommen pro Einkommenklasse in Mrd. DM Boltzmann Daten Einkommensklasse in DM/Monat Number of households per income class Income per income class

40 1964 Data Boltzmann N (Number of households in 1) K (Total income) 266,4 Mrd DM 351,8 Mrd DM G. Göseke, Verteilung und Schichtung der Einkommen privater Haushalte in der BRD Anzahl Haushalte pro Einkommensklasse BRD 1964 N=237, E=266,4 Mrd. DM Anzahl Haushalte pro Einkommensklasse BRD 1964 N=237, E=266,4 Mrd. DM Anzahl Haushalte pro Einkommensklasse in Boltzmann Daten Klasse: monatliches Einkommen in DM Anzahl Haushalte pro Einkommensklasse in Boltzmann Daten Klasse: monatliches Einkommen in DM Number of households per income class Income per income class

41 197 Data Boltzmann N (Number of households in 1) K (Total income) 424,9 Mrd DM 567,8 Mrd DM G. Göseke, Verteilung und Schichtung der Einkommen privater Haushalte in der BRD Anzahl Haushalte pro Einkommensklasse BRD 197 N=224, E=424,9 Mrd. DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM BRD 197 N=224, E=424,9 Mrd. DM Anzahl Haushalte pro Einkommensklasse in Boltzmann Daten Klasse: monatliches Einkommen in DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM Boltzmann Daten Klasse: monatliches Einkommen in DM Number of households per income class Income per income class

42 1975 Data Boltzmann N (Number of households in 1) K (Total income) 624,4 Mrd DM 757,6 Mrd DM G. Göseke, Verteilung und Schichtung der Einkommen privater Haushalte in der BRD Anzahl Haushalte pro Einkommensklasse in 1 Anzahl Haushalte pro Einkommensklasse BRD 1975 N=23525, E=624,4 Mrd. DM Boltzmann Daten Klasse: monatliches Einkommen in DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM Haushaltseinkommen pro Einkommenklasse in Mrd. DM BRD 1975 N=23525, E=624,4 Mrd. DM Boltzmann Daten Klasse: monatliches Einkommen in DM Number of households per income class Income per income class

43 US distribution of work after G. Willis 4,5 4 3,5 3 2,5 2 1,5 manufacturing Boltzmann 1,5 Income class (k) in US $ / week US income 1995 Distribution of workers N(k)=g(k) exp(-k/t)

44 Lorenz distribution of property data for Germany 1993 (DIW) amout of property property distribution in D ,8,6,4,2,5 1 number of households The distribution of wealth is a Boltzmann distribution up to 9 % of households. At very high incomes an additional function f(x) seems to indicate a two level wealth distribution. This may be explained by a Carnot production process. y Boltzmann = x + (1-x) ln (1-x) + f(x)

45 Lorenz distribution of world wealth GDP per Capita in k US $ USA Europa Asien World energy consumption / capita World GDP / capita

46 Distribution of world wealth 24 GDP per Capita in k US $ World GDP distribution G D P in U S $ / c a p ita Number in millions A two level distribution of world GDP per capita

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