Physics, Finance and Logistic Systems Boltzmann, Pareto, Levy and Malthus Sorin Solomon, Hebrew University of Jerusalem and ISI Torino

+ d X i = (rand i (t) Pierre Verhulst + c i x Boltzmann ()  P ( X i ) ~ X i –1-  At each time instance Vilifredo Pareto (X.,t)) X i +  j a ij X j

+ Pierre Verhulst x Boltzmann () P (  X ) ~ (  X ) –1 -  At short times Paul Levy  P (  X=0 ) ~ (  t ) –1 /  d X i = (rand i (t) (X.,t)) X i +  j a ij X j + c i

+ Pierre Verhulst x Boltzmann ()  d X i = ( rand i + c i (X.,t)) X i +  j a ij X j = XiXi X Andrzej Nowak+ +Kamil Rackozi Gur Ya’ari +Sorin Solomon Poland Russia Ukraine

Power Laws (Pareto- Zipf)

1 Gates, William Henry III 48,000, MicrosoftGates, William Henry III 2 Buffett, Warren Edward 41,000, BerkshireBuffett, Warren Edward 3 Allen, Paul Gardner 20,000, Microsoft,Allen, Paul Gardner 4-8Walton 5X18,000, Wal-MartWalton 9 Dell, Michael 14,200, DellDell, Michael 10 Ellison, Lawrence Joseph 13,700, OracleEllison, Lawrence Joseph GatesBuffett Allen Walton DellEllison Ln 2 Ln 4Ln 5Ln 6Ln 3 Ln 90 Ln 48 Ln 41 Ln 20 Ln 14.2 Ln 13.7 

~ population growth rate ~ average family size fixed income (+redistribution) / market returns volatility economic stability; Wealth Social Distribution Forbes 400 richest by rank Zipfplot of thewealthsof the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding model results are shown in the inset. Dell Buffet 20 ALLEN GATES WALMART Log INDIVIDUAL WEALTH Rank in Forbes 400 list 400 Wealth Social Distribution  Individual Wealth Distribution

red circles: Pareto for 400 richest people is USA The inset: the average wealth history and model fit blue squares: simulation results 

No. 6 of the Cowles Commission for Research in Economics, HAROLD T. DAVIS No one however, has yet exhibited a stable social order, ancient or modern, which has not followed the Pareto pattern at least approximately. (p. 395) Snyder [1939]: Pareto’s curve is destined to take its place as one of the great generalizations of human knowledge

Later POWER LAWS Connection: Wealth Inequality  Price Instability

~ population growth rate ~ average family size fixed income (+redistribution) / market returns volatility economic stability; Wealth Social Distribution Stock Index Stability in time Forbes 400 richest by rank Time Interval (seconds) 400 Probability of “No significant fluctuation” Time Interval Zipfplot of thewealthsof the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding model results are shown in the inset. Dell Buffet 20 ALLEN GATES WALMART Log INDIVIDUAL WEALTH Rank in Forbes 400 list 400 Time Interval (s) Probability of “no significant fluctuation” Stock Index Stability in time  

Logistic Equation

Malthus : autocatalitic proliferation/ returns : B+A  B+B+A death/ consumption B  Ø dw/dt = a  w a =(#A x birth rate - death rate) a =(#A x returns rate - consumption /losses rate) exponential solution: w(t) = w(0) e a t a < 0 w= #B a  TIME birth rate > death rate

Verhulst way out of it: B+B  B The LOGISTIC EQUATION dw/dt = a w – c w 2 c=competition / saturation Solution: exponential ==========  saturation w = #B

A+B-> A+B+B proliferation B->.   death B+B-> B   competition (radius R) almost all the social phenomena, …. obey the logistic growth. “ Social dynamics and quantifying of social forces ” E. W. Montroll I would urge that people be introduced to the logistic equation early in their education… Not only in research but also in the everyday world of politics and economics … Lord Robert May b. = ( a -  )  b –  b 2 + diffusion Simplest Model: A= conditions, B = plants/ animals WELL KNOWN Logistic Equation (Malthus, Verhulst, Lotka, Volterra, Eigen)

Phil Anderson “Real world is controlled … –by the exceptional, not the mean; –by the catastrophe, not the steady drip; –we need to free ourselves from ‘average’ thinking.”

Logistic Equation usually ignored spatial distribution, Introduce discreteness and randomeness ! w. = ( conditions x birth rate - death  x w + diffusion w - competition w 2 conditions is a function of many spatio-temporal distributed discrete individual contributions rather then totally uniform and static

+ = Multi-Agent stochastic prediction even for  a  

Instead: emergence of singular spatio-temporal localized collective islands with adaptive self-serving behavior => resilience and sustainability even for << 0 ! Multi-Agent Complex Systems Implications: one can prove rigorously that the LE prediction: Time Logistic Equations ( continuum << 0 approx ) Multi-Agent stochastic  a   prediction Is ALWAYS wrong !

Further Rigorous Theoretical Results: Even in non-stationary, arbitrarily varying conditions (corresponding to wars, revolutions, booms, crashes, draughts) Indeed it is verified: the list of systems presenting scaling fits empirically well the list of systems modeled in the past by logistic equations ! that stable Power Laws emerge generically from stochastic logistic systems The Theorem predicts :

VALIDATION: Scaling systems  logistic systems EXAMPLES Nr of Species vs individuals size Nr of Species vs number of specimens Nr of Species vs their life time Nr of Languages vs number of speakers Nr of countries vs population / size Nr of towns vs. population Nr of product types vs. number of units sold Nr of treatments vs number of patients treated Nr of patients vs cost of treatment Nr of moon craters vs their size Nr of earthquakes vs their strenth Nr of meteorites vs their size Nr of voids vs their size Nr of galaxies vs their size Nr of rives vs the size of their basin

Conclusion The 100 year Pareto puzzle Is solved by combining The 100 year Logistic Equation of Lotka and Volterra With the 100 year old statistical mechanics of Boltzmann

Market Fluctuations Scaling

Market Fluctuations in the Lotka-Volterra-Boltzmann model O

Paul Levy Gauss Levy DATA

        instead of Gauss    instead of Gauss     

Mantegna and Stanley The distribution of stock index variations for various values of the time interval   The probability of the price being the same after  as a function of the time interval  : P(0,  –  Market Index Dynamics Stock Index Stability in time  Probability of “no significant fluctuation”

 The relative probability of the price being the same after  as a function of the time interval  : P(0,  –   13

 Zipfplot of thewealthsof the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding model results are shown in the inset. Dell Buffet 20 ALLEN GATES WALMART  Stock Index Stability in time Time Interval (s) Probability of “no significant fluctuation” Rank in Forbes 400 list Log INDIVIDUAL WEALTH Theoretical Prediction  Forbes 400 richest by rank 400   Confirmed brilliantly Pioneers on a new continent: on physics and economics Sorin Solomon and Moshe Levy Quantitative Finance 3, No 1, C