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Power Laws are Boltzmann Laws in Disguise


  • Peter Richmond
  • Sorin Solomon


Using a model based on generalised Lotka Volterra dynamics together with some recent results for the solution of generalised Langevin equations, we show that the equilibrium solution for the probability distribution of wealth has two characteristic regimes. For large values of wealth it takes the form of a Pareto style power law. For small values of wealth, (w less then wmin) the distribution function tends sharply to zero with infinite slope. The origin of this law lies in the random multiplicative process built into the model. Whilst such results have been known since the time of Gibrat, the present framework allows for a stable power law in an arbitrary and irregular global dynamics, so long as the market is `fair', i.e., there is no net advantage to any particular group or individual. We show for our model that the relative distribution of wealth follows a time independent distribution of this form even thought the total wealth may follow a more complicated dynamics and vary with time in an arbitrary manner. In developing the theory, we draw parallels with conventional thermodynamics and derive for the system the associated laws of `econodynamics' together with the associated econodynamic potentials. The power law that arises in the distribution function may then be identified with new additional logarithmic terms in the familiar Boltzmann distribution function for the system. The distribution function of stock market returns for our model, it is argued, will follow the same qualitative laws and exhibit power law behaviour.

Suggested Citation

  • Peter Richmond & Sorin Solomon, 2000. "Power Laws are Boltzmann Laws in Disguise," Papers cond-mat/0010222,
  • Handle: RePEc:arx:papers:cond-mat/0010222

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    References listed on IDEAS

    1. Levy, Haim & Levy, Moshe & Solomon, Sorin, 2000. "Microscopic Simulation of Financial Markets," Elsevier Monographs, Elsevier, edition 1, number 9780124458901.
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    Cited by:

    1. Geoff Willis, 2004. "Laser Welfare: First Steps in Econodynamic Engineering," Microeconomics 0408003, EconWPA.
    2. Kaldasch, Joachim, 2012. "Evolutionary model of the personal income distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5628-5642.
    3. Kaldasch, Joachim, 2014. "Evolutionary model of the bank size distribution," Economics - The Open-Access, Open-Assessment E-Journal, Kiel Institute for the World Economy (IfW), vol. 8, pages 1-16.
    4. Kaldasch, Joachim, 2012. "Evolutionary model of the growth and size of firms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(14), pages 3751-3769.
    5. G. Yaari & D. Stauffer & S. Solomon, 2008. "Intermittency and Localization," Papers 0802.3541,, revised Mar 2008.

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