How skew distributions emerge in evolving systems

¹  Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University Seoul 151-747, Korea
²  Department of Mathematics, University of California - Berkeley, CA 94720, USA
³  Laboratoire de Physique des Matériaux, Université Henri Poincaré Nancy 1 BP 239, F-54506 Vandœuvre les Nancy, Cedex, France, EU
⁴  Department of Physics, Keimyung University - Daegu 704-701, Korea

Corresponding author: mychoi@snu.ac.kr

Received: 6 October 2008
Accepted: 15 January 2009

Abstract

Despite the ubiquitous emergence of skew distributions such as power law, log-normal, and Weibull distributions, there still lacks proper understanding of the mechanism as well as relations between them. It is studied how such distributions emerge in general evolving systems and what makes the difference between them. Beginning with a master equation for general evolving systems, we obtain the time evolution equation for the size distribution function. Obtained in the case of size changes proportional to the current size are the power law stationary distribution with an arbitrary exponent and the evolving distribution, which is of either log-normal or Weibull type asymptotically, depending on production and growth in the system. This master equation approach thus gives a unified description of those three types of skew distribution observed in a variety of systems, providing physical derivation of them and disclosing how they are related.