Space, Scale, and Scaling in Entropy Maximizing. 最大熵中的空间、尺度与标度 : Space and Scale in Entropy Maximizing

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Gibbs law does not hold.

It is shown, following Shockley [Shockley, W. (1957) Proc. IRE 45, 279-290], that, when a population is engaged in tasks whose completion requires the successful conclusion of many independent subtasks, the distribution function for successes in the primary task is log normal. It is also shown that, when the dispersion of the log-normal distribution is large, the distribution is mimicked by a 1/x distribution over a wide range of x. This argument provides a generic set of processes that yields the much observed 1/x distribution, and will also lead to a 1/f noise spectrum. It is commonly found that distributions that seem to be log normal over a broad range (say to the 95th percentile of a population) change to an inverse fractional power (Pareto) distribution for the last few percentile. Annual income distributions are examples with this structure. The very wealthy generally achieve their superwealth through amplification processes that are not available to most. We have introduced a simple amplification model to characterize the transition from a log-normal distribution to an inverse-power Pareto tail.