Power laws are disguised Boltzmann laws

Using a previously introduced model on generalized Lotka-Volterra dynamics together with some recent results for the solution of generalized Langevin equations, we derive analytically the equilibrium mean field solution for t he probability distribution of wealth and show that it has two characterist ic regimes. For large values of wealth, it takes the form of a Pareto style power law. For small values of wealth, w less than or equal to w(m), the d istribution function tends sharply to zero. 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 further show that the dynamics of relative wealth is independent of the specific nature of the agent interactions and exhibits a universal characte r even though the total wealth may follow an arbitrary and complicated dyna mics. In developing the theory, we draw parallels with conventional thermodynamic s and derive for the system some new relations for the "thermodynamics" ass ociated with the Generalized Lotka-Volterra type of stochastic dynamics. Th e power law that arises in the distribution function is identified with new additional logarithmic terms in the familiar Boltzmann distribution functi on for the system. These are a direct consequence of the multiplicative sto chastic dynamics and are absent for the usual additive stochastic processes .