Nonextensive statistical mechanics and economics

TL;DR

This paper discusses nonextensive statistical mechanics as an extension of Boltzmann-Gibbs theory, highlighting its foundations, applications, and relevance to economic phenomena like financial market distributions and risk aversion.

Contribution

It introduces the formalism of nonextensive statistical mechanics, bridging it to economic phenomena and demonstrating its ability to describe complex systems with power-law behaviors.

Findings

Nonextensive entropy $S_q$ generalizes BG entropy.

Power-law distributions emerge from nonextensive formalism.

Applications to financial markets and risk concepts.

Abstract

Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann-Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accomodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy $S_{BG}=-k \sum_i p_i \ln p_i$, and typically provides {\it exponential laws} for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form $S_q=k(1-\sum_ip_i^q)/(q-1)$ (with $S_1=S_{BG}$), and typically provides, for the same type of description, (asymptotic) {\it power laws}.