The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Ergodic Theory

  • William Parry
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_208

Abstract

To begin in the middle; for that is where ergodic theory started, in the middle of the development of statistical mechanics, with the solution, by von Neumann and Birkhoff, of the problem of identifying space averages with time averages. This problem can be formulated as follows: If xI(− ∞ < t < ∞) represents the trajectory (orbit) passing through the point x = x0 at time t = 0 of a conservative dynamical system, when can one make the identification
$$ \left({}^{\ast}\right) \lim \limits_{T\to \infty } \left(1/T\right){\int}_0^Tf\left({x}_t\right)\mathrm{d}t={\int}_{\Omega}f\;\mathrm{d}m/m\left(\Omega \right) $$
for suitable functions defined on the phase space Ω of the system?
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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • William Parry
    • 1
  1. 1.