The ergodic hierarchy, randomness and Hamiltonian chaos

Abstract

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called 'ergodic hierarchy' (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems. r

Evidence for the Deterministic or the Indeterministic Description? A Critique of the Literature About Classical Dynamical Systems

“…It is mathematically extremely hard to prove that descriptions satisfy the assumption of Theorem 1. Therefore, for many descriptions it is conjectured, but not proven, that they satisfy this assumption, e.g., for any finite number of hard balls in a box or for the Hénon description for many other parameter values (see Benedicks and Young 1993;Berkovitz et al 2006). So far we have focused on how to obtain stochastic descriptions when deterministic descriptions are given.…”

Notions of the ergodic hierarchy for curved statistical manifolds

“…The ergodic hierarchy ranks the chaos of a dynamical system according to a type of correlation C(T t A, B) between two subsets A and B of X that are separated by a time t. This is defined as [19,20] …”

“…Besides, in dynamical systems theory, the ergodic hierarchy (EH) characterizes the chaotic behavior in terms of a type of correlation between subsets of the phase space [19,20]. In the asymptotic limit of large times, the EH establishes that the dynamics is more chaotic when the correlation decays faster.…”

Probability and Randomness

“…A (discrete) dynamical system can be characterised by four parameters: a set of basic states , a σ-algebra Σ on (intuitively, the possible outcomes in the system, corresponding to regions of the basic space ), some measure such that = 1, and a deterministic evolution map from onto , which captures the lawlike evolution of states over one time step (Berkovitz and Frigg, 2006). Following Werndl (2009), a dynamical system is chaotic iff it is mixing: for all , ∈ Σ (where ' ! '…”