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A fundamental process for any given chaotic flow is the deterministic pointprocess (DPP) generated by any chaotic trajectory of the flow repeatedlycrossing a canonical surface-of-section (herein referred to as a sigma-typeDPP). This paper introduces the idea of using stochastic point process modelsto describe and understand the spatial statistical features of sigma-type DPPsin two-degree-of-freedom (2D) Hamiltonian systems. In the specific context of2D non-integrable systems possessing ergodic components, it is proposed that,in an ergodic region, the pertinent model for describing the spatialstatistical features of any typical sigma-type DPP is the two-dimensionalhomogeneous Poisson point process (herein denoted by P2). Of particularinterest in this paper are the (Euclidean) kth-nearest-neighbor distancecharacteristics (k=1,2,3,...) of a given sigma-type DPP. Employing thetwo-dimensional cardioid and semi-circular mushroom billiards as generic testcases, it is shown that typical sample sigma-type DPPs possesskth-nearest-neighbor distance characteristics consistent with model predictionsfor P2. Deviations from this observed Poissonian behavior are expected instrictly non-ergodic regions (i.e., the chaotic regions in generic Hamiltoniansystems), but in cases where the dynamics is strongly (but not fully) chaotic,such deviations are posited to be negligible. The validity of the latter claimis demonstrated in the case of the 2D Henon-Heiles system at the criticalenergy. The results of the numerical experiments are contextualized and theirsignificances to both classical and quantum chaos are discussed.

Distance characteristics of ergodic trajectories in two-dimensional non-integrable systems: Reduced phase space