Chaotic principle: An experimental test

Abstract

The chaotic hypothesis discussed in Gallavotti and Cohen (1995) is tested experimentally in a simple conduction model. Besides a confirmation of the hypothesis predictions the results suggest the validity of the hypothesis in the much wider context in which, as the forcing strength grows, the attractor ceases to be an Anosov system and becomes an Axiom A attractor. A first text of the new predictions is also attempted

Fluctuation Relation beyond Linear Response Theory

“…For this system, the Lyapunov exponents are known to be paired [25,31,32] like in Hamiltonian systems and the average of each pair is a constant equal to σ + /2N d. For E = 5, each pair is composed of a negative and a positive exponent. This means that the attractive set is dense in phase space [10,30] and the chaotic hypothesis is expected to apply to the system yielding a slope X = 1 in the fluctuation relation, as confirmed by our numerical data. The same happens up to E ∼ 20.…”

“…32, is due to the asymmetry of the distribution π τ (p) around the average value p = 1; consequently, it is proportional, at leading order in τ −1 , to ζ (3) ∞ which is indeed a measure of the asymmetry of ζ ∞ (p) around p = 1. This shift would be absent in the case of a symmetric distribution (e.g., a Gaussian) and for this reason it was not observed in previous experiments [6,10,11,15].…”

“…The fluctuation theorem predicts X = 1, but other values of X are possible under different hypothesis, see [7,10,29,30]. We define a function χ 2 (a, X) as…”

“…5 found that the measured distribution π τ (p) could not be distinguished from a Gaussian distribution in the interval of p accessible to the numerical experiment [6,10,11,15].…”

Generalized fluctuation relation and effective temperatures in a driven fluid

“…Gallavotti then showed that, close to equilibrium (σ + → 0), the FR implies the usual Fluctuation-Dissipation Relation (FDR) [3]. In the recent past, the FR has been tested under a wide class of different conditions, and is now believed to be a very general relation for chaotic systems [4,5,6,7]; recently, it has been also tested in some experiments [8].…”