Microscopic Chaos from Brownian Motion?

In a recent Letter in Nature, Gaspard et al. 1] claimed to present empirical evidence for microscopic chaos on a molecular scale from an ingenious experiment using a time series of the positions of a Brownian particle in a liquid. The Letter was preceded by a lead article 2] emphasising the fundamental nature of the experiment. In this note we demonstrate that virtually identical results can be obtained by analysing a corresponding numerical time series of a particle in a manifestly microscopically nonchaotic system. As in Ref. 1] we analyse the position of a single particle colliding with many others. We use the Ehrenfest wind-tree model 3] where the point-like (\wind") particle moves in a plane colliding with randomly placed xed square scatterers (\trees", Fig. 1a). We choose this model because collisions with the at sides of the squares do not lead to exponential separation of corresponding points on initially nearby trajectories. This means there are no positive Lyapunov exponents which are characteristic of microscopic chaos here. In contrast the Lorentz model used in 1] as an example similar to Brownian motion is a wind-tree model where the squares are replaced by hard (circular) disks (cf..1], Fig. 1) and exhibits exponential separation of nearby trajectories, leading to a positive Lyapunov exponent and hence microscopic chaos. Nevertheless, we now demonstrate that the nonchaotic Ehrenfest model reproduces all the results of Ref. 1]. A particle trajectory segment shown in Fig. 1b is strikingly similar to that for the Brownian particle, (cf..1], Fig. 2). Our subsequent analysis parallels that of Ref. 1], where more details may be found. Thus the microscopic chaoticity is determined by estimating the Kolmogorov-Sinai entropy h KS , using the method of Procaccia and others 4, 5] via the information entropy K(n; ;) obtained from the frequency with which the partical retraces part of its (previous) trajectory within a distance for n measurements spaced at a time interval. Since for the systems considered here h KS equals the sum of the positive Lyapunov exponents, the determination of a positive h KS would imply microscopic chaos. As in 1] we nd that K grows linearly with time (Fig 1c and 1] Fig. 3), giving a positive (non-zero) bound on h KS (Fig 1d and 1] Fig. 4). Indeed our Figs. 1b-d for a microscopically nonchaotic model are virtually identical with the corresponding gures 2-4 of 1]. Therefore Gaspard et al. …