Barnettet al.Reply:

Abstract

Barnett et al. Reply: Torcini et al. [1] raise some interesting issues. Their main point is that the diffusion coefficient of a dilute gas diverges with decreasing density, while the Lyapunov exponent tends to zero; surely they cannot be related by a 1 3 power rule. However, Torcini et al. overlook elementary dimensional analysis, which shows that the proportionality constant in the relationship would have to be density dependent. Equivalently, the Lyapunov exponent and diffusion constant must be rescaled by system parameters into dimensionless quantities before being compared by the 1 3 power rule. This they have not done. More specifically, the expressionl 1~D 1͞3 is obtained in the normalization ofl 1where v p ͑4pne 2 ͞m͒ 1͞2 and a ͓3͑͞4pn͔͒ 1͞3 . However, the normalization of the simulation results for a hand sphere gas in Fig. 1 [1] is not that of our expression.Unfortunately, for a dilute hard sphere gas, the hard sphere radius and number density can be combined into a dimensionless quantity by themselves. Consequently, any arbitrary relationship between the Lyapunov exponent and diffusion can be matched in the very dilute regime when density is the control parameter, which is not very useful.In addition to their main point, Torcini et al. allege that our example is outside the scope of our theory because it is a dense plasma, and the Coulomb force is "long range." However, the theory requires diluteness only in the sense that three-body and four-body interactions contribute negligibly to the autocorrelation integrals for the second derivative of the potential (i.e., binary collisions). This is met even for liquid plasmas. Indeed, with dense plasmas, the Debye length is shorter than the inter-ion spacing so the effective interaction is short range. Even in a sparse plasma, the range of the second derivative of the Coulomb potential goes as ͑1͞r͒ 3 which is still "short range."Our ab initio theory yields a fundamental result equating the Lyapunov exponent to a function of integrals of autocorrelations for fluctuations in the second derivative of the potential. The "diluteness" (in the sense of binary collisions) and equilibrium simplify the function to the 1 3 power of an autocorrelation integral, c 0 [Eq. (26)]. The expressions clearly yield the correct limiting behavior of a Lyapunov exponent which tends to zero with the density.We observed that the fluctuation-dissipation theorem (or Kubo formulas) [2] relates transport coefficients to time integrals of fluctuation autocorrelations for corresponding dynamical variables-a general result. This led to our suggestion that the Lyapunov exponent would be proportional to a positive power of the transport coefficients.We used self-diffusion as our example and had data (now published by Ueshima et al. [3]) for a relatively dense one-component plasma. c 1 and c 2 bear a simple relation to c 0 such that the solutions of the secular equation still scale as c 1͞3 0 . Barnett and Tajima [4] applied the ab initio theory in detail to a one-component plasma.The real subs...

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