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...

Under optimal interaction conditions ions can be accelerated up to relativistic energies by a petawatt laser pulse in both underdense and overdense plasmas. Two-dimensional particle in cell simulations show that the laser pulse drills a channel through an underdense plasma slab due to relativistic self-focusing. Both ions and electrons are accelerated in the head region of the channel. However, ion acceleration is more effective at the end of the slab. Here electrons from the channel expand in vacuum and are followed by the ions dragged by the Coulomb force arising from charge separation. A similar mechanism of ion acceleration occurs when a superintense laser pulse interacts with a thin slab of overdense plasma and the pulse ponderomotive pressure moves all the electrons away from a finite-diameter spot.

A method is proposed for generating collimated beams of fast ions in laser-plasma interactions. Two-dimensional and three-dimensional particle-in-cell simulations show that the ponderomotive force expels electrons from the plasma region irradiated by a laser pulse. The ions with unneutralized electric charge that remain in this region are accelerated by Coulomb repulsive forces. The ions are focused by tailoring the target and also as a result of pinching in the magnetic field produced by the electric current of fast ions. (C) 2000 MAIK "Nauka/Interperiodica"

Stability of a uniform contact surface is investigated in the case when a nonuniform shock wave passes through the surface. The nonuniform shock is generated by a rippled piston that moves with constant velocity. The amplitude of the shock oscillates and decreases as it propagates. A uniform contact surface is found to be unstable after the nonuniform shock passes across it. The growth rate depends sensitively on the phase of the oscillating shock wave at the time when the shock hits the contact surface. The physical mechanism of the instability is qualitatively discussed. The linear and nonlinear evolutions of the instability are studied. In particular, the dependence of the linear case on the Atwood number for a weak shock is investigated. Properties of this stability are found to exhibit differences from those of the standard Richtmyer-Meshkov instability in both the linear and nonlinear cases. ͓S1063-651X͑96͒50406-X͔ PACS number͑s͒: 52.35.ϪgIn inertial confinement fusion, fuel is required to be compressed approximately up to 1000 times solid density. An asymmetric implosion associated with hydrodynamic instabilities disturbs uniform high density compression and reduces fusion reaction yield. Nonuniform laser irradiation leads to nonuniform ablation and thus to the generation of nonuniform shock waves. This could seed density perturbations that will grow later due to the Rayleigh-Taylor ͑RT͒ instability in the acceleration phase. This happens even if the target surface is initially uniform.It is well known that a nonuniform contact surface becomes unstable when a ͑uniform͒ shock wave passes through the contact surface because of the so-called Richtmyer-Meshkov ͑RM͒ instability ͓1,2͔. In this paper we investigate the stability of a uniform contact surface when the nonuniform shock passes through it. It will be shown that the uniform contact surface becomes unstable and the growth rate depends on the phase of the oscillating shock wave at the time when the shock hits the material interface. Both linear and nonlinear evolutions of the instability are found to have differences from the RM instability.We consider a shock wave driven by a rippled piston as shown in Fig. 1͑a͒. In the figure, the piston is located at zϭ0 in a reference frame moving with the piston, and the shock propagates in the z direction with the speed of v s relatively to the piston. We consider the surface modulation of the piston to be given as a 0 exp͑ikx͒, where a 0 and k are the amplitude and the wave number, respectively. The surface modulation of the piston induces perturbations of velocity u, density , and pressure p in the shock compressed region.In the linear theory ͓1͔, the pressure perturbation in the shock compressed region satisfies the wave equation:where c 1 is the sound speed in the shock compressed region. The subscripts 0, 1, and 2 denote, respectively, the values ahead of the shock, behind the shock, and beyond the contact surface as shown in Fig. 1͑a͒. The superscripts 0 and 1 denote the unperturbed and first-order quantities...

A relativistically intense short laser pulse can produce a large flux of X rays through the interaction with electrons that are driven by its intense electromagnetic fields. Apart from X rays from the high-Z matter irradiation by an intense laser, two main processes, Larmor and Bremsstrahlung radiation, are among the most significant mechanisms for X-ray emission from short-pulse laser irradiation on low-Z matter in the regime of relativistic intensities. We evaluate the power, energy spectrum, brilliance, polarization, and time structure of these X rays. We suggest a few methods that significantly enhance the power of Larmor X rays. Because of the peakedness in the energy spectrum of Larmor X rays, Larmor X rays have important applications.