Carmeli and Nitzan Respond: Recently several workers' have attempted to extend Kramers's theory of activated rate processes to the whole friction range. Buttiker and Landauer express in a recent Comment5 their views on the relation between their work' and ours. Our view is stated here. The theories by Buttiker, Harris, and Landauer (BHL)' and by Carmeli and Nitzan use the same philosophymatching Kramers's energy solution which is valid in the low-friction limit inside the well to another solution valid in the barrier region. BHL use for the barrier region an energy equation similar to that used by Kramers inside the well, with an additional loss term due to escape out of the well. We use for this region Kramers's solution in coordinate-velocity space for an inverted parabolic potential. Both approaches involve approximations. In the BHL work these are as follows: (a) neglect of the flux of particles returning to the well from the outside region; (b) use of an expression [Ref. 1, Eq. (26)] for the flux on the energy axis which is strictly valid only when the phase-space density is uniform for constant energy, also for a case where this uniformity is broken (with an averaged phasespace density); and (c) introduction of an undetermined parameter [their n of Ref. 1, Eq. (3.2)] in calculating the outgoing flux. Our approximations are as follows: (a) use (following Kramers) of a solution in the barrier region which is appropriate for an inverted parabolic potential; and (b) the assumption that there is at least one point in phase space where Kramers's barrier solution and his well-energy equation are both valid. In addition both approaches (as well as Kramers's original work) have to assume that the well depth is large enough relative to kT so that a steady-state escape rate is well defined. Kramers's barrier solution does not account well for the high-velocity (v) tail of the velocity distri-bution of the escaping particles since at any position it approaches the Boltzmann thermal distribution as v~. This failure is most pronounced for vanishing friction y where Kramers's barrier solution yields a thermal distribution at the barrier for all v )0. This is the origin of the difference in the result for the most probable energy of escaping particles correctly pointed out in Ref. 5. Indeed neither Kramers nor we have attempted to use this approach to calculate the velocity distribution. Fortunately, this failure of Kramers's solution does not appreciably affect the result for the escape rate [Eq.(17) of Ref. 3a] because for moderately small ) the flux out of the well is dominated by low-velocity particles, while as y 0 the rate is dominated by the mean first-passage time for a particle in the well to reach the barrier energy, a result which does not depend on Kramers's barrier solution.

The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non-Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the original GLE. Non-Markovian effects are shown to have large effects on the rate of energy accumulation and relaxation within the well.

Particle swarm optimization (PSO) is a powerful metaheuristic population-based global optimization algorithm. However, when it is applied to nonseparable objective functions, its performance on multimodal landscapes is significantly degraded. Here we show that a significant improvement in the search quality and efficiency on multimodal functions can be achieved by enhancing the basic rotation-invariant PSO algorithm with isotropic Gaussian mutation operators. The new algorithm demonstrates superior performance across several nonlinear, multimodal benchmark functions compared with the rotation-invariant PSO algorithm and the well-established simulated annealing and sequential one-parameter parabolic interpolation methods. A search for the optimal set of parameters for the dispersion interaction model in the ReaxFF- lg reactive force field was carried out with respect to accurate DFT-TS calculations. The resulting optimized force field accurately describes the equations of state of several high-energy molecular crystals where such interactions are of crucial importance. The improved algorithm also presents better performance compared to a genetic algorithm optimization method in the optimization of the parameters of a ReaxFF- lg correction model. The computational framework is implemented in a stand-alone C++ code that allows the straightforward development of ReaxFF reactive force fields.

The Brownian motion of a general classical anharmonic oscillator is studied in the lowviscosity limit for a general non-Markoffian interaction with a heat bath. Memory effects are shown to have a profound influence on the rate of energy accumulation and relaxation.PACS numbers: 05.40. + J, 82.20.FdThe dynamics of activated rate processes plays a central role in many areas of physics and chemistry. Following Kramers, 1 most studies use as a model a particle moving in a potential well under the influence of a thermal bath and distinguish between three cases: The high-viscosity limit corresponds to a diffusive motion of the overdamped oscillator described by the Smoluchowski equation. The intermediate-viscosity case focuses on the diffusive motion near the potential barrier and yields transition-state theory as the low-viscosity limit. Finally, for very low viscosities, the dynamics of the energy accumulation by the particle becomes important and the rate approaches zero as the viscosity decreases. Obviously, the dynamics of energy accumulation is always important for nonsteady-state process-.es, e.g., when the kinetics is monitored following a temperature jump.The starting point in the Kramers model is the Langevin equation* = ~(l/M)dV(x)/Qx-yx +(1/ M)R(t) 9 where x is the coordinate of the particle of mass M moving in the potential V{x) and where y and R are the damping rate and the (assumed Gaussian) fluctuating force associated with the coupling to the thermal bath [y and R are related by the fluctuation-dissipation theorem (R(t 1 )R(t 2 j) = 2yMkT6{t 1 -t 2 ), k being the Boltzmann constant and T the temperature]. In many cases the correlation time associated with the bath is longer than the characteristic period of the particle (though still much shorter than the rate of energy exchange).In this case we use the model described by the generalized Langevin equation (1) For molecular dynamics problems we usually have (oo is the oscillator frequency) 2 y«l/r c «co.The existence of these vastly different time scales makes the problem very difficult to solve numerically though such solutions have been ob-where (R) = 0 and r <&(t 1 )R(t 2 ))=Z(t 1~t2 )MkT; £dtZ(t) = y.(2)The correlation function Z(t) is characterized by the correlation time r c . For specificity we shall refer to the simple model Z(t) = (y/T c )exp(-t/r c ).(3)

Kramers' theory of activated processes yields expressions for the steady-state escape rate in the large-and small-friction limits and for Markovian dynamics. The present work extends this theory to non-Markovian dynamics and to the whole friction range. Kramers results are recovered in the appropriate limits.