Carmeli and Nitzan Respond

Abstract

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.

Bistable flow driven by coloured gaussian noise: A critical study

“…Therefore, if the noise source has a nonvanishing correlation time z, the limiting (non-Markovian)master operators of the type in (3.3), (3.4) and (3.10) yield information about quantities which involve zero-modes only; e.g. the stationary probability if(x), or current carrying stationary non-equilibrium probabilities [9][10][11][12], which clearly also do not depend on the initial preparation.…”

“…Kubo [4,7] has shown that a very short correlation time of the fluctuating magnetic field yields a vanishing effect on the motion of the spin; on the contrary, if the fluctuations of the field are large and correlated over a long time scale, the motion of the spin is greately modified. Another important example is the relevant influence of the correlated noise on the activation rates in equilibrium systems [8][9][10][11] and in driven non-equilibrium systems [12]. Generally, the finite correlation of the noise will have an effect on the form of the stationary probability.…”

Modeling diffusive dynamics in adaptive resolution simulation of liquid water

“…͑15͒. In our simulation we employ a local Langevin thermostat with a position-dependent friction [49][50][51][52] …”

Transport properties controlled by a thermostat: An extended dissipative particle dynamics thermostat

“…Although the accelerated dynamics is advantageous in some cases it can turn out to be problematic if one is really interested in dynamics in situations where two levels of resolutions are used within one simulation, as in the case of AdResS. To overcome this problem one can couple different classes of DOFs to the Langevin thermostat with different friction constants 25,26,27,28 . We have shown in the example of liquid water that the coarse-grained dynamics can be slowed down by increasing the effective friction in the coarse-grained system 29 .…”