The low damping limit

In the simple two-level model we have distinguished between two limits The case k2 ki2 where thermal equilibrium prevails in the well and the reaction rate 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y - 0 limit of the Kramers model we are dealing with energy relaxation of a classical anharmonic oscillator. One may justifiably question the use of Markovian classical dynamics in this part of the problem, and we will come to this issue later. For now we focus on the solution of the mathematical problem posed by the low friction limit of the Kramers problem. 

To accomplish this we transform to action-angle variables (x, v — K, / ). Recall that the action K is related to the energy E by 

For a harmonic oscillator, where the frequency a is constant, Eq. (14.76) implies that E = Km so the action K is the classical equivalent to the number of oscillator quanta. The linear relationship between D and K is the classical analog of the fact that the rate of relaxation out of the nth level of a harmonic oscillator, Eq. (13.18), is proportional to n. 

Note that the pre-exponential factor in (14.80) is proportional in this onedimensional model to the density of states on the energy axis (the classical analog of the inverse of the quantum level spacing). 

This rate of energy exchange between an oscillator and the thermal environment was the focus of Chapter 13, where we have used a quantum harmonic oscillator model for the well motion. In the y 0 limit of the Kramers model we are dealing 

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In the low damping limit the rate determining step is again the energy accumulation in the well. The idea that one can average over the fast phase oscillations... 

For this to happen we know that f(y) = y in the low shear limit. As the shear stress is increased we also know that we want our viscosity to fall so we need to multiply our strain by a damping function that reduces from unity at low strains to a lesser value at high strains. A good candidate for... 

The subscript E on the right-hand side denotes fixed E (undamped trajectory). Computing x(t)x(0) without damping is consistent with the low-friction limit where damping is assumed to be small on the time scale associated with Z t) [see Eq. (5.29)]. Comparing the two results for (dE/dt)j.=o using Eq. (5.49) we get the result Eq. (5.57). Equation (5.57) provides a convenient numerical way to compute e(E) all one needs is to run a trajectory over the undisturbed molecular motion at the given E for a time of several t. ... 

The escape rate in this low damping limit can be found if we assume that Eq. (14.77) remains valid up to the barrier energy E-q, and that this energy provides... 

In the low viscosity limit, the analysis shows that is given by Eqnation 5.57, with pi the imposed freqnency. The waves travel ontwaid at a speed of pl /aR or ( pi y Pa) for short wavelengths where the effect of gravity is unimportant. For a free surface, the damping factor a, is found under the same circumstances to be... 

Figure 14.3 shows the Bode plots for overdamped (C>1), critically damped ( = 1), and underdamped (0< = 1) processes as a function of cot. The low-frequency limits of the second-order system are identical to those of the first-order system. However, the limits are different at high frequencies, cot 1. 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

A more robust method involves preheating the air 10-20°C above ambient. This increases the low adsorptive capacity of the air and ensures that drying continues throughout winter. Less basic versions recycle the heated air within an enclosed space and in effect function as a uninsulated, low-temperature kiln. Such an operation has attractions where damp winters make air-drying very slow. They are cheap, and capital is often limited for small companies. Low-temperature, low-cost driers provide some control over the drying elements that is lacking in air-drying. 

We may summarize the contents of this chapter in more detail as follows. In Section I we demonstrate how the explicit form of Gilbert s equation describing Neel relaxation may be written down from the gyromagnetic equation and how, in the limit of low damping, this becomes the Landau-Lifshitz equation. Next the application of this equation to ferrofluid relaxation is discussed together with the analogy to dielectric relaxation. 

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