Overdamped limit

The Monte Carlo (MC) method can be used to efficiently calculate thermal equilibrium properhes (see Fig. 3.2). However, since it is an energy-barrier-based method, it will fail to generate dynamic features such as the precession of the spins, and will be able to generate the dynamic magnetizahon in the overdamped limit (X —> oo) only if an appropriate algorithm is used [35]. 

Thus, the pressure difference on the two sides of the step is proportional to the difference of the inverse cubes of the terrace widths (neglecting possible intereactions with more distant steps). Again in the overdamped limit, the step velocity f)x/<5t is proportional to the pressure from the terrace behind the step minus the pressure from the terrace ahead of the step. Since the motion is again step diffusion, the prefactor ought to contain the same transport coefficient as that for equilibrium fluctuations, fa for EC or Ds Cs , for TD, in either case divided by keT. Alternatively, this can be described as a current produced by the gradient of achemicalpotential associated witheachstep(Rettori and Villain, 1988). 

Now in the strongly overdamped limit (8 -> 0) and for times longer than t = 4m/f(4 — 8), the first term of Eq. (1.3) dominates and the normal mode displacement decays in time according to ... 

In summary, our dimensidnal analysis suggests that in the overdamped limit —>0,(6) should be well approximated by the first-order system... 

We seem to have run into a paradox. Is (7) valid in the overdamped limit or not If it is valid, how can we satisfy the two arbitrary initial conditions demandedby (6) ... 

Equation (1) is a second-order system, but in the overdamped limit of extremely large b, it may be approximated by a first-order system (see Section 3.5 and Exercise 4.4.1), In this limit the inertia term mlrQ is negligible and so (1) becomes... 

Although quantum mechanics is required to explain the origin of the Josephson effect, we can nevertheless describe the dynamics of Josephson junctions in classical terms. Josephson junctions have been particularly useful for experimental studies of nonlinear dynamics, because the equation governing a single junction is the same as that for a pendulum In this section we will study the dynamics of a single junction in the overdamped limit. In later sections we will discuss underdamped junctions, as well as arrays of enormous numbers of junctions coupled together. 

We are not yet prepared to analyze (6) in general. For now, let s restrict ourselves to the overdamped limit fi 1. Then the term Ptfi" may be neglected after a rapid initial transient, as discussed in Section 3.5, and so (6) reduces to a nonuniform oscillator ... 

Find the current-voltage curve analytically in the overdamped limit. In other words, find the average value of the voltage (V) as a function of the constant applied current I, assuming that all transients have decayed and the system has... 

Current and voltage oscillations) Consider a Josephson junction in the overdamped limit j3 = Q. 

Previously, we could only treat the overdamped limit. The next four exercises deal with the dynamics more generally. 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

Here > 0 is proportional to the mass of the bead, and y > 0 is related to the spin rate of the hoop. Previously we restricted our attention to the overdamped limit f —> 0. 

The friction coefficient y defines the timescale, y of thermal relaxation in the system described by (8.13). A simpler stochastic description can be obtained for a system in which this time is shorter than any other characteristic timescale of our system. This high friction situation is often referred to as the overdamped limit. In this limit of large y, the velocity relaxation is fast and it may be assumed to quickly reaches a steady state for any value of the applied force, that is, v = x = 0. This statement is not obvious, and a supporting (though not rigorous) argument is provided below. If true then Eqs (8.13) and (8.20) yield... 

When the relaxation is not overdamped we need to consider the full Kramers equation (14.41) or, using Eqs (14.42) and (14.43), Eq. (14.44) forf. In contrast to Eq. (14.45) that describes the overdamped limit in terms of the stochastic position variable x, we now need to consider two stochastic variables, x and v, and their probability distribution. The solution of this more difficult problem is facilitated by invoking another simplification procedure, based on the observation that if the... 

Before comparing Eq. (3.53) with MD results, we note that the weak response limit co jp 0((Omip), is the converse of the strong overdamping limit i (5) 2 pmf of the diffusional Kramers model. Thus, Eq. (3.42) for... 

Kramers approach to rate theory in the underdamped and spatial-diffusion-limited regimes spurred extensions which were applicable to the much more complex STGLE. Grote and Hynes (23) used a parabolic barrier approximation to derive the rate expression for the GLE in the spatial diffusion limit. Carmeli and Nitzan derived expressions for the rate of the GLE (24) and the STGLE (25) in the underdamped limit. The overdamped limit for the rate in the presence of delta correlated friction was solved using the mean first passage time expression (26,27). A turnover theory, valid for space- and time-dependent friction, has only been recently presented by Haynes, Voth, and Poliak... 

The overdamped limit gives the golden rule rate, with k proportional to the square of the coupling, J, and inversely proportional to the damping strength, /. Note that the population relaxation rate in the eigenstate representation is simply f M in both regimes. Thus,... 

An interesting special case of Langevin dynamics is obtained by considering the large y limit, often referred to as the overdamped limit. In cases where the friction constant is scaled by the inverse mass (so y is replaced by yM in (6.32)-(6.33)) such as in [233], this is instead known as the zero-mass limit. In this model the inertial dynamics is assumed to be dominated by collisional effects. Let v = M p and assume that the acceleration is negligible, so that Langevin dynamics reduces to... 

The first order schemes of the form XYZ, as well as schemes such as XOYOX] are not consistent in this overdamped limit the errors in computed... 

In recent studies two groups (Nowik et al., 1983 Knapp et al., 1983) have shown that Mossbauer nuclei which are harmonically bound to a centre by a harmonic force mCl r, damped by a frictional force mfi dr/dt and acted upon by random forces will perform Brownian motion. In the overdamped limit this yields Mossbauer spectra (Nowik et al.,... 

V is the purely repulsive elastic energy (Equation 3.3, here with a = 2) between overlapping disks i and j. The viscous damping force is Pf = -bvi, with a damping coefficient hoslyfrne = 2 chosen to be in the overdamped limit. The force on... 

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