Unstable modes

For 0 < k < <3, is positive, and the corresponding Fourier mode fluctuations grow in time, i.e. these are the linearly unstable modes of the system. The maximally unstable mode occurs at k. , and overwhelms... 

Fig. 21. Integration contours in the complex plane for the unstable mode. Contours L, and Lj are used to calculate ImZ and the barrier partition function, respectively.

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

A similar expansion can be written in the vicinity of Q = 0. Path integration amounts to the Gaussian integration over the Q , whereas the integration over the unstable mode Qq is understood as described in section 3.3. In that section we also justified the correction factor (f) = T /T = X l2n which should multiply the Im F result in order to reproduce the correct high-temperature behavior. Direct use of the Im F formula finally yields... 

We see from both equations 8.32 and 8.33 that the most unstable mode is the mode and that ai t) = 1 - 1/a is stable for 1 < a < 3 and ai t) = 0 is stable for 0 < a < 1. In other words, the diffusive coupling does not introduce any instability into the homogeneous system. The only instabilities present are those already present in the uncoupled local dynamics. A similar conclusion would be reached if we were to carry out the same analysis for period p solutions. The conclusion is that if the uncoupled sites are stable, so are the homogeneous states of the CML. Now what about inhomogeneous states ... 

The linear instability theory of the behavior of a system near the bifurcation point can be successfully applied to many self-organization problems, such as thermal convection in hydrodynamics4 and crystal growth in solution.5 In these theories, various initial fluctuations play important roles. Occasionally the fluctuations arise from the thermal motion of atoms or molecules. If a system reaches an unstable mode over... 

The numbers on Fig. 7 refer to the number of unstable modes (corresponding to eigenvalues of the linearized form of (8) with positive real part) for shapes along segments of the families. Only the planar shape up the bifurcation point with the (lAe)>family and a portion of the (lAe)-family are stable to disturbances with the symmetry imposed by this sample size. 

This form, while less explicit than Eq. (21), allows one to treat stable and unstable modes on an equal footing and provides an efficient way to evaluate the TS trajectory numerically for a given instance of the noise. It also proves convenient to use this notation in the calculation of a TS trajectory for systems under the influence of nonwhite noise or deterministic driving. 

Gas-particle flows in fluidized beds and riser reactors are inherently unstable and they manifest inhomogeneous structures over a wide range of length and time scales. There is a substantial body of literature where researchers have sought to capture these fluctuations through numerical simulation of microscopic TFM equations, and it is now clear that TFMs for such flows do reveal unstable modes whose length scale is as small as ten particle diameters (e.g., see Agrawal et al., 2001 Andrews et al., 2005). 

They found a whole bunch of soft phonons, which are primarily horizontally polarized, near the zone boundaries between M and X. The most unstable mode they observed is the Mj phonon, the displacement pattern of which is shown in Fig. 40 note the similarity between this pattern and the reconstruction model in Fig. 39. According to Wang and Weber, these soft phonons are caused by electron-phonon coupling between the surface phonon modes and the electronic 3 surface states at the Fermi surface. They attributed the predominant Ms phonon instability to an additional coupling between d(x — y ) and d(xy) orbitals of the Zj states. 

Here (7, is the surface tension, and Sq is the permittivity of free space. The mode n = 0 corresponds to the equilibrium sphere, and n = 1 is a purely translational mode. The first unstable mode is n = 2. The critical charge for this mode is given by setting Eq. (37) to zero, which yields the Rayleigh limit of charge,... 

The main difference between the two approaches is that PGH consider the dynamics in the normal modes coordinate system. At any value of the damping, if the particle reaches the parabolic barrier with positive momentum i n the unstable mode p, it will immediately cross it. The same is not true when considering the dynamics in the system coordinate for which the motion is not separable even in the barrier region, as done by Mel nikov and Meshkov. In PGH theory the... 

The important quantity here, is A which is the average energy lost by the unstable p mode as it traverses from the barrier to the well and back. The equation of motion for the imperturbed unstable mode is p + V (p) = 0 and this defines the trajectory p(t) which at time —is initiated at the barrier top, moves to the well, reaches a turning point and then comes back to the barrier top at the time + °o. The force exerted by the imstable mode on the bath comes from the nonlinearity F(t) = -w([uooP(t)]- The average energy loss A, to first order in Ui is then found to be (see also Eq. 10) ... 

In many cases, when the damping is weak there is hardly any difference between the unstable mode and the system coordinate, while in the moderate damping limit, the depopulation factor rapidly approaches imity. Therefore, if the memory time in the friction is not too long, one can replace the more complicated (but more accurate) PGH perturbation theory, with a simpler theory in which the small parameter is taken to be for each of the bath modes. In such a theory, the average energy loss has the much simpler form ... 

Within the harmonic approximation, we observe that the lifetimes of the dominant resonances (with Vj = 0 for the unstable modes j) are therefore directly related to the sum of positive Lyapunov exponents,... 

The unstable mode at the saddle point has an imaginary frequency, and contributes negatively to the second sum, raising the transition rate. When this correction is applied to eq. 13, one still has an expression for the rate in terms of purely local properties at the saddle point and minimum. The size of this rather readily-calculated lowest-order correction can serve as a guide to whether more sophisticated quantum corrections are necessary. 

There is a one to one correspondence between the unperturbed frequencies CO, C0j j = 1,. .., N,. .. appearing in the Hamiltonian equivalent of the GLE (Eq. 3) and the normal mode frequencies. The diagonalization of the potential has been carried out explicitly in Refs. 88,90,91. One finds that the unstable mode frequency A is the positive solution of the Kramers-Grote Hynes (KGH) equation (7). This identifies the solution of the KGH equation as a physical barrier frequency. 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the unstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

The force exerted by the unstable mode on the bath comes from the nonlinearity F(t) = - [u00p(t). The average energy loss A, to first order in U is then found to be (see also Eq. 10) ... 

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