Fluctuation-initiated motion

Case c) Fluctuation Initiated Motion. If motion is started from a distribution concentrated around an unstable stationary point Xq where K(xo) = 0 and K (xq) = y > 0, the expansion of (2.82, 84,85) fails since the initial fluctuations are enhanced exponentially, see (2.92) before the drift dominated motion sets in. The appropriate approximation here, developed by Haake [2.2, 3] and by Suzuki [2.4] essentially consists of two steps 1) Solve the Fokker-Planck equation for the first fluctuation dominated stage and 2) Find a smoothly fitting solution for the second drift dominated stage. 

Case d) Fluctuation Dominated Motion. An initial distribution P x 0) concentrated at the potential minima x and x+ neighbouring the unstable point Xq will now be considered. In this case a slow equilibration process will take place by means of a probability flux between the modes until the stationary distribution Pst ( ) is established. Since the drift force K (jc) is directed towards the minima X- and x+, it cannot be primarily responsible for this process. In fact the motion... 

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... 

Consider reorientations of a diatomic surface group BC (see Fig. A2.1) connected to the substrate thermostat. By a reorientation is meant a transition of the atom C from one to another well of the azimuthal potential U(qi) (see Fig. 4.4)). The terminology used implies a classical (or at least quasi-classical) description of azimuthal motion allowing the localization of the atom C in a certain well. A classical particle, with the energy lower than the reorientation barrier Awhich does not interact with the thermostat cannot leave the potential well where it was located initially. The only pathway to reorientations is provided by energy fluctuations of a particle which arise from its contact with the thermostat. Let us estimate the average frequency of reorientations in the framework of this classical approach. 

Nonequilibrium effects. In applying the various formalisms, a Boltzmann distribution over the vibrational energy levels of the initial state is assumed. The rate constant calculated on the basis of the equilibrium distribution, keq, is the maximum possible value of k. If the electron transfer is very rapid then the assumption of an equilibrium distribution over the energy levels is not valid, and it is more appropriate to treat the nuclear fluctuations in terms of a steady-state rather than an equilibrium formalism. Although a rigorous treatment of this problem has not yet appeared, intuitively it seems that since the slowest nuclear fluctuation will generally be a solvent orientational motion, ke will equal keq when vout keq and k will tend to vout when vout keq (a simple treatment gives l/kg - 1/ vout + 1/keq). These considerations are... 

In general, oscillations may be oblate-prolate (H8, S5), oblate-spherical, or oblate-less oblate (E2, FI, H8, R3, R4, S5). Correlations of the amplitude of fluctuation have been given (R3, S5), but these are at best approximate since the amplitude varies erratically as noted above. For low M systems, secondary motion may become marked, leading to what has been described as random wobbling (E2, S4, Wl). There appears to have been little systematic work on oscillations of liquid drops in gases. Such oscillations have been observed (FI, M4) and undoubtedly influence drag as noted earlier in this chapter. Measurements (Y3) for 3-6 mm water drops in air show that the amplitude of oscillation increases with while the frequency is initially close to the Lamb value (Eq. 7-30) but decays with distance of fall. 

The velocity autocorrelation function (VAF) may be used to investigate the possibility of coupling between translational and rotational motions of the sorbed molecules. The VAF is obtained by taking the dot product of the initial velocity with that at time t. It thus contains information about periodic fluctuations in the sorbate s velocity. The Fourier transform of the VAF yields a frequency spectrum for sorbate motion. By decomposing the total velocity of a sorbate molecule into translational and rotational terms, the coupling of rotational and translational motion can be investigated. This procedure illustrates one of the main strengths of theoretical simulations, namely to predict what is difficult or impossible to determine experimentally. 

A further refinement of the harmonic oscillator model is possible, in which the lattice is put into contact with a heat bath at temperature T and remains in contact with the heat bath, so that the initial correlations decay not only through mutual interactions but also through random collisions with an external fluctuating field. Although it might appear that such a case would contain features of both the independent particle case and the harmonic oscillator model just analyzed, the resulting formalism is much closer to that required for the latter, and the results differ only in detail. The model to be discussed is specified by the equations of motion... 

A remarkable fact is that, in spite of all fluctuations and fractal properties exhibited by quantum motion, strong empirical evidence has been obtained that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions that is the very essence of classical chaos [2]. 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

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