Stochastic motion

We want to study the movement of a particle, which is in the deterministic sense force free, using the above equations. The kinetic energy during a stochastic process may change. We formulate the equation for the entire energy, as the Hamilton function with noise. 

The term at the left-hand side of Eq. (21.15) is the kinetic energy and at the same time the Hamilton function. We assumed here that the particle has on the average a constant energy, but this energy is subjected to a fluctuation, which has the character of a white noise. We can form the square root of this expression. If the noise is small in relation to the energy, then we obtain 

Here we expanded the root into a Taylor series and stopped after the first-order term. We notice that in the classical, deterministic mechanics the kinetic energy, which is 

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Lichtenberg A J and Lieberman M A 1983 Regular and Stochastic Motion (Berlin Springer)... 

Iicht83] Lichtenberg, A.J. and M.A. Lieberman, Regular and Stochastic Motion, Springer-Verlag (1983). 

A. J. Lichtenberg and M. A. Liebermann, Regular and Stochastic Motion, Springer, New York, 1982. 

The motion of polydispersed particulate phase is modeled making use of a stochastic approach. A group of representative model particles is distinguished. Motion of these particles is simulated directly taking into account the influence of the mean stream of gas and pulsations of parameters in gas phase. Properties of the gas flow — the mean kinetic energy and the rate of pulsations decay — make it possible to simulate the stochastic motion of the particles under the assumption of the Poisson flow of events. 

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, New York, 1978 A. J. Lieberman and A. J. Lichienberg, Regular and Stochastic Motion, Springer, New York, 1983 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. 

The stochastic motion of particles in condensed matter is the fundamental concept that underlies diffusion. We will therefore discuss its basic ideas in some depth. The classical approach to Brownian motion aims at calculating the number of ways in which a particle arrives at a distinct point m steps from the origin while performing a sequence of z° random steps in total. Consider a linear motion in which the probability of forward and backward hopping is equal (= 1/2). The probability for any sequence is thus (1/2). Point m can be reached by z° + m)/2 forward plus (z° m)/2 backward steps. The number of distinct sequences to arrive at m is therefore... 

The starting point is the well-known generalized Langevin equation (GLE) as adopted for stochastic motions involving coupling to a solvent coordinate. We employ the notation of Hynes [63] which is consistent with the discussion of solvation in Section II. 

Let us now consider stochastic motion in an OB system. In general, noise in an OB system may result from fluctuations of the incident field, or from thermal and quantum fluctuations in the system itself. We shall consider the former. The fluctuations of the intensities of the input or reference signals give rise respectively to either multiplicative or additive noise driving the phase. Both types of fluctuations can be considered within the same approach [108], Here we discuss only the effects of zero-mean white Gaussian noise in the reference signal ... 

The model (1)—(5) describes stochastic motion in a general OB system for white Gaussian noise in the low noise intensity limit. We now apply this model to the description of some experimental results on fluctuations and fluctuational transitions in some particular OB devices. 

It follows from the above discussion that an indicator of applicability of the description of stochastic motion in an OB system is an activation dependence of the transition probabilities VFnm on the noise intensity. Using level-crossing measurements (shown to be independent on the level positions), we found in our previous experiments [108] that the activation law applies over the whole range of noise intensities that we are using. 

Figure 3.91 Stochastic motion of a Brownian tracer particle in a laminar flow profile [129] (by courtesy of ACS).

The states of the macromolecule will be considered in points of time in a time interval At, so that the stochastic motion of Brownian particles of the chain can be described by the equation for the particle co-ordinates... 

The set of equations (3.43) describes the stochastic motion of a chain along its contour. The head and the tail particles of the chain can choose random directions. Any other particle follows the neighbouring particles in front or behind. The smaller the time interval At is the quicker moves the chain. Clearly, the time interval cannot be an arbitrary quantity and is specified by the requirement that the squared displacement of the entire chain by diffusion for the interval At is equal to 2, so that... 

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

We have thus far considered coherent processes that take place in RPs (which in some cases been have been modulated by stochastic motion). However, the common spin-lattice and spin-spin relaxation processes familiar from magnetic resonance also come to bear on the dynamics of RPs. Typical values of Ti and T2 for small organic radicals in homogeneous solution are on the microsecond timescale and as such are rather slow relative to coherent mixing and RP diffusion. Thus, for the most part, effects of incoherent spin relaxation are not manifest in such reactions. However, for reactions in which the RP lifetime is substantially extended, for instance, by constraining the RP inside a microreactor such as a micelle (many examples in Ref. 14), relaxation effects become significant. 

To recover the ideal case of Eq. (1.1) we would have to assume that (u ), vanishes. The analog simulation of Section III, however, will involve additive stochastic forces, which are an unavoidable characteristic of any electric circuit. It is therefore convenient to regard as a parameter the value of which will be determined so as to fit the experimental results. In the absence of the coupling with the variable Eq. (1.7) would describe the standard motion of a Brownian particle in an external potential field G(x). This potential is modulated by a fluctuating field The stochastic motion of in turn, is driven by the last equation of the set of Eq. (1.7), which is a standard Langevin equation with a white Gaussian noise defined by... 

Here (OjoU) accounts for precession arising from both the static field and the non-stochastic motion in the presence of the gradient, wJjf[r o(t), t]. 

In contrast to hyperbolic systems, the phase space structure in the mixed system is quite intricate and inhomogeneous, which brings about transport phenomena and relaxation processes essentially different from uniformly hyperbolic cases [3]. A remarkable fact is that qualitatively different classes of motions such as quasi-periodic motions on invariant tori and stochastic motions in chaotic seas coexist in a single phase space. The ordered motions associated with invariant tori are embedded in disordered motions in a self-similar way. The geometry of phase space then reflects the dynamics. 

At ambient temperatures, the stochastic motions of methyl groups are generally fast on the NMR time scale. The only exceptions repotted so far involve the methyl group in 9-methyltriptycene derivatives (see Fig. 13) in which the torsional energy barriers concerned can be very high they can exceed 40 k.l/mol. In the latter compounds, the stochastic dynamics of the methyl rotor can be frozen out on the time-scale of NMR experiments in liquids. The liquid-phase NMR investigations on a number of 9-methyltriptycene derivatives, performed in the period up to the early 1990s, were reviewed by Oki. The experimental spectra of the methyl protons were interpreted in terms of the AB theory there were no mentions to the possible inadequacies of the AB model in the description of the observed line shapes. 

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