Random perturbation

This is a random molecule s response to both random perturbations J(t) and AO(t), where AO(t) represents the shift in the vibrational frequency during collisions if in gas. 

The exact 1-electron Hamiltonian for a DBA can be written as the sum of the Hamiltonian for a translationally invariant solid plus that for the random perturbations, i.e.,... 

Pore size Geometric stability of surface against random perturbations... 

Change of profile of pore bottom due to random perturbations relative to pore size... 

The Markov chain Metropolis scheme [11] is by far the most common MC methodology. The system is randomly perturbed and the proposed move from microstate A to B is accepted with probability ... 

The Metropolis prescription dictates that we choose points with a Boltzmann-weighted probability. The typical approach is to begin with some reasonable configuration qj. The value of property A is computed as the first element of the sum in Eq. (3.33), and then qi is randomly perturbed to give a new configuration qa. In the constant particle number, constant... 

Thus, if the energy of point q2 is not higher than that of point qi, the point is always accepted. If the energy of the second point is higher than the first, p is compared to a random number z between 0 and 1, and the move is accepted if p > z. Accepting the point means that the value of A is calculated for that point, that value is added to the sum in Eq. (3.33), and the entire process is repeated. If second point is not accepted, then the first point repeats , i.e., the value of A computed for the first point is added to the sum in Eq. (3.33) a second time and a new, random perturbation is attempted. Such a sequence of phase points, where each new point depends only on the immediately preceding point, is called a Markov chain . 

In practice, MC simulations are primarily applied to collections of molecules (e.g., molecular liquids and solutions). The perturbing step involves the choice of a single molecule, which is randomly translated and rotated in a Cartesian reference frame. If the molecule is flexible, its internal geometry is also randomly perturbed, typically in internal coordinates. The ranges on these various perturbations are adjusted such that 20-50% of attempted moves are accepted. Several million individual points are accumulated, as described in more detail in Section 3.6.4. 

Example. Kubo constructed the following model to illustrate line broadening and narrowing due to random perturbations.510 In equation (1.2) suppose that co = (o0 + a (t), where a>0 and a are constants and (r) is the dichotomic Markov process (IV.2.3). As only takes two values we may abbreviate... 

Since H (t) is a stationary random perturbation , it is invariant under a change in the origin of the time, and the time (or ensemble) average will thus be... 

Below we show how the energy-optimal control of chaos can be solved via a statistical analysis of fluctuational trajectories of a chaotic system in the presence of small random perturbations. This approach is based on an analogy between the variational formulations of both problems [165] the problem of the energy-optimal control of chaos and the problem of stability of a weakly randomly perturbed chaotic attractor. One of the key points of the approach is the identification of the optimal control function as an optimal fluctuational force [165],... 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

We randomly perturb the ground state and evolve the perturbation with a given wave vector k in time numerically, searching for any exponential increase of its amplitude which would be a signature of the instability. An example of such a perturbation is presented in Fig. 21. For all velocity field models, the parameter space... 

There are two traditional approaches to the consideration of perturbed motions. One is the study of the motion in the presence of small continuously acting perturbations [24 30] the other is the investigation of fluctuations caused by small random perturbations [31-34]. Our results were obtained in terms of the former approach but using some ideas of the latter. 

The relationships, rather similar in sense, for smooth dynamic systems were introduced in ref. 34 (p. 220 etc.) for studying the random perturbations via a method of action functionals. Close concepts can also be found in ref. 39. 

A.D. Ventzel and M.I. Freidlin, Fluctuations in Dynamic Systems Caused by Small Random Perturbations, Nauka, Moscow, 1979 (in Russian). 

Genetics AcB Randomly perturbate the system looking for changes in a specific phenotype Target/pathway identification Assemble large diverse chemical library Assay for specific phenotype Identify targets of compounds that induce the phenotype... 

Random perturbations that are irreducible in principle, such as Heisenberg s Uncertainty Principle. 

The only difference between Froissart doublets for the noise-free c and noise-corrupted cn + rn time signals from Figures 4.10 and 4.11, respectively, is that the latter are more irregularly distributed than the former. This is expected due to the presence of the random perturbation rn in the noise-corrupted time signal. Flowever, this difference is irrelevant since the only concern to SNS is that noise-like or noisy information is readily identifiable by pole-zero coincidences. Note that the full auxiliary lines on each subplot in Figures 4.10 and 4.11 are drawn merely to transparently delineate the areas with Froissart doublets. 

The evaluation methodology must be capable of providing statistically reliable results at the levels in which we are interested. The measurement processes that are used to obtain responses need to be robust in the sense that they are resistant to both determinate accidental and random perturbations. This requirement for robustness applies to the processes for producing, representatively sampling, and evaluating products, be they materials, processes or otherwise. 

The vacuum interface is the source of all quantum effects. Interaction with the interface causes particles to make excursions into time and bounce back with time-reversal and randomly perturbed space coordinates. Different from classical particles, quantum objects can suffer displacement in space without time advance. They can appear to be in more than one place at the same time, as in a two-slit experiment. 

Modulations is a perturbation of the crystal lattice which, unlike random perturbations due to thermal motion and static disorder, has regular character and therefore creates sharp diffraction peaks, usually as satellites of ordinary reflections. The diffraction vector can be then expressed as (cf Section 2.2.1)... 

The stable solution is denoted by the superscript (0) the first perturbation by the superscript (1)/ the parameter e characterizes a random perturbation in the physical problem that is left unspecified. If the first perturbation decays with time, the displacement is stable to infinitesimal perturbations if it grows with time, the displacement is unstable to such perturbations. 

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