Force fluctuation

In this equation, m. is the effective mass of the reaction coordinate, q(t -1 q ) is the friction kernel calculated with the reaction coordinate clamped at the barrier top, and 5 F(t) is the fluctuating force from all other degrees of freedom with the reaction coordinate so configured. The friction kernel and force fluctuations are related by the fluctuation-dissipation relation... 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

M. G. Rozman, M. Urbakh, J. Klafter. Stick-slip motion and force fluctuations in a driven two-wave potential. Phys Rev Lett 77 683-686, 1996. 

This is a specific example of the fundamental fluctuation-dissipation theorem that relates the random force / (fluctuation) to the friction constant 7 (dissipation) to ensure that any initial state eventually evolves into a state in thermal equilibrium with the fluid at temperature T. With this requirement, we obtain from Eq. (11.12) the final result... 

The sum of the three forces fluctuate around zero, indicating that the ions remained close to their equilibrium positions. The interionic forces (a, b, e, and f) decrease with time when the crystal is deformed or if ions leave it. 

When 0 = 0 (smooth curve), Ij = 0 and when o oo, (extremely wide force fluctuations), 1. This is a simple effective index. Its only drawback when applied to normalized records is its diminishing sensitivity as the standard deviation and variance become large. 

When the tip was slid laterally at v= 100 m/s during indentation, the friction or cutting force fluctuated around zero as long as the substrate did not yield (Fig. 14). This nearly frictionless sliding can be attributed to the fact that the... 

Fig. 1 3.11 Fourier analysis of y-force fluctuations for 3.2 mm (0.13 in.) thick 2024-T3 sheet friction stir welded under position controls at (top) 200 rpm, 101 mm/min (4 in./min), and (bottom) 500 rpm, 202 mm/min (8 in./min). Note volumetric wormhole defect in top chart, showing large degree of low-frequency events.

It is also worth while to note that the ordinary derivation neglects the finite velocity of light and assumes that interactions between the two molecules are instantaneous. However, the dipoles involved in the dispersion forces fluctuate at optical frequencies so that, if the separation between the two molecules is not considerably less than an optical wavelength, there will be important phase effects which can be shown to reduce the interaction energy to an r- dependence (Longuet-Higgins, 1965). This effect will not need to concern us here. 

In dynamic situations forces are not balanced at every instant one special case of dynamic situations is the steady state, in which forces are constant but they are not balanced by opposing forces. In addition, dynamic situations may include states at equilibrium. In dynamic equilibria forces fluctuate at every instant, but the forces are balanced when they are averaged over finite durations and finite parts of the system. The relevant time and length scales may or may not be sensible or important to an observer. Moreover, these scales can differ substantially for systems in different phases of aggregation for example, property fluctuations in solids are typically orders of magnitude smaller than those in fluids. 

Equations 3.1-3.5 determine both the mean force of interphase interaction and the fluctuation of this force. The mean force will be used later when formulating conservation equations for mean suspension flow. Force fluctuation is sorely needed to study properties of pseudo-turbulent motion. It should be noted that the last term in Equation 3.5 had been omitted in a similar analysis of pseudo-turbulence [25]. 

The other constituent of the work done by fluctuations is connected with the action of the interphase interaction force fluctuation. It is described by the two terms in Equation 4,3 that contain (f ) and f. These terms give both the energy input into the pseudo-turbulent motion from the mean relative fluid flow and the dissipation of fluctuation energy by viscous forces, henceforth denoted by q and q, respectively. The sum of the two mentioned terms is equal to q - q Obviously,... 

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