Fluctuation coefficient

For all substances in the low-temperature range T, Tb the as>Tnptotic form of FEOS with constant excluded volume bo, zero fluctuation coefficient c=0 and T-dependent interaction coefficient a(T) is adequate for a liquid phase ... 

This result should be compared with (2.33 b, 34 b) where the short time mean deviation and mean square deviation of x (divided by At) are calculated on the basis of the Fokker-Planck equation for P (x 0 and where it turns out that they agree with the drift coefficient and fluctuation coefficient, respectively, of the Fokker-Planck equation. Equating the rhs of (2.33 b, 37) as well as those of (2.34b, 38), the relations between the coefficients of the Fokker-Planck equation (2.26) and the Langevin equation (2.35)... 

Case a) An Exact Solution. For the following drift and fluctuation coefficients ... 

If the trend parameters k are constants with time, the drift coefficients Ki(x) and the fluctuation coefficients Qij (x) become independent of time. Fokker-Planck equations of the form (3.32) with time independent coefficients are treated in textbooks on statistical physics [1.51-53]. 

In other words, the Fokker-Planck equation (3.34) and the current (3.35) adopt the standard forms (3.32,33) with fluctuation coefficients Qij(x) and drift coefficients Ki(x) given by (3.39, 40). Drift- and fluctuation coefficients of the form (3.39,40), however, do not represent the most general case since they satisfy the so called condition of detailed balance [3.2-4]. This condition follows by inserting (3.38, 39) in (3.40) ... 

W. Weidlich, G. Haag Quasi adiabatic solutions of Fokker-Planck equations with time-dependent drift- and fluctuation-coefficients. Z. Phys. B39, 81 (1980)... 

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of, and provides an expression for it in tenns of the friction coefficient, the thennodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strengtii of the fluctuations to the frictional forces acting on the particle [22]. 

Here is a friction coefficient which is allowed to vary in time 2 is a thennal inertia parameter, which may be replaced by v.j., a relaxation rate for thennal fluctuations g 3Ais the number of degrees of freedom. 

In a simulation it is not convenient to work with fluctuating time intervals. The real-variable formulation is therefore recommended. Hoover [26] showed that the equations derived by Nose can be further simplified. He derived a slightly different set of equations that dispense with the time-scaling parameter s. To simplify the equations, we can introduce the thermodynamic friction coefficient, = pJQ. The equations of motion then become... 

Other spectral densities correspond to memory effects in the generalized Langevin equation, which will be considered in section 5. It is the equivalence between the friction force and the influence of the oscillator bath that allows one to extend (2.21) to the quantum region there the friction coefficient rj and f t) are related by the fluctuation-dissipation theorem (FDT),... 

Where specialized fluctuation data are not available, estimates of horizontal spreading can be approximated from convential wind direction traces. A method suggested by Smith (2) and Singer and Smith (10) uses classificahon of the wind direction trace to determine the turbulence characteristics of the atmosphere, which are then used to infer the dispersion. Five turbulence classes are determined from inspection of the analog record of wind direction over a period of 1 h. These classes are defined in Table 19-1. The atmosphere is classified as A, B2, Bj, C, or D. At Brookhaven National Laboratory, where the system was devised, the most unstable category. A, occurs infrequently enough that insufficient information is available to estimate its dispersion parameters. For the other four classes, the equations, coefficients, and exponents for the dispersion parameters are given in Table 19-2, where the source to receptor distance x is in meters. 

The ratio to z depends only on (gag-, zjx, = 2/3 tga.g, and the ratio of x, /Xq has a constant value equal to 0.578. To clarify the trajectory equation of inclined jets for the cases of air supply through different types of nozzles and grills, a series of experiments were conducted. The trajectory coordinates were defined as the points where the mean values of the temperatures and velocities reached their maximum in the vertical cross-sections of the jet. It is important to mention that, in such experiments, one meets with a number of problems, such as deformation of temperature and velocity profiles and fluctuation of the air jet trajectory, which reduce the accuracy in the results. The mean value of the coefficient E obtained from experimental data (Fig. 7.25) is 0.47 0.06. Thus the trajectory of the nonisothermal jet supplied through different types of outlets can be calculated from... 

Lateral density fluctuations are mostly confined to the adsorbed water layer. The lateral density distributions are conveniently characterized by scatter plots of oxygen coordinates in the surface plane. Fig. 6 shows such scatter plots of water molecules in the first (left) and second layer (right) near the Hg(l 11) surface. Here, a dot is plotted at the oxygen atom position at intervals of 0.1 ps. In the first layer, the oxygen distribution clearly shows the structure of the substrate lattice. In the second layer, the distribution is almost isotropic. In the first layer, the oxygen motion is predominantly oscillatory rather than diffusive. The self-diffusion coefficient in the adsorbate layer is strongly reduced compared to the second or third layer [127]. The data in Fig. 6 are qualitatively similar to those obtained in the group of Berkowitz and coworkers [62,128-130]. These authors compared the structure near Pt(lOO) and Pt(lll) in detail and also noted that the motion of water in the first layer is oscillatory about equilibrium positions and thus characteristic of a solid phase, while the motion in the second layer has more... 

If the step is initially prepared to be straight, it relaxes to its fluctuating shape in the due course of time. This time evolution of step width depends on the relaxation kinetics, and can be used to determine the values of various kinetic coefficients [3,16-18,64-66], For example, if the attachment and detachment kinetics of adsorbed atoms at a step is rate limiting, the step width increases as [65]... 

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