Temperature distribution, Gaussian

With the assumption of a Gaussian temperature distribution (TEMoo-mode) on the laser affected zone, the temperature increase AT can be estimated by the following equation [63] ... 

If Restart is not checked then the velocities are randomly assigned in a way that leads to a Maxwell-Boltzmann distribution of velocities. That is, a random number generator assigns velocities according to a Gaussian probability distribution. The velocities are then scaled so that the total kinetic energy is exactly 12 kT where T is the specified starting temperature. After a short period of simulation the velocities evolve into a Maxwell-Boltzmann distribution. 

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

Figure 5.11. Decay time distribution for different temperatures recovered by fitting Gaussian lifetime distributions to the observed nonexponential decays of TB9ACN in PMMA.(26)x, 261 K , 243 K , 224 K +, 203 K O. 175 K , 146 K,...

For the case of D-RADP-20 we have assumed a Gaussian probability distribution of transition temperatures, and therefore used an error function to fit the transition region. For D-RADP-25 the transition temperatures range from 138 K down to 118 K. In this temperature range, it is not possible to separate... 

Range parameters, b (nm), for solvated electrons in various hydrocarbon solvents at room temperature, assuming a gaussian initial distribution of distances from the ionisation site to the thermalisation point... 

This assumes knowledge of the even derivatives of the DFCA which are obviously difficult to determine from the finite set of experimental points, without amplifying the experimental noise. Computer simulations, done for the bi-gaussian theoretical distributions, show that DFRJ provides a good approximation of the theoretical distribution, even for temperatures over the room temperature [32]. 

The longitudinal and transverse relaxation times for both samples 2 and 5 show in Figures 1 and 2 inverse temperature profiles typical of adsorbed systems for all the coverages except the 100% RH for sample 5. This implies a shallow Ti minimum, a drastic spreading out of the Ti curve, and may include a shoulder effect or maximum as T2 increases with temperature (20,21). Logarithmic Gaussian temperature-independent (B = constant) distributions have been used to model these systems and is discussed below. Because of the similarity of these profiles, the motional characteristics of the adsorbed water is probably similar for the different conditions listed in Figures 1 and 2. The data in Table II support this view since (except for sample 5 at 100% RH)... 

FIGURE 18.2 Steady-state temperature distributions in the x and y directions for an elliptical, CW, Gaussian irradiation of a moving, opaque material [18]. 

For the case of a Gaussian, pulsed beam, the solid s temperature distribution (for constant properties) may be estimated from Eq. 18.4, with Eq. 18.3 modified to describe the effects of pulsed beam irradiation. Specifically, Eq. 18.4 becomes [25, 26] ... 

Pulsed sources can be used to tailor the material s internal temperature distribution. Pulsing is typically used to sharpen spatial temperature gradients. Solutions to Eq. 18.9 involving a single pulse for Pe = 0, p = 1 have been obtained by a number of researchers, and consideration of the general case of pulsed irradiation of a moving material with elliptical Gaussian beams is presented by Sanders [27],... 

D. J. Sanders, Temperature Distributions Produced by Scanning Gaussian Laser Beams, Appl. Optics, 23, pp. 30-35,1984. 

We make a distinction between internal, or thermal, fluctuations and external flucma-tions, those due to variations in the environment of the system. Thermal flucmations are present in every system at nonzero temperature. For systems at equilibrium the absolute magnitude of the fluctuations, say in concentration of a chemical species, increases with the number N of molecules of that species in a given volume, and is proportional to The relative fluctuations therefore decrease proportional to At equilibrium the probability of a fluctuation of small magnitude is given by a Gaussian (normal) distribution. These relations lead to better understanding of the concept of a macroscopic system in which fluctuations are small compared to average quantities [1]. 

Fig. 8.62 Comparison of the simulated current-voltage characteristics for exponential and Gaussian trap distributions, as a log-log plot. The simulations were carried out for temperatures of 100 K, 200 K, and 300 K. The other parameters are ji. = 10 cm /Vs, Sf = A,d= 300 nm, and Nc = 2- 10 cm . The parameters for the trap distributions are given in the figure. From [38].

---