Thermal randomization events

The log-normal function is unique and does not deserve modification. It occupies a unique position in both botany and biology that is critically related to the processes involved in growth. There are four major classes of statistics of interest in vision. They are the normal, the log-normal, the Stefan-Boltzmann and the Fermi-Dirac statistics. The first is often spoken of as Gaussian Statistics. It relies on a totally random series of outcomes in a linear numerical space. Log-normal statistics rely on a totally random series of outcomes in a logarithmic space. This space is the logarithm of the linear space of Gaussian Statistics. The Stefan-Boltzmann class of statistics apply directly to totally random events constrained in their total energy. They explain the thermal radiation from a physical body. The Fermi-Dirac Statistics are also known as quantum-mechanical statistics. Fermi-Dirac Statistics represent totally random events constrained as to the amplitude of a specific outcome. While Fermi-... 

The application of an electric field E to a conducting material results in an average velocity v of free charge carriers parallel to the field superimposed on their random thermal motion. The motion of charge carriers is retarded by scattering events, for example with acoustic phonons or ionized impurities. From the mean time t between such events, the effective mass m of the relevant charge carrier and the elementary charge e, the velocity v can be calculated ... 

The random thermal motions of each B-particle will occasionally cause a vapor-phase molecule to be captured in the liquid phase (heavy arrow), or a liquid-phase molecule to escape into the vapor phase (light arrow). In the limit that solute molecules are so dilute that solute-solute interactions can be neglected, the probability of each such capture/escape event is simply proportional to the number density (or mole fraction) of solute particles in the originating phase. We therefore expect that... 

As described above, the electrons in a semiconductor can be described classically with an effective mass, which is usually less than the free electron mass. When no gradients in temperature, potential, concentration, and so on are present, the conduction electrons will move in random directions in the crystal. The average time that an electron travels between scattering events is the mean free time, Tm. Carrier scattering can arise from the collisions with the crystal lattice, impurities, or other electrons. However, during this random walk, the thermal motion is completely random, and these scattering processes will therefore produce no net motion of charge carriers on a macroscopic scale. 

Classical trajectory calculations for the reaction H2 + I2 HI + HI and its reverse have been carried out for two potential energy surfaces. Such calculations are not easy to perform because of the large number of possible states of reactant species, the mismatch of the masses of H and I atoms, and the low probability of reaction. The light H atoms require small time steps to avoid time-step error and the heavy I atoms require a long time for movement. The probability of reaction of two HI molecules, randomly selected from pairs at 700 K with sufficient total energy to react, is about 1 in 10 [15]. In a thermal system collisions of H2 with I2 (or 21) which result in reaction to form HI + HI are indeed rare events. 

In Chapters 4, 5 and 6, we shall be considering reactions initiated externally either by electromagnetic radiation (Chapters 4 and 6) or by pulses of high-velocity electrons (Chapter 5). Initiation is thus a very fast event. In consequence, the reacting system is initially not in thermal equilbrium, and the spatial distribution of reacting molecules is not random. The primary products — excited molecules or radicals — themselves undergo further reactions, which can often be treated in terms of diffusion. All these techniques are well suited to the study of ultrafast reactions. 

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