Gaussian distribution energy theory

The theory of thermal aspects of laser desorption has been developed for a substrate surface subjected to pulsed laser irradiation, assuming that the laser intensity has a Gaussian distribution [21], The given surface is covered with the organic layer, which does not absorb the laser energy. However, the heat flux in the substrate that absorbs the energy heats the sample to the same temperature as the substrate. For this case, the laser intensity flux /(r,t) is given by... 

The essential term is the exponential function of (here a gaussian distribution function of the empty energy states in the redox system), which is mathematically the same as in the original Marcus theory. Accordingly, the rate equation is almost identical for both theories although the basic models are conceptually different. The reason is that a harmonic oscillator type of fluctuation of the solvent molecules is assumed in both cases. 

Van Weperen et al. [1977] noted that in fluorite-structure materials the dielectric and ionic thermocurrent peaks broadened with increasing concentration of dipoles (for example Johnson et al. [1969]) and developed a theory of dipole-dipole interactions which predicted an almost Gaussian distribution of activation energies ... 

The Fisher information, reminiscent of von Weizsacker s [70] inhomogeneity correction to electronic kinetic energy in the Thomas-Fermi theory, charactoizes the compactness of the probability density. For example, the Fisher information in normal distribution measures the inverse of its variance, called invariance, while the complementary Shannon entropy is proportional to the logarithm of variance, thus monotonically increasing with the spread of Gaussian distribution. Therefore, Shannon entropy and intrinsic accuracy describe complementary facets of the probability density the former reflects the distribution s ( spread ( disorder , a measure of uncertainty), while the latter measures its narrowness ( order ). 

For events of a short duration compared with collisions times, the cumulant expansions have led to expressions showing that energy transfers into slow degrees of freedom are described by Gaussian distributions, with their parameters easily calculated from the theory. This was particularly useful in the description of rotational excitation, for example in the collisions of Li with CO and N, and of H with CO and CO,. 

These two models, however, are insufficient to handle disordered organic materials, where the DOS is a Gaussian distribution, with localized carriers and discrete hopping within a distribution of energy states. Arkhipov presented an analytical model based on hopping theory [31] and Wolf preformed detailed Monte-Carlo simulations of hopping injection from a metal into an organic semiconductor [32]. Here we will present two injection models one is based on drift-diffusion theory and the other on the master equation. 

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