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We present techniques to calculate statistical distributions in time-dependent off-equilibrium atomic physics. Starting from the master equation, a first method assumes a Gaussian distribution and deduces the time evolution of the means and correlations. Alternatively, the discrete probability distribution is written as the product of the known statistical factor with the exponential of an unknown function. This function of the electronic populations can then be fitted using a second-order polynomial. Another method sets up a continuous version of the master equation, then expands the probability around the most probable configuration. It is remarkable that the obtained equation set is the same as in the Gaussian approximation of the first method. A major property of all these models is that they recover the probability distribution of thermodynamical equilibrium, when external conditions make it possible. Numerical tests on a two-level system are presented.

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The microscopic structure of a tilt and a twist boundary in germanium are explored using a state-of-the-art total energy calculation. The structure of the tilt boundary (Σ = 5 (310)) is found to be simple as it exhibits a well defined minimum energy structure, consistent with previous experimental and theoretical results on this and other tilt boundaries. The structure of the twist boundary (Σ = 5 (100)), however, is found to be very complex. The boundary bonds are distorted and weak, and their weakness makes the twist boundary exhibit a wealth of local energy minima. The different types of energy minima for the twist boundaries are identified and studied in some detail. We summarize the consequence of the orientation of the two crystal grains upon the microscopic structure of the boundaries, and speculate about future applications.

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