Abstract
We consider partial distinguishability of micro particles in terms of averaging the cell probabilities by Dirichlet prior on k dimensional unit simplex with an added prior perturbation, where k is the number of states. The perturbation in uniform prior is such that the added term becomes negligible over progress in time; as the particles decay to a lower mass eventually. We compute Shannon’s measure of entropy for the ensemble of micro particles over time that converges to Shannon’s entropy of Bose-Einstein statistics for indistinguishable particles. Remainder in the expression of ensemble entropy of particles in intermediate state, from Shannon’s entropy of particles following Bose-Einstein (BE) statistics is examined to assess the evolution of the modeled system towards indistinguishability from partial indistinguishability. The rate of such convergence is seen to be polynomially decaying in terms of a controlling parameter in prior perturbation and the number of states k.
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Notes
- 1.
In general \(\alpha \) need not be an integer. The expression (11)) is of nice form for integer \(\alpha .\)
- 2.
An exciton is a bound state of an electron and an electron hole which are attracted to each other by the electrostatic Coulomb force. This occurs when an electron is displaced from its position leaving a positively charged ‘hole’. When a molecule absorbs a quantum of energy that corresponds to a transition from one molecular orbital to another molecular orbital, the resulting electronic excited state is also an exciton. This was proposed by Frenkel (1931), when he described the excitation of atoms in a lattice of insulators, and postulated that the excited state would be able to travel in a particle-like fashion through the lattice without the net transfer of charge. Molecular excitons are not stable, typically have characteristic lifetimes of small order, on the order of nanoseconds, after which the ground electronic state is restored and the molecule undergoes photon or phonon emission.
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Dasgupta, R. (2017). Growth Model for Micro-Particles Towards Indistinguishability and Dirichlet Prior. In: Dasgupta, R. (eds) Growth Curve Models and Applications. GCM 2016. Springer Proceedings in Mathematics & Statistics, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-63886-7_4
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DOI: https://doi.org/10.1007/978-3-319-63886-7_4
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