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Using the tools of random matrix theory we develop a statistical analysis ofthe transport properties of thermoelectric low-dimensional systems made of twoelectron reservoirs set at different temperatures and chemical potentials, andconnected through a low-density-of-states two-level quantum dot that acts as aconducting chaotic cavity. Our exact treatment of the chaotic behavior in suchdevices lies on the scattering matrix formalism and yields analyticalexpressions for the joint probability distribution functions of the Seebeckcoefficient and the transmission profile, as well as the marginaldistributions, at arbitrary Fermi energy. The scattering matrices belong tocircular ensembles which we sample to numerically compute the transmissionfunction, the Seebeck coefficient, and their relationship. The exact transportcoefficients probability distributions are found to be highly non-Gaussian forsmall numbers of conduction modes, and the analytical and numerical results arein excellent agreement. The system performance is also studied, and we findthat the optimum performance is obtained for half-transparent quantum dots;further, this optimum may be enhanced for systems with few conduction modes.

Statistical analysis of the figure of merit of a two-level thermoelectric system: a random matrix approach