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We investigate theoretically the energy exchange between electrons of twoco-propagating, out-of-equilibrium edge states with opposite spin polarizationin the integer quantum Hall regime. A quantum dot tunnel-coupled to one of theedge states locally injects electrons at high energy. Thereby a narrow peak inthe energy distribution is created at high energy above the Fermi level. Asecond downstream quantum dot performs an energy resolved measurement of theelectronic distribution function. By varying the distance between the two dots,we are able to follow every step of the energy exchange and relaxation betweenthe edge states - even analytically under certain conditions. In the absence oftranslational invariance along the edge, e.g. due to the presence of disorder,energy can be exchanged by non-momentum conserving two-particle collisions. Forweakly broken translational invariance, we show that the relaxation isdescribed by coupled Fokker-Planck equations. From these we find thatrelaxation of the injected electrons can be understood statistically as ageneralized drift-diffusion process in energy space for which we determine thedrift-velocity and the dynamical diffusion parameter. Finally, we provide aphysically appealing picture in terms of individual edge state heating as aresult of the relaxation of the injected electrons.

Statistical theory of relaxation of high energy electrons in quantum Hall edge states