# Definition of noise temperatures θ in three dimensions and the conjecture θ ≥ T in nonequilibrium systems

## Abstract

A system of electrons in a strong electric field obeying a linear transport equation is considered. Various possibilities to define a second-rank noise-temperature tensor θαβ by means of the velocity fluctuation spectrum Sαβ and the velocity-field response function μαβ are proposed. The noise-temperature conjecture θ ≥ T can obviously be generalized in three different forms to three dimensions: 1) weak form θ∥F (parallel to field) ≥ T, 2) intermediate form Sp θ ≥ 3T, 3) strong form θsym - T1 positive definite. They are supposed to hold for all electric fields F independently of band structure and collision operator (scattering mechanism). In the constant relaxation-time model of the collision operator forms 1) and 2) are shown to hold at least for small fields (warm electron), whereas for case 3) it is easy to find counter examples which are isotropic and therefore independent of the θ-tensor definition chosen. Noise temperatures are given for two examples with nontrivial collision operators: the LKDPYW model with deformation potential scattering and n-type GaAs using the data of Monte-Carlo calculations. © 1973.