Interrelation of electron diffusion tensor among position, velocity, and phase spaces in gases

Abstract The stochastic diffusion tensor of electrons D R defined by random walk theory in position space is investigated through the comparison with the expressions in velocity and phase spaces. We employ the Boltzmann equation of electrons with a small density gradient under a cold gas approximation. Our analysis confirms that the statistical expression of the electron diffusion tensor in position space coincides with that in phase space under a first-order approximation that includes the first two terms in the number density expansion, i.e. the number density n ( r , t ) and its gradient ∇ r n . In order to better understand the diffusion phenomena, we further decompose D R into physical stochastic components: the diffusion tensor in velocity space D V , and terms including the drift velocity in velocity space V dV , external field E , and non-conservative collisions J n − con , among others. It is noteworthy that the present argument is valid irrespective of the degree of anisotropy in the electron velocity distribution.