Finite orbit width effects in large aspect ratio stellarators

New orbit averaged equations for low collisionality neoclassical fluxes in large aspect ratio stellarators with mirror ratios close to unity are derived. The equations retain finite orbit width effects by employing the second adiabatic invariant $J$ as a velocity space coordinate and they have been implemented in the orbit-averaged neoclassical code KNOSOS. The equations are used to study the $1/\nu$ regime and the lower collisionality regimes. For generic large aspect ratio stellarators with mirror ratios close to unity, as the collision frequency decreases, the $1/\nu$ regime transitions directly into the $\nu$ regime, without passing through a $\sqrt{\nu}$ regime. An explicit formula for the neoclassical fluxes in the $\nu$ regime is obtained. The formula includes the effect of particles that transition between different types of wells. While these transitions produce stochastic scattering independent of the value of the collision frequency in velocity space, the diffusion in real space remains proportional to the collision frequency. The $\sqrt{\nu}$ regime is only recovered in large aspect ratio stellarators close to omnigeneity: large aspect ratio stellarators with large mirror ratios and optimized large aspect ratio stellarators with mirror ratios close to unity. Neoclassical transport in large aspect ratio stellarators with large mirror ratios can be calculated with the orbit-averaged equations derived by \cite{calvo17}. In these stellarators, the $\sqrt{\nu}$ regime exists in the collisionality interval $\epsilon |\ln \epsilon| \ll \nu_{ii} R a/\rho_i v_{ti} \ll 1/\epsilon$. In optimized large aspect ratio stellarators with mirror ratios close to unity, the $\sqrt{\nu}$ regime occurs in an interval of collisionality that depends on the deviation from omnigeneity $\delta$: $\delta^2 |\ln \delta | \ll \nu_{ii} R a/\rho_i v_{ti} \ll 1$.