Collective flux creep: beyond the logarithmic solution

Numerical studies of the flux creep in superconductors show that the distribution of the magnetic field at any stage of the creep process can be well described by the condition of spatial constancy of the activation energy $U$ independently on the particular dependence of $U$ on the field B and current $j$. This results from a self-organization of the creep process in the undercritical state $j<j_{c}$ related to a strong non-linearity of the flux motion. Using the spatial constancy of $U$, one can find the field profiles $B(x)$, formulate a semi-analytical approach to the creep problem and generalize the logarithmic solution for flux creep, obtained for $U=U(j)$, to the case of essential dependence of $U$ on $B$. This approach is useful for the analysis of dynamic formation of an anomalous magnetization curve (''fishtail''). We analyze the quality of the logarithmic and generalized logarithmic approximations and show that the latter predicts a maximum in the creep rate at short times, which has been observed experimentally. The vortex annihilation lines (or the sample edge for the case of remanent state relaxation), where B=0, cause instabilities (flux-flow regions) and modify or even destroy the self-organization of flux creep in the whole sample.