Boltzmann equation for fluctuation Cooper pairs in Lawrence-Doniach   model. Possible out-of-plane negative differential conductivity
  The differential conductivity for the out-of-plane transport in layered cuprates is calculated for Lawrence-Doniach model in the framework of time-dependent Ginzburg-Landau (TDGL) theory. The TDGL equation for the superconducting order parameter is solved in the presence of Langevin external noise, describing the birth of fluctuation Cooper pairs. The TDGL correlator of the superconducting order parameter is calculated in momentum representation and it is shown that the so defined number of particles obeys the Boltzmann equation. The fluctuation conductivity is given by an integral over the Josephson phase \theta of the particles distribution, depending on that phase, n(\theta) and their velocity v(\theta). It is demonstrated that in case of overcooling under T_c the transition from normal to superconducting phase while reducing the external electric field runs via annulation of the differential conductivity. The presented results can be used for analysis and experimental data processing of measurements of differential conductivity and fluctuation current in strong electric and magnetic fields. The possible usage of a negative differential conductivity (NDC) for generation of THz oscillations is shortly discussed. 
Todor M. Mishonov and Yana G. Maneva