A microscopic critical-state model of a hard superconductor is presented, in which the state of the system is described using the coordinates of individual vortices. After several model assumptions regarding the character of the pinning are introduced, the microscopic problem of the penetration of a field into a hard superconductor allows an exact solution. The interactions of the vortices with one another, as well as with their images, Meissner currents, and pinning centers, are considered. The existence of the Bean-Livingston barrier is taken into account. The coordinates of the vortices are calculated in both increasing and nonmonotonically varying external fields. The stability of the solution against small displacements of the vortices from their equilibrium positions is investigated. Strong pinning is investigated in detail. In this case, after going over to a macroscopic description in sufficiently strong fields, the model turns out to be completely equivalent to the phenomenological description within the macroscopic nonlocal critical-state model. A new effect, which is lost in the macroscopic description, has been detected in weak fields, viz., the formation of a macroscopic vortex-free region in a decreasing external magnetic field.
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