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An analytical description of a Cooper-pair box coupled to a dissipative cavity in a thermal bath is derived. The Wehrl entropy and the Wehrl density are used to investigate the dynamics of the mixedness and the information loss of the phase space for the state of the Cooper-pair box. We show that the Wehrl entropy and density are very sensitive not only to the damping rate and the number of thermal photons but also to the intensity of the coherent state and the qubit distribution angle of the initial state. The information of the phase space and the coherence are quickly lost due to the thermal photons.