Inevitable irreversibility in a quantum system consisting of many   non-interacting ``small'' pieces
  We review the recent result of Sato, Sekimoto, Hondou, and Takagi (cond-mat/0008393) on the irreversibility inevitably observed in systems consisting of many non-interacting ``small'' pieces. We focus on quantum models, and supply an explicit lower bound for the work required to complete a cyclic process. 
Hal Tasaki