Phtst~a 85A (1976) 1-17 ~, Xorth-Holland Pubhshmg Co ON THE STRUCTURE OF THE GENERALIZED MASTER EQUATION FOR OPEN SYSTEMS L A LUGIATO and M MILANI l~tttuto dt Ft~t~a dell'&'m~ e~stta, lta Celo~ta 16, 20133 ~ltlano, ltaha ReceJ~,ed 19 Januau, 1976 A quantum open s~stem S, interacting s~th a re~erso~r R ~ considered The kernel of the generahzed master equauons (G M E st, usualb deduced to describe the d.~nam~cs ol S, ~s ex- pressed m term,, of ume correlauon funcuon~ ol the bath operator~ msoked m the interaction betv, een S and R, calculated on a statable equdJbnum state of R Such eqmhbrmm state enters" into the kernel ~n a rather a~ uficml ssa~, e g s m the projectmn operator used to deduce the G M E In thl+ paper the +tructure of the kernel of the G M E for open ~+tem+ ~+jusufied lrom first principles, ~ho++mg that the presence ot the eqmhbrmn+ state ol R m the kernel an+es m qmte a natural ma} The thermod,, nam~c I.mt of R pla}s an essentml role m connecuon x~ ~th this result 1. Introduction Let us constder an open quantum s.~stem S, mtetactmg xsJth a re~erso~r R One ha~ to handle the problem of deducing the d.~nam]cs of S from the d3namlcs of the full s~,stem S + R To this aim a commonly used method ts the projection techntque ~) ~lth the projection of ~.rg3 res-Kelley 2) Recently one of us 3) has proposed a nes~ method, based on a suitable h]erarch3, which has the ad~,antage of mtroducmg no per- turbatton expansion Jn the coupling constant bet~xeen S and R Thus this latter method alio~xs us to treat also cases m ~hlch a xseak couphng approximation does not hold Both the method of ref 2 and that of ref 3 are approxmlatmn schemes, each ~tep of sshtch leads to a generalized master equation (G M E I for the statistical operator of S The kernel of such G M E Js expressed m terms of time correlation functtons of the bath operators tnsol~ed m the interaction between S and R, cal- culated on art eqmhbrtttm state (canonical or grandcanontcal) of R The presence of the eqmhbrJum state m the kernel arises m a rather artificial ssa3 from purely technical featutes ut the method of ref 2 tt comes from the mtroductton of the eqmhbrtum state into the projection operator sshde Jn the method of ref 3, it