l- This Week’s Citation Classic @ JAEEEL

ZaamigREnsemblemethodinthetheoryofineversibility. J. chem Phvs. 33:133841,1960.

AnefMentmethodwasintmducedformakingthe tfansMnlmmexactdynamlcalequationsofmotlon to irreversible tranqtt equations. The method was based on use of projection operators to separate ‘releva& from -irreievacr vafiebles. [The SC/@ indicates that this paper hes been cited in more than 575 pubiiitions.]

Robsrtzwsnxlg Lsborstory at chsnlical Physics

NIDDK Nstlonal InsMutes of Health

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Inthe19X&theprlmsrytoolsofnon- equlllbrium statfstlcal mechanics wsre the Golkmann equation (low denslty gsseo), the Paullmasterequatlon(weaklycoupledquan- turn systems), and the Langevln equatlon for Brown&n motion (particles in fluids). It was recognized that these equations were cor- rect only In llmlted clrcumstancea, and a lti of effort went Into attempts to extend thelr valldlty. Further, they areall dynamically lrre verslble equations, and their Mlatlon oftlme revemal symmetry led to a general unease about their consistency wlth exact dynaml- cal equations of motion.

Two attacks on this question were due to I. Prlgoglnel and L. van Hove.* lhelr ap preach was a fashionable one at the tlmr+ useoflnRniteorderparturtwtionexpansion8, as In quantum field theory. The re8ultlng inflnite series were rearranged, artfully cho- wn classes of terms were dropped, and what was left was Mummed to arrive at the de- sired result. Prlgoglne vlslted the National Bureau of Standards In 1999 to give a series of lectures on his dlagrammetlc version of the petturbatlon expansion method. Many experts In statistical mechanics illtended his lectures; some were skeptical about his method, perhaps because of its complexity.

Idecldedtobytounderstandanappar- sntly simpler approach, van Hove’s derlv& tlon of the Paul1 master equation. On study- lnghlswork,ltoccurredtomethatbothvan Hove’s and Prlgoglne’s methodscouMbe summarlxed as the partftlonlng of a dynami- calpmblemklto mlevantandIrmkwantparts, followedbyellmlnatfonofthelrrelevantparts. Ilmmedlatelysawthatthlscouldbedoneln a qulte general way by intrcduclng projec- tlonoperatorstoaccountforthepartltkMng. lwasthenabletoarrlveatthemainresultsof Prlgoglna and van Hove In a remarkably simple and direct way, wlthout dolng inftnlte order expansions and -mmatlons. Fur- ther, my approech could clearty be general- ized in many dlrectlona, for example, to In- elude non-Markovlan or “memory” effec& flnally,Icouldseethattherewasnol- Bistwcy between tha requhements of time reversal symmetry and the actual behavlor ti many body systems.

Whenlwasinvltedtosubmltapapertoa Bpeclal issue of the Journa/ of ChemkiM Physics (JCP), honorlng the memory of my mentor, the late John G. Klrkwood, I deckfed to write a short summary of my Ideas about projection operatom. An extended version of the paper was publlshadg as my 1999 ‘Boul- der lectwes.” That paper has received even more citations than the JCP paper.

R took many years for this work to gain ~ubstantlal recognition. An early appllcatlon by H. MOI? led to what is now commonly referred to as the Zwanxig-Morl formalism. Physical chemlets in pertlcular became In- terested in my work beceuse lt provided a conceptual framework for dealing with the kinds of complex sltuatlons that come up so often in thelr world. A monograph, In 19fR5 and a volume of Advances in Chemical Phys- its, in 1999,E were devoted entirely to appll- cations of the projection operator method. And, a recent application In solid-state phys- its was discussed by Lax et al.’

1. Rlgogk I. Nonequilibrium sraristicd mechadcs. Nm York: Intenciencc. I%2. (Cited 700 tims.) 2. van Hove L Quantum mechanical pemubations giving rise to a statistical tnutsport equation. Physica 21:5174 i9S5.

(Cited 510 times.) 3. Zwmwig R Stadstical mchanics of irrcvchbility. Lecnues in theonficdphysicr (Bodder). New YorL: Interwicocc. 1961.

(Cited 665 times.) 4. Mod Ii. Tran~pon collstive motion and Blotian m&m. Prog. Theor. Phys. SuppL 33:423-55.1%5. (Cited 1.755 limes.) 5. Gnbert Il. Projection opxatw tcdmiqvcs in nonequilibrium systems. Springer tracts in modem physics, vdnme 95.

Berlin, Germany: Springs. 1982. 6. &mm M W, Grigolhi P & ~PanwIchi G, cds. Memory function approaches to stcchastic problems in condensed

matter. (whole volume.) Advm. Chcn Physics 62.1985.

7.LuM,CaiW,HuP,ZbengTF& MS&& M C. Coupling bcovcen 2-D elecoons in quantum wells and 3-D phonon~. AM. MAcod. Sci. 581:195-206. IWO.

Received August 2. 1991

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