Shuhov, A. G.

Suhov, Yu. M.

English

DIAS Access to Institutional Repository

DIAS-STP-9Q-Q5Towards Time - Dynamicsfor Bosonic Systems in Quantum Statistical Mechanics, 2A.G.Shuhovl)and Yu..Suhov1’2)Abstract Tnis is a continuation of a study of tiae—evolutonof a state of an infinitely—extended one—dimensional boson latticeaysteni, which was initiated in the oaper under the sane titleWe consider here finite range, pair interaction potentials andprove txiat the family of the time—evolved states provides asolution to the corresponding Iaouville equation and BBGKY hierarchy.Key wor.s and phrases: one—dimensional lattice boson system,diagonal state, finite—range interaction potential, time—evolutiorLiouville’s equation, BEGKY hierarchynstitute for Problems of Information Transmission, USSR Acadeof Sciences, 19 Yermolova Str., GSP — 4 Moscow, 101447 USSR2Dublin Institute for Advanced Studies, 10 Eurlinton Road, Duol:4 Ireland-z -.This paper is a continuation of 11 . 7e studied in the paper1 3 the non—equilibrium time—evolution problem for a ?1ocallyunboundedH phase space quantum model where the hltraditionaPt aproac1C2) based on a limiting * — auomorphism group of an underlyingC*_algebra does not work. The model was on the one—dimensional lat‘1tice and was given by its formal Hamiltonian± (4.1)3were . and are, respecti—OJvely, th’reation, annihilation and particle number operators inUthe Hubert space which is realized as t, ( + ) with4the standard orthonormal basis e € Z Letdenote the C*_algebra of bounded operators in andbe the corresponding quqsilocal C*_algebra,4de have constructed in 1 3 a family ‘ ‘f. , ± € [. ofstates of which describes the time-evolution of en initialstate—The states e given byL‘÷ (A), 4€ (1,2)where( ) (1.3and is the Hamiltonian (1,1) confined to the finite volut\ C (bounded interval on the lattice).In the paper t 1 3 we assumed that the (locally normal) initial state is diagonal. This condition means that the densitILaurix of the state in any bounded interval A C 2—‘-S(A5)is Jia nl in es casis € i tneHere x is ri arii:rary occups—A EA- i A , i.e ., a f ction A >çss n C I 3 v snal uce ie notation ‘x A). Who Jia3ons1iici a s;ae ‘:f necns in hysica1 terms tflEi it ciescriesclissicaii5 particle syste on ne latuice. Of course, this property is desroed by ne thas—evolutiofl, but it nakes possibleto foruuiate fartnor 5ssu:L1tiofls on the initial state in naturprobabilistic terms. ‘or example, the basic assumption that thestate has lon space “ho1es’ which are free of artic1esis related to orel — Cante lii type assertions (see 1 3).in the situation under consideration ve are able to provethat t staes -b given by (1,2), provide asolution to tne iiffinie—volwne Louville equationci—r- (t4) —o4\A L(1.4(Pu’fl‘ i— L h / -,, ‘€Zfri(MquctiOn (1.4) is on a family { of density matrices4in finite volumes A C Z, vjnon obey the standard consistency condiion ti , A D A The term—E b in he .?JiS of (1.4) correspondS ;o the kine5,tic enery: mere A4 is v— i, V + 4 for Afor a enoral case is defined s54 } ha:ielj,Q -(1.5)(ihe notion is used to stress tnet no bounry con—involved Ln the HS of (.5)), ihe partial trace, A C is taken in in the cSe AØ(whion occurs vinen { , j C t\ in the second—teru sumin the HS of’ (1.4) no trace is taken.quivalentiy, equation (1.4) ay be rewritten iii the folio’d . — (1.6)JHere and below4{€7Ls }, fltheprcise for—rnulation of the result is given in the next section.nother ooect related to the Lne—evo1utiQn in statistic—1 LilC oh ariics i a the BEGKY hie rarchy:(1.7)his is a systeLa of equations for a sequence { ,of operators where(4) )is tne snetrizad tensor product of copies o the uncier—(1) fl i1in Hubert space 7L i is set to oetne one—d ensiLia] coipier space) . Dhereby the riatrx e1e.1entOf e 1bel’ed b: pairs (, jx.’) of occupationuuoer oiguratiOfls , C withrr‘‘ r -I.U) 0 C,) 0 ctC HI-;,C)-L’:oC,Lb()C)oCDCDc-i-3HccCtDC>)CD C)U) ctU)ci CCci‘t;Z5CD Cl,U) CDC)C)HC ci-CD CD CI) C ct 4- I—J.U)OiU) ct0) ci-CDCH SII‘I)—Sr)Ict—i-j—ciF-.??cit’(1.L.•I—,U)0)k—aCL,LbH0’HCI)ii)c1ç[—>ct0LI)C)tzq-I—Cl)‘C-CD Fr>)I—C)—ci-)- ctCt-F--CDCHLbHCDC CDCU.F- C.) C) 4—) Ftci- H 0QC) ci-()U)0H-CcF-LIH.H-%COc-ir: tx,,,Cc0.cC)___Y()CDcH-iiCDa.C’)ci-ci0C)Q.“ChiHFl—Lbci•C•i—_,F-(_)F-’C)‘-.--‘I—•CDC>cCi)bFI-’-Cr’)‘_k-- :C>’4--4--i I-, I- CLC)z-..‘—S+mCH H) 0 C) C)“C 0) c-IF 0 4—)C LI) HIi‘I 0 0 ci ‘I) CD CD‘1 NI.4-ciCDCD ci(rCD CD Cr)“C CD H c-IH-C) CD C) 0 Fto.1I-I-”-HCi-zCDCY ccH-“C,-.0CI-’F-CDH H U)0’1—S(‘,,—‘--:3C, CD F— r-1 CL>I-.I U) C C) I— 0 [-CC-.c-I SC-.+%——ci’‘IC)e CX—0I( ( ,J(call Xt cotS Il tefore it:flctrlX aJ ents ee labeled o’ eirS (, ‘)\- \‘j-= k-*1. Lika7Je, UcD is aa oacor init(u), - 2if oocapetDia n’aaber cn at ions x end‘coide,( 0 ,In OL1 situation e prove that a solutioll to (1,7) isyen by the fi1y oi operators withx- () (fl c fl ftaz1, , -0± courSe, tne existence of tne CXpCO;btlOflS in theiS of (1shoilJ be ‘ aranteed. For the precise fornu1etiDnof tns re—Jt, see Section 2’No;ice that e do not stt t ; our solaionS tothe Lovll and B( utinS are aniue: one rast ce aul fu:tnto constract ie the ovolution for a much lrier fauilyofLlt1Ol StS 3., ih Liouv±.l.L3 JOIliheoroLa I. Sppoe tSE . Jual stte f hs proper—and Lroin [ I ] . ihaI, for &ny oou.nded in0 A 0orv1 ooor looallzed n A(A €. tre functj°fl - J’ ( i s:iJJtn ad thefu1lowiii equation teS LcO(2,1)wnere the RHS of the equality is defined as the limit—(fl H n ) (2.2)s is the ortboonal projection in onto the subspacenerated by the occupation number coflfigLwatiOflSwith x1 < S.quation(Z.j)naY be considered s a v:ak orm of in—finit—vO1urae Liouville equation. From this point of view, afamily of states E. R, is called souetimeS a weaksolution of tne Liouville equation.Proof of Jheorem 1. t he fist step we shall checc equa—(tion (2.1) for ALemi .i. ive bounded D the coller3nco(t E] )is a ifor Zo /‘\ D Li conpacts, snthe1”V.f\4(1)C)ct>0it>ci()t3ri”,— l— ç:C)rt”C ;CD “2 ctC) ct C-) CI CD CD ci C ciCD ctCD I-:1>C) 1-2 C’ ci‘4 (p÷0LI)%IC) C F--’C-.co1olC’>>.1’H’(I) ‘-I)I— C) CD U) ct C P C) ci 0) ci-C>C) I— Ii Cl) U) 1-U.H)0 J-Jci 0 C) 0 CU 0 I— I—’ 1:;r1 1% Cr•.0I>0‘S >0•ro> L-J1’0In4. 1’1>C)‘4C F-b1%>I—,.(PIQq:izi>01’ ‘50Gf 1’tlh.1’ii(.,____•%.‘.5“0rnASd112(1’>(--‘—F—L-JIiI-L_._JH2.C1Cji—.—1-304.itI—-’>I2:‘.0‘SH)rnC-:)> QcCD F) Cl) I-- C) CD CD 0 I—”H)0 H) 0 H’ C) 0 13 C) ciI---ci‘-“ci’cit%>(I)r1’1*I7.”irn•C) CD C9 0’)0Cl)I-.Q,f0ci:p 0D÷fJHCDC)k- 4(1)0c_i0FzCD0’tCcrCDI--C)U-C)to0H-ctI—cI’--:,-I-”<CtU)eo-—H:z;ciHSD 00,SDctCDH,H,0“aIIH)00I-y’Ijd‘:(I)CDH(1’Cl))Dcib’hc<J-CD()0)C1)C-’ctH.c-f-cttD:oCL)H,CDCDrn H’0SDEC’ci‘JH,I—-’HHCD(ct-H)0-f(r10fuH’0)‘-‘c-i-•H’l—J00.C)UCrCDcic)JC)SD-’H)U)2ct.:J CD U) CD CD“-ii 0’)IUC)_,0cCDC-’ H)SD.H)I__sH), Lb•ocx,i- CDH’Cl ctH)CDC) SDto SD CD 0 (IS 0 H)C) 0OsU)H o0 0”(I’.CtCDcic-f-I—’-CDDCDCDciSD-’Li)iI’IN>-c?r+ci-•H’ctIS H 0’ Cl)cttI) 0 F H’ Cl)rnCTU,oriU,,4,0—‘%‘7+CD I-s-Cl) çNini——”,)CL)(I’C,—‘a(PL-t1C‘a—‘cN(J)LCLSI—‘—““T‘-S—,—I—I‘—It’__rf3__J___•‘%1‘‘.‘rU%,0‘—(15+0U55—1’V’U—a=%_.__I5—>trI.>.5t,0-_—t 5_LW..>I-,rt-4-*0 Cl) H’Ci-H <5 C’) 0 0 U)0 CD‘a-ci CD1’’Co 0” H LI)S--atI1—’55.5C.;) CD SD CD Cl) C) CD SDit’I55 (P (js I-’4-0—5%(s3to00-“H0rtC)>5HCD—cTr-1C)>[jcjCH-H1>>o0CtCDC)Cti-OfJ,(0CtH-CDCl)Ci--U)F(.h>ct,CYc-I-“.c’H(_(LiCDH)F-1’1Hc1C)>JU)0)HCDs—’—ciiH)H)--rflu’H.)10.:-U)ICDCH-C])CDCL)Cl).NHf\iCl)CDHICjFwH1H-‘C)HCt>.oIr\)C)—0o0F:Dl\)CDfriCDrr1jçI(‘0_CDe-CI)0r(Ycc-i-DcciCl)F-tCD—L_—JCi)CD ciCDCl)C—’Ct1Z)1C)U)H0Ct•r0H-ctI--iii0h-0H)Ci)Lbo0Hct3HCD>IEc-h(CH)1F-C)CDHCi)H)H00CDFH1ti H C) 0 H) 9 FL H C:, CD U, H 9 H 0>1:1+OLDICi) Ci) CtFJt:5-HCDoCtE-.CD,H-0rtYOrIiciCD.—C00raYI>C)CDJ)0c-I-IU)H”,-,-HEHI—i-€__—.HC:0‘dlf.1-s(I)LP4. .rLb Cf- :‘-H)400IH*—r4I— 0fT9H •cCD CD CDH H C>0Ci+ IC’: c-hEl-,,___‘—C H H0 CD H H N CD IiC) 0 CD H (I) C) CDc÷H-Cl)U)Ii 0 CD0 H CDwHF-,HI—,— II —he prooi. of Le.a l. is riven j a oooination o aru.ients fro:a the pruJS OJ L3iJs 1.1 and oiinia 1.. Here one usesn:e folioinb boundI AILAU [ nTne details are omitted. D‘rom Lenma 1. one obtans+l( O÷r A ) ‘ (2.1for amy localized in A he last remark isthat (2.l) is equivalent to (2,1). Lheorem I is proven.D1. ssuLne th&t a diaonai state ‘ satisfiestne conditions of Theorem 1. t be t densitymatrix of the state in a volume A . ie family of() •1 *Lnair1ces , ± € provides a soluion to the infiniue—volume Liouville equation written in the operator form(ihe derivative in tne LHS is understood in the wea operatortopology..quivaient1y, this equation maj ce rewrt en in the form4-’•cic’JCl)ClCu•j)Vci)C)S-‘‘-I-’>r—14--1-’4Cu00-f’iHJ4-)HCl‘U,HClU,L3f)0r-4C)C!)Cici+Hr4i0-pIIzz•—1,.orCl42j70‘COH0IC)CICI)COIs—’L_-J‘C’>-i-’m0Cif—fitr-14)Ci1J.L(i)-pt,s:IrCl-m‘Ci+JCuCI)‘°•ci>o>C\I(>-l(\jr—LJJ(0C)]Hs—’—_Cicj-I-’Cl—Ci4C!)C)-iHICI)-ct,0d4-’C”0r4C’)0(I)iv0L)ciciItoaciCiCiCu(JI<C)°LJJccir-______1%-__-,--0H42<(Y)•CIC’)1.C-Iz1•C)]\jCC).4.(1)HL._J0-j-U)H1t)H0iIiL—JTATVC)WCii4)ClciCi0CDi—i-i-- l. —. The .bJ3GKY hierarchy.i’heoreu . at . u.1a3011si st0e 3 have opertes (j )ond from e Then, for iny 0 and anypair X, of occupation nwber confi’urations withx’ j .t unc-IJ_Ontis smooth end the foliowi1iequation takes place:(1% (_(x,,c’)U\ )(x,c.’) (3.1)Here is defined by(°) x.(k’(O)y fl o fl )‘S(x÷z’),the projectors fl were introduced before. R’is given by(l%’ (is (%) (4_R(x,z) Z (R(x,&(N )9+4)and P4 is given by(+1)(A\ (3(ti-x’Ck)R€7L’1quation (3.1) t.aay be considered as a week form of theiniinie—voluuao s3GK hierarchy for the tiue—evolation uiiderConsieration,1\c_CE..ctiiCI—’FSI>‘-coT0CDIiLI)+‘mCl)&0C) 0oLi)fpCDCI)HFsr)ctctCF0I-)CCHCrrjcs::‘FU-riHizI0I_ICDCr1(fHct‘a+‘aLI)C)Li)Ifl•0“JP0—i>•oP÷:>HrnT,f.1.,ci’p>I—H•rIpP<•CDoiIcv—i ‘H010CDHCoi—’ccH(p>Cl)_ip—t-U)C)‘aL.—y‘aCD0£2>U)•(l)C)4..40C)•10r10I_IIjlUoctIP11FlCi•HPV00H‘.‘C.-’HH.C)°cvHf’—’QC)H>C)g_..F)0‘.0U’r4“-dIEl__.__JHU’F-ft-o+IctHI(fHCl)....-F)CCC)°F)1—C)‘00CCi)C)Li)-H H)UH,i-•ctC)Li)1$[)CL)0C)Li)•IctdU) 0)0—1512he proof .1 LOCCCdS in tn as Loniaroi ena .1 one tis tnt ior ixedifiO io11oviin q iiULY iS vo1i(ccc ‘)÷ (5.7)=A,tLenaa .2. Uniformly for in compacts the following1Liics exist:(&o(nflLf7L4aiid4( , ).The prooz of Lemma 5.2 repeats that of Lemma 2.2.As in Section 2, from (3.7) &iid ama 5.2 one obtains(.8)Hence, tne fmnctior - ‘e— x.) is smooth, aid direcCoLaputioII diVas tnat the eqaality (3.8) may rewritten in thform (5.1). 2heorem is proven. D030•1ElLUci-PliD11rØ0303I0riV4a-I0’Qicito•rIhO-II43cici•.0‘4UIa,rItc,S01LU£0COr-IV-q43Sr°Oo0-1‘riCiAci43rT•rI•(IILU—Io‘-I,&I4303ci.00a,P43i—I01a:LU•031131a,wao‘ci•430Co03••—Itocn4iP0a,acii-a0•rIjJ)ci03.ci00to03“-1tOricicm-I•cod)•-l-0(0i-‘<,o3G”.•(1300a--I5>)%-qQtob-iwflaaa0•niC.i,r4‘3ori.i30id.COç-‘-I•rJ:503..-v-443a.ci0!.CDCltO(0,0Ø43‘v-I0)i—IOL‘U‘o•0s%I43CD.q1LLqa,•‘33

Towards Time - Dynamics for Bosonic Systems in Quantum Statistical Mechanics, 2

http://dair.dias.ie/762/1/DIAS-STP-90-05.pdf

Towards Time - Dynamics for Bosonic Systems in Quantum Statistical Mechanics, 2

Abstract

Similar works

Full text

Available Versions

DIAS Access to Institutional Repository