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JIeHHH B Ta KKX COCTORHRRX 06ycnOBJIeHH B03paoT~e~ poJIhm 3JIeMeHTapHHX

BOs 6YQeJmit no llepe Ilpll6Jnmemm K TOlKe ifJa30B01O nepexolla

ornpaBHHM nyHKTOM BCllOJlhBO~ MPr B TeOplm KpHTHleOmlX JlBJIeshy

JDlA mwmCh n peo6pasoBaImfI KanaHOBa III2 B npHMeHewm It 3~atJe HSIDITa CYTD 3TIDr upeo6pasoBaHHtt COCTODT B TOM ITO TaR BaR Hltampe RPHshy

TKie CRoft TODG BIilOnBll Te KOHiliHrYPa tlBll B ROTOPbIX COceItHHe CIIHHH BS8shy

mmo rrapalJIeJl1iliH MOlIHO YRpymmTlgt n CXOnHYJ) p elUeTKY C nOCTommo~ a

u epelwI R 6JIOIHoit pemeTRe C pa3MepaMH MOROB 2a 3a KaJotOMY

6lIOKy c onOCTaBruI6T CJl aqxpeRr1UlHHtt cmrn ~ H npefIOJIaraeTCSl ITO

KOppeJliImm M~ Jli H j1j B YRpyIllleHHo pemeTKe n0JIJrHll mre Th

TOT Jle ~ ITO 11 cmmoBile KOppeJIJlIJlnt B BOXOnHOit HO rrPH COOTBeTCTshy

BYDlIeM H9MeRemm 6e3pa3Io4ePHokt KOHCTaHTIi CBJl3l1I K bull B6JmSl RplWHleCshy

HOB TOIKH wurna ROPPeJlJlIUlf ~ raquo Q B n08TOMY BSMeHeBlle MaolllTs6a

o o=at K - K=K(t~K) (5)

f4OllH0 xapaKTeplt90BaTlgt n paltlBl6CKJ1 HenpeplIBHIiM napaMelpoM t 1laquo t laquo ~ Q bull llpeo6paaoB8HM (5) no CYlII8CTBY IIIlT TOT ae xapax-

Tep HO B ( I) bull B BlJJL COH B CBo e nepBO pa60Te I3 nocTYJIlfpoBWl

JVlH HIIX CnpaBen11HB06lb In$IJepeHIJB8JIlHoro y paaHeHIDllIH (2) llpJ 9TOM

ItpllTHqecK8J TOlKa I e OIlpeneJIJleTCH JltaK HenOnmDO8H TOlKa 81010

y paBRemm 8 JlIDleapaSaIllII ypaBHeltBJI B6mSII f3eo H llCnOJ1hSOBaHHe

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CMIIlCJI npoCleDme CTeneHHoit aBTOMOn6JIlgtHOC11J rrpKieM pr-rrpeo6pasoBBHHR

Sal8IlTCSl KaK H Y IeJlJI - MaHH8 H JIoy Ha MHOlIteCTBe MOneneA H lW81lT

CMIlCJI npe06pasoBaHldl lJlR8JIIl MKJ(poCROIIHleCKHX paCOTomm nplil OnHOBpeMeHshy

HOi peHOIilpoBRe 6eSpaSMePRoA ROHOTaHlll CMSH K bull CleJlaeM B CBJl3H C IlOCTpoeHBeM ~WiOBa CJIe)lYlltBe saMeUUmll

I) llapaMeTp t SneCD JmlIJIeTOJl TOJIlgtIltO rrpll6JnmeHHo HerrpeplllHHM

l nocneoOBaTeJIDHOCTh IIpe06pa30B8BliI Q a I Q t a = a I It Jrol[eT IIPHBeOTH He R nOSBpaIIemm Ha 0nHY HS yae paCCMaTpJmaBllIHXOJl

6JIOiHHX pemeTOl( a R TlOCTPOeH1ll) HOBOtt p emeTKH ( MOlteJIJII) IDtaie 1OBOpll

~ rrpR6nBxeHHO RenpepuBH01O MBCmla6H01O npeoopaSOBaHHJl Kal(aHOBa n e

cyIlleCTByeT 06paTB01O flpe06pasoB8HHJI R CTano 6ulD pe MOlIteT 1IlTl

TQJ[LltO 0 peHOPMMBS8DH0HII01 TIOJy1pynne (PilI) 2) CoxpaHeBlle BlUta KOPpeJIJIIJIdt CrImiOB ~ MHOJteCTBa Mon6Jle JJOJIEshy

HO n cmry onpeoeneHllUi IltOPpeJUm)ilOHlOOC cpyKItItltll rrpenn0JIaI81Igt coxpaHeshy

5

HlIe BIma laMIllJlblOHllaJiOB H ( K Il) ~ H (k a ) lTO JJIIJl MO~eJm H3ltHshy

r s H8BepHO Dpl YRpyaHeHIDl B BOBOM laMHJThTOIDIaIlC 1l0Jm1JRJ0TCR UIeHH

B3alJMOneHCTBRR 6olee BllCOKOro llOpanK8 16M B lCXOI[ROM lJ IDe BIOlan B

TepMOllIDiaMlUltY OIlta3lmae1 CR llp eHe6p ellmMO M8JI JJliIIlIIgt IIPJI f3 ~c 3) CnpaBeJt11liBOc1b yprumeHRR PI TlIDa (2) IIOCTYJmIlY8 1CJl

1almM 06paSOM 1l00Tpoelme Ka)aHOBa OJIeJtYeT PSOCMBTpKBaTlgt TOJ11gtRO

K8K rrpB6JmJJteHHjlO ~A ( B upoCTel1meD tPoPMe CA) rrpJIBOIlfDllY1l R PlIT

5 lIonxon BwrbCOHS

BwrbOOH npeJ(JlPJlHampll nOI1HTKY o6o~lJ TPY~HOCTB RMeDllHeCR B nocshy

Tpoemm RanaHOBa B nepBYJO olepelUgtonpel(eJlHTlo Knace raMWIloTOHlIffilOB

HITBapJIrurTHHX OTaOOBTe~bHO MaCillTadHoro rrpe06pasoB8llJJA CCTeCTBeHHHX

rrepeMeHllHlt wm KOTOPOro YpallHemm PIAlKe He rroCTYJmPYXlTCJl a BJiTeshyKaDT aenocpenCTBeHHO HS paCCMOTpeRHH 415

IIepBJdl lUar CBJI3aH C n epexonoM 01 MO~eJm 3lUlra HMeDlle1 npocTOA

JI Har~ qmsKleC1tlfil CMliCJI K qleHOMellOJlOIWIecKoll MCmeJm B 1ltOTOPO~

BMeCTO JlOl8J[LHOro eIIlIHa HMfJe1CR CrnUlOBOe TIOJIe S (x) K KpoMe 1l8PHoro

B3UHMoneicTBWl c onePJmTCR Bille B npoJISBOllbliJil WHJltUltOIUIJI 01 cmmoBbIX

n()]e~ Q[S (XC)] ITO OYT11 I1M8 YGTIffiWOIlllJt 6eCROHelliO MUOro nOllOJI-

HHTeJIloliHX BSalIMO~e ilcTImA B TaRO sanale eCTeCTBelllioK nepeMBHHott (ruta-Jlor x B ( I raquo nBlUieTCH cmmOBoe nOJIe nepeHoPMHPYeMOtI BeJllJDDlOit

( aHaJlor 9 B ( I) - lPYHKUROHWI Q H8H60JIee IIPOCTaH KHTeprrpe 1~ MOampeT 6uT norryqeHa ITpR nepexone

K HMIIYJlbCHOt) npenCT8BJ1emm B KOTOPOM upoCTpaJrCTBBHlwe BlOIana cmmo-

BOre 110J1R MOMHO x apaRTepa30BBTL ~eKCOM t 3~~ nocne~oBa-TMLliOCTlgt c(gte~le ClOOC o6onOleK P8nKyca k= l- (l- 2) B lIMTTYJllgtCHOM npoCTshy

paHCTBB ( CM 6) B aTOM CJIyqae B BHpaJKeHlle lVlJI CT81IICTlleCKOJ CYM-

MIi BXOWlT RHTerpEpoB8Jille no Sl II npOY3Be~eHlfe no BceM bull lb-TerpEpoBBlllle 110 Sf C 4mKc~poBaHHliMR = 01 Ha KalIinOM ware

YMeHLII8eT nICJIO uepeM8HlilD Ha elmlUW 1orna JtalC C16THCTKleOKWI CYMshy

Ma npn aTOM coxpawteT CBO~ BHli C TOUIOCThll no peHoprmpollKH fJYR1Jll10ItBshy

Jla Q Ha -lit ware KHTerpllpoB8IDlSl ynaeTCR 3armC8Th PBKYPpeHTshy

Roe COOTHOlUeRHe THTI8

Q(llQ) Q( Q ( ~raquo ) (6)

corJIaCVm1ee CR c ypaBHeHHeM tM (2) CJlenaeM ItBB 3BMelaHIDl Bo-ne PBIlX B3HTHe HB K8lI)tOM mare no t

llITerpana j 01 Sl bullbull bull MelUaeT BOSMOllUOCTR nooTpoeHrul 06paTHOro npeo6shy

pa30BaHM~aK I(8f( orrpeneneHHoMY lIHTerpany ( lltCJIY ) y1laquolJ JI eJ1Jgt3H conooshy

1aIDfTlgt n OJ1)lHTerpaJIbHY1O ltPYKKUIm bull Cle~oBBTenhHO lI8IlRoe nOCTpoeHHe

BBJIICOHa TaKlKe o6JlanaeT OJ30JtOTBaMll JIHIIIb nonyrpyTIml Bo-BTOPllX BIlJI()]shy

HeBRe TOtIHOro lffiTerpapoBaHIDI OTBTOywfii npoBeCTl He ynaeTCR a ICnonLshy

6

3y eMlle B ITOlOCone BMCOHB arrrrpoKCHMalllDl He JiMeJOT CTpOroro 0600110mllUfR

TaKHM 06paSOM ~ B BWIhCOHOBClCOH pr 3anaBaeMBfl OOOTHOlUBlHe

(6) JllIJllle1 CJl 1QilhlCO llpH6mmeHBoi IltaK B B CJIyqae llo cTpoeHl1lI KnnIUlOshy

-sa s~eMoro rrpeo6p830B8HlleM (5) Cl8nynmui mar B l1onxone BRnLeoH8 COCTOBT B npB~eHHoM pemeHKB

YpaBHeHlli PITI (6) JrBJIJmItID(OR Hemm8tbmMH HenoIlIDll(H8JI TOlJta npl 3TOM

COOTB6TClByeT He 3aIDlcmneMY OT pelUeamo ypaBH~ ( 6) bull Jl7IR rrpgt16 shy

XHKeHHOro pemeHHH YpaBHeHKa (6) BHKbCOH Ii ~ep I4 HcnQnb30B8JIH

nOCT8TOtIHO cTapy( ~e~ 0 BOSMOXHOCTY BBeneHKll HenpepHBHo Me~ettCR

pasMepHOeTll d 16 If IIOOTPORJU Teopmo B03iWlUeHldi ITO MaJOloY napaMeTshy

py Eo X8paKTepESYDI1eMy OTM OHelme pa 3MepHOCTlI OT cJ = 4 BIl6op

pasMepHOCTlI d = 4 CBa3aH C BOSMCMHOCTL~ rrocTpoe~~ TeopHH BOSMy-

Illem B6JmSH lIISBeCTHoro ( llpH c1 = 4) peweHRll

6 bull Me Ton JtOJIJJe RlllBHllX TIepeMeHHHX KlxHo BC KOro

HHTepecH1lfi aapHaJiT 7ir rrroo~8 paSpa60TaH B PffIte pa60T 1JxshyHOBCROro Ii era y~eRlKOB 17-20 B 3TOM nOl(XOI1e B MOnBJIH HSBHra rrapSMeTp Ba8HMOneaCTBRll paeCMaTPKBBeTCR K8K ~yHKUKH KOOp~8T yanoraquo

llMElJWIWl wypLG-06pas

] ( k ) elkmiddot r = ) r - r J(r) 1 ~ ( - - d ) )

N k

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TDCTRleCKaR CYMMB aallRCNBaeTCH B BHne ~

7 = r(df)N I (f)egtltL f]O) J J ~ 4 -lc

rne I (f) - ruco6BaR nepexo~a OT CI1HHOBllX nepeMeHHHX S K j(0JIshy

JIeKTlIBmIM llepeMeHHliM f BIImCJIe~e T8KOa CTaTHOTHtleCROil ClYMMll

OCROBllB8e1CH Ha JlBHOM onp eneJlerum I (f) i ITpenCTaMemm CT8TKCshy

TEieCKO~ cyMMll B BKne cyMMU MOM8HTOB 0 1 raYCCOB8 paonpenBJIeHHH

nPll~eM nOBenelme B KpJITlJIeCKOA TOQKe OITllCllBaeTCH C ITOMOmrD cneuaBJIloshy

RUM 06paSOM KOHC1pYJIPyeMoro 6aa lfcHoro pacrrpeneneHRR K0JIJI8KTKBHHX neshy

peMeHHHX npe~oTa~ero 00603 sCKlloReHTY OT nOHRBOMa no QeTHHM CTeshy

n eHIM KOJLIIeKTlmHllX nepeMemna laJ1IgtHelmWi npouenypa COC10111 B HRTe~

PHPOBBJUm no CJIOfIM ltPaSOBOIO rrpocTpaHOTBB ROJIJJeKTHBHllX u epeM81UUDC

TO KalC lI Y BwIbcoHa npKBol(BT It PITI B rronxOT(e lOxHOBCROro 0IUJ8shy

KO 0 rrOMOIlblO p eKyppeH1H1lX Coo1HolllelmA nOJIYte lIIIHX npn paCCMOTpBshy

rum lSCTKTnID CT8Tl1CTll1i8ClOOC CYM1~ Ylac1 CR 1l0JIYlltTb npaBHJr1gtHoe SHashy

7

leIrne RpHTHI8CKIDC mme KCOBOCTaBaJICb B peamUOM rrpoCTpaRCTB8 C d 3

11 He npl10eraH K HORjCCTBeHHOMY pa CCMOTp eHmO CHTYatnm C d = 4 - pound bull

7 l~aco~ XHRBe pCaxbBOCTH MOne~e~

YfllB8P Ca1IbHOCTb KPlflHleCKHX 5ffiJIemdl OUpellM8WlIlfI KanaHomolI2 B OOlleM CJlfae nOImMaeT CII KBK 9 KBKBa1leHTHOCTlgt KpWllIIeCKoro n OBsneHIDI

p a3Jm1ffiWt qms1fl e c IGX CHOTSM TO 80 Tb rrpennOJIaraeTCSI ITO I(PlfTIfI80Klf8

HHneKC~ xapaRTep lISyJWHe TepMOnmr8MIKY CRCTelWl B6Jm31I KPRTIfI8CK0i1 TO1shy

KII ororna KOBIl wm pa3JlnlHX MOnenetl B p aMRaX HeKOTOporo Macc a

B PHne cnyqa eB n pH OnpellM 8HHB Kna Cca YHRBepO~HOOTR OKaSHB8e Tshy

OR ~06H1lM HCn~30BaTJgt 8HaJIOr HB B qoPMY~POBK8 sana~ CT8TRCTlfle Oshy

ROit MeX8IfHKB H KBaHTOBOH T80Plrn rrOJIJI PaCCMOTPRM B KEIl8CTBe IIJlHM8pa

S8nam CT8TliCTHle CKOR MeXBHHKB lIIOTHOCTJgt n OTeH1]JaJIbHOA SHeprKK B KOshy

TOPID( lIMeeT BHD

r T + cI U (x)f JI(X) rex) CP(x x) Cf (X) I(x) x ) (7)

rlle laquo(x) - nOJIeBoii 0 11SpaTOp ibop18 ( 7) c ooTBeT cTByeT ~OMY PIClY qKshy

3H1eCRllIX 1Irlllenei upmreMeMInC IJ)UI OIIlfCamuJ mmRQCTei KpHCT8JDIOB

M8IHeTHKOB R T ll B tJaCTHOCTli yrrOMHlfaBweitCSI BliIIIe TpeXMePHOI MOlleJll

l1aKHra B TeoPRi KpHTHleCKHX lIB1le~ npellllOJla raeTCH iTO KB8IlTOBHe CBoitshy

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RBaHTOBOra rrp eItCTaMemm raMWllgtTOHHaHa MOJlIiO nepem K era ltJIaocllshy

18 CROMY 8R1llIB81J eHTY SaMeJtflJt n OJIeBHe on epSTOpN r ( It) un cKaJlJlpHHe

cpym anm V (1) bull B p esymTaTe 1era

U (X) f JltPClLX l cp()t) (I2e x) cJX (8 )

ECJIJIl B3aHMOIIeicTBlle JIOKaJIbHO

l ( XIt ) = C d(x- r) J

(9) TO nnOTHOCTlgt nOTe~lgtHOa aHeprKH (8) nepexoItHT B BHpaa6HHe

U ltx) S If ~(X) J == C 1 ( ID)

COOTBeTOTByDee n OTeliIJl8JIJgtHoll 1a c TlI MOTllOOTll JIaIpaHDEUiB B T60pHH

nOJlJl f ~ CJiellOB8TemHo CTaTllCTHleCKHe CBCTew 0 1ltOHeDtHM ~shyCOM nByxtJaCTHllIOro B3amOIIeaCTBRII n0JlltHH IIpHItaWI8ampaTlgt TOMY Ite KllBCshy

cy ymmep caJIbHOCTil lT0 l MOIteJI1gt TeopBH ttOJlJt I ~ B Tp8XMElpml

8

J1Hrur CHTYaIlHH HM8e T M8CTO IIpH BS8HMoneitCTBID 6 e CKOHellIoro p azmyshy

c a ROr lla Hanp HM8p

ltfgt (x x) =rIV eIO)

1It8 V - o6beM CHCT8MJ1 B 3TOM CJYIa e RBK XOPOlllO l3B8ClHO MOIeJTh

cpe~era norut rrpH V - 00 nseT TepMOItHHSMHleCme pemeHIDl aommTOTKshy

1eCKH COBuan8llJtK8 C TOUlilM TO eCTb WlOlHOCTlgt nOT8liDaaJIJgtHOii 3HepIHB

(8) np B38i1MOlleKCT1lID1 (10) If V 4 00 a cmmTOTlfIeCla paBlla

~ v I() 2 (X) U = pound j h (It) 04 XU ( )() I ) 0 7V 0 ( II)

2 ne n (x ) == lt If ( x) gt O1Cllla BIIWIO ITO CTaTHCTHIeCRaJI MOnem 0

llarrbHOlleicTBKeM COOTBeTcT~yeT TeOPRH ITQnR ~l B ~KTKBROM BRewshy

HeM nOJIe Boo6ue roBOpg SaBHCJ3111eM 01 TepMOl(HltaMllecHmC nepeMeHHliX

CymecTBOBaHKe 9KBHBt~eHTHOCTK Me~ CTSTHCTxqeCKE1ID H no~eBW~

MOneJlJIMH nOSDOJIJleT ~~CJIJiTlgtItPllTKIeCroe mmeRC~ nOJIb3YfiCr MeTonaMr

KBawrOBOA TeopD noM 2I22 lis IIDYIlX HHTepeClHHX CTaTHCTxqecroX rrpo6JIeM ~OIlYCK8DlIDC n epetfJopshy

MYJllIlPOB KY HS Jl3lgt1l(e Teopmr ITOJIJI ynoMflReM nrsret KSOTpom(O~ TYpc1yshyJleHTROCll lieCllBM8eMoA M3lltoi i JmlIKOCTR 23- 5

B nplBeneHHIDC BIDIle UPHMepax rrplf nepexoll8 01 (7) K (8 ) HMMCR

~ rrpOCTSilimlli CJIYlall OllHORormoneH THOJd OHCTSMU COClomnea H3 llCshy

Tnn O~oro COp Ta 6es BHYTpeHlUIX c TeneHea CBOOOIIli IoiMeHllo n03ToMY non

If ( It) l101mMBJlaClgt CRaJIIfPHrut rpytlKUJm B oOmeM lite cnyqae Macc yml-

BepCarrhBOCTE Ha RSHKe TeopHE l1CAR onpellenReTCR aHe~ xapaKTepncshy

TKKaMl1 lHOJIOM pa3lrnlHHX nOJlea mCJIOM ROMlIOHellT 1ltBXJlora fS uOJIen

cTapuoom CTerreHfIMK nOJIel paSII1epHOCThlO npocTpaHCTBa If xapaKTepoM B3lshy

RMolleacTBIDI (~SRO- R4H narrbHoneHcTBH8 HarrKtJKe KnK OTCYTCTBHO W1Hshy

sOTpomm) bull

BBelleHKe KnaCCOB YHKBepC8JlhHOCTH O~eCTB8HBO ynpo~aeT rrpo6JIs~

0III4C8IllUl RPUTlfI8CKKX JIB1IeHHii TBK KaK BMeCTO 60JIblI10ro mOJla MOll8JIeA

C p8SHWII cImsHle cKHM COlleJlll8Hl1eM ORB31iB8eTCR BOSMOXJ3lIM UOJIJgtSOBaTJgtCfi

JIKlIIb He60JThllHM 1DIIcnOM HX MaTeMaTJlfleeKHX 9KBKBaJIeHTOB CJrezcreT nonshy

IepRHY Tb ITO IlpmmurrelKHOCTb ROHKpeTl1HX MOlleJIeit It o n pellMetniOMy KJIacshy

ey YHKBepC~HOCTR 06~o n ocTYJllIlpyeTCRanpH0pHO a He nOKa3USaSTOR

nO)lTBepllleIm8M WUl npennOJfollteHID1 0 I tJIaccax ymmepcamHocTll CJYJl1lT

Ha~aeMoe B 9RcnePKMeHTBX YHKBep eanAHOe nOB8IteHKe paarrHIHHX Be~ecTB n PH aCHMITTOTlfleCKOM nPR~eHKH R RpRTKtJeCROI TO~

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ROe IIOBelleHKe MHOIHX BemeCTB He UOSBOJIJieT OTIle CTll RX j KaKOMY-n60

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x amum n OlliITlie M aeeon yrure8PCaJIbHOCTJd T8 JOJte JImtIeHO CMI1CJIa B KaqeCTshy

Be np~p8 MOXHO y KaS8Tb JdSBeCTHYID TOtjHO pemaet~~ MOnarrb EaxOTepa

8 ]pPlllrIeCKHe Jm1Iel1IDl H reTepoifJasHIoIe COCTOHlIIDI

KaK YYBSIpoundB8JlOClgt Bwre MPr no Ha CTOJnI(ero BpeMerm rrpRMeHiIe T(Jl K

paccMoTp emno TOJIloIO roMlt$SIDlX RpHTlNeClGDC KBJIsIDrli Torn8 RaP BO MHOshy

IRX fjMSlfIeCRHX CPICrSMaX 9KCrrepKMeHTaJIbHO JUl6JllJIaHlTGfI reTepo4lasHHe

COCTOmmH rrpencTa1lJiJIIW(I8 c0601 TepUOIllm8MleuroCKH paBHOBecHYIO CMeOh

MBKpocKOrmieORHX COCTOHlIiIIl c paSHoR cm1MeTpleit TBK B CsepxnpoBOJtIIHshy

ampaX cymeCTByeT npmtecL HeCBeptIpoBO~el ~Il B qeppo- l aJllHqiepposhy

MarHeTJlIlt8X - MaKpoCKOtIPlecKHe napalltarI-tIlTHJre odJracTK 13 cerneTOaJIeRshy

TPIKaX - napasneKTp~ecKHe saponarng JI T n OTJIHIRTeJIDHOH qepTol reTeshy

potlJaSHWC ffBJIelmi CJJJlnT TO qTO Olm Ha6JrnnaJOTCJI B umpoKHX HHlepBa-JIax TeMITepaTYP ( n8GfITKlI r paIYCOB) BhIDle B mue lOlKH rpaSOBOro nepeshyXona rrplIteM Raf( IV1t nepexonOB nepBoro TaR lJ BTOparo pona B STOn

CBJlSJII OTMeTlIM ITO noIlXon BwThcoHa OllpaBn8H WlfI nepexonOB nTOparo poshynB ~oTfi HMe~TCH rrOITHTK3 era ododmeHRR Ba cnyqaa rrcpexona rrepBoro poshyua 126

Mn~pocKom~ecKoe OITAC8Hle reT8po~sHHX cocTOHHKa 27-30 OCHoshy

BUBaIDmeeCfl Ha rrocnenOBaT8JInHOM HcnO~bSOB8HHB KOHnerr~ KBa3HCpe~

l3oIOJ1llX5oBa l1pe)U10JIaraeT pacumpeIrne lIPOClpaHCIB-3 COCToflHlIR M8RPOCKOshy

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1 Stueckelberg EG G Petermann AHelv bull Phys Acta 195) 26 499 2 Gell - Mann H Low F E Phys Rev 195495 1)00

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7 illHPKOB ~B nAB CCCP 1982 263 63

8 IIIHPKOB ~B TllHi 1984 60 218

9 IIIHPKOB ~B JUUI COCP 1955 105 972

10 IrnamKalLH M A UAH CCCP 1982 262 856 II Kadanoff L P Phys1cs 1966 2 26 ) 12 Kadanoff L P et al Rev Mod Pbys 1967 39 395 13 Wil s on K Phys Rev 1971 B4 3174 3184 14 Wilson K Fisher M Phys Rev Let t 1972 28 240 15 Wi lson K Phys Rev Let t 1972 ~8 548 16 Fisher ME Gaunt D S lhys Rev bull bull 1964 133 A224

17 ~OBCRBH MP ~3T~ 1958 34 379

18 KlxHoBCKZti HP Pyn8Bcm Kl K HN-74-17IPY1TJ1 AH YCCP KIElB 974 19 lDxHoBcKHil HP bull bullLAlI CCCP 1977 232 3 12

20 DxHOBCKKA M P n~~ 1978 36 373

21 bull Brezin E et al Pbys Let t 197) 44A 227 lhys Bev 1974 D9 II2 I

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22 BlanmmPOB AAbullbull lasa1ltOB nH bullbull TapacoB OB lim 1979 7 1035 23 Martin P C S1gg1a EDHose HA Ibyl Rev I97J A8 42J 24 De Dominic1s Mart i n PC Phys Rev 1979 AIq 419 25 AnlKewDI JIUbull BaCRII1gteB AH bullbull IDCbMaK KlM M 1983 57 268 26 Bpyc A KayJIH P CTPYKryplllB qE30Bble lTepeXO~ 14 MBp 1984 27 IDKanOB BH n~ 1976 26 40328 92 28 YUkalov VI Pbysi ca A 1981 108 402

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Shirkov DV bullbull Shumovsky AS Yukalov VI P2-85-529 Interrelation of Different Renormal ization Groups

The functional self-similarity property of a number of phys ical sysshytems is considered Different formulations of renonmal i zation group and semi-group transformations are analysed Their qeneral and particular proshyperties are discussed From the mathematical point of view a renormali zashytion group can be Introduced in two ways One is based on the elucidation of an internal symmetry of a concrete physical system and on the descripshytion of the symmetry given using the language of varIables t hat are natural for the problem considered Such a method of construct l nq r enormali zation groups has been used by Stueckelberg and Pe te rman and by Bogolubov and Shirlltov Another ~ay i s connected with the separation of a cl as s of models that have similar properties and di f fer one from another by the scale of some variable This method is illustra ted by the approaches due to Gel l shyMann and Low Kadanoff and Wilson and Yukhnovsky Both the way s mentioned are particular sequences of the general principle of the functiona l selfshysimi larity

The investigation has been performed at t he Laboratory of Theoretical Physics JINR

Preprint of the Joint Institute for Nuclear Research Dubna 1985