- Email: [email protected]
Relaxing properties of Hamiltonian systems
I n ~ r ~ X ] ~ G PROPERT/ES OF H A b I ~ T O N I A N SYSI'EMS*
EL SponN F~hlm, c~s Pto,~ da" L u d ~ - ~ , T - ~ - U m ~ n r ~ , M ~
Fcdazl ~publlc of
"I~emecgpm ofm-~lic tbem~ tm ggnenJi~ ba su~ha way that arban~ sub.o.Wlds of iuvarisnt s~sam admitted.Throe mnccpet naturally apply to ~ systmm fattm who/~phasespsce. Weshowdmt " m o s t ' s e p a r a b l e ~ * , , ~ a r e ~ ( m
L
~
.w
In recent ycar~ there has been a revived interest in ergodic theory (to mention just a few: [2], [3], [7D. Essentially, the physics community expects the clarfft~t;on of two pmb~ from erlPxfic theory. (1) The ] x u t i f ~ o n o f the use of the microcanonkal ensemble. There, one has to show that the H a ~ l t o n i a a system considered is ergodi¢ on every surface of constant encr~. (1I) The approach to eqml~rium of systems of cW-,:dcal stads~cal mechanics. Traditionally, fh;~ is equated with the fzcz thnt the Ham~onian sys:em considered is rn;~"S or is even a ~-system or a ~.raonlli system on every surface of constant energy. By the very am%;6on of t/lose concepts, tim two probler~s are coupled, since (I1) implies (I~ To put it in slightly more ph)~icnl terms: In order to show the approach to equilibrium, as defined in 0T), of a classical sy~em, one must at bast prove that ~he e ~ c r ~ is its only constaat of the motion- = problem which seems to be exceedingly difficult. The purpose o r this paper is twofokL (I) By a generalization o f t h e concepts of mixing. ~Y-mixing, etc. we show that, in fact, both I~robl~m~ can be studied completely independen:ly. Thg basic idea of the generaliza, tion is to treat the elements of an arbitrary sub.c-field as invaria~, sets and not only, as in the ~ t / o n of an ergodic s)~tera, the trivial subfield cons/sting of the empty set and the whole (2) It then becomes possible to study the genera "~lLzedergodic properties of H=m;Itonlan systems in the whole phase space. As an application, we investigate the Class of separ• Part o~"the IwJ~otl thetis ('MO~h~ 1974). m Postal address: Thmr. Ph~'k. Ther~emtr. 37, $ Mllacbas, PRO.
[3631
264
H. sPOm,~
able ('mtegable) H a m ~ o a i a n systems. Since for those systems the s~rfaces d~ined by all constants of the motion arc On general) distorted N-tori, their ergodic properties on those surfaces are rather triviaL Howerer, studied as systems in the whole phase space, they show i n ~ g properties: Under relatively weak conditions they arc relaxing (which is a gener=l;-~ m i ~ g property), but not 2-fold relaxing, let alone K-relaxing. Their Kolmogorov-S'mal entropy is zero. It should be noted that the spectrum of the induced one-lmrmneu~ unitary group can be countable Lebesgue on the subspacc orthog-
ona/to ~ c i n ~
Re]Jnd~ d~s==zrx=as-yst== Since we are primarily interested in applying our concepts to statistical mechaak~ we conccntr-~ on f l o ~ rather thaa on amomorpb~ms. The following d ~ n M o n s can e a s y b¢ m ~ to be appIkable to automorphisms and/or ~ n ~ c mca.m~ spaces.
A d~vums/ca/s~em is a quadruple (.1".~..,.. S,). where (X, ~ , ~) is a Lebesgue space, Le. isomorphic te the unit interval with thn usual L ~ b e s ~ measure, and ,.~, a 0ne-para meter group of mm_qu~pre~'ving automorphisms. $, is measurable in t. t ~[ = ~ denotes the sub-a-field o f all ins~:qam mcasu~bl¢ sets: ~ =. {B • ~l 5'~B ~ B a.¢. for all t ~ ~}.
For~ f~-~'(Z,~.~)
Pofdenotes thecondi~ioaal ~ o , .
of f #yen ~. It" f ~
c ~ (X', ~ , p), tlum Po is O~ projection operator projecting onto the subspa~ ~z(X, ~ , p).
Let ~ ~: ;Lj ~ ~, be a fanm:,- of sub-~-fic~ of ~. Then 'V' ~s is the unanest ~:b..e.-fteid o f ~ containing all tim listed ~b-~fields and 6
~1 is the largest ~mb-o-~eld of~: contained
~n all the lhzed sub-~fields. We us~ the same notation for dosed sub, paces of . ~ ( X , ~ , p).
< "l "> denotes th~ ,.~I=, ~-oduct in -~"(X, ~,~). DE~'~TnO.'~ 2.1. A dynamical system (X, ~ ' ( x , ;~, ~,)
t-.~. t
~, p ,
$~) is ~,eakly relaxing, i f for all f , g
f'2-D
J
It :_sr ~ t : d . t , if for all f , z e -~2(X,
~:,/+)
ma ( z l f ,S,) = ~ l J+of) .
It~ll
It is
m-fold relaxing, m e~.. if for all .~ ..... f , • - ~ " ( X , ~ , ~ )
•
as t: -~ : , t:--t~ -4 : ,
s F-or mo~ ~
S (fo
J-I
..., t.--t._ a -boo.
dcr.,"" . ~ z c [IOL [13].
S
z
(2.a) J-o
~
Renw'k.
G
PROPEgTI]~ OF HAMII,TONIAJq SYSTt34S
365
(a) According to this definition, relaxing is l-fold relaxing.
(b) Vor all f , f , , ..., f,, e -°~(X, ~, p) we have .m
lit
lS(]-[~)d~l, ]-I Ira,. J,,,I J-I by genes=n,~4 H~lder's inequality, ill-$,ll,ILfll. since $, is measure.preserving and P o l e .z "(X, ~,/s). Thus all integrals in Defudtion I are ~m~ae. -
~o1¢
2.2. A dynandcaI system (X, ~:, p, Sf) is called K-re/axing, if there
a. sub.~fleld ~9 ~ 8: ~
that
(0 • (: S , ~ for all t > O,
c~ V ~ s = ~ , (m~ A ~ = ~ . U
Remm'/,. (a) In ('dO it is sufficientto require
A s,~ =
~t.Suppo~
t m ~
A s.~ ~ ~. t,m~I
I
tin--9)
(See proof or L~mma 2.3 below.) (b) K..relar2ng implies re.fold relaxing for all m • N. Proof: we have
Let (X, ~, p, $0 he g-rel~xing. Then for all m 6 N and all f ¢ .~'+'(Xo ;~o p)
m4.l
as t -* ao, where the m p r m u r a is taken over aUg • .~ = (X, S _ , ~ , t') ~dth I L f l l ~ ~ l am
t~ of D~-;do. 2). We have
sinceg is measurable with respect to 5_,93. Since by (iii) $_,fSJ,~ as t -~ oo, it follows from the martingale convergence theorem that F.(.flS_,~)-.PJa.e. and, since the family {c"(I'1S_,93)1 t e R} is re+l-fold uniformly integrable, we obtain llE(flS-,~)-Pofll.,÷,
--. o a s t --. ~ .
366
14. SPOH~
To prove m-fold relaxing we observe that for a='lfo, .... f= e .-~"÷~(Z, ~ , p ) II
Js&e
j-I
~IB
j-.o
J.,@
m
"(IS ( 1-[ /..0
J,,4
as
Go-,o:o)
J--I
s;,)",'I *
J,~
Let fo . . . . . f . b¢ measurable with respect to S , ~ , a E R. Then ~ . . ~ j is measurable with respect to S,_,j~. Thus it follows from (2.4) that For t=, t a - t s .... , t=--t,,_= large enough, we ~.n m . ~ every term of the sum (2.5) smaller than ~Im. Hence the ~rni~ (2.2) holds for all f. .....f. e {re m'÷'(i'. ~.:)Ifis measurable with ~ m s o m c $,~,s=X} -9 . Since by definition S = ~ as t--- oo, 9 lles dense in -.~=*s(X, ~ , p ) and therefore the limit (2.3) can b¢ extended to all .-<~'+=(X, ~ , p). • By the last r~mmr.k we obtain the following hierarchy of implicatiom: X'-mlaxing =,, (-I+ l)-fold relaxing =,. m-fold relaxing ~. weakly relaxing. Obv/ously, if ~ = {~,X~, then ....relying co/ncidm with ... Roughly sr~,~ng, all the theorems proven in ergodJ¢ theory can be generalized ia the sense o f D~f~nltions I and 2 simply by replacing "...-mixing" by "_.-r,'~,~ug" and the tr/vial subfield by .~f and carrying through esscntiagy the ~szm=proof. In fact, many of the theorems axe already proven for an arbitrary sub-c-field ofinvariant sets. In tlm remaind=r of this section we list just a few lemmata needed for out further investigations. Let U(t) be the one-paz-ameter unltary group induced by ..~ on -~=(X, ~ , p):
~(t)f- f . ~
~03
for a l l f ~ -~=(X, ~ , p). We collect the spectra/propcr6es of U(t) (or, by Stone's theorem, rather those of the generating self-adjoint operator A) in *.he foilowin;
13. Let (X, ~, p, $,) be a dynamical system with the indu¢~ unitary group U(t) ,. ~ .
Then
(i) (X, ~,/=, ~) ix weakly re~axing i~" the apectram of AI(I--Po).S'=(X, U,/~) ¢ont~
Oi) l f the spectrum of.4 [ (1 - Po).~=(X, 3 , P) is absolutely ¢ontinuous, tlum (X, ~, I~, S,) C~J) If(X, ~, p, S,) ia E-relaxing, then A [ ( I - P , ) M ' ( X , ~, l~)ka¢¢owttable L ~ apectra~.
R~.AXINO PROF~TIi~ OF HAMILTONIAN ~
367
Ronark. It is not known, whether m-fold .,da.~ng, m • I, is a ~ property of dynamical syu~-uu, d'. [I0]. The re~lU of S e ~ o n 4 ~ a ~trong iudicafio, thr,t t h ~ , not tl~ ca~. Proof: 0")"and ('u') follow from a m'aishtforward g e n e ~ i o n of the usual proof. 0i0 follows from [IlL Theorem 3, where it is proven that A r.~,2(x, ~ , @ ) ~ I.
e - f "(X. b ,Ar - - O
t') Iris co..table Lct sguc spe ,'.m
of
•
We note that the relaxing propenies of a dynanfical system do not get worse by making the original space smaller. 2.4. Let 0", ~ , p , ~ ) /~ a a y n ~ n ~ d -9"~em and J e K , p ( S ) ~ 0.
i f (X, ~, p, ~) iS X-relaxing (weak/7 relaxing, m~rold relaxing), their (a, ~ rl B. p./p(B), ~ ) is K-relaxing (weakly relaxing, re.fold feinting). The proof follows immediately from the definitions and the invat/ance of B. 3. Hamiltonian systems and their s-ab
d_p,(t )
=B
eH
d
(3.1)
x - , (¢~, ...,qs, P, .... ,Ps) G/`, where /4:/'--* R is the (twice differvntiabl©) Hamiltonian function of the system. We assume that for every initial value xo G/'there exists a unique global solution t~,x(t, xo), - o o < t < oo, of (3.1). Then the mappings Tt: ;re,-. x(t, .re), t G1£, form a one-pm'ameter group of ¢anoniccJ tran~formatior,s a of F Onto/`. We call the triple (/`, p, Tt) a Homiitoninn system. Up to a constant,/4' is uniquely deter=
by r . U(t) be the unitary ~Foup ir.duced by r , . By Stone's theore-~ U(t) , , • u~. L is the L/ouriUeoperator of the Hantiltonian system (/`, p, T~). A theorem due to Hunziker [5] asserts that L with domain D(L)= .Z2(F) is the closure o f the operator i[H,.] s de. Let
o. C'(/`). The principal aim is to relate relaxing properties of a certain Hamiltoniaa system to properties of its Hamiltonian function. According to Section 2, we should first investigate the structure of the sub-c-field of invariant sets e r a Hamiltonian system (/`, p, T,). Let us call the union/"1 of all bound: All canonk~i tramformatlonsare assumed to be twice diff~'~mtiable.~ is twicediffm'entiablein t. • [" • "1 is the P o t u e e b n g t a .
2~l
H. seotn~
ed trajectories of 0r',/Z, T,) the bounded part of (/'o g , Tt) and its complement the unbounded part of ( r , / z , T,). The bounded part/'~ is strictly invaxiant 0-e. T, Pz " r , for all t e R) and measurable, in fact, even a countable union of compa~ sets. 3.1. Let ( r , /z, ~ ) be a Hamiltonian x)'xt~m ,~th bounded part r j and unbounded part r a. Then (Fi, ~[~I'I, /Z) is c-finite, l f A ~ ~ t~Fa, then either p(zJ) = 0 or p(zJ) = oo. Proof: Let K, be the 2N-dimensional dosed ball with centre zero and radius n ¢ N and let ,4, - {x • / ' i {T, xJ t ¢ R} c K,}. Then zJ, is strictly invariant, i.e. ,4, • oR, It is easy to see that ,4. is closed which implies that ,4. is compact and has finite measure. By definition we have I,J J . : Tl which proves that J , t r~ ~ n --, ao. ,,s l e t ..J e ~ r ~ r = and/Z(A) < oo. By Poincar~'s recurrence theorem, almost every point
of J is recurrent, i.e.,/z(J ~/'z)~" 0 =/~(,4).COROt2~Y. Pof[r~ a= 0 a.e. for ail f e L=(r, dp).
The relaxing properties of the unbounded part of a Hamihonian system are easily
anab~ecL 3.2. /let (]",/Z, Tt) be a Hamiltonian x3"xtem with Liourille operator L and unbounded part ['a c I', /z(rz) # O. Then L[.~a(/'a, dp) has countable Lebexgue spectrum. O
Proof: Let X" be an open ball with /Z(K~Pz) # O, A -- ~J T. X,~ Pz and ~ ( t ) Sm~al f
c ~J
be the su~-¢-field generated by the Borel sets of ~
7",Kta,4. (By the corol-
lary to I.emma 3.1 we cou;d add the sub-c-field ~t'~d to ~ ( t ) wkhout invalidating the proof.) Then we have Tt~(0) =- ~(t), ~($) c ~ ( I ) for s < t, ~ / ~ ( t ) = ~ r ~ 4 and tam~
~
"
/ ~ ~ ( t ) = , {I~,A}, since, for almost all x f J , lT,.vJ --, o0as t - - * - c o [4]. Let ,~f'o ~,.~a(zJ, ~(0), /Z). Then with U(t) = e "
we obtain U(x)Jf'o = U(t)o'f'e for x < t,
~ / U(t)Jf" o I, 2a(A, d/z)and / ~ U(t),~o 1, {0}, since p(,4) - co by Lcmma 3.1. By a theorem of Sinai ([13], Proposition 12.1), L[-~"(,~, dH) has a homogeneous Lebesgtm spectrum. Since ~ is not one-dimensional, there exists a partition o f zr', into countably many invariant sets ~ " with p ( ~ " ) ¢t 0, i.e. -~=(/'a, dp) = ~ v ~ a ( l ~ ", ~dp~ Covering each ~ , ~ t h open balls shows that L [ ~ ' ( ~ a '~, d/z) has a homogeneous Lebesgue spectrum for all n e N. • Thus we see that the unbounded part o f a Hamiltonian system is relatively uninteresting from the ergodic point of view. The remark in brackets in the proof above shows that the unbounded part of every Hamiltonian system is even K-relaxing in the sense of Defini-
RELAX1NG P R O P E g ~ OF HAMILTONIAN SYSTEMS
.N69
don 2 with the condition p ( A ' ) - 1 omitted. We have an essentially "free p a r d c k ' l i ~ behaviour. Since (/'s, A r~Pt, p) is e-finite, there always exists a partition o[ Fs into countab~ many invariant sets/~s"~ with finite measure. Thus we can naturally apply the conccl~ defined in S¢~tion 2 to the bounded, part of a Hamiltonian system.
4. ~
pro~
of s ~ l ~ b / e ]Ram~tonhut wj'~as
Studying the relaxing prolx.rties of an arbitrary Hamiltonian system is a formidabk task. However, for the class of separable Hamiltonian systems, this can be done ¢omph:tg~. For this class we also compute the Kolmogorov-Sinai entropy, which by a theorem o f Abramov ([13], Theorem 12.2) can be d e ~ e d u h({T,l t a a } ) = h(T,).
(4j)
In our context of poss~ly non-ergodic systems it is worth remembering that the K.-S entropy of a dynam/cal system can alwa)~ be represented as an integral over the IC-S entropies of the crgt~¢ subsystems. (See [9] and proof below.) DEFI~qnION 4.1. Let T ~ be the N-dlmensional torus. A Hamiltonian system (F, p , T,) with Hamiltonian fcncdon /it is called separable if zhere exists an open subset .0 ¢: and a canon/ca/ mapping $ such that s r - D x T N and ~/.,S-z does not depend ms w e t w. (.Q x T m, px xpH, So ~ . S - ' ) iS the transformed ,~Tstem o/" ( ] ' , p , Tf). '~ TW~pJ~ 4.2. Let (F, p , ~ ) be a zepm'~le l~amiltonia~ system and (O x T m, p# x p s , • S o l o S - t ) be its transformed syxtem with Hamiitonian function H: ~ x T ~ ~ R. CO xf fo, ~s n ,~z"X{o} grad(n-gradH) # 0 pw-a.e-
(4.2)
on Q s, their (F, p, TO is relax£.flr.
~u') (P, p, ~ ) i" not 2-fold relaxing and ther~ore not X-relaxing. Oh") The ~Imogaroc-SbTai entropy o f (I", p, Tt) is zero.
Proof: ad (T): Let L be the Liouvige operator of (F, p, ~) and Z, be tl~ Liouvillc operator of ( Q x T x, p x x p x , SoT,~S-'). Since S is canon/cal, L and r~ are unitarJy Since { ~ ' H n e Z " } is an orthogonal basis in .~'(T"), we have
~2(.o x T") = ~ m,gN
" m t b the N'-dlmendoa~ Lebesgu~n~asue~ ,
I~.
....
,'NP~VJ
c-'
.....
--~'C~)~"'.
(4J)
370
H. S3OHN
I.at a°~m be the orthogonal projection on -~a(D)e~'% U ( t ) - - ~ ~., f . ( v ) e " " e ~ ' ( O x TD, ~ e O, . e T". Sino'. by defmidon
and f(v, w ) -
(4.4) we have U(t)P"/(t,, w) - f.0,)e~-+~,~m - P'~U(0f(~, w)
(4~
for all t ~ R. Thus for all n e Z #, L is reduced by P~'~M2(.0 x T~. Let us define a unitary eransformadou
U: ~ " ~ 2 ( O x T'D --.- ~ ( ~
by
U: f(Oe ~ " ~..f(~X'~Y'~
Then for all f e UD(£,| }~'22(.0 x T~))U -~
U ( L | ~ " ( 2 z ( O x T-)) U - ' : f(~) ~ (n- BradH)(~)f(e).
(4.6)
Thus L|~'}M2(.O x T -v) is unitan'ly equivalent to multiplication by n - gradH ~ R'2(.O). Since., for all n e Z'~{0}, grad (,I-gradH) ~ 0 a.e. on O, the spectrum of n-gradH, # 0, is absolmdy continuous ([6], Chap. X, Exa.mple 1.9). Thus the st>e~cum o f L rcstrioted to (I-Po)-~'2(I ", dp) is absolutely condnuous and by ~ 2.3 ~ the - ~ o n followl. aa (,): I.~ . o ( n ) - { ~ 1 (n-~p-~aa)(O-0}. T h ~ aa 6) shows po.Z'(n x T') - _~ .z~(.O(~))~..
Suppose (F. # . TJ and hence (D x Tx, ~x xpx. So T,o$-~) is 2-~old rclaxin~ Choose n e Z " such that p.~L~x.Q(n)) ~ O, f e -~'(~) a n d f o =,f~ ,=fe-~"",ft = , f ~ ' % Then we obtain
Let tz -- 2tz, width is a~J~ssible; then[ Sf~dp. =, $ fSdp.. If f is chosen properly, this is a contradic6o~. ad (iii): Let h,($oTa*$ -~) be the K-S entropy of the dynamical s ~ m ( O x 7 ~, 0, x ~ , So 7, • S " ~ where ~o is the measure on 0 with unit w_~s~ a~ ~ • L?. Then by a theorem due to ROhlin [9] and by the invarinnc~ of the K-S e ~ r o l ~ under c~o~ica] tra~formadoos °
~ ( s . 7", , s - ' ) = ~ h.CSo ~'~ oS-')d/,.(~) - ~(TJ.
(4.8)
"Since for all oe.O t1¢ induced unitary operator of ( ~ x T ~, ~oxp~,S°T~°$ -~) 1 ~ a pure point spectrum, by Roldin'$ theorem ([10], Chap. 15, Corollary 6) the K-S entropy of (OxT~,~.xps, SoT~oS -t) is zero. •
RELAXING PROPERTIES OF HAMILTOI~[AN SY~i'F.MS
371
R~awk. By a different mg~od, Pro~er [8] proved that the system of a single particle in a central f o r ~ field is rela~ng prov/ded tl~t the potential is analyt/c~i and ,h~t ~d third derivative doe; not vanish. This result is a special case of Theorem 4 2 0)The proof ad (i) makes it plausible that the spectrum of the LiouviHe operator o f a arable Hami/tonian system can be studied in much greater d ~ i l A~t~tough there is no iwincipal technical dit~cu]ty, the analysis is rather lengthy and we d~fer it to a subsequent paper. One o f th~ results is that the point spec'mun is always the union of at most countably many free additive groups with rank ~ N. Every eigeavalue i.~ infinitely degenerate and arises from a linear part of the transformed Hamiltonian fuactiou. It should be noted that, Mthougtx the class of separable HamiltoMan systems is not very typieal from the point of view of statistical mechanics, it contains a few interesting m~ny-body systems: the ideal gas in a container with arbitrary wall potential, the harmonic chain and the exponential chain ('l'oda ta~tice) w/th fixed boundary condit/o~ [12].
Ac~.o~tgU I would I ~
to th-nk Dr W. Ochs for his helpful discussions and careful reacting of
the manusmp. REFI[RE4CZS [1] ~ R . . , a m d L E . Marsdq: ¥ o m ~ a o n s o f M ~ W. A. kn~unin, Re~linll, ~ 1967. [21 Farqubar. I. ~ : F ~ and Related T o ~ In: l r r e w n ~ l ~ ~- ~ M~>,.bod't l ~ , l a n , od. L Bid, L ~ ~ Press, New York, 19r~.. [31 Fm'4L L: Tt~ 7emattm from Analytical lbctomies to $,atlraad M ~ In: Aagatea m
0973}. [41 ~Opf, e.: Math. Ann. 103 093O), 710. [51 ~ , w.: Comm. ~tttSL Pt~ys. | (I968). 2g2. [6] ~ T.: r,,nw~som ~ for ~ Op,,,.ato,.a. ~ . ~ 1966. t'71 ~ J. I . : F-7o~ ~7~7 and $tat~kat ~ ( ~ n ~ In: T~,tort P ~ n e u . lacturt Norm ha Physic, 3t, ~ q ~ r . Berlin, 1974. [11 Prosur, R. 1".: L Math. Phys. 10 (1969), [91 ~ v . A . : Utp. M . N . 15 (4) 0960). ] -- Rms. Math. ~mrv. lS (4) ( 1 ~ 1. [10] SLaa;,ft. G., Thmor7ofD.mnm/ca/S~'enu, Aarhus UniversiU~l.ectm~ Note SeriesNo. 23, Ma.-ch 1570, [11] ~ : Izv. Akad. l'{auk S~lt, Sin'. Mat. 25 (1961), $99 ,, AMS Tr-d.qsLStw. 2, 39 (1964), g3. [12] Toda. M.: S r a , ~ z o w a N ~ L a n l e e , ArkJvforDetF-ysiskeSeminariTroad~imNo.2,1974. [13] Totoki, H.: ~ TXmr~, Aarhus UnJvmJtct ~ Note Series No. 14, Ftbmaw 1970.
EL SponN F~hlm, c~s Pto,~ da" L u d ~ - ~ , T - ~ - U m ~ n r ~ , M ~
Fcdazl ~publlc of
"I~emecgpm ofm-~lic tbem~ tm ggnenJi~ ba su~ha way that arban~ sub.o.Wlds of iuvarisnt s~sam admitted.Throe mnccpet naturally apply to ~ systmm fattm who/~phasespsce. Weshowdmt " m o s t ' s e p a r a b l e ~ * , , ~ a r e ~ ( m
L
~
.w
In recent ycar~ there has been a revived interest in ergodic theory (to mention just a few: [2], [3], [7D. Essentially, the physics community expects the clarfft~t;on of two pmb~ from erlPxfic theory. (1) The ] x u t i f ~ o n o f the use of the microcanonkal ensemble. There, one has to show that the H a ~ l t o n i a a system considered is ergodi¢ on every surface of constant encr~. (1I) The approach to eqml~rium of systems of cW-,:dcal stads~cal mechanics. Traditionally, fh;~ is equated with the fzcz thnt the Ham~onian sys:em considered is rn;~"S or is even a ~-system or a ~.raonlli system on every surface of constant energy. By the very am%;6on of t/lose concepts, tim two probler~s are coupled, since (I1) implies (I~ To put it in slightly more ph)~icnl terms: In order to show the approach to equilibrium, as defined in 0T), of a classical sy~em, one must at bast prove that ~he e ~ c r ~ is its only constaat of the motion- = problem which seems to be exceedingly difficult. The purpose o r this paper is twofokL (I) By a generalization o f t h e concepts of mixing. ~Y-mixing, etc. we show that, in fact, both I~robl~m~ can be studied completely independen:ly. Thg basic idea of the generaliza, tion is to treat the elements of an arbitrary sub.c-field as invaria~, sets and not only, as in the ~ t / o n of an ergodic s)~tera, the trivial subfield cons/sting of the empty set and the whole (2) It then becomes possible to study the genera "~lLzedergodic properties of H=m;Itonlan systems in the whole phase space. As an application, we investigate the Class of separ• Part o~"the IwJ~otl thetis ('MO~h~ 1974). m Postal address: Thmr. Ph~'k. Ther~emtr. 37, $ Mllacbas, PRO.
[3631
264
H. sPOm,~
able ('mtegable) H a m ~ o a i a n systems. Since for those systems the s~rfaces d~ined by all constants of the motion arc On general) distorted N-tori, their ergodic properties on those surfaces are rather triviaL Howerer, studied as systems in the whole phase space, they show i n ~ g properties: Under relatively weak conditions they arc relaxing (which is a gener=l;-~ m i ~ g property), but not 2-fold relaxing, let alone K-relaxing. Their Kolmogorov-S'mal entropy is zero. It should be noted that the spectrum of the induced one-lmrmneu~ unitary group can be countable Lebesgue on the subspacc orthog-
ona/to ~ c i n ~
Re]Jnd~ d~s==zrx=as-yst== Since we are primarily interested in applying our concepts to statistical mechaak~ we conccntr-~ on f l o ~ rather thaa on amomorpb~ms. The following d ~ n M o n s can e a s y b¢ m ~ to be appIkable to automorphisms and/or ~ n ~ c mca.m~ spaces.
A d~vums/ca/s~em is a quadruple (.1".~..,.. S,). where (X, ~ , ~) is a Lebesgue space, Le. isomorphic te the unit interval with thn usual L ~ b e s ~ measure, and ,.~, a 0ne-para meter group of mm_qu~pre~'ving automorphisms. $, is measurable in t. t ~[ = ~ denotes the sub-a-field o f all ins~:qam mcasu~bl¢ sets: ~ =. {B • ~l 5'~B ~ B a.¢. for all t ~ ~}.
For~ f~-~'(Z,~.~)
Pofdenotes thecondi~ioaal ~ o , .
of f #yen ~. It" f ~
c ~ (X', ~ , p), tlum Po is O~ projection operator projecting onto the subspa~ ~z(X, ~ , p).
Let ~ ~: ;Lj ~ ~, be a fanm:,- of sub-~-fic~ of ~. Then 'V' ~s is the unanest ~:b..e.-fteid o f ~ containing all tim listed ~b-~fields and 6
~1 is the largest ~mb-o-~eld of~: contained
~n all the lhzed sub-~fields. We us~ the same notation for dosed sub, paces of . ~ ( X , ~ , p).
< "l "> denotes th~ ,.~I=, ~-oduct in -~"(X, ~,~). DE~'~TnO.'~ 2.1. A dynamical system (X, ~ ' ( x , ;~, ~,)
t-.~. t
~, p ,
$~) is ~,eakly relaxing, i f for all f , g
f'2-D
J
It :_sr ~ t : d . t , if for all f , z e -~2(X,
~:,/+)
ma ( z l f ,S,) = ~ l J+of) .
It~ll
It is
m-fold relaxing, m e~.. if for all .~ ..... f , • - ~ " ( X , ~ , ~ )
•
as t: -~ : , t:--t~ -4 : ,
s F-or mo~ ~
S (fo
J-I
..., t.--t._ a -boo.
dcr.,"" . ~ z c [IOL [13].
S
z
(2.a) J-o
~
Renw'k.
G
PROPEgTI]~ OF HAMII,TONIAJq SYSTt34S
365
(a) According to this definition, relaxing is l-fold relaxing.
(b) Vor all f , f , , ..., f,, e -°~(X, ~, p) we have .m
lit
lS(]-[~)d~l, ]-I Ira,. J,,,I J-I by genes=n,~4 H~lder's inequality, ill-$,ll,ILfll. since $, is measure.preserving and P o l e .z "(X, ~,/s). Thus all integrals in Defudtion I are ~m~ae. -
~o1¢
2.2. A dynandcaI system (X, ~:, p, Sf) is called K-re/axing, if there
a. sub.~fleld ~9 ~ 8: ~
that
(0 • (: S , ~ for all t > O,
c~ V ~ s = ~ , (m~ A ~ = ~ . U
Remm'/,. (a) In ('dO it is sufficientto require
A s,~ =
~t.Suppo~
t m ~
A s.~ ~ ~. t,m~I
I
tin--9)
(See proof or L~mma 2.3 below.) (b) K..relar2ng implies re.fold relaxing for all m • N. Proof: we have
Let (X, ~, p, $0 he g-rel~xing. Then for all m 6 N and all f ¢ .~'+'(Xo ;~o p)
m4.l
as t -* ao, where the m p r m u r a is taken over aUg • .~ = (X, S _ , ~ , t') ~dth I L f l l ~ ~ l am
t~ of D~-;do. 2). We have
sinceg is measurable with respect to 5_,93. Since by (iii) $_,fSJ,~ as t -~ oo, it follows from the martingale convergence theorem that F.(.flS_,~)-.PJa.e. and, since the family {c"(I'1S_,93)1 t e R} is re+l-fold uniformly integrable, we obtain llE(flS-,~)-Pofll.,÷,
--. o a s t --. ~ .
366
14. SPOH~
To prove m-fold relaxing we observe that for a='lfo, .... f= e .-~"÷~(Z, ~ , p ) II
Js&e
j-I
~IB
j-.o
J.,@
m
"(IS ( 1-[ /..0
J,,4
as
Go-,o:o)
J--I
s;,)",'I *
J,~
Let fo . . . . . f . b¢ measurable with respect to S , ~ , a E R. Then ~ . . ~ j is measurable with respect to S,_,j~. Thus it follows from (2.4) that For t=, t a - t s .... , t=--t,,_= large enough, we ~.n m . ~ every term of the sum (2.5) smaller than ~Im. Hence the ~rni~ (2.2) holds for all f. .....f. e {re m'÷'(i'. ~.:)Ifis measurable with ~ m s o m c $,~,s=X} -9 . Since by definition S = ~ as t--- oo, 9 lles dense in -.~=*s(X, ~ , p ) and therefore the limit (2.3) can b¢ extended to all .-<~'+=(X, ~ , p). • By the last r~mmr.k we obtain the following hierarchy of implicatiom: X'-mlaxing =,, (-I+ l)-fold relaxing =,. m-fold relaxing ~. weakly relaxing. Obv/ously, if ~ = {~,X~, then ....relying co/ncidm with ... Roughly sr~,~ng, all the theorems proven in ergodJ¢ theory can be generalized ia the sense o f D~f~nltions I and 2 simply by replacing "...-mixing" by "_.-r,'~,~ug" and the tr/vial subfield by .~f and carrying through esscntiagy the ~szm=proof. In fact, many of the theorems axe already proven for an arbitrary sub-c-field ofinvariant sets. In tlm remaind=r of this section we list just a few lemmata needed for out further investigations. Let U(t) be the one-paz-ameter unltary group induced by ..~ on -~=(X, ~ , p):
~(t)f- f . ~
~03
for a l l f ~ -~=(X, ~ , p). We collect the spectra/propcr6es of U(t) (or, by Stone's theorem, rather those of the generating self-adjoint operator A) in *.he foilowin;
13. Let (X, ~, p, $,) be a dynamical system with the indu¢~ unitary group U(t) ,. ~ .
Then
(i) (X, ~,/=, ~) ix weakly re~axing i~" the apectram of AI(I--Po).S'=(X, U,/~) ¢ont~
Oi) l f the spectrum of.4 [ (1 - Po).~=(X, 3 , P) is absolutely ¢ontinuous, tlum (X, ~, I~, S,) C~J) If(X, ~, p, S,) ia E-relaxing, then A [ ( I - P , ) M ' ( X , ~, l~)ka¢¢owttable L ~ apectra~.
R~.AXINO PROF~TIi~ OF HAMILTONIAN ~
367
Ronark. It is not known, whether m-fold .,da.~ng, m • I, is a ~ property of dynamical syu~-uu, d'. [I0]. The re~lU of S e ~ o n 4 ~ a ~trong iudicafio, thr,t t h ~ , not tl~ ca~. Proof: 0")"and ('u') follow from a m'aishtforward g e n e ~ i o n of the usual proof. 0i0 follows from [IlL Theorem 3, where it is proven that A r.~,2(x, ~ , @ ) ~ I.
e - f "(X. b ,Ar - - O
t') Iris co..table Lct sguc spe ,'.m
of
•
We note that the relaxing propenies of a dynanfical system do not get worse by making the original space smaller. 2.4. Let 0", ~ , p , ~ ) /~ a a y n ~ n ~ d -9"~em and J e K , p ( S ) ~ 0.
i f (X, ~, p, ~) iS X-relaxing (weak/7 relaxing, m~rold relaxing), their (a, ~ rl B. p./p(B), ~ ) is K-relaxing (weakly relaxing, re.fold feinting). The proof follows immediately from the definitions and the invat/ance of B. 3. Hamiltonian systems and their s-ab
d_p,(t )
=B
eH
d
(3.1)
x - , (¢~, ...,qs, P, .... ,Ps) G/`, where /4:/'--* R is the (twice differvntiabl©) Hamiltonian function of the system. We assume that for every initial value xo G/'there exists a unique global solution t~,x(t, xo), - o o < t < oo, of (3.1). Then the mappings Tt: ;re,-. x(t, .re), t G1£, form a one-pm'ameter group of ¢anoniccJ tran~formatior,s a of F Onto/`. We call the triple (/`, p, Tt) a Homiitoninn system. Up to a constant,/4' is uniquely deter=
by r . U(t) be the unitary ~Foup ir.duced by r , . By Stone's theore-~ U(t) , , • u~. L is the L/ouriUeoperator of the Hantiltonian system (/`, p, T~). A theorem due to Hunziker [5] asserts that L with domain D(L)= .Z2(F) is the closure o f the operator i[H,.] s de. Let
o. C'(/`). The principal aim is to relate relaxing properties of a certain Hamiltoniaa system to properties of its Hamiltonian function. According to Section 2, we should first investigate the structure of the sub-c-field of invariant sets e r a Hamiltonian system (/`, p, T,). Let us call the union/"1 of all bound: All canonk~i tramformatlonsare assumed to be twice diff~'~mtiable.~ is twicediffm'entiablein t. • [" • "1 is the P o t u e e b n g t a .
2~l
H. seotn~
ed trajectories of 0r',/Z, T,) the bounded part of (/'o g , Tt) and its complement the unbounded part of ( r , / z , T,). The bounded part/'~ is strictly invaxiant 0-e. T, Pz " r , for all t e R) and measurable, in fact, even a countable union of compa~ sets. 3.1. Let ( r , /z, ~ ) be a Hamiltonian x)'xt~m ,~th bounded part r j and unbounded part r a. Then (Fi, ~[~I'I, /Z) is c-finite, l f A ~ ~ t~Fa, then either p(zJ) = 0 or p(zJ) = oo. Proof: Let K, be the 2N-dimensional dosed ball with centre zero and radius n ¢ N and let ,4, - {x • / ' i {T, xJ t ¢ R} c K,}. Then zJ, is strictly invariant, i.e. ,4, • oR, It is easy to see that ,4. is closed which implies that ,4. is compact and has finite measure. By definition we have I,J J . : Tl which proves that J , t r~ ~ n --, ao. ,,s l e t ..J e ~ r ~ r = and/Z(A) < oo. By Poincar~'s recurrence theorem, almost every point
of J is recurrent, i.e.,/z(J ~/'z)~" 0 =/~(,4).COROt2~Y. Pof[r~ a= 0 a.e. for ail f e L=(r, dp).
The relaxing properties of the unbounded part of a Hamihonian system are easily
anab~ecL 3.2. /let (]",/Z, Tt) be a Hamiltonian x3"xtem with Liourille operator L and unbounded part ['a c I', /z(rz) # O. Then L[.~a(/'a, dp) has countable Lebexgue spectrum. O
Proof: Let X" be an open ball with /Z(K~Pz) # O, A -- ~J T. X,~ Pz and ~ ( t ) Sm~al f
c ~J
be the su~-¢-field generated by the Borel sets of ~
7",Kta,4. (By the corol-
lary to I.emma 3.1 we cou;d add the sub-c-field ~t'~d to ~ ( t ) wkhout invalidating the proof.) Then we have Tt~(0) =- ~(t), ~($) c ~ ( I ) for s < t, ~ / ~ ( t ) = ~ r ~ 4 and tam~
~
"
/ ~ ~ ( t ) = , {I~,A}, since, for almost all x f J , lT,.vJ --, o0as t - - * - c o [4]. Let ,~f'o ~,.~a(zJ, ~(0), /Z). Then with U(t) = e "
we obtain U(x)Jf'o = U(t)o'f'e for x < t,
~ / U(t)Jf" o I, 2a(A, d/z)and / ~ U(t),~o 1, {0}, since p(,4) - co by Lcmma 3.1. By a theorem of Sinai ([13], Proposition 12.1), L[-~"(,~, dH) has a homogeneous Lebesgtm spectrum. Since ~ is not one-dimensional, there exists a partition o f zr', into countably many invariant sets ~ " with p ( ~ " ) ¢t 0, i.e. -~=(/'a, dp) = ~ v ~ a ( l ~ ", ~dp~ Covering each ~ , ~ t h open balls shows that L [ ~ ' ( ~ a '~, d/z) has a homogeneous Lebesgue spectrum for all n e N. • Thus we see that the unbounded part o f a Hamiltonian system is relatively uninteresting from the ergodic point of view. The remark in brackets in the proof above shows that the unbounded part of every Hamiltonian system is even K-relaxing in the sense of Defini-
RELAX1NG P R O P E g ~ OF HAMILTONIAN SYSTEMS
.N69
don 2 with the condition p ( A ' ) - 1 omitted. We have an essentially "free p a r d c k ' l i ~ behaviour. Since (/'s, A r~Pt, p) is e-finite, there always exists a partition o[ Fs into countab~ many invariant sets/~s"~ with finite measure. Thus we can naturally apply the conccl~ defined in S¢~tion 2 to the bounded, part of a Hamiltonian system.
4. ~
pro~
of s ~ l ~ b / e ]Ram~tonhut wj'~as
Studying the relaxing prolx.rties of an arbitrary Hamiltonian system is a formidabk task. However, for the class of separable Hamiltonian systems, this can be done ¢omph:tg~. For this class we also compute the Kolmogorov-Sinai entropy, which by a theorem o f Abramov ([13], Theorem 12.2) can be d e ~ e d u h({T,l t a a } ) = h(T,).
(4j)
In our context of poss~ly non-ergodic systems it is worth remembering that the K.-S entropy of a dynam/cal system can alwa)~ be represented as an integral over the IC-S entropies of the crgt~¢ subsystems. (See [9] and proof below.) DEFI~qnION 4.1. Let T ~ be the N-dlmensional torus. A Hamiltonian system (F, p , T,) with Hamiltonian fcncdon /it is called separable if zhere exists an open subset .0 ¢: and a canon/ca/ mapping $ such that s r - D x T N and ~/.,S-z does not depend ms w e t w. (.Q x T m, px xpH, So ~ . S - ' ) iS the transformed ,~Tstem o/" ( ] ' , p , Tf). '~ TW~pJ~ 4.2. Let (F, p , ~ ) be a zepm'~le l~amiltonia~ system and (O x T m, p# x p s , • S o l o S - t ) be its transformed syxtem with Hamiitonian function H: ~ x T ~ ~ R. CO xf fo, ~s n ,~z"X{o} grad(n-gradH) # 0 pw-a.e-
(4.2)
on Q s, their (F, p, TO is relax£.flr.
~u') (P, p, ~ ) i" not 2-fold relaxing and ther~ore not X-relaxing. Oh") The ~Imogaroc-SbTai entropy o f (I", p, Tt) is zero.
Proof: ad (T): Let L be the Liouvige operator of (F, p, ~) and Z, be tl~ Liouvillc operator of ( Q x T x, p x x p x , SoT,~S-'). Since S is canon/cal, L and r~ are unitarJy Since { ~ ' H n e Z " } is an orthogonal basis in .~'(T"), we have
~2(.o x T") = ~ m,gN
" m t b the N'-dlmendoa~ Lebesgu~n~asue~ ,
I~.
....
,'NP~VJ
c-'
.....
--~'C~)~"'.
(4J)
370
H. S3OHN
I.at a°~m be the orthogonal projection on -~a(D)e~'% U ( t ) - - ~ ~., f . ( v ) e " " e ~ ' ( O x TD, ~ e O, . e T". Sino'. by defmidon
and f(v, w ) -
(4.4) we have U(t)P"/(t,, w) - f.0,)e~-+~,~m - P'~U(0f(~, w)
(4~
for all t ~ R. Thus for all n e Z #, L is reduced by P~'~M2(.0 x T~. Let us define a unitary eransformadou
U: ~ " ~ 2 ( O x T'D --.- ~ ( ~
by
U: f(Oe ~ " ~..f(~X'~Y'~
Then for all f e UD(£,| }~'22(.0 x T~))U -~
U ( L | ~ " ( 2 z ( O x T-)) U - ' : f(~) ~ (n- BradH)(~)f(e).
(4.6)
Thus L|~'}M2(.O x T -v) is unitan'ly equivalent to multiplication by n - gradH ~ R'2(.O). Since., for all n e Z'~{0}, grad (,I-gradH) ~ 0 a.e. on O, the spectrum of n-gradH, # 0, is absolmdy continuous ([6], Chap. X, Exa.mple 1.9). Thus the st>e~cum o f L rcstrioted to (I-Po)-~'2(I ", dp) is absolutely condnuous and by ~ 2.3 ~ the - ~ o n followl. aa (,): I.~ . o ( n ) - { ~ 1 (n-~p-~aa)(O-0}. T h ~ aa 6) shows po.Z'(n x T') - _~ .z~(.O(~))~..
Suppose (F. # . TJ and hence (D x Tx, ~x xpx. So T,o$-~) is 2-~old rclaxin~ Choose n e Z " such that p.~L~x.Q(n)) ~ O, f e -~'(~) a n d f o =,f~ ,=fe-~"",ft = , f ~ ' % Then we obtain
Let tz -- 2tz, width is a~J~ssible; then[ Sf~dp. =, $ fSdp.. If f is chosen properly, this is a contradic6o~. ad (iii): Let h,($oTa*$ -~) be the K-S entropy of the dynamical s ~ m ( O x 7 ~, 0, x ~ , So 7, • S " ~ where ~o is the measure on 0 with unit w_~s~ a~ ~ • L?. Then by a theorem due to ROhlin [9] and by the invarinnc~ of the K-S e ~ r o l ~ under c~o~ica] tra~formadoos °
~ ( s . 7", , s - ' ) = ~ h.CSo ~'~ oS-')d/,.(~) - ~(TJ.
(4.8)
"Since for all oe.O t1¢ induced unitary operator of ( ~ x T ~, ~oxp~,S°T~°$ -~) 1 ~ a pure point spectrum, by Roldin'$ theorem ([10], Chap. 15, Corollary 6) the K-S entropy of (OxT~,~.xps, SoT~oS -t) is zero. •
RELAXING PROPERTIES OF HAMILTOI~[AN SY~i'F.MS
371
R~awk. By a different mg~od, Pro~er [8] proved that the system of a single particle in a central f o r ~ field is rela~ng prov/ded tl~t the potential is analyt/c~i and ,h~t ~d third derivative doe; not vanish. This result is a special case of Theorem 4 2 0)The proof ad (i) makes it plausible that the spectrum of the LiouviHe operator o f a arable Hami/tonian system can be studied in much greater d ~ i l A~t~tough there is no iwincipal technical dit~cu]ty, the analysis is rather lengthy and we d~fer it to a subsequent paper. One o f th~ results is that the point spec'mun is always the union of at most countably many free additive groups with rank ~ N. Every eigeavalue i.~ infinitely degenerate and arises from a linear part of the transformed Hamiltonian fuactiou. It should be noted that, Mthougtx the class of separable HamiltoMan systems is not very typieal from the point of view of statistical mechanics, it contains a few interesting m~ny-body systems: the ideal gas in a container with arbitrary wall potential, the harmonic chain and the exponential chain ('l'oda ta~tice) w/th fixed boundary condit/o~ [12].
Ac~.o~tgU I would I ~
to th-nk Dr W. Ochs for his helpful discussions and careful reacting of
the manusmp. REFI[RE4CZS [1] ~ R . . , a m d L E . Marsdq: ¥ o m ~ a o n s o f M ~ W. A. kn~unin, Re~linll, ~ 1967. [21 Farqubar. I. ~ : F ~ and Related T o ~ In: l r r e w n ~ l ~ ~- ~ M~>,.bod't l ~ , l a n , od. L Bid, L ~ ~ Press, New York, 19r~.. [31 Fm'4L L: Tt~ 7emattm from Analytical lbctomies to $,atlraad M ~ In: Aagatea m
0973}. [41 ~Opf, e.: Math. Ann. 103 093O), 710. [51 ~ , w.: Comm. ~tttSL Pt~ys. | (I968). 2g2. [6] ~ T.: r,,nw~som ~ for ~ Op,,,.ato,.a. ~ . ~ 1966. t'71 ~ J. I . : F-7o~ ~7~7 and $tat~kat ~ ( ~ n ~ In: T~,tort P ~ n e u . lacturt Norm ha Physic, 3t, ~ q ~ r . Berlin, 1974. [11 Prosur, R. 1".: L Math. Phys. 10 (1969), [91 ~ v . A . : Utp. M . N . 15 (4) 0960). ] -- Rms. Math. ~mrv. lS (4) ( 1 ~ 1. [10] SLaa;,ft. G., Thmor7ofD.mnm/ca/S~'enu, Aarhus UniversiU~l.ectm~ Note SeriesNo. 23, Ma.-ch 1570, [11] ~ : Izv. Akad. l'{auk S~lt, Sin'. Mat. 25 (1961), $99 ,, AMS Tr-d.qsLStw. 2, 39 (1964), g3. [12] Toda. M.: S r a , ~ z o w a N ~ L a n l e e , ArkJvforDetF-ysiskeSeminariTroad~imNo.2,1974. [13] Totoki, H.: ~ TXmr~, Aarhus UnJvmJtct ~ Note Series No. 14, Ftbmaw 1970.