Content-Type: multipart/mixed; boundary="-------------0009021730299" This is a multi-part message in MIME format. ---------------0009021730299 Content-Type: text/plain; name="00-332.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-332.comments" Conference proceedings. "Visions in Mathematics, Towards 2000", A. Alon, J. Bourgain, A. Connes, M. Gromov and V. Milman, eds., GAFA 2000, no. 1, Birkhauser, p. 334-358 (2000). ---------------0009021730299 Content-Type: text/plain; name="00-332.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-332.keywords" entropy, second law ---------------0009021730299 Content-Type: application/postscript; name="Lieb-Yngvason3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Lieb-Yngvason3.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: Lieb-Yngvason.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips.bin -h/tmp/DVF8039 -Pqms11 Lieb-Yngvason -o %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.04.27:1409 %%BeginProcSet: /tmp/DVF8039 %%EndProcSet %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 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b(but)g(full)f(details,)h(and)g(a)f(discussion)h (of)f(related)h(previous)f(w)o(ork)g(on)h(the)0 1693 y(foundations)10 b(of)h(classical)e(thermo)q(dynamics,)g(are)i(giv)o (en)f(in)h([LiY1].)18 b(The)11 b(literature)0 1749 y(on)g(this)g(sub)s (ject)h(is)f(extensiv)o(e)g(and)h(it)f(is)g(not)g(p)q(ossible)g(to)g (giv)o(e)g(ev)o(en)h(a)f(brief)h(accoun)o(t)0 1805 y(of)k(it)g(here,)i (except)f(for)f(men)o(tioning)f(that)h(the)h(previous)f(w)o(ork)g (closest)g(to)g(ours)h(is)0 1862 y(that)11 b(of)h([Gi)o(],)g(and)h ([Bu],)f(\(see)g(also)f([Co)o(],)h([D)o(])g(and)h([RL]\).)18 b(These)13 b(other)f(approac)o(hes)0 1918 y(are)f(also)e(based)j(on)f (an)f(in)o(v)o(estigation)f(of)h(the)h(relation)f Fl(\036)p Fp(,)i(but)f(the)g(o)o(v)o(erlap)e(with)i(our)0 1975 y(w)o(ork)k(is)g(only)g(partial.)20 b(In)c(fact,)f(a)h(ma)s(jor)d(part) i(of)g(our)h(w)o(ork)f(is)g(the)h(deriv)m(ation)f(of)0 2031 y(a)f(certain)h(prop)q(ert)o(y)f(\(the)h(\\comparison)d(h)o(yp)q (othesis")j(b)q(elo)o(w\),)e(whic)o(h)i(is)f(tak)o(en)h(as)0 2088 y(an)f(axiom)e(in)j(the)f(other)g(approac)o(hes.)19 b(It)c(w)o(as)e(a)h(remark)m(able)f(and)i(largely)e(unsung)0 2144 y(ac)o(hiev)o(emen)o(t)h(of)g(Giles)h([Gi)n(])g(to)g(realize)f (the)i(full)e(p)q(o)o(w)o(er)h(of)g(this)f(prop)q(ert)o(y)l(.)68 2201 y(Let)i(us)f(b)q(egin)g(the)h(story)e(with)h(some)e(basic)i (concepts.)68 2269 y Fl(\017)23 b Fm(Thermo)n(dynamic)16 b(System)s Fp(:)j(Ph)o(ysically)l(,)13 b(this)i(consists)f(of)g (certain)g(sp)q(eci\014ed)114 2325 y(amoun)o(ts)k(of)j(certain)f(kinds) g(of)g(matter,)g(e.g.,)h(a)f(gram)f(of)h(h)o(ydrogen)g(in)h(a)114 2382 y(con)o(tainer)13 b(with)g(a)g(piston,)h(or)f(a)h(gram)e(of)h(h)o (ydrogen)h(and)g(a)f(gram)f(of)i(o)o(xygen)114 2438 y(in)i(t)o(w)o(o)e (separate)i(con)o(tainers,)f(or)h(a)g(gram)e(of)h(h)o(ydrogen)h(and)h (t)o(w)o(o)e(grams)f(of)p eop %%Page: 5 5 5 4 bop 0 191 a FB(GAF)l(A2000)307 b(SECOND)12 b(LA)l(W)h(OF)f(THERMOD) o(YNAMIC)q(S)310 b(5)114 299 y Fp(h)o(ydrogen)13 b(in)h(separate)f(con) o(tainers.)19 b(The)14 b(system)e(can)i(b)q(e)g(in)g(v)m(arious)f (states)114 355 y(whic)o(h,)18 b(ph)o(ysically)l(,)f(are)h Fm(e)n(quilibrium)g(states)p Fp(.)28 b(The)18 b(space)g(of)g(states)f (of)g(the)114 412 y(system)e(is)h(usually)g(denoted)h(b)o(y)f(a)g(sym)o (b)q(ol)f(suc)o(h)i(as)f(\000)h(and)g(states)e(in)i(\000)f(b)o(y)114 468 y Fk(X)q(;)8 b(Y)s(;)g(Z)q(;)k Fp(etc.)68 537 y(Ph)o(ysical)17 b(motiv)m(ation)f(aside,)i(a)g(state-space,)g(mathematicall)o(y)l(,)e (is)i(just)g(a)g(set)0 593 y({)g(to)f(b)q(egin)h(with;)g(later)f(on)h (w)o(e)f(will)g(b)q(e)h(in)o(terested)g(in)f(em)o(b)q(edding)h (state-spaces)0 649 y(in)h(some)e(con)o(v)o(ex)h(subset)h(of)f(some)g Fj(R)681 633 y Fi(n)p FB(+1)750 649 y Fp(,)h(i.e.,)f(w)o(e)g(will)g(in) o(tro)q(duce)g(co)q(ordinates.)0 706 y(As)f(w)o(e)g(said)g(earlier,)g (ho)o(w)o(ev)o(er,)f(the)i(en)o(trop)o(y)e(principle)h(is)g(quite)g (indep)q(enden)o(t)i(of)0 762 y(co)q(ordinatization,)13 b(Carath)o(\023)-21 b(eo)q(dory's)13 b(principle)i(not)o(withstanding.) 56 843 y(2.)22 b Fm(Comp)n(osition)15 b(and)i(sc)n(aling)e(of)h(states) t Fp(:)j(The)d(notion)e(of)h(Cartesian)f(pro)q(duct,)114 899 y(\000)142 906 y FB(1)165 899 y Fl(\002)s Fp(\000)231 906 y FB(2)263 899 y Fp(corresp)q(onds)e(simply)d(to)i(the)h(t)o(w)o(o) e(\(or)h(more\))f(systems)g(b)q(eing)i(side)f(b)o(y)114 956 y(side)h(on)h(the)g(lab)q(oratory)e(table;)h(mathematicall)o(y)e (it)i(is)g(just)g(another)h(system)114 1012 y(\(called)f(a)h Fm(c)n(omp)n(ound)i(system)p Fp(\),)d(and)i(w)o(e)f(regard)g(the)g (state)g(space)g(\000)1306 1019 y FB(1)1333 1012 y Fl(\002)6 b Fp(\000)1402 1019 y FB(2)1436 1012 y Fp(as)114 1068 y(the)12 b(same)f(as)i(\000)382 1075 y FB(2)407 1068 y Fl(\002)5 b Fp(\000)475 1075 y FB(1)495 1068 y Fp(.)19 b(P)o(oin)o(ts)11 b(in)h(\000)742 1075 y FB(1)767 1068 y Fl(\002)5 b Fp(\000)835 1075 y FB(2)868 1068 y Fp(are)12 b(denoted)h(b)o(y)g(pairs)e(\()p Fk(X)q(;)d(Y)h Fp(\),)k(as)114 1125 y(usual.)19 b(The)13 b(subsystems)f(comprising)f(a)h(comp)q(ound)h (system)f(are)g(ph)o(ysically)114 1181 y(indep)q(enden)o(t)h(systems,)e (but)h(they)g(are)f(allo)o(w)o(ed)f(to)i(in)o(teract)f(with)g(eac)o(h)h (other)114 1238 y(for)i(a)h(p)q(erio)q(d)h(of)e(time)g(and)h(thereb)o (y)h(alter)e(eac)o(h)h(other's)f(state.)114 1294 y(The)k(concept)h(of)e (scaling)g(is)h(crucial.)28 b(It)18 b(is)g(this)f(concept)i(that)e(mak) o(es)g(our)114 1351 y(thermo)q(dynamics)d(inappropriate)h(for)g (microscopic)g(ob)s(jects)g(lik)o(e)h(atoms)e(or)114 1407 y(cosmic)h(ob)s(jects)h(lik)o(e)g(stars.)24 b(F)l(or)17 b(eac)o(h)g(state-space)f(\000)h(and)h(n)o(um)o(b)q(er)e Fk(\025)f(>)h Fp(0)114 1464 y(there)f(is)g(another)f(state-space,)h (denoted)h(b)o(y)f(\000)951 1447 y FB(\()p Fi(\025)p FB(\))1016 1464 y Fp(with)g(p)q(oin)o(ts)g(denoted)g(b)o(y)114 1520 y Fk(\025X)r Fp(.)31 b(This)18 b(space)h(is)f(called)g(a)g Fm(sc)n(ale)n(d)h(c)n(opy)f Fp(of)h(\000.)30 b(Of)19 b(course)g(w)o(e)f(iden)o(tify)114 1577 y(\000)142 1560 y FB(\(1\))202 1577 y Fp(=)13 b(\000)e(and)g(1)p Fk(X)k Fp(=)e Fk(X)t Fp(.)18 b(W)l(e)10 b(also)g(require)h(\(\000)925 1560 y FB(\()p Fi(\025)p FB(\))975 1577 y Fp(\))993 1560 y FB(\()p Fi(\026)p FB(\))1056 1577 y Fp(=)i(\000)1132 1560 y FB(\()p Fi(\025\026)p FB(\))1214 1577 y Fp(and)e Fk(\026)p Fp(\()p Fk(\025X)t Fp(\))g(=)114 1633 y(\()p Fk(\026\025)p Fp(\))p Fk(X)t Fp(.)17 b(The)c(ph)o(ysical)e(in)o (terpretation)g(of)h(\000)902 1617 y FB(\()p Fi(\025)p FB(\))965 1633 y Fp(when)h(\000)g(is)f(the)g(space)h(of)f(one)114 1689 y(gram)i(of)h(h)o(ydrogen,)h(is)f(simply)g(the)h(state-space)f(of) h Fk(\025)f Fp(grams)f(of)i(h)o(ydrogen.)114 1746 y(The)21 b(state)g Fk(\025X)j Fp(is)d(the)h(state)f(of)g Fk(\025)g Fp(grams)f(of)h(h)o(ydrogen)g(with)g(the)h(same)114 1802 y(`in)o(tensiv)o(e')11 b(prop)q(erties)i(as)g Fk(X)t Fp(,)g(e.g.,)f(pressure,)i(while)e(`extensiv)o(e')h(prop)q(erties)114 1859 y(lik)o(e)h(energy)l(,)h(v)o(olume,)e(etc.,)i(are)g(scaled)g(b)o (y)g(a)g(factor)f Fk(\025)h Fp(\(b)o(y)f(de\014nition\).)68 1939 y(F)l(or)f(an)o(y)g(giv)o(en)g(\000)h(w)o(e)f(can)h(form)d (Cartesian)i(pro)q(duct)g(state)g(spaces)h(of)f(the)g(t)o(yp)q(e)0 1996 y(\000)28 1979 y FB(\()p Fi(\025)63 1984 y Fh(1)80 1979 y FB(\))105 1996 y Fl(\002)c Fp(\000)177 1979 y FB(\()p Fi(\025)212 1984 y Fh(2)229 1979 y FB(\))253 1996 y Fl(\002)g(\001)f(\001)g(\001)f(\002)i Fp(\000)431 1979 y FB(\()p Fi(\025)466 1985 y Fg(N)495 1979 y FB(\))510 1996 y Fp(.)20 b(These)15 b(will)e(b)q(e)i(called)f Fm(multiple)i(sc)n (ale)n(d)e(c)n(opies)g Fp(of)g(\000.)68 2052 y(The)21 b(notation)e(\000)382 2035 y FB(\()p Fi(\025)p FB(\))453 2052 y Fp(should)i(b)q(e)g(regarded)f(as)g(merely)f(a)i(mnemonic)d(at)i (this)0 2108 y(p)q(oin)o(t,)14 b(but)i(later)e(on,)g(with)h(the)g(em)o (b)q(edding)g(of)f(\000)i(in)o(to)e Fj(R)1034 2092 y Fi(n)p FB(+1)1103 2108 y Fp(,)h(it)f(will)g(literally)f(b)q(e)0 2165 y Fk(\025)p Fp(\000)g(=)g Fl(f)p Fk(\025X)i Fp(:)d Fk(X)k Fl(2)d Fp(\000)p Fl(g)i Fp(in)g(the)g(usual)g(sense.)56 2245 y(3.)22 b Fm(A)n(diab)n(atic)14 b(ac)n(c)n(essibility)t Fp(:)j(No)o(w)12 b(w)o(e)h(come)g(to)f(the)i(ordering.)k(W)l(e)13 b(sa)o(y)g Fk(X)j Fl(\036)d Fk(Y)114 2302 y Fp(\(with)h Fk(X)k Fp(and)d Fk(Y)25 b Fm(p)n(ossibly)15 b(in)g(di\013er)n(ent)p 651 2309 165 2 v 16 w(state-sp)n(ac)n(es)p Fp(\))e(if)i Fk(Y)25 b Fp(is)14 b Fm(adiab)n(atic)n(al)r(ly)114 2358 y(ac)n(c)n(essible)f Fp(from)g Fk(X)18 b Fp(according)d(to)g(the)g (de\014nition)g(b)q(elo)o(w.)68 2438 y(What)f(do)q(es)h(this)f(mean?)20 b(Mathematicall)o(y)l(,)12 b(w)o(e)i(are)h(just)f(giv)o(en)g(a)g(list)g (of)g(pairs)p eop %%Page: 6 6 6 5 bop 0 191 a FB(6)388 b(E.H.)12 b(LIEB)f(AND)i(J.)f(YNGV)l(ASON)390 b(GAF)l(A2000)0 299 y Fk(X)23 b Fl(\036)c Fk(Y)11 b Fp(.)32 b(There)20 b(is)e(nothing)h(more)f(to)h(b)q(e)g(said,)h(except)g(that)e (later)g(on)i(w)o(e)f(will)0 355 y(assume)h(that)g(this)h(list)f(has)h (certain)f(prop)q(erties)h(that)g(will)e(lead)i(to)g(in)o(teresting)0 412 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y(adiabatic)i(accessibilit)o(y;)g(other)g(de\014nitions)h(are)g (conceiv)m(ably)g(p)q(ossible.)25 b(W)l(e)17 b(em-)0 1372 y(phasize)e(also)e(that)h(w)o(e)g(do)g(not)h(care)f(ab)q(out)g (the)h(temp)q(oral)d(dev)o(elopmen)o(t)h(in)o(v)o(olv)o(ed)0 1428 y(in)k(the)f(state)g(c)o(hange;)i(w)o(e)e(only)h(care)f(ab)q(out)h (the)g(net)g(result)f(for)g(the)h(system)e(and)0 1485 y(the)g(rest)g(of)g(the)g(univ)o(erse.)114 1566 y Fm(A)g(state)h Fk(Y)26 b Fm(is)15 b(adiab)n(atic)n(al)r(ly)g(ac)n(c)n(essible)f(fr)n (om)i(a)f(state)h Fk(X)t Fm(,)f(in)g(symb)n(ols)114 1623 y Fk(X)23 b Fl(\036)e Fk(Y)10 b Fm(,)22 b(if)e(it)h(is)f(p)n(ossible)f (to)i(change)f(the)h(state)f(fr)n(om)h Fk(X)j Fm(to)d Fk(Y)31 b Fm(by)114 1679 y(me)n(ans)20 b(of)i(an)f(inter)n(action)g (with)h(some)g(devic)n(e)f(c)n(onsisting)e(of)j(some)114 1736 y(auxiliary)g(system)f(and)h(a)f(weight,)i(in)f(such)f(a)h(way)g (that)g(the)g(auxil-)114 1792 y(iary)17 b(system)f(r)n(eturns)g(to)h 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Fk(X)17 b Fp(adiabatically)11 b(w)o(e)i(w)o(ould)g (ha)o(v)o(e)g(to)g(use)0 525 y(some)j(mac)o(hinery)g(and)h(a)g(w)o (eigh)o(t,)f(with)h(the)g(mac)o(hinery)f(returning)h(to)g(its)f (initial)0 581 y(state,)c(and)i(there)f(is)g(no)g(w)o(a)o(y)f(this)h (can)g(b)q(e)h(done.)20 b(Using)13 b(a)g(w)o(eigh)o(t)f(w)o(e)h(can,)g (indeed,)0 638 y(recompress)h(the)h(gas)g(to)f(its)h(original)e(v)o (olume,)g(but)j(w)o(e)f(will)e(\014nd)j(that)f(the)g(energy)0 694 y(is)g(then)g(larger)f(than)h(its)g(original)e(v)m(alue.)68 752 y(Let)j(us)f(write)148 831 y Fk(X)h Fl(\036\036)e Fk(Y)56 b Fp(if)48 b Fk(X)16 b Fl(\036)d Fk(Y)56 b Fp(but)15 b(not)45 b Fk(Y)23 b Fl(\036)13 b Fk(X)19 b Fp(\(written)14 b Fk(Y)23 b Fl(6\036)13 b Fk(X)t Fp(\))8 b Fk(:)0 910 y Fp(In)14 b(this)f(case)h(w)o(e)f(sa)o(y)g(that)f(w)o(e)i(can)f(go)g (from)f Fk(X)17 b Fp(to)c Fk(Y)24 b Fp(b)o(y)13 b(an)g Fm(irr)n(eversible)h(adiab)n(atic)0 966 y(pr)n(o)n(c)n(ess)p Fp(.)34 b(If)20 b Fk(X)25 b Fl(\036)c Fk(Y)31 b Fp(and)20 b Fk(Y)32 b Fl(\036)21 b Fk(X)j Fp(w)o(e)c(sa)o(y)f(that)h Fk(X)j Fp(and)e Fk(Y)30 b Fp(are)20 b Fm(adiab)n(atic)n(al)r(ly)0 1023 y(e)n(quivalent)31 b Fp(and)15 b(write)659 1080 y Fk(X)719 1055 y Fh(A)713 1080 y Fl(\030)e Fk(Y)18 b(:)68 1147 y Fp(Equiv)m(alence)e(classes)f(under)599 1121 y Fh(A)592 1147 y Fl(\030)h Fp(are)f(called)g Fm(adiab)n(ats)p Fp(.)56 1218 y(4.)22 b Fm(Comp)n(ar)n(ability)t Fp(:)d(Giv)o(en)c(t)o (w)o(o)f(states)g Fk(X)k Fp(and)e Fk(Y)25 b Fp(in)15 b(t)o(w)o(o)f(\(same)g(or)g(di\013eren)o(t\))114 1275 y(state-spaces,)j(w)o(e)h(sa)o(y)f(that)g(they)h(are)g(comparable)e(if) i Fk(X)i Fl(\036)d Fk(Y)29 b Fp(or)17 b Fk(Y)28 b Fl(\036)17 b Fk(X)114 1331 y Fp(\(or)e(b)q(oth\).)21 b(This)15 b(turns)h(out)f(to) g(b)q(e)h(a)g(crucial)f(notion.)21 b(Tw)o(o)14 b(states)h(are)h(not)114 1387 y(alw)o(a)o(ys)d(comparable;)f(a)j(necessary)g(condition)f(is)g (that)g(they)h(ha)o(v)o(e)f(the)h(same)114 1444 y(material)h(comp)q (osition)i(in)h(terms)f(of)i(the)f(c)o(hemical)f(elemen)o(ts.)32 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y(or)k(not)g(they)h(are)f(comparable.)k(Giv)o(en)c(the)h(relation) e Fl(\036)i Fp(for)e(all)h(p)q(ossible)g(states)g(of)0 2023 y(all)e(p)q(ossible)g(systems,)f(w)o(e)h(can)h(ask)f(whether)h (this)f(relation)f(can)i(b)q(e)g(enco)q(ded)h(in)e(an)0 2079 y(en)o(trop)o(y)i(function)g(according)f(to)h(the)g(follo)o(wing:) 0 2164 y Fj(En)o(trop)o(y)22 b(principle)p Fp(.)32 b Ff(There)21 b(is)e(a)h(real-v)m(alued)f(function)h(on)g(all)f(states)g (of)g(all)0 2220 y(systems)14 b(\(including)g(comp)q(ound)i(systems\),) d(called)h Fj(en)o(trop)o(y)g Ff(and)i(denoted)g(b)o(y)f Fk(S)0 2277 y Ff(suc)o(h)h(that)51 2348 y Fp(a\))22 b Fe(Monotonicity:)c Ff(When)d Fk(X)k Ff(and)c Fk(Y)26 b Ff(are)15 b(comparable)e(states)h(then)407 2427 y Fk(X)i Fl(\036)d Fk(Y)56 b Fp(if)15 b(and)g(only)g(if)45 b Fk(S)s Fp(\()p Fk(X)t Fp(\))10 b Fl(\024)j Fk(S)s Fp(\()p Fk(Y)d Fp(\))e Fk(:)234 b Fp(\(1\))p eop %%Page: 8 8 8 7 bop 0 191 a FB(8)388 b(E.H.)12 b(LIEB)f(AND)i(J.)f(YNGV)l(ASON)390 b(GAF)l(A2000)48 299 y Fp(b\))23 b Fe(Additivity)j(and)h(extensivity:) 38 b Ff(If)24 b Fk(X)k Ff(and)d Fk(Y)35 b Ff(are)24 b(states)g(of)g (some)114 355 y Fp(\()p Ff(p)q(ossibly)d(di\013eren)o(t)p Fp(\))g Ff(systems)f(and)i(if)g Fp(\()p Fk(X)q(;)8 b(Y)h Fp(\))22 b Ff(denotes)g(the)g(corresp)q(ond-)114 412 y(ing)c(state)g(in)g(the)h(comp)q(ound)f(system,)g(then)h(the)f(en)o (trop)o(y)g(is)g(additiv)o(e)g(for)114 468 y(these)d(states,)f(i.e.,) 542 530 y Fk(S)s Fp(\()p Fk(X)q(;)8 b(Y)g Fp(\))k(=)h Fk(S)s Fp(\()p Fk(X)t Fp(\))c(+)h Fk(S)s Fp(\()p Fk(Y)g Fp(\))e Fk(:)368 b Fp(\(2\))114 599 y Fk(S)19 b Ff(is)e(also)g (extensiv)o(e,)g(i.e.,)f(for)h(or)g(eac)o(h)g Fk(\025)f(>)h Fp(0)g Ff(and)g(eac)o(h)h(state)e Fk(X)21 b Ff(and)c(its)114 656 y(scaled)e(cop)o(y)g Fk(\025X)g Fl(2)e Fp(\000)505 639 y FB(\()p Fi(\025)p FB(\))556 656 y Ff(,)h(\(de\014ned)j(in)e(2.)k (ab)q(o)o(v)o(e\))621 743 y Fk(S)s Fp(\()p Fk(\025X)t Fp(\))10 b(=)j Fk(\025S)s Fp(\()p Fk(X)t Fp(\))8 b Fk(:)446 b Fp(\(3\))68 830 y(A)16 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Fp(.)11 1365 y Fj(A6.)22 b(Stabilit)o(y)p Fp(.)35 b(If)20 b(\()p Fk(X)q(;)8 b("Z)537 1372 y FB(0)556 1365 y Fp(\))21 b Fl(\036)h Fp(\()p Fk(Y)s(;)8 b("Z)772 1372 y FB(1)791 1365 y Fp(\))20 b(for)g(some)g Fk(Z)1055 1372 y FB(0)1074 1365 y Fk(;)8 b(Z)1126 1372 y FB(1)1166 1365 y Fp(and)21 b(a)f(sequence)114 1422 y(of)d Fk(")p Fp('s)h(tending)g(to) g(zero,)g(then)h Fk(X)h Fl(\036)e Fk(Y)11 b Fp(.)28 b(This)18 b(axiom)e(is)i(a)f(substitute)h(for)114 1478 y(con)o(tin)o(uit)o(y)l(,) 12 b(whic)o(h)h(w)o(e)g(cannot)g(assume)g(b)q(ecause)h(there)g(is)f(no) g(top)q(ology)f(y)o(et.)114 1534 y(It)21 b(sa)o(ys)g(that)g(`a)g(grain) g(of)g(dust)h(cannot)f(in\015uence)i(the)f(set)f(of)h(adiabatic)114 1591 y(pro)q(cesses'.)68 1657 y(An)i(imp)q(ortan)o(t)d(lemma)f(is)j (that)g(\(A1\){\(A6\))e(imply)h(the)h Fm(c)n(anc)n(el)r(lation)g(law)p Fp(,)0 1713 y(whic)o(h)15 b(is)g(used)h(in)f(man)o(y)e(pro)q(ofs.)20 b(It)15 b(sa)o(ys)f(that)h(for)f(an)o(y)h(three)h(states)e Fk(X)q(;)8 b(Y)s(;)g(Z)456 1784 y Fp(\()p Fk(X)q(;)g(Z)s Fp(\))j Fl(\036)i Fp(\()p Fk(Y)s(;)8 b(Z)s Fp(\))j(=)-8 b Fl(\))14 b Fk(X)i Fl(\036)d Fk(Y)18 b(:)397 b Fp(\(5\))68 1855 y(The)16 b(next)f(concept)h(pla)o(ys)e(a)h(k)o(ey)g(role)f(in)h (our)g(treatmen)o(t.)-2 1921 y Fj(CH.)22 b(De\014nition:)28 b Fp(W)l(e)20 b(sa)o(y)f(that)g(the)g Fm(Comp)n(arison)h(Hyp)n (othesis,)g Fp(\(CH\),)f(holds)114 1977 y(for)14 b(a)h(state-space)g (\000)g(if)g(all)f(pairs)h(of)g(states)f(in)h(\000)g(are)g(comparable.) 68 2043 y(Note)j(that)f(A3,)i(A4)f(and)g(A5)g(automaticall)o(y)e (extend)i(comparabilit)o(y)d(from)i(a)0 2100 y(space)e(\000)g(to)f (certain)g(other)h(cases,)f(e.g.,)f Fk(X)j Fl(\036)d Fp(\(\(1)8 b Fl(\000)i Fk(\025)p Fp(\))p Fk(Y)s(;)e(\025Z)s Fp(\))k(for)i(all)g(0)e Fl(\024)h Fk(\025)g Fl(\024)f Fp(1)j(if)0 2156 y Fk(X)h Fl(\036)d Fk(Y)24 b Fp(and)14 b Fk(X)i Fl(\036)d Fk(Z)s Fp(.)20 b(On)14 b(the)g(other)f(hand,)i (comparabilit)n(y)c(on)j(\000)g(alone)g(do)q(es)g(not)0 2212 y(allo)o(w)f(us)i(to)f(conclude)i(that)e 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FA(0)1044 2056 y Fp(w)o(e)h(put)h Fk(D)q Fp(\(\000)p Fk(;)8 b Fp(\000)1329 2040 y FA(0)1340 2056 y Fp(\))k(=)h Fl(1)p Fp(.)0 2113 y(Next,)k(for)g(an)o(y)g(giv)o (en)g(\000)h(and)f(\000)571 2096 y FA(0)600 2113 y Fp(w)o(e)g(consider) h(all)e(\014nite)h(c)o(hains)g(of)g(state)g(spaces,)0 2169 y(\000)c(=)g(\000)117 2176 y FB(1)137 2169 y Fk(;)8 b Fp(\000)186 2176 y FB(2)205 2169 y Fk(;)g(:)g(:)g(:)d(;)j Fp(\000)335 2176 y Fi(N)381 2169 y Fp(=)13 b(\000)457 2153 y FA(0)484 2169 y Fp(suc)o(h)j(that)e Fk(D)q Fp(\(\000)770 2176 y Fi(i)784 2169 y Fk(;)8 b Fp(\000)833 2176 y Fi(i)p FB(+1)892 2169 y Fp(\))k Fk(<)h Fl(1)j Fp(for)f(all)f(i,)g(and)i(w)o(e) e(de\014ne)244 2247 y Fk(E)s Fp(\(\000)p Fk(;)8 b Fp(\000)376 2228 y FA(0)387 2247 y Fp(\))k(:=)h(inf)541 2210 y Fd(\010)567 2247 y Fk(D)q Fp(\(\000)652 2254 y FB(1)672 2247 y Fk(;)8 b Fp(\000)721 2254 y FB(2)740 2247 y Fp(\))i(+)g Fl(\001)e(\001)g(\001) g Fp(+)j Fk(D)q Fp(\(\000)1007 2254 y Fi(N)t FA(\000)p FB(1)1086 2247 y Fk(;)d Fp(\000)1135 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