%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: hifi3.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -ohifi3.ps hifi3 %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.02.15:1959 %%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 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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y(When)e(all)f(the)h(terms)e(are)i(presen)m(t)h(it)d(is)i (clear)f(that,)g(in)h(general)f(the)h(equations)f(k)m(eep)h(the)g (distri-)0 3490 y(bution)h Fl(F)48 b Fp(a)m(w)m(a)m(y)33 b(from)f(equilibrium.)120 3665 y(Here,)41 b(w)m(e)f(neglect)g Fj(B)s Fp(\()p Fl(F)14 b Fp(\),)41 b(as)f(electron{electron)g(in)m (teractions)g(under)h(the)f(assumption)f(of)h(lo)m(w)0 3784 y(densit)m(y)34 b(lev)m(els)f(are)g(not)g(relev)-6 b(an)m(t.)120 3904 y(In)27 b(fact,)h(as)g(a)e(\014rst)i(appro)m (ximation,)e(w)m(e)i(shall)f(assume)g(\(for)g(instance,)i(Mark)m(o)m (wic)m(h)e(et)h(al.)41 b([MRS],)0 4024 y(\(&)33 b(1\),)g(or)g(F)-8 b(erry)33 b(and)g(Grondin)h([F)m(G],)f(\(&)h(2\))e(\))h(the)h (non{degenerate)h(case)e(and)h(tak)m(e)0 4293 y(\(1.5\))960 b Fj(L)p Fp(\()p Fl(F)14 b Fp(\))27 b(=)h Fj(\000)1612 4226 y Fl(M)p 1612 4270 108 4 v 1639 4362 a(\034)1782 4158 y Fi(Z)1898 4293 y Fl(F)14 b Fp(\()p Fl(v)2067 4252 y Fh(0)2095 4293 y Fp(\))p Fl(dv)2238 4252 y Fh(0)2287 4293 y Fp(+)2401 4226 y(1)p 2399 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5400 y Fp(,)33 b(where)h Fl(k)798 5415 y Fk(B)900 5400 y Fp(is)f(the)h(Boltzmann)e (constan)m(t)h(and)h Fl(T)47 b Fp(the)34 b(temp)s(erature)f(of)g(the)h (lattice.)p eop %%Page: 5 5 5 4 bop 3858 -139 a Fn(5)120 100 y Fp(While)30 b(this)h(appro)m (ximation)e(is)h(not)g(su\016cien)m(t)i(to)e(mo)s(del)g(all)f(the)i (inhomogeneities)f(of)h(an)f(actual)0 219 y(device,)25 b(our)f(primary)e(goal)g(is)i(to)f(in)m(v)m(estigate)g(a)g(b)s(oundary) h(v)-6 b(alue)24 b(problem)e(corresp)s(onding)j(to)e(strong)0 339 y(forcing)34 b(regimes,)e(since)i(they)f(will)f(tak)m(e)h(o)m(v)m (er)h(some)e(region)h(of)h(the)f(device.)120 521 y(W)-8 b(e)25 b(obtain)g(the)h(\015uid)g(lev)m(el)f(appro)m(ximation)e(b)m(y)j (t)m(w)m(o)f(di\013eren)m(t)i(metho)s(ds:)39 b(one)26 b(uses)g(a)g(Chapman{)0 641 y(Ensk)m(og)32 b(expansion)g(tec)m(hnique)h (and)g(the)f(other)h(giv)m(es)e(a)h(momen)m(t)f(expansion)h(where)h (the)f(closure)h(is)0 760 y(de\014ned)42 b(in)d(a)h(w)m(a)m(y)f(that)g 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5330 y Fg(\()p Fk(N)7 b Fg(+2\))1735 5372 y Fj(\000)1846 5304 y Fl(N)33 b Fp(+)23 b(1)p 1846 5349 263 4 v 1918 5440 a Fl(")17 b(\034)2154 5372 y(u)p Ff(\015)-57 b(_)13 b Fl(\032)2334 5330 y Fg(\()p Fk(N)7 b Fg(\))2494 5372 y Fp(+)2640 5304 y(1)p 2606 5349 118 4 v 2606 5440 a Fl(")17 b(\034)2752 5372 y(\032)2804 5330 y Fg(\()p Fk(N)7 b Fg(+1\))3070 5372 y Fp(=)28 b(0)p Fl(:)p eop %%Page: 20 20 20 19 bop 0 -139 a Fn(20)120 100 y Fp(Replace)34 b Fl(\032)544 63 y Fg(\()p Fk(N)7 b Fg(+2\))815 100 y Fp(in)33 b(equation)g(\(4.4\))f (b)m(y)i Fl(\032)1767 63 y Fg(\()p Fk(N)7 b Fg(+2\))2032 100 y Fp(=)38 b(~)-59 b Fl(\032)2190 63 y Fg(\()p Fk(N)7 b Fg(+2\))2461 100 y Fp(from)33 b(\(4.3\).)42 b(That)33 b(yields)854 340 y Fl(\021)21 b(@)977 355 y Fk(t)1012 340 y Fl(\032)1064 299 y Fg(\()p Fk(N)7 b Fg(+1\))1324 340 y Fp(+)23 b Fj(r)1507 355 y Fk(x)1579 340 y Fj(\001)1629 229 y Fi(\020)1688 340 y Fl(\032)17 b(m)1844 288 y Fg(\()p Fk(N)7 b Fg(+2\))1844 369 y Fk(M)2105 340 y Fp(+)22 b(\()p Fl(N)33 b Fp(+)23 b(2\))p Fl(u)p Ff(\015)-57 b(_)13 b Fl(\032)2725 299 y Fg(\()p Fk(N)7 b Fg(+1\))2963 229 y Fi(\021)876 580 y Fj(\000)988 513 y Fl(N)33 b Fp(+)22 b(1)p 988 557 263 4 v 1060 648 a Fl(")17 b(\034)1295 580 y(u)p Ff(\015)-56 b(_)13 b Fl(\032)1476 539 y Fg(\()p Fk(N)7 b Fg(\))1636 580 y Fp(+)1781 513 y(1)p 1747 557 118 4 v 1747 648 a Fl(")17 b(\034)1894 469 y Fi(\020)1970 580 y Fl(\032)2022 539 y Fg(\()p Fk(N)7 b Fg(+1\))2282 580 y Fj(\000)23 b Fl(\032)17 b(m)2538 528 y Fg(\()p Fk(N)7 b Fg(+1\))2538 609 y Fk(M)2776 469 y Fi(\021)2863 580 y Fp(=)28 b(0)p Fl(:)-3046 b Fp(\(4.5\))120 824 y(No)m(w,)44 b(the)g(closure)f(consists)h(in)f(taking)f(the)i(Chapman{Ensk)m(og)e (pro)s(cedure)i(\(3.16\){\(3.17\))c(to)0 943 y(expand)855 1077 y Fl(\032)907 1036 y Fg(\()p Fk(N)7 b Fg(+1\))1173 1077 y Fp(=)28 b Fl(\032)1330 1025 y Fg(\()p Fk(N)7 b Fg(+1\))1330 1103 y(0)1590 1077 y Fp(+)23 b Fl(")17 b(\032)1805 1025 y Fg(\()p Fk(N)7 b Fg(+1\))1805 1103 y(1)2071 1077 y Fp(=)28 b Fl(\032)17 b(m)2332 1025 y Fg(\()p Fk(N)7 b Fg(+1\))2332 1106 y Fk(P)2592 1077 y Fp(+)23 b Fl(")17 b(\032)2807 1025 y Fg(\()p Fk(N)7 b Fg(+1\))2807 1103 y(1)0 1277 y Fp(and)48 b(replace)h(in)f(equation)f(\(4.5\))g(in)h (order)g(to)f(obtain)h(an)g(expression)h(for)f Fl(\032)3100 1225 y Fg(\()p Fk(N)7 b Fg(+1\))3100 1304 y(1)3338 1277 y Fp(.)88 b(Therefore,)0 1418 y(replacing)33 b Fl(\032)478 1381 y Fg(\()p Fk(N)7 b Fg(+1\))750 1418 y Fp(in)33 b(\(4.5\))f(b)m(y)h Fl(\032)17 b(m)1398 1366 y Fg(\()p Fk(N)7 b Fg(+1\))1398 1447 y Fk(P)1659 1418 y Fp(+)23 b Fl(")17 b(\032)1874 1366 y Fg(\()p Fk(N)7 b Fg(+1\))1874 1444 y(1)2145 1418 y Fp(w)m(e)34 b(obtain)694 1662 y Fl(\021)20 b(@)816 1677 y Fk(t)868 1552 y Fi(\020)927 1662 y Fl(\032)d(m)1083 1610 y Fg(\()p Fk(N)7 b Fg(+1\))1083 1692 y Fk(P)1322 1552 y Fi(\021)1403 1662 y Fp(+)23 b Fj(r)1586 1677 y Fk(x)1658 1662 y Fj(\001)1708 1552 y Fi(\020)1768 1662 y Fl(\032)17 b(m)1924 1610 y Fg(\()p Fk(N)7 b Fg(+2\))1924 1692 y Fk(M)2184 1662 y Fp(+)23 b(\()p Fl(N)33 b Fp(+)22 b(2\))p Fl(u)p Ff(\015)-57 b(_)13 b Fl(\032)k(m)2908 1610 y Fg(\()p Fk(N)7 b Fg(+1\))2908 1692 y Fk(P)3147 1552 y Fi(\021)874 1902 y Fj(\000)985 1835 y Fl(N)33 b Fp(+)23 b(1)p 985 1879 263 4 v 1057 1971 a Fl(")17 b(\034)1260 1902 y(u)p Ff(\015)-57 b(_)13 b Fl(\032m)1527 1851 y Fg(\()p Fk(N)7 b Fg(\))1527 1932 y Fk(P)1687 1902 y Fp(+)1833 1835 y(1)p 1799 1879 118 4 v 1799 1971 a Fl(")17 b(\034)1945 1792 y Fi(\020)2021 1902 y Fl(\032m)2160 1851 y Fg(\()p Fk(N)7 b Fg(+1\))2160 1932 y Fk(P)2421 1902 y Fj(\000)23 b Fl(\032)17 b(m)2677 1851 y Fg(\()p Fk(N)7 b Fg(+1\))2677 1932 y Fk(M)2915 1792 y Fi(\021)874 2146 y Fp(+)988 2079 y(1)p 985 2123 55 4 v 985 2214 a Fl(\034)1052 2146 y(\032)1104 2094 y Fg(\()p Fk(N)g Fg(+1\))1104 2173 y(1)1364 2146 y Fj(\000)1476 2079 y Fl(N)33 b Fp(+)22 b(1)p 1476 2123 263 4 v 1580 2214 a Fl(\034)1750 2146 y(u)p Ff(\015)-56 b(_)13 b Fl(\032)1931 2094 y Fg(\()p Fk(N)7 b Fg(\))1931 2173 y(1)874 2369 y Fp(+)22 b Fl(")17 b(\021)t(@)1142 2384 y Fk(t)1194 2369 y Fl(\032)1246 2317 y Fg(\()p Fk(N)7 b Fg(+1\))1246 2395 y(1)1506 2369 y Fp(+)23 b Fl(")17 b Fj(r)1752 2384 y Fk(x)1819 2258 y Fi(\020)1878 2369 y Fp(\()p Fl(N)33 b Fp(+)23 b(2\))17 b Fl(u)p Ff(\015)-58 b(_)13 b Fl(\032)2415 2317 y Fg(\()p Fk(N)7 b Fg(+1\))2415 2395 y(1)2654 2258 y Fi(\021)2741 2369 y Fp(=)28 b(0)p Fl(:)-2924 b Fp(\(4.6\))120 2638 y(F)-8 b(rom)31 b(\(3.12\),)h(\()p Fl(N)h Fp(+)22 b(1\))17 b Fl(u)p Ff(\015)-57 b(_)13 b Fl(m)1269 2587 y Fg(\()p Fk(N)7 b Fg(\))1269 2668 y Fk(P)1435 2638 y Fp(=)28 b Fl(m)1627 2587 y Fg(\()p Fk(N)7 b Fg(+1\))1627 2668 y Fk(P)1889 2638 y Fj(\000)22 b Fl(m)2075 2587 y Fg(\()p Fk(N)7 b Fg(+1\))2075 2668 y Fk(M)2314 2638 y Fp(,)33 b(then)h(\(4.6\))e(giv)m(es)414 2883 y Fl(\032)466 2831 y Fg(\()p Fk(N)7 b Fg(+1\))466 2910 y(1)732 2883 y Fp(=)23 b Fj(\000)g Fl(\034)k(m)1090 2831 y Fg(\()p Fk(N)7 b Fg(+1\))1090 2912 y Fk(P)1329 2883 y Fl(\021)21 b(@)1452 2898 y Fk(t)1487 2883 y Fl(\032)g Fj(\000)i Fl(\034)k Fj(r)1814 2898 y Fk(x)1887 2883 y Fj(\001)1937 2773 y Fi(\020)1996 2883 y Fl(\032)2065 2773 y Fi(\020)2124 2883 y Fl(m)2211 2831 y Fg(\()p Fk(N)7 b Fg(+2\))2211 2912 y Fk(M)2472 2883 y Fp(+)23 b(\()p Fl(N)32 b Fp(+)23 b(2\))p Fl(u)p Ff(\015)-57 b(_)13 b Fl(m)3127 2831 y Fg(\()p Fk(N)7 b Fg(+1\))3127 2912 y Fk(P)3367 2773 y Fi(\021\021)832 3092 y Fj(\000)23 b Fl(\034)11 b(\021)t(\032@)1145 3107 y Fk(t)1179 3092 y Fl(m)1266 3040 y Fg(\()p Fk(N)c Fg(+1\))1266 3121 y Fk(P)1527 3092 y Fp(+)23 b(\()p Fl(N)33 b Fp(+)22 b(1\))p Fl(u)p Ff(\015)-57 b(_)14 b Fl(\032)2148 3040 y Fg(\()p Fk(N)7 b Fg(\))2148 3118 y(1)832 3286 y Fj(\000)23 b Fl(")995 3176 y Fi(\020)1054 3286 y Fl(\021)t(@)1160 3301 y Fk(t)1195 3286 y Fl(\032)1247 3234 y Fg(\()p Fk(N)7 b Fg(+1\))1247 3313 y(1)1507 3286 y Fp(+)23 b Fj(r)1690 3301 y Fk(x)1757 3176 y Fi(\020)1816 3286 y Fp(\()p Fl(N)33 b Fp(+)23 b(2\))p Fl(u)p Ff(\015)-57 b(_)13 b Fl(\032)2337 3234 y Fg(\()p Fk(N)7 b Fg(+1\))2337 3313 y(1)2575 3176 y Fi(\021)q(\021)2711 3286 y Fl(:)-2739 b Fp(\(4.7\))120 3556 y(Replacing)36 b Fl(\021)20 b(@)705 3571 y Fk(t)740 3556 y Fl(\032)35 b Fp(b)m(y)h Fj(\000r)1128 3571 y Fk(x)1179 3556 y Fp(\()p Fl(\032u)p Fp(\))22 b(+)i Fj(O)s Fp(\()p Fl(")p Fp(\))36 b(and)g Fl(m)2014 3504 y Fg(\()p Fk(N)7 b Fg(+2\))2014 3585 y Fk(M)2277 3556 y Fp(+)24 b(\()p Fl(N)34 b Fp(+)24 b(2\))p Fl(u)p Ff(\015)-57 b(_)14 b Fl(m)2937 3504 y Fg(\()p Fk(N)7 b Fg(+1\))2937 3585 y Fk(P)3207 3556 y Fp(=)32 b Fl(m)3403 3504 y Fg(\()p Fk(N)7 b Fg(+2\))3403 3585 y Fk(P)3643 3556 y Fp(,)35 b(then)0 3680 y(\(4.7\))d(giv)m(es)h(the)h(second)g(appro)m(ximation)e(of)h(the) h(\()p Fl(N)f Fp(+)22 b(1\))2265 3643 y Fk(th)2380 3680 y Fp(momen)m(t)744 3920 y Fl(\032)796 3868 y Fg(\()p Fk(N)7 b Fg(+1\))796 3946 y(1)1062 3920 y Fp(=)p Fl(\034)28 b(m)1298 3868 y Fg(\()p Fk(N)7 b Fg(+1\))1298 3949 y Fk(P)1553 3920 y Fj(r)1636 3935 y Fk(x)1686 3920 y Fp(\()p Fl(\032u)p Fp(\))22 b Fj(\000)g Fl(\034)11 b Fj(r)2132 3935 y Fk(x)2199 3809 y Fi(\020)2258 3920 y Fl(\032m)2397 3868 y Fg(\()p Fk(N)c Fg(+2\))2397 3949 y Fk(P)2636 3809 y Fi(\021)1161 4128 y Fj(\000)23 b Fl(\034)11 b(\021)t(\032@)1474 4143 y Fk(u)1525 4128 y Fp(\()p Fl(m)1651 4077 y Fg(\()p Fk(N)c Fg(+1\))1651 4158 y Fk(P)1890 4128 y Fp(\))p Fl(u)1986 4143 y Fk(t)2043 4128 y Fp(+)23 b(\()p Fl(N)33 b Fp(+)22 b(1\))p Fl(u)p Ff(\015)-57 b(_)13 b Fl(\032)2663 4077 y Fg(\()p Fk(N)7 b Fg(\))2663 4155 y(1)2823 4128 y Fp(+)23 b Fj(O)s Fp(\()p Fl(")p Fp(\))p Fl(:)-3157 b Fp(\(4.8\))120 4373 y(See)25 b(that)f(this)h(expression)h(coincides)g(up)f(to)f(order) h Fl(")h Fp(with)e(the)h Fl(\032)2571 4321 y Fg(\()p Fk(n)p Fg(\))2571 4400 y(1)2712 4373 y Fp(giv)m(en)f(b)m(y)h(the)g(C{E) f(pro)s(cedure)0 4493 y(in)33 b(\(3.18\))f(and)i(the)f(remark)f(b)s (elo)m(w)i(\(3.21\).)120 4612 y(Hence,)f(the)h(\()p Fl(N)f Fp(+)22 b(1\))955 4576 y Fk(th)1070 4612 y Fp(momen)m(t)31 b(is)j(giv)m(en)f(b)m(y)673 4842 y Fl(\032)725 4801 y Fg(\()p Fk(N)7 b Fg(+1\))991 4842 y Fp(=)p Fl(\032)17 b(m)1224 4790 y Fg(\()p Fk(N)7 b Fg(+1\))1224 4871 y Fk(P)1090 5066 y Fp(+)23 b Fl(")1236 4926 y Fi(\032)1311 5066 y Fl(\034)11 b(m)1453 5014 y Fg(\()p Fk(N)c Fg(+1\))1453 5095 y Fk(P)1692 5066 y Fj(r)1775 5081 y Fk(x)1825 5066 y Fp(\()p Fl(\032u)p Fp(\))22 b Fj(\000)g Fl(\034)11 b Fj(r)2271 5081 y Fk(x)2338 4956 y Fi(\020)2397 5066 y Fl(\032m)2536 5014 y Fg(\()p Fk(N)c Fg(+2\))2536 5095 y Fk(P)2775 4956 y Fi(\021)1190 5345 y Fj(\000)23 b Fl(\034)11 b(\021)t(\032@)1503 5360 y Fk(u)1554 5345 y Fl(m)1641 5293 y Fg(\()p Fk(N)c Fg(+1\))1641 5374 y Fk(P)1880 5345 y Fl(u)1937 5360 y Fk(t)1994 5345 y Fp(+)23 b(\()p Fl(N)33 b Fp(+)23 b(1\))p Fl(u)p Ff(\015)-57 b(_)13 b Fl(\032)2615 5293 y Fg(\()p Fk(N)7 b Fg(\))2615 5372 y(1)2752 5205 y Fi(\033)2849 5345 y Fp(+)23 b Fj(O)s Fp(\()p Fl(")3116 5304 y Fg(2)3161 5345 y Fp(\))p Fl(:)-3228 b Fp(\(4.9\))p eop %%Page: 21 21 21 20 bop 3815 -139 a Fn(21)120 100 y Fp(Clearly)-8 b(,)31 b(the)i(momen)m(t)e(v)-6 b(ariables)32 b(obtained)h(b)m(y)g(the)g (Chapman{Ensk)m(og)e(expansion)j(agree)e(with)0 219 y(the)i(ones)g (from)e(these)i(closures)g(to)f(\014rst)h(order)f(in)h(the)f(scaled)h (mean)f(free)h(path)g(parameter)e Fl(")p Fp(.)120 339 y(Notice)g(that)h(for)h Fl(N)k Fp(=)28 b(0)33 b(this)h(b)s(ecomes)0 578 y(\(4.10\))341 b Fl(\032)649 537 y Fg(\(1\))783 578 y Fj(\031)29 b Fl(\032)17 b(u)k Fp(+)h Fl(")17 b(\034)28 b Fj(f\000r)1480 593 y Fk(x)1553 578 y Fj(\001)21 b Fp(\()p Fl(\032)p Fp(\()p Fl(\022)s(I)29 b Fp(+)22 b(2)p Fl(u)p Ff(\015)-56 b(_)13 b Fl(u)p Fp(\)\))22 b(+)g Fl(u)p Fj(r)2529 593 y Fk(x)2601 578 y Fj(\001)g Fp(\()p Fl(\032u)p Fp(\))f Fj(\000)i Fl(\021)t(\032@)3117 593 y Fk(t)3152 578 y Fl(u)p Fj(g)16 b Fl(;)0 817 y Fp(whic)m(h)34 b(reco)m(v)m(ers)g (\(3.22\),)e(with)h Fl(u)27 b Fp(=)i Fj(\000)p Fl(\034)e(\021)21 b(E)6 b Fp(.)120 936 y(See)43 b(that,)g(b)m(y)g(Lemma)d(3,)k Fj(\000)p Fl(u)1328 951 y Fk(t)1406 936 y Fp(is)e(appro)m(ximated)f(to) h(order)h Fl(")f Fp(b)m(y)h Fl(\034)2878 892 y Fk(\015)p 2858 914 86 4 v 2858 972 a(\021)2902 952 y Fe(d)2956 936 y Fp(\()p Fl(\032)17 b(u)27 b Fp(+)h Fl(!)t Fp(\).)71 b(Hence)43 b(the)0 1077 y(momen)m(t)32 b(equations)h(for)h(this)f (closure)h(are)g(up)g(to)f(an)g(order)h(of)g Fl(")2503 1041 y Fg(2)2581 1077 y Fp(the)g(same)f(as)g(those)h(for)g(the)f(C{E)0 1196 y(expansion,)g(i.e.)44 b(system)32 b(\(\(3.21\).)0 1435 y Fq(Higher)45 b(order)h(momen)m(ts)40 b Fp(F)-8 b(or)39 b Fl(N)50 b Fp(=)39 b(1)h(w)m(e)h(obtain)e(an)h(appro)m (ximation)f(to)g(the)h(energy)h(as)f(the)0 1555 y(second)35 b(momen)m(t)c(of)j Fl(F)14 b Fp(.)44 b(It)32 b(b)s(ecomes)45 1829 y Fl(W)42 b Fp(=)28 b Fl(\032)338 1788 y Fg(\(2\))472 1829 y Fp(=)g Fl(\032)p Fp(\()17 b Fl(\022)s(I)28 b Fp(+)23 b(2)p Fl(u)p Ff(\015)-57 b(_)14 b Fl(u)j Fp(\))k(+)i Fl("\034)1421 1688 y Fi(\032)1518 1829 y Fj(\000)f Fp(3)p Fj(r)1750 1844 y Fk(x)1823 1829 y Fj(\001)f Fp(\()p Fl(\032)c(u)p Ff(\015)-57 b(_)30 b Fp(\()o Fl(\022)s(I)g Fp(+)22 b(2)p Fl(u)p Ff(\015)-57 b(_)14 b Fl(u)p Fp(\)\))i(+)1512 2108 y(\()p Fl(\022)j(I)30 b Fp(+)23 b(2)p Fl(u)p Ff(\015)-57 b(_)13 b Fl(u)p Fp(\))j Fj(r)2164 2123 y Fk(x)2237 2108 y Fj(\001)22 b Fp(\()o Fl(\032)17 b(u)p Fp(\))k Fj(\000)i Fp(4)p Fl(\034)11 b(\021)20 b(\032u@)2947 2123 y Fk(t)2982 2108 y Fl(u)3039 1967 y Fi(\033)3136 2108 y Fp(+)i Fl(")p Fp(2)p Fl(u)p Fp(\()p Fl(\032)3479 2067 y Fg(\(1\))3608 2108 y Fj(\000)h Fl(\032u)p Fp(\))472 2387 y(=)28 b Fl(\032)p Fp(\()17 b Fl(\022)s(I)28 b Fp(+)23 b(2)p Fl(u)p Ff(\015)-57 b(_)14 b Fl(u)j Fp(\))k(+)i Fl("\034)1421 2246 y Fi(\032)1518 2387 y Fj(\000)f Fp(2)p Fj(r)1750 2402 y Fk(x)1823 2387 y Fj(\001)f Fp(\()p Fl(\032)c(u)p Ff(\015)-57 b(_)30 b Fp(\()o(2)p Fl(\022)s(I)f Fp(+)23 b(3)p Fl(u)p Ff(\015)-57 b(_)14 b Fl(u)p Fp(\))o(\))1518 2666 y(+)22 b(2)p Fl(\032)17 b Fp(\()o Fl(\022)i(I)30 b Fj(\000)23 b Fl(u)p Ff(\015)-57 b(_)13 b Fl(u)p Fp(\))k Fj(r)2338 2681 y Fk(x)2410 2666 y Fj(\001)22 b Fl(u)g Fj(\000)h Fl(\032)17 b(u)p Fj(r)2848 2681 y Fk(x)2897 2666 y Fp(\()p Fl(\022)s(I)8 b Fp(\))21 b Fj(\000)h Fp(6)p Fl(\021)f(\032u@)3479 2681 y Fk(t)3513 2666 y Fl(u)3570 2525 y Fi(\033)3645 2666 y Fl(:)-3673 b Fp(\(4.11\))120 2994 y(Similarly)-8 b(,)35 b Fl(\032)620 2958 y Fg(\(3\))761 2994 y Fp(=)g(3)p Fl(\032)p Fp(\()p Fl(\022)s(I)8 b Ff(\015)-58 b(_)13 b Fl(u)25 b Fp(+)g(2)p Fl(u)p Ff(\015)-57 b(_)14 b Fl(u)p Ff(\015)-57 b(_)13 b Fl(u)p Fp(\))25 b(+)g Fl(\032)1952 2942 y Fg(\(3\))1952 3021 y(1)2059 2994 y Fp(,)38 b(with)f Fl(\032)2408 2942 y Fg(\(3\))2408 3021 y(1)2552 2994 y Fp(\(4.8\))f(for)i Fl(N)45 b Fp(=)35 b(2,)j(and)g Fl(\032)3554 2942 y Fg(\(2\))3554 3021 y(1)3698 2994 y Fp(from)0 3114 y(the)c Fl(")f Fp(order)h(term)e (in)h(\(4.11\).)120 3233 y(W)-8 b(riting)36 b(the)h(equations)g(for)g (the)g(momen)m(ts)f(up)h(to)g(the)g(second)h(one)g(w)m(e)f(obtain)g(a)f (system)g(that)0 3353 y(mo)s(dels)31 b(mass,)g(momen)m(tum)f(and)i (energy)-8 b(,)32 b(where)g Fl(\032)1994 3317 y Fg(\(3\))2133 3353 y Fp(pro)m(vides)g(the)g(closure)h(for)f(the)g(energy)g(equa-)0 3472 y(tion.)120 3592 y(This)g(closure)h(yields)f(drift{di\013usiv)m(e) 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1428 3385 a Fl(j)1469 3400 y Fg(0)p Fh(\000)1598 3385 y Fj(\001)h Fl(n)2031 3317 y Fp(and)233 b(\010)28 b(=)g(constan)m(t)133 b(on)67 b(\000)3364 3332 y Fh(D)3433 3317 y Fl(;)329 3599 y Fp(where)29 b Fl(j)653 3614 y Fg(1)p Fh(\000)760 3599 y Fp(,)h Fl(j)859 3614 y Fg(0)p Fh(\000)995 3599 y Fp(and)f Fl(\032)1236 3614 y Fg(0)p Fh(\000)1371 3599 y Fp(are)g(giv)m(en)g(b)m(y)g(the)g(kinetic) f(computation)g(on)h(the)g(left)g(side)h(of)f(the)329 3718 y(b)s(oundary)k(la)m(y)m(er.)44 b(Then)34 b(the)g(condition)f(for) h Fl(!)i Fp(can)e(b)s(e)g(computed)f(as)0 3949 y(\(5.12.4\))132 b Fl(!)26 b Fj(\001)c Fl(s)27 b Fp(=)h(\()p Fl(E)896 3964 y Fk(t)953 3949 y Fp(+)22 b Fl(j)1093 3964 y Fg(0)1138 3949 y Fp(\))g Fj(\001)g Fl(s)27 b Fp(=)h(\()p Fj(r)p Fp(\010)1622 3964 y Fk(t)1679 3949 y Fp(+)23 b Fl(\026\032E)6 b Fp(\))21 b Fj(\001)g Fl(s)28 b Fp(=)g(\()p Fj(r)p Fp(\010)22 b Fj(\001)g Fl(s)p Fp(\))2612 3964 y Fk(t)2668 3949 y Fj(\000)h Fl(\026\032)p Fj(r)p Fp(\010)e Fj(\001)h Fl(s)28 b Fp(=)g(0)132 b(on)34 b(\000)3669 3964 y Fh(D)3738 3949 y Fl(:)0 4925 y Fq(Ac)m(kno)m(wledgemen)m(t:)47 b Fp(The)27 b(researc)m(h)h(of)f(the)g(\014rst)g(author)f(is)h(supp)s(orted)h(b)m (y)e(M.U.R.S.T.)g(of)h(Italy)-8 b(.)0 5045 y(The)32 b(researc)m(h)g(of) f(the)h(second)g(author)f(is)g(supp)s(orted)h(b)m(y)f(the)h(National)d (Science)k(F)-8 b(oundation)31 b(under)0 5164 y(gran)m(t)i(DMS)h (9623037.)1660 5400 y Fa(References)p eop %%Page: 26 26 26 25 bop 0 -139 a Fn(26)0 100 y Fp([BCN])32 b(C.)h(Bardos,)f(R.)i(E.)e (Ca\015ish)i(and)g(B.)e(Nicolaenk)m(o,)g Fo(The)i(Milne)f(and)h (Kramers)f(Problems)g(for)0 219 y(the)39 b(Boltzmann)d(Equation)i(of)g (a)g(Hard)g(Sphere)i(Gas,)f Fc(Comm.)60 b(Pur)-5 b(e)40 b(Appl.)59 b(Math.)p Fp(,)39 b Fq(39)p Fp(,)h(323-352)0 339 y(\(1986\).)0 467 y([BW])46 b(H.U.)g(Barenger)i(and)g(J.W.)f (Wilkins,)j Fo(Ballistic)45 b(structure)k(in)e(the)g(electron)h (distribution)0 587 y(function)26 b(of)f(small)e(semiconducting)i (structures:)41 b(General)25 b(features)h(and)f(sp)s(eci\014c)h(trends) g Fp(,)g(Ph)m(ysical)0 706 y(Review)33 b(B,)g(v)m(ol.)43 b(36,)33 b(No)f(3,)h(pp.)45 b(1487{1502)31 b(\(1987\).)0 835 y([BD])f(N.)g(Ben)h(Ab)s(dallah)f(and)h(P)-8 b(.)30 b(Degond,)i Fo(On)g(a)e(hierarc)m(h)m(y)i(of)f(macroscopic)f(mo)s(dels) g(for)h(semicon-)0 954 y(ductors)p Fp(,)j(J.)f(Math.)44 b(Ph)m(ys.)g Fq(37)34 b Fp(\(7\))e(3306{3333)g(\(1996\).)0 1083 y([BDG])49 b(N.)g(Ben)h(Ab)s(dallah,)i(P)-8 b(.)49 b(Degond)h(and)h(S.)e(Genieys,)54 b Fo(An)c(Energy-T)-8 b(ransp)s(ort)49 b(mo)s(del)g(for)0 1202 y(semiconductors)e(deriv)m(ed) f(from)g(the)g(Boltzmann)f(Equation,)j Fp(J.)d(Statist.)82 b(Ph)m(ys.)h Fq(84)46 b Fp(no.)83 b(1{2,)0 1322 y(202{231)32 b(\(1996\).)0 1450 y([C1])g(C.)h(Cercignani,)h Fo("The)g(Boltzmann)d (Equation)i(and)h(its)f(Application")p Fp(,)f(Springer,)i(New)f(Y)-8 b(ork.)0 1570 y(\(1988\).)0 1698 y([C2])48 b(C.)h(Cercignani,)k Fo("Half-space)d(problems)f(in)g(the)g(Kinetic)h(Theory)f(of)h(Gases")p Fp(,)j Fc(T)-8 b(r)j(ends)50 b(in)0 1818 y(Applic)-5 b(ations)47 b(of)g(Pur)-5 b(e)48 b(Mathematics)g(to)g(Me)-5 b(chanics)p Fp(,)48 b(E.)e(Kr\177)-50 b(oner)48 b(and)f(K.)g(Kirc)m (hg\177)-50 b(assner)48 b(Eds.,)0 1937 y(Springer,)34 b(Berlin)f(\(1986\).)0 2066 y([CGL])e(C.)h(Cercignani,)g(I.M.)f(Gam)m (ba)g(and)i(C.L.)e(Lev)m(ermore,)g Fo(A)h(High)f(Field)h(Appro)m (ximation)e(to)h(a)0 2185 y(Boltzmann{P)m(oisson)e(System)g(in)h (Bounded)h(Domains)p Fp(,)e(Applied)h(Math)g(Letters,)h(V)-8 b(ol)29 b(\(4\),)g(111{118)0 2305 y(\(1997\).)0 2433 y([CIP])36 b(C.)h(Cercignani,)i(R.)e(Illner)h(and)f(M.)h(Pulv)m(eren)m (ti,)g Fo("The)g(Mathematical)e(Theory)h(of)h(Diluted)0 2553 y(Gases")p Fp(,)c(Springer{V)-8 b(erlag)32 b(\(1994\).)0 2681 y([DSR])e(P)-8 b(.)30 b(Dmitruc)m(k,)g(L.)g(Reyna)g(and)h(A.)f (Saul,)h Fo(High)f(Electric)g(Field)g(Appro)m(ximation)f(to)g(Charged)0 2801 y(T)-8 b(ransp)s(ort)29 b(in)h(Semiconductor)f(Devices.)44 b Fp(Appl.)e(Math.)h(Lett.)g(V)-8 b(ol.)42 b(5,)29 b(No)g(3,)h(pp.)43 b(99-102,)30 b(\(1992\).)0 2929 y([F)m(G])36 b(D.K.)h(F)-8 b(erry)36 b(and)h(R.O.)g(Grondin,)h Fo("Ph)m(ysics)f(of)g(submicron)f (devices",)i Fp(Plen)m(um)e(Press,)i(New)0 3049 y(Y)-8 b(ork)32 b(\(1991\).)0 3177 y([FV])37 b(G.)h(F)-8 b(rosali,)38 b(C.V.M.)f(v)-6 b(an)38 b(der)h(Mee)g(and)g(S.L.)f(P)m(a)m(v)m(eri)f(F) -8 b(on)m(tana,)39 b Fo(Conditions)f(for)h(run{a)m(w)m(a)m(y)0 3297 y(phenomena)g(in)e(the)i(kinetic)e(theory)h(of)g(particle)g(sw)m (arms)p Fp(,)g(J.)g(Math.)59 b(Ph)m(ysics)38 b Fq(30)p Fp(\(5\),)h(1177{1186)0 3416 y(\(1989\).)0 3545 y([GK])34 b(F.)f(Golse)h(and)g(A.)f(Klar,)h Fo(A)f(Numerical)f(Metho)s(ds)j(for)f (Computing)f(Asymptotic)f(States)h(and)0 3664 y(Outgoing)j (Distributions)g(for)g(Kinetic)h(Linear)f(Half)f(Space)i(Problems,)f Fp(J.)g(Statist.)51 b(Ph)m(ys.)i Fq(80)36 b Fp(no.)0 3784 y(5{6,)c(1033{1061)g(\(1995\).)0 3912 y([GP])52 b(F.)g(Golse)g(and)h(F.)f(P)m(oupaud,)58 b Fo(Limite)51 b(\015uide)j(des)f(e)q(\023)-51 b(quations)52 b(de)h(Boltzmann)e(des)j (semi-)0 4032 y(conducteurs)35 b(p)s(our)f(une)g(statistique)e(de)i(F) -8 b(ermi-Dirac,)32 b Fp(Asymptotic)f(Analysis)i Fq(6)g Fp(135-160)g(\(1992\).)0 4160 y([Kl])c(A.)g(Klar,)h Fo (Asymptotic{Induced)f(Domain)g(Decomp)s(osition)f(Metho)s(ds)i(for)g (Kinetic)g(and)g(Drift{)0 4280 y(Di\013usion)k(Semiconductor)f (Equations)p Fp(,)f(to)h(app)s(ear)h(in)f(SIAM)h(Scien)m(ti\014c)g (Comput.)43 b(\(1996\).)0 4408 y([MRS])h(P)-8 b(.)42 b(Mark)m(o)m(wic)m(h,)k(C.A.)c(Ringhofer)i(and)g(C.)f(Sc)m(hmeiser,)92 b Fo(\\Semiconductor)43 b(Equations",)0 4528 y Fp(Springer,)34 b(Wien-New)g(Y)-8 b(ork,)32 b(\(1989\).)0 4656 y([D])h(B.)f(Da)m (vison,)h Fo(\\Neutron)g(T)-8 b(ransp)s(ort)33 b(Theory")p Fp(,)g(Oxford)h(Univ)m(ersit)m(y)e(Press,)i(Oxford)g(\(1957\).)0 4785 y([M])e(J.)g(C.)f(Maxw)m(ell,)h Fo("On)g(stresses)i(in)e (rari\014ed)g(gases)h(arising)f(from)f(inequalities)g(of)i(temp)s (erature")0 4904 y Fc(Phil.)45 b(T)-8 b(r)j(ans.)46 b(R)-5 b(oyal)35 b(So)-5 b(c.)44 b Fq(170)p Fp(,)34 b(231-256,)e(App)s(endix)h (\(1879\).)0 5032 y([P1])40 b(F.)g(P)m(oupaud,)j Fo(Runa)m(w)m(a)m(y)f (Phenomena)f(and)g(Fluid)g(Appro)m(ximation)d(Under)k(High)e(Fields)h (in)0 5152 y(Semiconductor)34 b(Kinetic)f(Theory)-8 b(,)33 b Fp(Z.Angew.)43 b(Math.)i(Mec)m(h.)g Fq(72)39 b Fp(8,)32 b(359-372,)g(\(1992\).)0 5280 y([P2])41 b(F.)g(P)m(oupaud,)k Fo(Deriv)-6 b(ation)41 b(of)h(a)g(h)m(ydro)s(dynamic)e(systems)i (hierarc)m(h)m(y)g(from)f(the)h(Boltzmann)0 5400 y(equation,)32 b Fp(Appl.Math.Lett.)44 b(v)m(ol)33 b(4,)f(75-79,)h(\(1991\).)p eop %%Page: 27 27 27 26 bop 3815 -139 a Fn(27)0 100 y Fp([Rg])29 b(L.)g(Reggiani,)h Fo("Hot{Electron)e(T)-8 b(ransp)s(ort)29 b(in)g(Semiconductors")h Fp(T)-8 b(opics)30 b(in)f(Applied)h(Ph)m(ysics,)0 219 y(Springer{V)-8 b(erlag,)33 b(New)g(Y)-8 b(ork)32 b(\(1985\).)0 339 y([SS])i(P)-8 b(.C.)33 b(Stic)m(hel)i(and)g(D.)f(Strothmann)f Fo(Asymptotic)g(analysis)h(of)g(the)h(high)f(\014eld)h(semiconductor)0 458 y(Boltzmann)d(equation)p Fp(,)g(Ph)m(ysica)i(A)e Fq(202)p Fp(,)i(553{576,)e(\(1994\).)0 578 y([TT])j(S.)g(A.)g(T)-8 b(rugman)34 b(and)i(A.)e(J.)i(T)-8 b(a)m(ylor,)34 b Fo(Analytic)g (solution)h(of)h(the)f(Boltzmann)f(equation)h(with)0 697 y(applications)43 b(to)g(electron)h(transp)s(ort)f(in)h (inhomogeneous)f(semiconductors,)p Fp(Ph)m(ys.)75 b(Rev.)g(B)43 b Fq(33)p Fp(,)0 817 y(5575{5584)32 b(\(1986\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF