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Text File  |  1995-11-25  |  5KB  |  73 lines
  1. bop 17 -105 a Fk(FINITE)19 b(LA)-5 b(TTICE)20 b(SYSTEMS)g(WITH)f(TR)n(UE)h
  2. (CRITICAL)f(BEHA)-6 b(VIOR)767 -45 y(OUTLINE)-7 59 y Fj(I.)25
  3. b(Ov)o(erview:)133 139 y([a.])32 b(Lattice)21 b(mo)q(dels)e(are)i(widely)e
  4. (used)i(in)e(ph)o(ysics)i(to)f(either)h(mimic)c(or)k(appro)o(ximate)50
  5. 199 y(ph)o(ysical)14 b(systems.)23 b(In)14 b(statistical)j(mec)o(hanics,)d
  6. (for)g(example,)g(the)h(system)g(under)g(study)h(ma)o(y)50
  7. 259 y(b)q(e)e(a)h(crystal)h(lattice.)22 b(In)14 b(quan)o(tum)f(\014eld)h
  8. (theory)l(,)h(lattices)h(are)f(used)g(as)g(discrete)i(appro)o(xima-)50
  9. 319 y(tions)g(to)f(spacetime.)133 399 y([b.])28 b(Lattice)20
  10. b(systems)g(are)g(generally)f(w)o(ell)g(suited)g(for)g(n)o(umerical)e(sim)o
  11. (ulation,)g(but)i(the)50 459 y(adaptation)g(of)h(the)h(system)g(to)g(the)g
  12. (computer)f(requires)i(that)f(the)g(lattice)h(b)q(e)e(made)f(\014nite)50
  13. 519 y(and)c(that)i(appropriate)g(conditions)f(b)q(e)h(imp)q(osed)e(at)i(the)g
  14. (b)q(oundaries.)133 599 y([c.])j(Once)15 b(these)g(systems)g(are)f(reduced)g
  15. (to)g(manageable)d(sizes,)16 b(ho)o(w)o(ev)o(er,)d(they)i(no)e(longer)50
  16. 659 y(displa)o(y)f(the)i(sharp)g(critical)g(b)q(eha)o(vior)f(whic)o(h)f(is)i
  17. (represen)o(ts)h(a)f(phase)f(transition)h(in)f(statistical)50
  18. 719 y(mec)o(hanics)h(or)i(sp)q(on)o(taneously)g(brok)o(en)f(symmetry)g(in)h
  19. (\014eld)f(theory)l(.)22 b(This)16 b(is)g(unfortunate,)f(if)50
  20. 779 y(y)o(our)h(purp)q(ose)h(is)g(to)g(study)h(the)e(phase)h(structure)i(of)c
  21. (a)i(theory)l(.)133 859 y([d.])41 b(Giv)o(e)22 b(example)g(of)h
  22. (magnetization.)40 b(Second)23 b(order)h(phase)f(transition)h(at)g(Curie)50
  23. 919 y(temp)q(erature.)e(Result)17 b(gets)g(sharp)q(er)i(as)e(lattice)g(size)g
  24. (increases.)133 999 y([e.])22 b(It)17 b(is)h(p)q(ossible,)f(ho)o(w)o(ev)o
  25. (er,)f(to)h(construct)i(\014nite)d(lattice)i(systems)g(whic)o(h)e(exhibit)g
  26. (true)50 1059 y(critical)f(b)q(eha)o(vior.)22 b(These)16 b(systems)g(ma)o(y)d
  27. (b)q(e)j(useful)e(when)h(the)g(phase)g(structure)j(of)c(a)h(theory)50
  28. 1119 y(is)20 b(of)e(primary)h(in)o(terest.)33 b(The)19 b(metho)q(d)g(emplo)o
  29. (ys)f(a)i(sp)q(ecial)g(t)o(yp)q(e)g(of)f(b)q(oundary)g(condition,)50
  30. 1180 y(whic)o(h)11 b(w)o(e)h(call)h Fi(self-adjusting)e Fj(or)i
  31. Fi(adaptive)p Fj(.)22 b(Increasingly)13 b(larger)g(lattices)h(constitute)g(a)
  32. e(series)50 1240 y(of)h(non-p)q(erturbativ)o(e)h(appro)o(ximations,)f(the)i
  33. (\014rst)h(of)d(whic)o(h)g(is)i(equiv)m(alen)o(t)g(to)f(the)h(mean)e(\014eld)
  34. 50 1300 y(appro)o(ximation.)133 1380 y([f.])20 b(So,)c(I)g(will)g(b)q(egin)g
  35. (with)g(a)h(discussion)g(of)e(the)i(mean)d(\014eld)i(appro)o(ximation,)e(and)
  36. h(sho)o(w)50 1440 y(ho)o(w)h(it)h(is)h(related)f(to)g(\014nite)g(lattice)h
  37. (systems)g(with)f(sp)q(ecial)h(b)q(oundary)e(conditions.)23
  38. b(Most)18 b(of)50 1500 y(the)f(discussion)h(will)f(b)q(e)g(in)g(the)g(con)o
  39. (text)h(of)f(d-dimensional)d(Euclidean)i(lattice)i(\014eld)e(theory)l(,)50
  40. 1560 y(but)i(the)g(translation)i(to)e(statistical)j(mec)o(hanics)c(is)i
  41. (straigh)o(t-forw)o(ard.)27 b(Then)18 b(I)h(will)f(discuss)50
  42. 1620 y(a)e(few)g(example)f(applications.)-27 1719 y(I)q(I.)26
  43. b(Lattice)17 b(Mean)f(Field)h(Appro)o(ximation:)133 1799 y([a.])k(Consider)15
  44. b(\014eld)f(theoretic)i(mo)q(del)e(with)h(one)g(or)g(more)f(scalar)i
  45. (\014elds)f(describ)q(ed)h(b)o(y)e(an)50 1859 y(action)i Fh(S)s
  46. Fj([)p Fh(\036)p Fj(])e(=)356 1824 y Fg(R)392 1859 y Fh(d)418
  47. 1841 y Ff(d)437 1859 y Fh(x)p Fe(L)p Fj(\()p Fh(\036)p Fj(\).)133
  48. 1939 y([b.])33 b(Appro)o(ximate)19 b(this)i(theory)g(on)f(a)h(d-dimensional)c
  49. (cubic)j(lattice)h(with)g Fh(N)26 b Fj(in)o(terior)50 1999
  50. y(sites)c(in)e(eac)o(h)g(direction,)h(spacing)f Fh(a)p Fj(,)i(and)e(length)g
  51. Fh(L)g Fj(=)h(\()p Fh(N)f Fj(+)14 b(1\))p Fh(a)p Fj(.)35 b(Lab)q(el)20
  52. b(the)h(sites)h(with)50 2059 y Fh(x)13 b Fj(=)h(\()p Fh(n)193
  53. 2066 y Fd(1)212 2059 y Fh(;)8 b(:::;)g(n)328 2066 y Ff(d)347
  54. 2059 y Fj(\))p Fh(a)p Fj(,)51 b(0)13 b Fe(\024)h Fh(n)578 2066
  55. y Ff(i)606 2059 y Fe(\024)f Fh(N)k Fj(+)12 b(1.)133 2139 y([c.])25
  56. b(The)17 b(lattice)i(action,)e Fh(S)673 2146 y Ff(N)705 2139
  57. y Fj([)p Fh(\036)p Fj(])e(=)833 2106 y Fg(P)877 2149 y Ff(x)906
  58. 2139 y Fh(a)932 2121 y Ff(d)952 2139 y Fe(L)p Fj(\()p Fh(\036)p
  59. Fj(\),)j(can)g(usually)f(b)q(e)h(de\014ned)f(to)h(con)o(tain)f(at)50
  60. 2199 y(most)f(next-neigh)o(b)q(or)g(in)o(teractions.)22 b(Av)o(erage)c(v)m
  61. (alues)f(of)f(observ)m(ables)i(are)f(calculated)f(with)699
  62. 2327 y Fe(h)p Fh(A)p Fe(i)774 2334 y Ff(N)820 2327 y Fj(=)878
  63. 2258 y Fg(R)906 2294 y Fj([)p Fh(d\036)p Fj(])p Fh(A)p Fj([)p
  64. Fh(\036)p Fj(])p Fh(e)1108 2275 y Fc(\000)p Ff(S)1154 2279
  65. y Fb(N)1180 2275 y Fd([)p Ff(\036)p Fd(])p 878 2316 343 2 v
  66. 925 2326 a Fg(R)953 2361 y Fj([)p Fh(d\036)p Fj(])p Fh(e)1060
  67. 2347 y Fc(\000)p Ff(S)1106 2351 y Fb(N)1132 2347 y Fd([)p Ff(\036)p
  68. Fd(])50 2453 y Fj(with)661 2513 y([)p Fh(d\036)p Fj(])d(=)811
  69. 2472 y Fg(Y)817 2560 y Ff(i;x)872 2513 y Fh(d\036)928 2493
  70. y Ff(i)942 2513 y Fj(\()p Fh(x)p Fj(\))p Fh(;)22 b Fj(1)14
  71. b Fe(\024)f Fh(i)h Fe(\024)g Fh(n:)888 2678 y Fj(1)p eop
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