TP 0:Importnce Smpling-The Metropoli Algorithm-The Iing Model-The Jckknife Method June, 200 The Cnonicl Enemble We conider phyicl ytem which re in therml contct with n environment. The environment i uully much lrger thn the phyicl ytem of interet nd conequence energy exchnge between the two of them will not chnge the temperture of the environement. The environement i clled het bth or het reervoir. When the ytem reche equilibrium with the het bth it temperture will be given by the temperture of the het bth. A ytem in equilibrium with het bth i decribed ttiticlly by the cnonicl enemble in which the temperture i fixed. In contrt n iolted ytem i decribed ttiticlly by the microcnonicl enemble in which the energy i fixed. Mot ytem in nture re not iolted but re in therml contct with the environment. It i fundmentl reult of ttiticl mechnic tht the probbility of finding ytem in equilibrium with het bth t temperture T in microtte with energy E i given by the Boltzmnn ditribution The normliztion conntnt Z i the prtition function. It i defined by P = Z e βe, β = k T. () Z = e βe. (2) The um i over ll the microtte of the ytem with fixed N nd V. The Helmholtz free energy F of ytem i given by F = kt lnz. (3) In equilibrium the free energy i minimum. All other thermodynmicl quntitie cn be given by vriou derivtive of F. For exmple the internl energy U of the ytem which i the expecttion vlue of the energy cn be expreed in term of F follow The pecific het i given by U =< E >= 2 Importnce Smpling E P = Z E e βe = β lnz = (βf). (4) β C v = U. (5) T In ny Monte Crlo integrtion the numericl error i proportionl to the tndrd devition of the integrnd nd i inverely proportionl to the number of mple. Thu in order to reduce the error we hould either reduce
the vrince or incree the number of mple. The firt option i preferble ince it doe not require ny extr computer time. Importnce mpling llow u to reduce the tndrd devition of the integrnd nd hence the error by mpling more often the importnt region of the integrl where the integrnd i lrget. Importnce mpling ue lo in crucil wy nonuniform probbility ditribution. Let u gin conider the one dimenionl integrl F = We introduce the probbility ditribution p(x) uch tht dx f(x). (6) We write the integrl = dx p(x). (7) F = dx p(x) f(x) p(x). (8) We evlute thi integrl by mpling ccording to the probbility ditribution p(x). In other word we find et of N rndom number x i which re ditributed ccording to p(x) nd then pproximte the integrl by the um F N = N f(x i ) p(x i ). (9) The probbility ditribution p(x) i choen uch tht the function f(x)/p(x) i lowly vrying which reduce the correponding tndrd devition. 3 The Metropoli Algorithm The internl energy U =< E > cn be put into the form < E >= E e βe. (0) e βe Generlly given ny phyicl quntity A it expecttion vlue < A > cn be computed uing imilr expreion, viz < A >= A e βe. () e βe The number A i the vlue of A in the microtte. In generl the number of microtte N i very lrge. In ny Monte Crlo imultion we cn only generte very mll number n of the totl number N of the microtte. In other word < E > nd < A > will be pproximted with < E > < E > n = = E e βe = e βe. (2) < A > < A > n = = A e βe = e βe. (3) The clcultion of < E > n nd < A > n proceed therefore by ) chooing t rndom microtte, 2) computing E, A nd e βe then 3) evluting the contribution of thi microtte to the expecttion vlue 2
< E > n nd < A > n. Thi generl Monte Crlo procedure i however highly inefficient ince the microtte i very improbble nd therefore it contribution to the expecttion vlue i negligible. We need to ue importnce mpling. To thi end we introduce probbility ditribution p nd rewrite the expecttion vlue < A > A p < A >= e βe p. (4) p e βe p Now we generte the microtte with probbilitie p nd pproximte < A > with < A > n given by = A p < A > n = e βe =. (5) p e βe Thi i importntce mpling. The Metropoli lgorithm i importnce mpling with p given by the Boltzmnn ditribution, i.e p = e βe = e βe. (6) We get then the rithmetic verge < A > n = n n A. (7) The Metropoli lgorithm in the ce of pin ytem uch the Iing model cn be ummrized follow ) Chooe n initil microtte. 2) Chooe pin t rndom nd flip it. 3) Compute E = E tril E old. Thi i the chnge in the energy of the ytem due to the tril flip. 4) Check if E 0. In thi ce the tril microtte i ccepted. 5) Check if E > 0. In thi ce compute the rtio of probbilitie w = e β E. = 6) Chooe uniform rndom number r in the inetrvl [0, ]. 7) Verify if r w. In thi ce the tril microtte i ccepted, otherwie it i rejected. 8) Repet tep 2) through 7) until ll pin of the ytem re teted. Thi weep count one unit of Monte Crlo time. 9) Repet etp 2) through 8) ufficient number of time until thermliztion (i.e equilibrium) i reched. 0) Compute the phyicl quntitie of interet in n thermlized microtte. Thi cn be done periodiclly in order to reduce correltion between the dt point. ) Compute verge. We kip here the proof tht thi lgorithm led indeed to equence of tte which re ditributed ccording to the Boltzmnn ditribution. It i cler tht the tep 2) through 7) correpond to trnition probbility between the microtte { i } nd { j } given by W(i j) = min(, e β E ), E = E j E i. (8) 3
Since only the rtio of probbilitie w = e β E i needed it i not necery to normlize the Boltzmnn probbility ditribution. It i cler tht thi probbility function tifie the detiled blnce condition W(i j) e βei = W(j i) e βej. (9) Any other probbility function W which tifie thi condition will generte equence of tte which re ditributed ccording to the Boltzmnn ditribution. Thi cn be hown by umming over the index j in the bove eqution nd uing j W(i j) =. We get e βei = j W(j i) e βej. (20) The Boltzmnn ditribution i n eigenvector of W. In other word W leve the equilibrium enemble in equilibrium. A it turn out thi eqution i lo ufficient condition for ny enemble to pproch equilibrium. 4 The Het-Bth Algorithm The het-bth lgorithm i generlly le efficient lgorithm thn the Metropoli lgorithm. The cceptnce probbility i given by W(i j) = min(, + e β E ), E = E j E i. (2) Thi cceptnce probbility tifie lo detiled blnce for the Boltzmnn probbility ditribution. In other word the detiled blnce condition which i ufficient but not necery for n enemble to rech equilibrium doe not hve unique olution. 5 The Iing Model We conider d dimenionl periodic lttice with n point in every direction o tht there re N = n d point in totl in thi lttice. In every point (lttice ite) we put pin vrible i (i =,..., N) which cn tke either the vlue + or. A configurtion of thi ytem of N pin i therefore pecified by et of number { i }. In the Iing model the energy of thi ytem of N pin in the configurtion { i } i given by E I { i } = <ij> ǫ ij i j H i. (22) The prmeter H i the externl mgnetic field. The ymbol < ij > tnd for neret neighbor pin. The um over < ij > extend over γn 2 term where γ i the number of neret neighbor. In 2, 3, 4 dimenion γ = 4, 6, 8. The prmeter ǫ ij i the interction energy between the pin i nd j. For iotropic interction ǫ ij = ǫ. For ǫ > 0 we obtin ferromgnetim while for ǫ < 0 we obtin ntiferromgnetim. We conider only ǫ > 0. The energy i E I { i } = ǫ <ij> i j H i. (23) The prtition function i given by Z =... e βei{i}. (24) 2 N There re 2 N term in the um nd β = k BT. 4
In d = 2 we hve N = n 2 pin in the qure lttice. The configurtion { i } cn be viewed n n n mtrix. We impoe periodic boundry condition follow. We conider (n + ) (n + ) mtrix where the (n + )th row i identified with the firt row nd the (n + )th column i identified with the firt column. The qure lttice i therefore toru. 6 The Jckknife Method Any et of dt point in typicl imultion will generlly tend to contin correltion between the different point. In other word the dt point will not be ttiticlly independent nd conequence one cn not ue the uul formul to compute the tndrd devition of the men (i.e the probble error). The im of the Jckknife method i to etimte the error in et of dt point which contin correltion. Thi method work follow. ) We trt with mple of N meurement (dt point) {X,..., X N }. We compute the men < X >= N X i. (25) 2) We throw out the dt point X j. We get mple of N meurement {X,..., X j, X j+,..., X N }. Thi mple i clled bin. Since j =,..., N we hve N bin.we compute the men < X > j = N N ( X i X j ). (26) 3) The tndrd devition of the men will be etimted uing the formul The Jckknife error i σ. It i not difficult to how tht Thu σ 2 = σ 2 = N N (< X > j < X >) 2. (27) j= < X > j < X >= < X > X j. (28) N N(N ) (X j < X >) 2 = σmen. 2 (29) j= However in generl thi will not be true nd the Jckknife etimte of the error i more robut. 4) Thi cn be generlized by throwing out z dt point from the et {X,..., X N }. We end up with n = N/z bin. We compute the men < X > j over the bin in n obviou wy. The correponding tndrd devition will be given by σ 2 z = n n n (< X > j < X >) 2. (30) j= 5) The z tke the vlue z =,..., N. The error i the mximum of σ z function of z. 5