ࡱ> ~} v bjbjT~T~ +66v||  J($B"%%% %%%%0~a"%0J%d"d%d%0"%:V_"Jd| :  Improvements in Monte Carlo Thermodynamic Algorithms Mike Motyka Advisor: Dr. Nolan When studying a thermodynamic system, obtaining an analytic solution is often either time-consuming or impossible. Therefore a method to study the physics in the system is to use statistical approximation methods. In statistical physics, one of the more popular methods is called the Metropolis Algorithm which is based one the Monte Carlo simulation methods. Monte Carlo simulations require the experimenter to know the temperature of the system in order to simulate the canonical partition function of the system. Also, Monte Carlo simulations experience a critical slowing down when evaluating a temperature near a critical point. A correction to the Metropolis Algorithm called the Wang-Landau Algorithm solves these problems. This temperature independent algorithm not only avoids this critical slowing down, but can easily provide any desired thermodynamic values. For this study, the studied thermodynamic quantities were the internal energy and the heat capacity. As a preliminary test, the one-dimensional Ising model spin system was studied and the Wang-Landau method predicted the energy per spin that agreed with the known analytical solution to within  EMBED Equation.3  and the heat capacity to within an average relative percent error of  EMBED Equation.3 .   % ) <      , . ] ^ q r s t u v ȽȽȽȵȦȗȽȦwf[hz-h :CJaJ!j-hz-h;&CJEHUaJjI h;&CJUVaJ!jhz-h;&CJEHUaJjI h;&CJUVaJjhz-h(CJUaJhyCJaJhz-h.CJaJhz-h(CJaJhz-h65CJaJhz-h5 5CJaJ(jhz-h(CJUaJmHnHu     DPdefg v `gd.gd($a$gd(gd5 $a$gd :,1h/ =!"#$% F5KrSJFIFC    #%$""!&+7/&)4)!"0A149;>>>%.DIC;C  ;("(;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;" }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz w!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz ?E8Ȧ>x\Tr:݈r}iKJ#əTeO,ӚvcӼȁZ9f;rM~ϋ+y V(r=:T$~5"s Ro!ܛ3Ms5P07FĹ8԰% b0!4X\pI= /E.$<0?杘\xF?=r*"$ E+0$8)pJ}BRSI9C~n@8<~H>Ǔ!,z)@$ 0I3^OEـf;b ~>g&G\&˥C'bL9 Nu'?Js_4b< {0NQ CPhEjXX`Tt?JE @;r=Ê ی\s\in?AIB?X R>gƁj}b~[&~ B_ʊd'GH-$?Vw]d4f?GmLx/,2My61D+ * m#bHڦEnK?($Zp, 8SFIN2> c{rqtar%sJ᪆fPQ Jk]OMtX+Ec?Z+ӔhOַ;s4hZ8Rd9ޓ>7?Z3`b{~&3H2hi(ؤɣ\Rw֎}h{4ܟJ94g Ҕr3E&=qI1G4Rgϵ <=6!#ҘqKf@e zM H*7Cc~肏6($6Z+e)r*GT}qH X:S$Y ,lnlSUX271gΗɄ9؞բ/;m5\t\(C27m?:?ѿuAڜO$ʏD>#x&msċ}V D覕NO$tGOQz~mOф#Gբ?7YaN,B@?(E{} oegϘ??*O-(C:1qb9Bc?B1)# ƴXɥX&nB˴iP'ұ<ʔGGՠX5 ,\ӆe=<f@>3oBA QV0z]M?@>3cF̶6?5 `53!&jiZIk| F{h&jiZzpV$hEu+]31n!ҶqFS܎:Ѵh 4NlIo P`|ƗS&_t46`k?k֗a&_:4 F|Ѵ4be oBMQqրu4}Zơn9>l5@)֏>2m;CKl R h.׼u9i=}^o֤ xKڗXoB2_QO/ڍ}^Z7Ϝ/m?:Sڏ,cxh[gjT}zU|P"_ _W?k}QOU%I?J(?*>ĉO1AR?*¶!<ƆEGGP!}?违/#?.?ʃwQR&[֘ב?8#Tb(?*wpR;! dsUncO,⦗9jԔ,J3{R (J 8K#f=ԭrqק''ָrG]",»rdюxbsK8(t=ڔJL-)hQ1I/AH<R12};h? ? :PO򰿳ݟMʶ*:ӭ!ﲀ1rz9s7 PIi3LW43&y5KɃѺ\J GA4&5jw]G:,2) َ6!S dB 4WBi>iD&T[\ g9R=g3Lq ?4]̡W1T?^(Vl|nQ2Bi<Q(I2i;UAkpYGSr1'`$5=F i1zƀ2iUU5;\~m@#/z5 Iy:] Se9Sg>%ʹBЏcP& CQ攜R#FGd%H"jmAvA\}h`IC!Jw#}j ՗wh(f+ߓ{RNUa# {?vhaHp5ed*ݿ&Xn%cPC8w.Xrqڀ$FOjPxl=3HcJOF 4< R5(< tJ8!8#ڭYŒq0}{ H$`zP1q>-3v)@+;**Ē5Ȯ\92ܭ< 8Myoszg-F-!@qX*LOAI҂}k8ŤȥZ:P;P3iM ֌sI@|ց҂Fh ^ <R`PG4u;PMA6 B)sINQKI@&E:Kڐ c=S k=b Us g??lNzHB]g19icPTS\`Wy0PǚB}HZҬZi> Yv-XLeX-};U-7ש4oV FSPpz\e -y<\,/4rURGSjFHk97:m"\w\{j0cLOmOƨKi5í?4l-'v{P3-{$UuLVI,]Kr,SFyqӰd/adjߗfcNj&tqL>€.h#)1ǭ3{ӑ8'j1rY2H횗ʴ2J25ia#YKoտ€.BF>I$l3x/,[%+ l'%s% 'R*'CND28d"zA8X})*ӊrEFKq@:~\sAƜ qJOA@ =9Jb$V]3,h 徔Ko4# AnAc*0(U|ks[˕y04d 2\lqK4*dy&36-mu ehGPyNFzc{37bKG Z*$H#Y.8Hc_Z}H{Mbs*gؠɪd~XǢi5a0x݂P [ur~ΩE2e<[7dtAȅw>ҥ-=ҷlϠ6԰< ZM( ʍ&% ǍŏQycYoaCC }z@TcJ`HV.2;nq\ӯοku2S!<yvƸWUۨWZsF? 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