Monte Carlo methods for magnetic systems

Transcription

1 Monte Carlo methods for magnetc systems Zoltán Néda Babeş-Bolya Unversty Dept of Theoretcal and Computatonal Physcs Cluj-Napoca, Romana Man objectve of the lecture: To gve an ntroducton for basc Monte Carlo methods n some smple models of magnetsm. uropean School on Magnetsm, Tmşoara, Sept. -, 2009

2 Syllabus About Monte Carlo methods Determnstc versus stochastc smulaton methods lements of Stochastc Processes Markov chans Monte Carlo ntegraton Theoretcal approach to magnetc models What we are nterested n? The Metropols MC method for magnetc systems Implementng the Metropols MC method for the 2D Isng model Fnte-sze effects ffcent MC technques applet made by R. Sum

3 What are Monte Carlo methods? Computer smulaton methods: - Molecular dynamcs determnstc smulatons, based on the ntegraton of the equaton of moton - Monte Carlo methods Stochastc smulaton technques, where the random number generaton plays a crucal role - In general we speak about Monte Carlo smulaton methods whenever the use of the random numbers are crucal n the algorthm! MC: the art of usng pseudo random numbers - Monte Carlo technques are wdely used n studyng models of: statstcal physcs, soft condensed matter physcs, materal scence, many-body problems, comple systems, flud mechancs, bophyscs, econo-physcs, nonlnear phenomena, partcle physcs, heavy-on physcs, surface physcs, neuroscence etc.

4 Determnstc versus stochastc smulatons the Galton table - used to eemplfy the normal dstrbuton Molecular dynamcs approach: ntegratng n tme the equaton of moton of the partcles. advantage the realstc dynamcs dsadvantage slow even on supercomputers, only short tme-scales or small systems can be smulated Monte Carlo approach: the result of many determnstc effects s handled as a stochastc random force. advantage fast, easy to mplement dsadvantage less realstc, many elements of the real phenomena are not n the model t t Random number: wth p=/2 and - wth p=/2 Molecular dynamcs MC

5 Some necessary elements e of Stochastc Processes Markov processes/ Markov chans Stochastc process: let label the element of any state-space. A process that randomly vsts n tme these possble states s a stochastc process: ample the D random walk: 2D random walk: P=/2 P=/ Markov processes chan are characterzed by a lack of memory.e. the statstcal propertes of the mmedate future are unquely determned from the present, regardless of the past ample: random walk --> Markov process; self-avodng walk s NOT a Markov process Let be the state of the stochastc system at step, a stochastc varable The tme- evoluton of the system s descrbed by a sequence of states: 0,,.., n,. P,... The condtonal probablty that n s realzed f prevously we had: 0,,.., n- : n n 0 Defnton: For a Markov process we have: P,,..., P P a 0,..., n P n n. P n n2... P, 0. n n n2 0 n n 0 P m, j P m j Pm, j one-step transton probabltes, elements of the stochastc matr

6 Defnton: A probablty dstrbuton over the possble states w k s called nvarant or statonary for a gven Markov chan f satsfy: wk The probablty that =k durng an nfntely long process w m m w s 0; w m ; m w m P ms - A Markov chan s rreducble f and only f every state can be reached from every state! the stochastc matr s rreducble -A Markov chan s aperodc, f all states are aperodc. A state has a perod T> f P n =0 unless n=zt z: nteger, and T s the smallest nteger wth ths property. A state s aperodc f no such T> est. ere we denoted by P k n the probablty to get from state to state k through n steps Defnton: An rreducble and aperodc Markov chan s ergodc The basc theorem for Markov processes: An ergodc Markov chan posses an nvarant dstrbuton w k over the possble states

7 One dmensonal Monte Carlo ntegraton Problem: gven a functon f, compute the ntegral: b I f d The ntegral can be computed by choosng n ponts randomly on the [a,b] nterval, and wth a unform dstrbuton : / b a Const probablty densty: P, d d normalzaton: d Straghtforward samplng I f b a a b a n The strong law of large numbers guarantees us that for a suffcently large sample one can come arbtrary close to the desred ntegral! Let, 2,, n be random numbers selected accordng to a normalzed probablty densty, then :! the above affrmaton s also true f the random numbers are correlated, or the nterval s fnte ow rapdly the sum converge? --> for I Plm n n n f d; n f f very badly!!! Central lmt theorem the convergence mprove f the shape of appromates f we are samplng n the neghborhood where f s bg Const b a I

8 Important samplng The mportant samplng MC method wll calculate the I ntegral by samplng on random ponts on the [a,b] nterval accordng to a dstrbuton whch appromates the shape of f If one generates n ponts,,accordng to an arbtrary I b a f d b a f d n N f the convergence s nfntely fast f f Before gettng to ected. one cannot smply choose f, snce n ths case one cannot normalze normalzaton of s equvalent wth the ntal problem one cannot generate thus random numbers smply accordng to the desred f dstrbuton

9 Theoretcal approach to a magnetc orderng Usually canonc ensemble s used T, N, h s fed T temperature, h eternal magnetc feld, N partcle number relevant energes [nternal nteractons + nteracton wth eternal magnetc feld, h, +knetc terms] heat bath s a stochastc effect favor randomness a determnstc effect could favor orderng T or / kt Statstcal thermodynamcs amltonan,,= approach Z labels the mcrostates ep kt F kt lnz Z ep d kt F the free energy; k the Boltzmann constant assumng that the densty of state-space ponts s constant

10 What are we nterested n? The prmary goal of the MC type smulatons n magnetc systems s to estmate some averages at varous T, h and N values M 2 2 n canoncal ensemble average magnetzaton average energy average square energy X X Z Z X M ep X ep[ ] d average square magnetzaton X M we are also nterested n measurable quanttes lke: ; V C V T M h V, N 0 2 kt N NkT M 2 2 M 2 2 a sum wth huge number of terms number of terms ncreasng eponentally wth system sze e: 2 N or very hgh dmensonal ntegrals C 2 2 ; M ; ; the problem s that these sums cannot be usually analytcally calculated MC methods!! eact enumeraton s possble for small N not of thermodynamc nterest heat capacty at constant volume susceptblty

11 The Metropols MC method for magnetc systems ] ep[ ] [ ] [ ] [ u d u Z d u A Z A We want to compute ntegrals sums lke: -->elements of the state-space --> the entre state-space --> the amltonan of the system Very hgh dmensonal ntegral whch s eactly computable only for a lmted number of problems!!! Basc dea: to use the mportant samplng for calculatng these ntegrals IF n the MC ntegraton we choose the states wth probablty : n n u u A d u d u A A ] [ ] [ ] [ ] [ by choosng Z u ] [ the sum converges rapdly and: n A n A Problem: we stll don t know Z!

12 The Metropols et al. dea... an algorthm has to be derved that generates states accordng to the desred! Basc dea: usng a Markov chan, such that startng from an ntal state 0 further states are generated whch are ultmately dstrbuted accordng to for ths Markov s an nvarant chan need dstrbuton to specfy over the the P possble ' states transton probabltes from state to state. In order that the nvarant lmtng dstrbuton be we need:. The Markov chan should be ergodc any state pont should be reachable from any other state-pont through the Markov chan 2. For all possble mcrostates: 3. For all possble mcrostates : ' estence of the lmtng dstrbuton ' N. Metropols et. al; J. Chem. Phys., vol. 2, P P ' ' ' condton for the Instead of 2. and 3. a stronger but smpler condton can be used, the so called detaled balance: P ' P ' ' Result: We can construct Markov chans leadng to the desred wthout the pror knowledge of Z!!! dstrbuton,

13 ] ep[ ] [ kt u In the canoncal ensemble: Detaled balance satsfed Algorthm for MC smulatons:. Desgn an ergodc Markov process on the possble mcrostates each state should be reachable from each other 2. Specfy an ntal mcrostate for startng 3. Choose randomly a new mcrostate preferably so that 4. Compute the value of 5. Generate a unformly dstrbuted random number r between [0,]. 6. If --> jump to the new state, and return to 3. If --> count the old state as new and return Average the quantty A for the generated states. Repeat steps -6 untl the average converge The Metropols algorthm: 0 ', 0 ', ] ', ep[ ' for for P ' ', another possblty Glauber algorthm: kt kt P ', ep ', ep ' 0 ' P ' P P P r P r

14 The Isng spn system J, j j h { } - spontaneous magnetzaton s possble M0 for h=0 - frst model for understandng ferro- and ant-ferromagnetsm for localzed spns - for J>0 --> ferromagnetc order - for J<0 --> ant-ferromagnetc order - no phase transton n D - ferro-paramagnetc phase transton for D> - second order phase transton order-dsorder - In D and 2D eactly solvable! - Due to the local nteractons calculatng Z s dffcult. - eact soluton very dffcult n 2D - no eact soluton n 3D - Appromaton methods: mean-feld theory, renormalzaton, hgh and low temperature epanson

15 Implementng the Metropols MC for the 2D Isng model Implementng the Metropols MC for the 2D Isng model Problem: Study mt, <T>, C v T, T and T c for 2D Isng model We consder h=0, and f J=. The temperature unts are consdered so that k=. Square lattce topology s consdered, k j j -Let us assume a lattce L L wth free boundary condtons -We consder a canoncal ensemble and f thus N and T We plan to calculate: N N N M m T j j T, } { T T T Nk T C B v 2 2 T M T M NkT T T c from the mama of C v T and T

16 The Metropols MC algorthm for the problem:. F a temperature T 2. Consder an ntal spn confguraton { }. For eample for all 3. Calculate the ntal value of and M 4. Consder a new spn confguraton by vrtually flppng one randomly selected spn 5. Calculate the energy of the new confguraton, and the energy change due to ths spn-flp 6. Calculate the Metropols P=P--> probabltes for ths change 7. Generate a random number r between 0 and 8. If r P accept the flp and update the value of the energy to and magnetzaton to M If r P reject the spn flp and take agan the ntal and M values n the needed averages 9. Repeat the steps 4-8 many tmes drve the system to the desred canoncal dstrbuton of the states 0. Repeat the steps 4-8 by collectng the values of, 2, M, M 2, for the needed averages. Compute ths average for a large number of mcrostates 2. Calculate the value of mt, <T>, C v T and T usng the gven formulas 3. Change the temperature and repeat the algorthm for the new temperatures as well. 4. Construct the desred mt, <T>, C v T, T curves, N

17 Fnte-sze effects -The bggest problem wth computer smulatons s that t can be performed for relatvely small systems far from the ones needed n thermodynamcs N - Real phase-transton real dvergences n the thermodynamc quanttes dervatves of the thermodynamc potental s possble only n nfnte systems! In a fnte-sze system the correlaton length cannot dverge and t s cut by the sze of the system nstead of dvergences rounded mamum or contnuous behavor s obtaned. -The results obtaned by MC smulatons for fnte systems has to be carefully evaluated and etrapolated for nfnte systems! fnte sze scalng s needed! - Important quanttes that have to be scaled: mt, CvT, T curves and the value of T c N A N L L N L L L=0 L=5 L=20 L=30 L=40 L=0 L=5 L=20 L=30 L=40 The order parameter mt as a functon of T for dfferent system szes Specfc heat C v T as a functon of T for varous system szes

18 Observatons and techncal ponts: the consdered P--> transtons leads to an ergodc Markov process one MC step s defned as N spn flp trals! By applyng the above algorthm for T<T c one can also follow how the order arses n the system. Ths dynamcs mght not necessarly be the real one. The Metropols MC method s ntended to yeld equlbrum propertes and not dynamcal smulaton of the system! It s beleved that the Glauber probabltes gves a realstc pcture for the dynamcs as well! One way of makng the system quas-nfnte s to mpose perodc boundary condtons see the eercse n the computer codes! however ths cuts also the correlaton length The smple Metropols and Glauber algorthm can be further mproved, desgnng more clever and faster methods

19 ffcent MC technques T T c I. At low temperatures the Metropols and Glauber algorthm s neffcent. After equlbrum s reached spns are ordered most of the spn-flps are rejected, and computer tme s wasted very long smulatons are needed to get a reasonable estmate for the averages. Ths drawback s elmnated by the BKL MC algorthm, see A. B. Bortz, M.. Kalos and J.L. Lebowtz, J. Comp. Phys. Vol. 7, II. In the neghborhood of T c the Metropols and Glauber algorthm s neffcent due to the crtcal slowng down the relaaton tme s lnked z to the correlaton length by ~ the dynamcal crtcal eponent, z. as T-->Tc we have --> and get that --> The bg problem: for the Metropols or the Glauber algorthm z=2!!! ---> There are many MC steps necessary to generate ndependent uncorrelated confguratons --> the samplng s restrcted only to a small part of the state-space The system has a long memory. Ths problem s partally solved by flppng together clusters of correlated spns cluster algorthms see: U. Wolff, PRL vol. 62, ; R.. Swendsen and J-S. Wang, PRL vol. 58, III. Quantum-statstcal models ubbard, Stoner, T-J, etc can be studed by Quantum MC methods, see: J. Tobochnk, G. Batroun and. Gould, Computers n Physcs, vol. 6, IV. Frustrated, spn-glass type models dward-anderson, Potts glass, etc can be studed also by MC methods. One of these s the smulated annealng method, see: S. Krckpatrck, G.D. Gelatt and M.P. Vecch, Scence vol. 220,

20 Conclusons -MC methods are powerful tools for numercally studyng varous models of magnetsm. -MC methods can be mplemented on normal PC type computers, no supercomputers are needed. - MC methods are easy to learn however some basc programmng eperence s needed -Masterng the MC method opens possbltes for studyng many other models n sold-state physcs, bophyscs, ecology, economcs, socology, nuclear and medcal physcs, etc. -The most cted paper n statstcal physcs s the paper of Metropols et. al!