Evaluation of an Efficient Monte Carlo Algorithm to Calculate the Density of States

Transcription

1 Internatonal Journal of Machne earnng and Computng, Vol., No., Aprl valuaton of an ffcent Monte Carlo Algorthm to Calculate the Densty of States Seung-Yeon Km, Member, IACSIT Abstract Phase transtons and crtcal phenomena are the most unversal phenomena n nature. To understand the phase transtons and crtcal phenomena of a gven system as a contnuous functon of temperature and to obtan the partton functon zeros n the complex temperature plane ndcatng most effectvely phase transtons and crtcal phenomena, we need to calculate the densty of states. Currently, Wang-andau Monte Carlo algorthm s one of the most effcent Monte Carlo methods to calculate the approxmate densty of states. Usng Wang-andau Monte Carlo algorthm, the densty of states for the Isng model on x square lattces ( = ~ 3) wth perodc boundary condtons s obtaned, and the partton functon zeros of the Isng model are evaluated n the complex temperature plane. y examnng the behavor of the frst partton functon zero (partton functon zero closest to the postve real axs), phase transtons and crtcal phenomena can be much more accurately analyzed. The approxmate frst zeros of the Isng ferromagnet, obtaned from Wang-andau algorthm, are qute close to the exact ones, ndcatng that t s a relable method for calculatng the densty of states and the frst partton functon zeros. Index Terms Phase transton, densty of states, Isng model, partton functon zeros. I. INTRODUCTION Phase transtons and crtcal phenomena are the most unversal phenomena n nature. The two-dmensonal Isng model s the smplest system showng phase transtons and crtcal phenomena at fnte temperatures. Snce the Onsager (Nobel prze wnner n 96) soluton [] of the square-lattce Isng model wth perodc boundary condtons n the absence an external magnetc feld, the two-dmensonal Isng model has played a central role n our understandng of phase transtons and crtcal phenomena []. To understand phase transtons and crtcal phenomena, varous theoretcal methods (such as mean-feld theory, power-seres expanson and analyss, renormalzaton group, and transfer matrx) have been developed. Recently, computer smulatons, n partcular, Monte Carlo computer smulatons have been the most popular method n studyng phase transtons and crtcal phenomena. To understand the phase transton and crtcal phenomena of a gven system as a contnuous functon of temperature, to obtan the partton Manuscrpt receved March 9, ; revsed Aprl 3,. Ths research was supported by asc Scence Research Program through the Natonal Research Foundaton of Korea (NRF) funded by the Mnstry of ducaton, Scence and Technology (grant number -99). Seung-Yeon Km s wth the School of beral Arts and Scences, Korea Natonal Unversty of Transportaton, Chungju 3-7, South Korea (e-mal: sykmm@ut.ac.kr). functon zeros ndcatng most effectvely phase transtons and crtcal phenomena, and to perform mcrocanoncal analyss of phase transtons and crtcal phenomena, we need to calculate the densty of states. Currently, Wang-andau Monte Carlo algorthm [3] s one of the most effcent Monte Carlo methods to calculate the approxmate densty of states. In Wang-andau Monte Carlo algorthm, the nverse of the densty of states s employed as the samplng probablty functon, and the real values for the densty of states can be obtaned quckly due to ts modfcaton factor. Phase transtons and crtcal phenomena can also be understood based on the concept of partton functon zeros. Fsher ntroduced the partton functon zeros n the complex temperature plane utlzng the Onsager soluton of the square-lattce Isng model n the absence of an external magnetc feld []. Fsher also showed that the partton functon zeros n the complex temperature plane of the square-lattce Isng model determne ts ferromagnetc and antferromagnetc crtcal temperatures at the same tme n the absence an external magnetc feld. y calculatng the partton functon zeros and examnng the behavor of the frst partton functon zero (partton functon zero closest to the postve real axs) n the thermodynamc lmt, phase transtons and crtcal phenomena can be much more accurately analyzed than examnng the behavor of the specfc heat per volume for real values of the temperature, whch s plagued by the nose due to the subleadng terms contanng zeros other than the frst ones [5 ]. In the next secton, the densty of states and the partton functon of the square-lattce Isng model are defned. In Secton III, Wang-andau Monte Carlo algorthm to calculate the approxmate densty of states s brefly explaned. In Secton IV, the concept of the partton functon zeros n the complex temperature plane s ntroduced, the partton functon zeros of the square-lattce Isng model are evaluated n the complex temperature plane usng the exact densty of states and the approxmate densty of states, and both results are compared. II. XACT DNSITY OF STATS The Isng model [, ] on a lattce wth Ns stes and Nb bonds s defned by the Hamltonan H = J σσ, () <, j> where J s the couplng constant between two neghborng magnetc spns (postve value of J for a ferromagnetc nteracton and negatve value of J for an antferromagnetc j

2 Internatonal Journal of Machne earnng and Computng, Vol., No., Aprl nteracton), <, j > ndcates a sum over all nearest-neghbor pars ( and j) of lattce stes, and σ =± (postve value for the upward magnetc spn on a lattce ste and negatve value for the downward magnetc spn). For the Isng model on square lattce wth perodc boundary condtons, the number of spns s Ns = and the number of bonds s Nb =. If we defne the densty of states, Ω ( ), wth a gven energy = ( σσ j), () <, j> where s a postve nteger ( N ), the Hamltonan b can be wrtten as H J = ( ) (3) for the square-lattce Isng model. Fnally, the partton functon of the square-lattce Isng model (a sum over possble spn confguratons) Z = exp( β H ), () { σ n } where β = / kt, k s the oltzmann constant, and T s temperature, s expressed as Z ( y) = exp( β J ) Ω( ) y, (5) where y = exp( β J). For a ferromagnetc nteracton, the physcal nterval s y ( T ). That s, y s a convenent temperature varable, confned to a short nterval [, ]. Gven the densty of states Ω ( ), the partton functon s a polynomal n y. Naturally, we obtan the entropy as a functon of energy accordng to the oltzmann formula S( ) = k ln Ω ( ) (6) from the densty of states. Therefore, the partton functon of the square-lattce Isng model can be wrtten as Z ( y) = exp( β J ) exp[ S( ) / k ] y. (7) The states wth are the ferromagnetc ground states (where all spns algn n the same drecton), whereas the states wth = correspond to the antferromagnetc ground states (where all nearest-neghbor spns of any spn on the lattce are oppostely orented to t). Also, the densty of states satsfes the followng relaton Ω ( ) = Ω( ) () due to the symmetry of the square-lattce Isng model. Table I shows the exact nteger values for the densty of states of the Isng model on the square lattce wth perodc boundary condtons, obtaned from the Onsager soluton []. TA I: XACT INTGR VAUS FOR TH DNSITY OF STATS Ω ( ) OF TH ISING MOD ON TH X SQUAR ATTIC WITH PRIODIC OUNDARY CONDITIONS, AS A FUNCTION OF NRGY (= ~ 6) Ω ( ) The values of the densty of states for 66 are easly obtaned usng the relaton Ω ( ) = Ω( ). As shown n the table, we notce that Ω ( ) = (9) for = odd numbers. It s a general result for perodc boundary condtons, ndependent of the system sze. For other knds of boundary condtons, we obtan non-zero values for the densty of states even n the case of = odd numbers. Also, we have Ω ( ) = = () for =. 5

3 Internatonal Journal of Machne earnng and Computng, Vol., No., Aprl TA II: APPROXIMAT VAUS FOR TH DNSITY OF STATS Ω ( ) OF TH ISING MOD ON TH X SQUAR ATTIC WITH PRIODIC OUNDARY CONDITIONS, AS A FUNCTION OF NRGY, OTAIND FROM WANG-ANDAU MONT CARO AGORITHM Ω ( ) III. APPROXIMAT DNSITY OF STATS One of the most mportant methods n studyng phase transtons and crtcal phenomena s computer smulaton, n partcular, Monte Carlo computer smulaton. The mportance samplng Monte Carlo method, Metropols Mote Carlo algorthm [], has been used extensvely n scence and engneerng. In Metropols Mote Carlo algorthm, the natural canoncal dstrbuton functon, exp( β H ), () where H s the Hamltonan of a gven system, s employed as the samplng probablty functon at a gven temperature T. The canoncal dstrbuton functon can be wrtten as Ω( )exp( β ), () as a functon of energy, where Ω ( ) s the densty of states and exp( β ) s the oltzmann-gbbs factor. As energy ncreases, the densty of states ncreases sharply and the oltzmann-gbbs factor decreases sharply. Therefore, the canoncal dstrbuton functon s a needle-shaped functon around T, whch becomes the delta functon n the thermodynamc lmt. Metropols Mote Carlo algorthm s the most effcent method for understandng the propertes of a gven system at a fxed temperature. However, f we want to understand the phase transtons and crtcal phenomena of a gven system as a contnuous functon of temperature, to obtan the partton functon zeros n the complex temperature plane ndcatng most effectvely phase transtons and crtcal phenomena, and to perform mcrocanoncal analyss of phase transtons and crtcal phenomena, Metropols Mote Carlo algorthm s not useful. To understand the transton propertes of a gven system as a contnuous functon of temperature, to obtan the partton functon zeros, and to perform mcrocanoncal analyss, we need to calculate the densty of states. Currently, Wang-andau Monte Carlo algorthm s one of the most effcent Monte Carlo methods to calculate the approxmate densty of states [3]. In Wang-andau Monte Carlo algorthm, the recprocal of the densty of states, / Ω ( ), s also employed as the samplng probablty functon. That s, the transton probablty from a state wth energy to another state wth energy s defned by p( ) = mn[ Ω( )/ Ω ( ),], (3) where and are energes before and after a random spn flp. ecause ths defnton means a random walk n energy space wth a probablty proportonal to the recprocal of the densty of states, / Ω ( ), a truly flat hstogram for the energy dstrbuton can be obtaned after an nfnte number of random Monte Carlo moves. The densty of states s a pror unknown. Therefore, at the frst tme, we generate a crude verson of the densty of states, Ω ( ) 6, by acceptng all Monte Carlo steps. Now, 6

4 Internatonal Journal of Machne earnng and Computng, Vol., No., Aprl the ntal densty of states Ω ( ) s employed as the samplng probablty functon, and then the densty of states s changed after a Monte Carlo step. ach Monte Carlo step, the densty of states s updated accordng to the followng rule Ω( ) f Ω ( ), () where f ( > ) s the modfcaton factor [3]. Owng to ths factor, an mproved verson of the densty of states can be obtaned quckly. At the frst tme, the modfcaton factor s convenently chosen as f = e (5). Durng a fnte number of Monte Carlo steps wth f, the ntal densty of states Ω ( ) evolves nto the new densty of states Ω ( ). Next, the modfcaton factor s reduced accordng to the followng rule IV. PARTITION FUNCTION ZROS In the thermodynamc lmt, the specfc heat (per volume) of the square-lattce Isng ferromagnet becomes nfnte at the crtcal temperature where the transton between the paramagnetc phase and the ferromagnetc phase emerges. In fnte systems, the specfc heat per volume shows a sharp peak but s not nfnte. At the same tme, the locaton (the so-called effectve crtcal temperature) of the sharp peak of the specfc heat n a fnte system s dfferent from the crtcal temperature at the nfnte system. As the system sze ncreases, the effectve crtcal temperature approaches the crtcal temperature. Gven the densty of states Ω ( ), the free energy (per volume) of the square-lattce Isng model s gven by f y k T Z y ( ) = ( )ln ( ) = ( kt )[β J+ ln Ω( ) y ]. Therefore, the specfc heat (per volume) s expressed as (9) f f. = (6) Therefore, durng a fnte number of Monte Carlo steps wth C y k T Z y β ( ) = ( ) ln ( ) = k y < > < > ( )(ln ) ( ). () f = exp(/ ), (7) the densty of states Ω ( ) evolves nto Ω ( ). If we repeat these processes thrty tmes, we have the fnal modfcaton factor 9 f 3 exp( ).6, = = () and reach an accurate verson of densty of states Ω ( ), 3 qute close to the true densty of states, for example, the exact nteger values n Table I. 6 very Monte Carlo steps, we check the flatness of the energy hstogram dstrbuton h ( ). If the maxmum hstogram value s less than. tmes the average value of hstogram < h ( ) > and the mnmum hstogram value s larger than. < h ( ) > (the so-called % flatness crteron), we reduce the modfcaton factor from f to f +, and reset the hstogram to zero. If not, we repeat 6 Monte Carlo steps agan and agan untl the % flatness crteron s satsfed wthout changng the value of the modfcaton factor. Table II shows the approxmate values for the densty of states of the Isng model on the square lattce wth perodc boundary condtons, obtaned from Wang-andau Monte Carlo algorthm. As shown n the table, the approxmate values are qute close to the exact nteger ones n Table I. In the thermodynamc lmt, the specfc heat (per volume) of the square-lattce Isng ferromagnet dverges at the crtcal temperature yc = exp( J / ktc) = = () The ordered ferromagnetc phase appears below y c, whereas the dsordered paramagnetc phase appears above y c. The propertes of the phase transton of the Isng model are completely equvalent to those of the gas-lqud phase transton for a smple system [, 3]. Phase transtons and crtcal phenomena can also be understood based on the concept of partton functon zeros. Yang and ee (Nobel prze wnners n 957) proposed a rgorous mechansm for the occurrence of phase transtons n the thermodynamc lmt and yelded an nsght nto the unsolved problem of the ferromagnetc Isng model at arbtrary temperature (T) n an external magnetc feld () by ntroducng the concept of the zeros of the partton functon Z(T,) n the complex magnetc-feld plane []. They also formulated the celebrated crcle theorem, whch states that the partton functon zeros of the Isng ferromagnet le on the unt crcle n the complex fugacty plane [3]. Followng Yang and ee's dea, Fsher ntroduced the partton functon zeros n the complex temperature plane utlzng the Onsager soluton of the square-lattce Isng model n the absence of an external magnetc feld []. Fsher also showed that the partton functon zeros n the complex temperature plane of the square-lattce Isng model determne ts ferromagnetc and antferromagnetc crtcal temperatures at the same tme for =. In fnte systems no zero cut the 7

5 Internatonal Journal of Machne earnng and Computng, Vol., No., Aprl postve real axs n the complex temperature plane, but some zeros for a system showng a phase transton approach the postve real axs as the system sze ncreases, determnng the crtcal temperature and the related crtcal exponents n the thermodynamc lmt. Snce the propertes of the partton functon zeros of a gven system provded the valuable nformaton on ts exact soluton, the earler studes on partton functon zeros were manly performed n the felds of mathematcs and mathematcal physcs. Nowadays, the concept of partton functon zeros s appled to all felds of scence from elementary partcle physcs to proten foldng, and they are used as one of the most effectve methods to determne the crtcal temperatures and exponents [5 ]. The partton functon Z ( y ) of the square-lattce Isng model can be expressed as ts zeros {y } n the complex temperature (y) plane: Usng the Onsager soluton of the square-lattce Isng ferromagnet wth perodc boundary condtons, we can obtan the exact partton functon zeros of the square-lattce Isng model n the complex y plane. Among the partton functon zeros, the frst zero (y ) s most mportant because t determnes the crtcal temperature and the crtcal exponents. Table III shows the exact frst zeros y () of the Isng ferromagnet on x square lattces ( = ~ 3) wth perodc boundary condtons. As shown n the table, as the system sze ncreases, the real part of the frst zero Re[y ()] approaches the exact crtcal temperature y c = + followng the fnte-sze scalng law Δ = (5) Re[ y( )] Re[ y( )] yc ~. Also, the magnary part of the frst zero Im[y ()] decreases quckly followng the smlar fnte-sze scalng law ( ) = exp( β ) Ω ( ) = ( ), = () Z y J y A y y where A s constant. In terms of the partton functon zeros, the free energy s gven by ( ) = ( )[ln + ln( )], = f y k T A y y (3) and the specfc heat by y y C( y) = ( k )(ln y). () = y y y y For a system wth the phase transton at the crtcal temperature y c, the loc of the partton functon zeros close n toward the real axs to ntersect t n the thermodynamc lmt, and the sngularty of the specfc heat (per volume) C(y) appears n ths lmt. It s clear from C(y) that the leadng behavor of such a sngularty s due to the par of partton functon zeros closest to the postve real axs, called the frst zeros (y ). Therefore, by calculatng the partton functon zeros and examnng the behavor of the frst zero n the thermodynamc lmt, the crtcal behavor can be much more accurately analyzed than examnng the behavor of the specfc heat per volume for real values of the temperature, whch s plagued by the nose due to the subleadng terms contanng zeros other than the frst ones. TA III: XACT FIRST ZRO y ( ) OF TH ISING FRROMAGNT ON X SQUAR ATTIC ( = ~ 3) WITH PRIODIC OUNDARY CONDITIONS y () Im[ ( )] ~. y (6) However, t s mpossble to calculate the partton functon zeros by usng popular Metropols Monte Carlo computer smulatons. That s why the concept of the partton functon zeros has not been used popularly and extensvely n scence and engneerng. Now, wth new Wang-andau Monte Carlo computer smulatons, t s possble to calculate the partton functon zeros. We have calculated the partton functon zeros of the square-lattce Isng ferromagnet wth perodc boundary condtons from the densty of states Ω ( ), generated by Wang-andau Monte Carlo computer smulatons wth the % flatness crteron for hstograms. We have used one core of a nux PC wth one Intel 7-6K CPU for Wang-andau Monte Carlo computer smulatons. The CPU tme for the Isng model on x square lattce s just 3 mnutes and 9 seconds. Also, the CPU tme s mnutes and seconds on x square lattce, and mnutes and 6 seconds on 3 x 3 square lattce. Therefore, Wang-andau Monte Carlo algorthm s qute fast wth a modern computer. TA IV: APPROXIMAT FIRST ZRO y ( ) OF TH ISING FRROMAGNT ON X SQUAR ATTIC ( = ~ 3) WITH PRIODIC OUNDARY CONDITIONS, OTAIND FROM WANG-ANDAU MONT CARO AGORITHM y () Table IV shows the approxmate frst zeros y () of the Isng ferromagnet on x square lattces ( = ~ 3) wth perodc boundary condtons, obtaned from Wang-andau Monte Carlo computer smulatons. As shown n the table, the approxmate frst zeros are qute close to the exact ones n Table III.

6 Internatonal Journal of Machne earnng and Computng, Vol., No., Aprl TA V: RRORS OF TH APPROXIMAT FIRST ZROS FROM TH XACT ONS FOR TH ISING FRROMAGNT ON X SQUAR ATTIC rror (%) for rror (%) for rror (%) for Re[y ()] ΔRe[y ()] Im[y ()] Table V shows the errors (%) of the approxmate frst zeros from the exact ones. The second column of the table shows the error of the real part of the approxmate frst zero from the exact one, and the errors are less than.6%. The thrd column shows the error for the dfference, as n (5), between the real part and the crtcal temperature, and the errors are less than.%. The fourth column of Table V shows the error of the magnary part of the approxmate frst zero from the exact one, and the errors are less than.6%. Overally, the errors are qute small, ndcatng that Wang-andau Monte Carlo algorthm s a relable method for calculatng the densty of states and the frst partton functon zeros. RFRNCS []. Onsager, Crystal statstcs. I. A two-dmensonal model wth an order-dsorder transton, Physcal Revew, vol. 65, pp. 7-9, 9. [] C. Domb, The Crtcal Pont; ondon, U.K.: Taylor and Francs, 996. [3] F. Wang and D. P. andau, Multple-range random walk algorthm to calculate the densty of states, Physcal Revew etters, vol. 6, pp. 5-53,. [] M.. Fsher, The nature of crtcal ponts, n ectures n Theoretcal Physcs, vol. 7C, W.. rttn, d. oulder, CO: Unversty of Colorado Press, 965, pp [5] S.-Y. Km and R. J. Creswck, Yang-ee zeros of the Q-state Potts model n the complex magnetc feld plane, Physcal Revew etters, vol., pp. -3, 99. [6] S.-Y. Km, Partton functon zeros of the Q-state Potts model on the smple-cubc lattce, Nuclear Physcs, vol. 637, pp. 9-6,. [7] S.-Y. Km, Densty of the Fsher zeros for the three-state and four-state Potts models, Physcal Revew, vol. 7, artcle: 6, pp. -5,. [] S.-Y. Km, Yang-ee zeros of the antferromagnetc Isng model, Physcal Revew etters, vol. 93, artcle: 36, pp. -,. [9] S.-Y. Km, Fsher zeros of the Isng antferromagnet n an arbtrary nonzero magnetc feld plane, Physcal Revew, vol. 7, artcle: 7, pp. -, 5. [] S.-Y. Km, Densty of Yang-ee zeros and Yang-ee edge sngularty for the antferromagnetc Isng model, Nuclear Physcs, vol. 75, pp. 5-5, 5. [] S.-Y. Km, Honeycomb-lattce antferromagnetc Isng model n a magnetc feld, Physcs etters A, vol. 35, pp. 5-5, 6. [] S.-Y. Km, Densty of Yang-ee zeros for the Isng ferromagnet, Physcal Revew, vol. 7, artcle: 9, pp. -7, 6. [3] M.. Monroe and S.-Y. Km, Phase dagram and crtcal exponent ν for nearest-neghbor and next-nearest-neghbor nteracton Isng model, Physcal Revew, vol. 76, artcle: 3, pp. -5, 7. [] S.-Y. Km, C.-O. Hwang, and J. M. Km, Partton functon zeros of the antferromagnetc Isng model on trangular lattce n the complex temperature plane for nonzero magnetc feld, Nuclear Physcs, vol. 5, pp. -5,. [5] S.-Y. Km, Partton functon zeros of the square-lattce Isng model wth nearest- and next-nearest-neghbor nteractons, Physcal Revew, vol., artcle: 3, pp. -7,. [6] S.-Y. Km, Partton functon zeros of the honeycomb-lattce Isng antferromagnet n the complex magnetc-feld plane, Physcal Revew, vol., artcle: 7, pp. -7,. [7] C.-O. Hwang and S.-Y. Km, Yang-ee zeros of trangular Isng antferromagnets, Physca A, vol. 39, pp ,. [] J. H. ee, H. S. Song, J. M. Km, and S.-Y. Km, Study of a square-lattce Isng superantferromagnet usng the Wang-andau algorthm and partton functon zeros, Journal of Statstcal Mechancs, vol., artcle: P3, pp. -9,. [9] J. H. ee, S.-Y. Km, and J. ee, xact partton functon zeros and the collapse transton of a two-dmensonal lattce polymer, Journal of Chemcal Physcs, vol. 33, artcle: 6, pp. -6,. [] J. H. ee, S.-Y. Km, and J. ee, Collapse transton of a square-lattce polymer wth next nearest-neghbor nteracton, Journal of Chemcal Physcs, vol. 35, artcle:, pp. -,. [] N. Metropols, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and. Teller, quaton of state calculatons by fast computng machnes, Journal of Chemcal Physcs, vol., pp. 7-9, 953. [] C. N. Yang and T. D. ee, Statstcal theory of equatons of state and phase transtons. I. Theory of condensaton, Physcal Revew, vol. 7, pp. -9, 95. [3] T. D. ee and C. N. Yang, Statstcal theory of equatons of state and phase transtons. II. attce gas and Isng model, Physcal Revew, vol. 7, pp. -9, 95. Seung-Yeon Km receved the.sc. and M.Sc. degrees n Physcs from Yonse Unversty, Seoul, South Korea, n 99 and 99, respectvely. He receved the Ph.D. degree n Physcs from Unversty of South Carolna, Columba, South Carolna, USA n. Currently, he s an assocate professor at Korea Natonal Unversty of Transportaton. Hs research nterests are mathematcal physcs, computatonal physcs, computatonal chemstry, computatonal bology, and bonformatcs. 9