Brazilian Journal of Physics, vol. 29, no. 1, March,

Size: px

Start display at page:

Download "Brazilian Journal of Physics, vol. 29, no. 1, March,"

Darleen Long
5 years ago
Views:

1 Brazilian Journal of hysis, vol. 29, no., Marh, Computational Methos Inspire by Tsallis Statistis: Monte Carlo an Moleular Dynamis Algorithms for the Simulation of Classial an Quantum Systems John E. Straub ;2 an Ioan Anriioaei ;3 Department of Chemistry, Boston University, Boston, MA Institute for Avane Stuies, The Hebrew University of Jerusalem, Givat Ram 9904 Israel 3 resent aress: Department of Chemistry an Chemial Biology, Harvar University, 2 Oxfor Street, Cambrige, MA 0238 Reeive 07 Deember, 998 Tsallis's generalization of statistial mehanis is summarize. A moiation of this formalism whih employs a normalize expression of the -expetation value for the omputation of euilibrium averages is reviewe for the ases of pure Tsallis statistis an Maxwell-Tsallis statistis. Monte Carlo an Moleular Dynamis algorithms whih sample the Tsallis statistial istributions are presente. These methos have been foun to be eetive in the omputation of euilibrium averages an isolation of low lying energy minima for low temperature atomi lusters, spin systems, an biomoleules. A phase spae oorinate transformation is propose whih onnets the stanar Cartesian positions an momenta with a set of positions an momenta whih epen on the potential energy. It is shown that pure Tsallis statistial averages in this transforme phase spae result in the -expetation averages of Maxwell-Tsallis statistis. Finally, an alternative novel erivation of the Tsallis statistial istribution is presente. The erivation begins with the lassial ensity matrix, rather than the Gibbs entropy formula, but arrives at the stanar istributions of Tsallis statistis. The result suggests a new formulation of imaginary time path integrals whih may lea to an improvement in the simulation of euilibrium uantum statistial averages. I Introution Ten years ago, Tsallis publishe his seminal work on a possible generalization of the stanar Gibbs- Boltzmann statistial mehanis []. His intriguing theory began with a reexpression of the Gibbs-Shannon entropy formula S = lim S = lim!, =,k! k p (,)(, [p (,)], ), () p(,)lnp(,), ; (2) where, = r N p N is a phase spae inrement. On the right of this expression, the ientity lnx = lim n!0 (x n, )=n has been use to transform the logarithm an is a real number [,2]. A similar result was previously presente in the ontext of generalize information entropies but ha apparently not been applie to esribe the physial worl [3]. Tsallis's bol move was to o just that.

2 80 John E. Straub an Ioan Anriioaei Tsallis note a number of properties of S, whih he referre to as a \generalize entropy," an the assoiate statistial istributions. He foun that muh of the stanar mathematial struture of Gibbs- Boltzmann statistial mehanis is preserve. This is interesting in itself. However, even more interesting was what is not preserve. Thermoynami state funtions suh as the entropy an energy were no longer extensive funtions of the system. This prompte the use of a generalize formalism base on the non-aitive funtion S to erive, for non-extensive systems, a variety of results of the stanar statistial mehanis (see [4] an referenes therein). II \ure" Tsallis statistis Tsallis erive the probability of ning the system at a given point in phase spae, =(r N ; p N )by extremizing S subjet to the onstraints p (,), = an [p (,)]H(,), =E (3) where H(,) is the system Hamiltonian for N istinguishable partiles in imensions. The result is where p (,) = [, (, )H(,)] hn, (4) = h N [, (, )H(,)],, (5) plays the role of the anonial ensemble partition funtion. Using the ientity lim n!0 (+an) =n = exp(a), in the limit that =, the stanar probability istribution of lassial Gibbs-Boltzmann statistial mehanis p(,) = exp(,h(,)) (6) hn is reovere. However, there is a problem. For ertain values of an a harmoni potential, the istribution p (,) has innite variane an higher moments. II. Introution of the \-expetation" value To aress this problem, Tsallis erive the statistial probability by extremizing S subjet to the moie onstraints p (,), = an [p (,)] H(,), =E (7) where the average energy is ene using a \expetation" value. The result is p (,) = h N [, (, )H(,)], (8) with the \partition funtion" = [, (, )H(,)] h N,,: (9) In the limit that = the lassial anonial ensity istribution is reovere. To be onsistent, the -expetation value is also use to ompute the average of an observable A hai NN = ( h N ) A(,) [, (, )H(,)],,: (0) However, sine the averaging operator is not normalize, in general hi NN 6= for 6=. Moreover, it is neessary to ompute to etermine the average. To avoi this iulty, a ierent generalization of the anonial ensemble average was propose [5] R A(,)[, (, )H(,)],, R [, (, )H(,)],, : () It is obviously normalize an onvenient to apply. II.2 Monte Carlo estimates of Tsallis statistial averages The uestion arises, how mightwe simulate systems that are well esribe by the Tsallis statistial istributions when 6=? A point in phase spae or onguration spae an be sai to have the probability p (r N ) exp[, U] (2)

3 Brazilian Journal of hysis, vol. 29, no., Marh, where U is the eetive potential energy [] U(r N )= (, ) ln, (, )U(r N ) : (3) From this expression it is possible to simply state a Monte Carlo algorithm whih samples the euilibrium Tsallis statistial istribution. () A new onguration is hosen within a region with uniform onstant probability. (2) The point in onguration spae is aepte or rejete aoring to the riterion p = min ; exp[, U] (4) where the hange in the eetive potential energy is U. (3) Repeat the previous steps. This Monte Carlo algorithm will satisfy etaile balane an eventually sample the euilibrium Tsallis istribution. II.3 Monte Carlo estimates of Gibbs-Boltzmann statistial averages While this Monte Carlo algorithm will sample the generalize statistial istributions, it an also be use to eetively sample the onguration spae that is statistially important to the stanar anonial ensemble probability ensity [6]. The rst metho of this kin was the \-jumping Monte Carlo" metho. () At ranom a hoie is mae to make a uniformly istribute \loal" move, with probability, J,ora global \jump" move, with probability J. (2a) The loal trial move is sample from a uniform istribution, for example from a ube of sie. (2b) The jump trial move is sample from the generalize statistial istribution T (r! r 0 )=p (r 0 ) (5) at >. (3a) The loal move is aepte or rejete by the stanar Metropolis aeptane riterion with probability p = min ; exp[, U] : (6) (3b) The jump trial move is aepte or rejete aoring to the probability p = min ; exp[, U p (r) p (r 0 ) ] (7) where the bias ue to the non-symmetri trial move has been remove. This algorithm satises etaile balane. The eloalize Tsallis statistial istributions are sample to provie long range moves between well-separate but thermoynamially signiant basins of onguration spae. Suh long range moves are ranomly shue with short range moves whih provie goo loal sampling within the basins. However, the Monte Carlo walk samples the euilibrium Gibbs-Boltzmann istribution. When >, it has been shown that the generalize -jumping Monte Carlo trajetories will ross barriers more freuently an explore phase spae more eiently than stanar Monte Carlo (without jump moves). (For a review of reent methos for enhane phase-spae sampling see [7].) We have shown how this property an be exploite to erive eetive Monte Carlo methos whih provie signiantly enhane sampling relative to stanar methos. II. 4 roblems for many-boy systems Consier a system of N partiles in imensions. Using the stanar proeure of integrating over the momenta in Cartesian oorinates, we an write the average of a mehanial property A(r N ) using E. () as [5] R A(r N ), (, )U(r N ), + N2 r N R [, (, )U(rN )], + N 2 r N : (8) This enition of the normalize statistial average is base on an proportional to the -expetation value. However, it is more useful sine it is not neessary to evaluate the partition funtion to ompute the average. Nevertheless, this result is unsatisfatory. It is poorly behave in the N!thermoynami limit. We will

4 82 John E. Straub an Ioan Anriioaei aress this problem in the next setion in the ontext of Maxwell-Tsallis statistis. III Maxwell-Tsallis statistis For many systems for whih it is iult to ene effetive Monte Carlo trial moves, Moleular Dynamis methos an be use to ompute euilibrium averages. The esire to reate suh a moleular ynamis algorithm to sample the Tsallis statistial istribution le to the use of Maxwell-Tsallis statistis [6,5]. The euilibrium istribution is taken to be a hybri prout of () a Tsallis statistial istribution (for the ongurations) an (2) a Maxwell istribution (for the momenta) as where p (r N ; p N ) exp[,(k(p N )+ U(r N ))] (9) U(r N )= (, ) ln, (, )U(r N ) (20) is the transforme potential an K(p N )= NX k 2m k p 2 k (2) is the kineti energy of the N-boy system ene in the usual way. III. Ensemble averages Consier a system of N partiles in imensions. Using the stanar proeure of integrating over the momenta in Cartesian oorinates, we an write the average of a mehanial property A(r N ) using E. () as R A(r N ), (, )U(r N ), r N R [, (, )U(rN )], r N : (22) This enition is base on an proportional to the -expetation value. This result laks the o N- epenene in the exponent of the ongurational weighting funtion foun for the ase of pure Tsallis statistis in E. (8). While it is not lear whih result is \right," this expression is ertainly more satisfying in the N!thermoynami limit. III.2 Moleular Dynamis estimates of Maxwell- Tsallis statistial averages Stanar Moleular Dynamis for an ergoi system generates time averages whih agree with averages taken over a miroanonial istribution. To ompute averages suh as E. (22) for the generalize anonial ensemble probability ensity using MD, we will employ a trik. We ene a moleular ynamis algorithm suh that the trajetory samples the istribution (r N )by having the trajetory move on a temperature epenent, but stati, eetive potential [6]. The euation of motion takes on a simple an suggestive form m k 2 r k t 2 =,r r k U =, [, (, )U(r N )] r r k U(r N ) (23) for a partile of mass m k, at position r k, an U ene by E. (3). It is known that in the anonial ensemble a onstant-temperature moleular ynamis algorithm generates samples from the onguration spae aoring to the Boltzmann probability. As a result, this generalize moleular ynamis will sample the Tsallis statistial istribution (r N ). The eetive fore employe is the \exat" fore for stanar moleular ynamis,,rr k U, sale by (r N ; ) =, (, )U(r N ) (24) whih is a funtion of the potential energy. This saling funtion is unity when = but an otherwise have a strong inuene on the ynamis. Assume that the potential is ene to be a positive funtion. In the regime >, the saling funtion (r N ;) is largest near low lying minima of the potential. In barrier regions, where the potential energy is large, the saling funtion (r N ;) is small. This has the eet of reuing the magnitue of the fore in the barrier regions. A partile attempting to pass over a potential energy bar-

5 Brazilian Journal of hysis, vol. 29, no., Marh, rier will meet with less resistane when > than when =. At euilibrium, this leas to more eloalize probability istributions with an inrease probability of sampling barrier regions. III.3 Moleular ynamis estimates of Gibbs- Boltzmann statistial averages The generalize moleular ynamis esribe above generates trajetories whih, average over time, sample the Tsallis statistial istribution. To ompute averages over the Gibbs-Boltzmann istribution we reweight eah measurement as hai = * A e,h(,) [, (, )H(,)], + * e,h(,) [, (, )H(,)], +, : (25) Using this expression, the stanar ( = ) Gibbs- Boltzmann euilibrium average properties may be alulate over a trajetory whih samples the generalize statistial istribution for 6= with the avantage of enhane sampling for >. This metho leas to an enhane sampling of onformation spae. However, it suers a bit from the ease with whih the trajetory moves through high energy regions in the > regimes [6]. Most of those regions of high potential energy are not thermoynamially important. It is goo to visit them, as the -jumping Monte Carlo oes, but only on the way to another thermoynamially important region. The -jumping Monte Carlo metho has the avantage that the trajetory samples the Gibbs-Boltzmann istribution (no reweighting is neessary). The walk is ompelle to spen time in thermoynamially signiant regions. This euation of motion is onsistent with Tsallis statistis in as muh as a long time ynamial average for an ergoi system will provie results iential to the average over the Tsallis statistial istribution. However, it annot be sai to tell us about the true ynamis of the system. In the next setion, we present an alternative interpretation of the origin of Maxwell- Tsallis statistis. III.4 From Newton's euation to Tsallis statistis through a eformation of spae The Tsallis statistial istribution is typially erive from an extremization of the reformulate Gibbs entropy. Aswehave shown, it is possible to erive a mirosopi ynamis that generates time averages whih are in agreement with statistial averages over the Tsallis istribution. However, there is no uniue way to o this. We have presente one metho base on the Maxwell-Tsallis moel. The momenta are treate in the stanar way an the ynamis is a normal Hamiltonian ow over an eetive potential energy funtion U. A seon possibility is to begin with the Hamiltonian H T = X k 2m k (p T k ) 2 + U T (r T ) (26) in the phase spae, T =(r T ;p T ). We ouple this expression with two enitions. First, the moie potential energy U T is ene through the euality U T (r T )=U(r): (27) For example, if U(r) =r 2 an r T = r 2 then U T (r T )= r T. Seon, the transformation between the oorinate x k an x T k is T j = p [, (, )U(r N )] =2 jk ; k where jk is the Kroneker elta, an T j = p [, (, )U(r N )] =2 jk : k As these enitions iniate, the Jaobian transformation matrix is iagonal an in fat an be written M = p [, (, )U(r N )] =2 I: (30)

6 84 John E. Straub an Ioan Anriioaei The eterminant of this Jaobian matrix is simply et M = N [, (, )U(rN )] N =, V T =V N (3) where is the imension of the spae in whih the N partile system is foun. This eterminant esribes how the inremental volume of onguration spae in the stanar Cartesian oorinates (V ) N = x y :::y N z N is transforme to the inrement (V T ) N is the Tsallisian spae. How oes this metri eform the orinary Cartesian spae? When =,we reover the stanar metri. When >, in regions of high potential energy, the istane between two points will be eetively ialate. This leas to a ialation of the barrier an sale regions relative to regions of lower potential energy an a reution in the assoiate fore. Note that the usual aveats apply. The egree of both the absolute an relative ontration will epen on the zero of energy, an we must ignore those ases where the saling fator is negative. With these enitions it is possible to rewrite the Hamiltonian as H T = X T 2 k p 2 k + U(r): (32) 2m k The euations of motion for the kth atom are _x T k = m k p T k ; _pt T k : (33) whih results T k _p @p T : (34) k Rewriting this expression using the metri transformation ene above we obtain p [, (, )U(r N )] =2 _p k =, p [, (, )U(r )] k whih we an reognize as _p k k : (36) This is preisely Newton's euation that we foun using Maxwell-Tsallis statistis { the stanar momentum for a partile moving over an eetive potential U. So this oorinate transformation (r; p)! (r T ; p T ) maps the Hamiltonian ynamis of r on U(r) onto a Hamiltonian ynamis of r T on U T (r T ). If we insert this Hamiltonian in our enition of the expetation value of a property A(r) we n R A(r)[, (, )H T (,)],, T R [, (, )HT (,)],, T (37) whih, after a hange of variables for the integration, beomes R A(r)[, (, )HT (,)], et M, R [, (, )HT (,)], et M, : (38) erforming the integral over the momenta we n R A(rN ), (, )U(r N ), r N R [, (, )U(rN )], r N (39) whih we foun earlier using Maxwell-Tsallis statistis. However, the Maxwell-Tsallis statistis was an \impure" appliation of the formalism sine the kineti energy was assume to have a Gaussian Maxwell istribution. Reall that in the appliation of \pure" Tsallis statistis we foun R A(r N ), (, )U(r N ), + N2 r N R [, (, )U(rN )], + N 2 r N : (40) This result was eeme to be unsatisfatory ue to the unwelome epenene on N in the thermoynami limit (N!). Our new result allows us to follow a pure appliation of the Tsallis statistis using an eetive Hamiltonian that both satises Newton's euations of motion on the orinary potential an leas to an intuitively satisfying enition of a statistial average.

7 Brazilian Journal of hysis, vol. 29, no., Marh, IV Another path to the Tsallis statistial istributions We begin with the lassial ensity matrix exp(,h). Rather than writing e,h = e,h= ; (4) as is ommonly one in isussions of path integral representations of the ensity matrix, suppose that we express the exponential as a limit e,h = lim : (42)! +H= Now suppose that we remove the kernel of the limit (43) +H= an onsier it for arbitrary.if we substitute =, (44) we n that = ; 2; 3; 4 ::: beomes = 0; 2 ; 2 3 ; 3 ::: an 4 e,h (, (, )H), : (45) The right han sie of this expression is the Tsallis statistial istribution whih was originally erive by extremizing the \entropy" S subjet to the onstraints that the istribution is normalize an that the average energy is ompute in the stanar way. Now suppose that we instea ene =, (46) so that =; 2; 3; 4 ::: beomes =2; 3 2 ; 4 3 ; 5 4 :::. The resulting istribution is e,h (, (, )H), : (47) The right han sie of the expression is preisely the Tsallis statistial istribution p (,) erive by extremizing S subjet to the onstraints that the istribution is normalize an the average energy is ene in terms of the \-expetation" value. How an we interpret these results? Tsallis showe how a generalize statistis an originate from the Gibbs entropy formula an the ientity p ln p = lim n! n p(pn, ). He then strippe away the limit an interprete the kernel where n =, p ln p!, p(p,, ): (48) However, it is possible to reah the same istribution from a ierent starting point { the euilibrium ensity matrix. We rewrite the ensity matrix using the ientity exp(,h) = lim! ( + h= ),. We then strip away the limit an interpret the kernel where ==(, ) e,h! (, (, )H), (49) in the spirit of Tsallis statistis. For the later expression, erive using the onstraint base on the -expetation value, when = we have the interesting ase of = 2. In the limit that =, we reover the Gibbs-Boltzmann statistial istribution. Intermeiate values of provie ases in between these limits. IV. Reovering Maxwell-Tsallis statistis Think of the lassial ensity matrix e,h = e,h= = e,k= e,u= (50) where we have separate the kineti energy an potential energy ontributions. Now suppose that we arry out the approximation esribe above for the Boltzmann fator alone. We n e,h e,k : (5) +U= This is preisely the expression for the Maxwell-Tsallis statistial istribution onsiere earlier where = =(, ). IV.2 Tsallis statistis an Feynman path integral uantum mehanis The uantum mehanial ensity matrix an be written

8 86 John E. Straub an Ioan Anriioaei e,h = e,(k+u)= = lim e,k= e,u= : (52)! If restrit ourselves to nite, the kernel is only approximate sine K an U o not, in general, ommute. In the path integral formalism, is interprete as the number of pseuopartiles in the isomorphi lassial system. When =, one reovers the fully lassial system. When =, one obtains an exat representation of the uantum mehanial system. For intermeiate values of we have a semi-lassial representation of the system. Suppose that we write e,h= e,k= = e,k= e, U= : (53) +U= This result provies us with a path integral representation akin to the lassial Maxwell-Tsallis statistis where the pseuopartile \neklae" of beas, eah harmonially restraine to its nearest neighbors, interat in imaginary time through the logarithmi potential Sine U(r) = ln + U(r) : (54) e,h = lim e,k= (55)! +U= this is an exat expression for the ensity matrix. It is possible to employ this expression in path integral simulations of onense phase systems. Orinarily, in systems where uantum eets are most important, one must approah large values of before there is onvergene to the exat uantum mehanial average. As we have shown, if the neklae samples the Tsallis statistial istribution it shoul visit regions of higher potential energy more freuently an the istribution shoul be signiantly more eloalize than the stanar representation for the same number of beas in the neklae. This implies that this representation might provie faster onvergene to the uantum mehanial limit than the stanar form. We gratefully aknowlege the National Siene Founation for support (CHE ), the Center for Sienti Computing an Visualization at Boston University for omputational resoures, Troy Whitel for helpful isussions, an the Institute for Avane Stuies at The Hebrew University of Jerusalem for support an hospitality. Referenes. C. Tsallis, J. Stat. hys. 52, 479 (988). 2. E.M.F. Curao an C. Tsallis, J. hys. A: Math. Gen. 24, L69 (99). 3. A. Wehrl, Rev. Mo. hys. 50, 22 (970);. Darozy, Inf. Control 6, 36 (970). 4. C. Tsallis, hys. Rev. Lett. 75, 3589 (995). See also 5. J.E. Straub an I. Anriioaei. Exploiting Tsallis statistis. In. Deuhar, J. Hermans, B. Leimkuhler, A. Mark, S. Reih an R. D. Skee, eitors, Algorithms for Maromoleular Moeling (Leture Notes in Computational Siene an Engineering), vol. 4, page 89. Springer Verlag, I. Anriioaei an J.E. Straub, J. Chem. hys. 07, 97 (997). 7. B.J. Berne an J.E. Straub, Curr. Opin. Stru. Bio. 7, 8 (997).

Chapter 2: One-dimensional Steady State Conduction

Chapter 2: One-dimensional Steady State Conduction 1 Chapter : One-imensional Steay State Conution.1 Eamples of One-imensional Conution Eample.1: Plate with Energy Generation an Variable Conutivity Sine k is variable it must remain insie the ifferentiation