Topics on Statistical Mechanics TCMM Lecture Notes

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1 Topcs on Statstcal Mechancs TCMM Lecture otes Fernando Fernandes Department of Chemstry and Bochemstry, Faculty of Scences, Unversty of Lsbon, Portugal Emal: A short revew of basc concepts and formulae of statstcal mechancs s presented, pavng the way to lqud state theory and computatonal methods. The Legendre- Laplace transformaton of statstcal ensembles s llustrated. These notes support the lectures on Lqud State and Phase Transtons of the TCMM Intensve Course. It s presupposed that the partcpants are famlar wth the fundamentals of thermodynamcs and statstcal mechancs at the level of an undergraduate course on physcal chemstry. 1. Introducton The equlbrum state of a bulk system s defned by a relatvely small number of thermodynamc varables. For example, a one-component gas at equlbrum, free of external appled felds or chemcal reactons, can be dentfed by three varables such as: (V, T, ) (1) where V, T and are, respectvely, the volume, the absolute temperature and the number of molecules. The other thermodynamc propertes of the gas are functons of those varables. Ths contrasts wth the huge number of varables requred to defne a dynamcal mcroscopc state (mcrostate) of the system, whch s of the order of (Avogadro s number). In ths revew on basc concepts and formulae of statstcal mechancs, we shall consder, for the sake of smplcty, one-component systems wth monoatomc molecules (translatonal degrees of freedom only) nteractng through central forces, and free of external appled felds, surface effects and chemcal reactons (desgnated as smple systems, from now on). Indeed, the ncluson of more general stuatons would add consderable analytcal complexty though not alterng the essental underlyng concepts. We also assume that Born-Oppenhemer s approxmaton and Boltzmann s statstcs are vald for the systems under study. A classcal mcrostate,, can then be defned by: 1, 2,...,, 1, 2,...,, r r r p p p r p (2) The r s are the poston vectors and the p s are the lnear momentum vectors of the molecules, relatvely to a gven referental, both functons of tme. As each vector has three components, the mcrostate s defned by a total of 6 varables. 1

2 A quantum mechancal mcrostate,, s defned by the wave functon: r ; t (3) Wave functons are vectors of Hlbert s space. The coordnates are not functons of tme. Thus, r ; t1 and r ; t2 should be understood as two dfferent vectors at tmes t 1 and t 2 respectvely. The complete specfcaton of a thermodynamc state leaves the mcrostates undefned. Indeed, although n an equlbrum state the macroscopc propertes are nvarant on tme, the system, durng ts tme evoluton, goes through very many mcrostates, that s, t s subjected to thermal fluctuatons. In other words, there are very many mcrostates consstent wth a gven thermodynamc state (macrostate). 2. Sate Space, Trajectores, Ensembles and Averages The mcrostate of a smple classcal many-body system may be geometrcally specfed by locatng a pont n a 6 dmensonal state space, the coordnates of the pont beng the 3 poston coordnates and the 3 momentum components whch, as we have seen before, defne the dynamcal state. The space and the pont are called phase space and phase pont respectvely. Once the forces actng on each molecule and the phase pont are known at tme t, the future (and the past) trajectory of the pont through phase space s determned by the classcal equatons of moton, whch can be wrtten n the Hamltonan form: H r, p r p H p r r, p 1, 2..., (4) where H 2 r p p U r (5) 2m, 1 s the Hamltonan (total energy) of the system, that s, the sum of the knetc and potental energes, and m s the mass of molecule. Assumng that the potental energy s parwse addtve, t can be wrtten as: j U urj j r (6) where u r s the effectve par potental and rj r rj s the dstance between molecules and j. For example, the Lennard-Jones potental: 2

3 12 6 u r j 4 (7) rj rj s an mportant effectve par potental for computer smulaton and theory (ε s the depth of the potental-well, and σ the molecular dameter also called the collson dameter). The ewtonan or Lagrangan forms of the moton equatons are, of course, equvalent and one can choose whchever s more convenent for the problem under study. The ewtonan form s often the preferred one: dr m dt F (8) F (the force actng on partcle of mass m by the remanng -1 molecules) s, n turn, the gradent of the potental: F U r r (9) In a quantum mechancal context, the state space s Hlbert s vector space. Once a vector at tme t and the Hamltonan operator are known, the tme evoluton of the system s determned by the tme-dependent Schrödnger s equaton: Hˆ (10) t It s mpossble, of course, to ntegrate, analytcally, the moton equatons of many-body systems. So, to follow ther dynamc evoluton numercal methods are requred. The state space, and the path of a system through t, can represented by the twodmensonal projecton n the followng fgure: Each box corresponds to a phase space hypervolume,r dr, p dp, or to a Hlbert s space vector. ote that, even n the classcal case, although the poston and momentum spaces are contnuous, the molecular postons and momenta are always affected by the uncertantes of Hesenberg s prncple. Therefore, the phase space s decomposed nto an arbtrary number of hypervolumes, each representng a mcrostate, nstead of takng a mcrostate as an exact phase pont. 3

4 The geometry of the state space spanned by the system depends on the macroscopc constrants mposed upon t. For example, f the total energy, E, the volume, V, and the number of molecules,, are fxed, then the path of the system s constraned to a hypersurface of constant energy. If the temperature, T, s constraned nstead of the total energy (system n a thermostat), the energy can fluctuate and the trajectory of the system s free to flow through hypersurfaces of dfferent energes. Ether the classcal equatons of moton or Schrödnger s equaton are symmetrc n tme, that s, are nvarant on tme reversal. Thus, there s nothng ntrnsc to the mcroscopc laws of moton that can prevent the system to repeatedly pass, after a long enough tme, arbtrarly close to all mcrostates consstent wth the macroscopc constrants mposed on the system. Ths s the so-called prncple of ergodcty, whch s assumed to hold for all real many-body systems. Another basc assumpton of statstcal mechancs s that a measurable property, G, of a gven system s generally assocated to a mechancal property,g, defned for each mcrostate,, such that the observed value, Gobs, s equal to the tme average, Gtm, of G over the number of nstantaneous mcrostates, M, spanned by the system durng the observaton tme: G obs 1 M G G tm (11) 0 M where G s the value of the mechancal property at each nstantaneous mcrostate. Examples of mechancal propertes are the nstantaneous nternal energy, temperature and pressure, respectvely: 2 p r (12) Enst t U m 1 2 T nst 2 p / m 1 () t 3k (13) p nst t kt V nst rj f j 3V j (14) The last two equatons come from the energy equpartton and the vral theorems, respectvely. s the force on atom due to atom j. f j A general computatonal algorthm to work out tme averages: a) Defne the ntal mcrostate of the system. b) Gve the potental energy, U(r ), or the Hamltonan operator, Ĥ. 4

5 c) Calculate the trajectory (mcrostates) of the system, durng a long enough tme, by numercally ntegratng the moton equatons, takng nto account the macroscopc constrants mposed upon the system. d) Calculate the mechancal property,g, at each of the M generated nstantaneous mcrostates. e) Calculate the tme average of the mechancal property by equaton (11) Ths s the basc procedure of the well-known Molecular Dynamcs method (MD). There s, however, another scenaro where tme apparently takes no role. Assumng the ergodc hypothess, equaton (11) can be rewrtten as: G obs n G (15) M where n s the number of tmes that mcrostate s vsted. In the case of a quantum system, G Gˆ, that s the expectaton value of the operator Ĝ when the system s n mcrostate. The last equaton can be recast n the form: obs en (16) G PG G where P n / M s the probablty or fracton of tme spent n mcrostate, and Gen s the so-called ensemble average. The equaton suggests the followng abstract construct: at a gven nstant, we can assocate to any macroscopc system the ensemble of all possble mcrostates, consstent wth the constrants, each mcrostate havng the weght P. In other words, an ensemble s a set of very many mental copes (ghosts) of the macroscopc system, each ghost wth the same macroscopc constrants, but not necessarly n the same mcrostate. It should be emphaszed that as the ensemble s an nstantaneous mental construct, ther elements (mcrostates) have not a tme sequence, contrary to the mcrostates generated by ntegratng the moton equatons. However, f the system s ergodc, then, t seems plausble to assume that the ensemble and tme averages are equal. The last statement consttutes one of the postulates of statstcal mechancs. Supposng that the system s not ergodc, t does not sample, along ts tme evoluton, all the avalable mcrostates consstent wth the constrants, whlst the correspondng mental ensemble contans, by defnton, all those mcrostates. In such a case the two averages would not be equal. Yet, as we have referred to before, all real many-body systems are assumed to be ergodc. How can we set up an ensemble? One possble computatonal procedure s to generate, from a unform probablty dstrbuton, an ensemble of very many dfferent random mcrostates consstent wth the macroscopc constrants of the system. Then, the ensemble average s calculated by 5

6 equaton (16), weghtng the values of the mechancal property for each mcrostate wth P. As we shall see, ths procedure has a poor computatonal effcency as far as molecular systems are concerned. Instead of generatng the mcrostates unformly and weghtng each one wth ts probablty, a much more effcent procedure s to generate the mcrostates wth a probablty P and weghtng them evenly. Then, the ensemble averages are calculated by a smple arthmetc mean of the mechancal propertes over the mcrostates, as n equaton (11). Ths way, denoted by mportance samplng, guarantees that the mcrostates that most contrbute to the ensemble averages have frequences consstent wth ther probabltes. Ths s the basc procedure of Metropols s Monte Carlo method (MC). To put all ths nto acton t s necessary to prevously establsh the probablty dstrbutons, P, from frst prncples, whch s one of the objectves of statstcal mechancs. The ensemble mmcs, n a certan way, the trajectory of the system n state space along ts tme evoluton. There s, however, an essental dfference. The tme evoluton s determnstc, calculated by ntegratng the moton equatons, so the mcrostates have a sequence on tme. On the contrary, the ensemble s a mental nstantaneous construct, operatonally generated n a stochastc way. o ntegraton of moton equatons s nvolved, tme takes no role and the mcrostates have no determnstc tme sequence. Ether n a dynamc or n a stochastc process an assembly of mcrostates s always obtaned. Orgnally, the term ensemble was reserved for the mental construct, but as tme and ensemble averages are assumed to be equal, the term s often used for both cases. From the context, the partcular meanng s always clear. The ensembles are classfed accordng to the macroscopc constrants mposed upon the system. For example: Ensemble Constrants Mcrocanoncal E, V, Canoncal T, V, Isothermal - Isobarc T, p, Grand - Canoncal T, μ, V (μ s the chemcal potental) The concept of ensemble and ensemble average was nvented manly to get round the dffcult operatonal problem of calculatng tme averages, faced by the fathers of statstcal mechancs (Boltzmann, Maxwell, Gbbs and Ensten). It turns out, however, that ensemble averages are realzed n measurements on real systems. In fact, a bulk system can be pctured as composed by many macroscopc sub-systems (an ensemble) whose szes are greater than the correlaton lengths. Therefore, an nstantaneous (shorttme) observaton on a real system corresponds, after all, to many ndependent measurements over the members of the ensemble, that s, to an ensemble average. Moreover, ths concept also allows the operatonal defnton of tme dependent averages whch are of the utmost mportance n non-equlbrum systems. As we have sad before, one basc assumpton of statstcal mechancs s that all real many-body systems are ergodc, at least n the range of normal energes, temperatures or pressures. In computatonal studes, however, we use models and smulaton technques, such as MD and MC, nstead of real systems. It s noteworthy that we frequently encounter ergodc problems n computer smulatons. In fact, the tme 6

7 trajectores or/and the ensembles generated for the models may reman stuck n a restrcted regon of phase space, not properly samplng all the avalable space. In such cases, the tme and ensemble averages are not equal. Ths s not ntrnsc to the real system. It s due to the technques themselves, whch have to be used wth care n order to avod ergodc problems. One crucal problem of statstcal mechancs, as a molecular theory, s to establsh operatonal ways for calculatng tme and ensemble averages of mechancal propertes over the mcrostates (consstent wth the partcular thermodynamc state of nterest) n order to put the averages nto correspondence to the observable macroscopc propertes. When the system s not n equlbrum, the observable macroscopc propertes are also averages over the dynamcal states of the system. In such cases, the averages are tme dependent and statstcal mechancs also has to ratonalze how the tme dependent propertes rreversbly approach ther equlbrum values. All based on the mcroscopc nformaton gven by classcal and/or quantum mechancs. In other words, statstcal mechancs ams to establsh a brdge between the macroscopc observable propertes and the underlyng mcroscopc propertes of a gven system. 3. Probablty Dstrbutons, Partton Functons and Thermodynamcs The mcrocanoncal ensemble s characterzed by the fxed constrants E, V and. As the total energy underles the dynamcs of the system (see the Hamltonan equatons of moton) and s constant n ths case, t seems plausble to assume that all the avalable mcrostates, Ω (E, V, ), have equal probablty: 1 P E, V, (17) Ths s one of the postulates of statstcal mechancs. We can recast the last equaton n a sem-classcal form, once the poston and momenta spaces are contnuous: P E, V,, 1 3!, H r p E dr dp h H E r p dr dp (18) where δ s the Drac s delta functon (see Appendx). Also:! 1 3,! 1 3, r p r p r p r p h H E d d h H E d d 1 (19) Therefore, the normalzng factor of the probablty dstrbuton s: 3 1,,!,,, Q E V h H r p E dr dp E V (20) the so-called mcrocanoncal partton functon. It s drectly related to the entropy of the system, S(E,V, ), by the well-known Boltzmann s equaton: 7

8 S E, V, k ln Q E, V, (21) where k s Boltzmann s constant. S(E,V,)/k, s called the thermodynamc dmensonless characterstc functon of the mcrocanoncal ensemble. It s noteworthy that the entropy s a property of the total number of avalable mcrostates (the so-called multplcty of the macrostate), not defned for each mcrostate. The same s true for other propertes, for example, the Helmholtz and Gbbs free energes. As such they are called statstcal propertes, contrastng wth mechancal propertes (total energy, pressure, etc.) that are defned for each mcrostate. Every ensemble has a thermodynamc dmensonless characterstc functon, TCF, whch s related to the correspondng partton functon, Q, by a generalzed Boltzmann s equaton. TCF ln Q (22) Ths s one of the key equatons of statstcal mechancs, whch establshes a drect brdge between the macroscopc descrpton of a gven system (thermodynamcs) and ts mcroscopc descrpton. 3 All sem-classcal partton functons are affected by the factor! h 1.! corrects the molecular dstngushablty ntroduced by the ntegrals; h 3 (where h s Planck s constant) makes the functons consstent wth the lmtatons mposed by Hesenberg s 3 uncertanty prncple ( dr dp h ). The mcrocanoncal ensemble s, theoretcally, very mportant. For example, t s the ensemble underlyng the basc method of molecular dynamcs. Under a practcal standpont, however, t has some dsadvantages. On the one hand, the constrant of fxed total energy s not usual n expermental work. On the other hand, the calculaton of the mcrocanoncal partton functon mples ether dffcult countng technques or dealng wth Drac s delta functon, both not drectly amenable to numercal calculatons. As such, we should be able to change to other ensembles accordng to the probed expermental envronment. The canoncal ensemble, for example, s a sutable one, snce fxed T, V and are common expermental constrants. There are varous ways, descrbed n the lterature, to move from one ensemble to another. We shall adopt a powerful postulatonal approach, whch conveys the beautful structure of thermodynamcs and statstcal mechancs producng, n turn, a toolkt n a straghtforward manner: P1. There exsts a functon, called thermodynamc dmensonless characterstc functon, TCF, whch contans all macroscopc nformaton and depends upon the expermental constrants (natural varables) mposed on the system P2. There exsts a functon, called partton functon, Q, whch contans all the mcroscopc nformaton and depends upon the Hamltonan and the natural varables of the system. The unversal relaton between those functons s the generalzed Boltzmann s equaton: TCF ln QH; X X (23) 8

9 where H s the Hamltonan, and X represents the vector of natural varables. The partton functon s the normalzng factor of the probablty dstrbuton functon. P3. The relatons between other thermodynamc functons and the mcroscopc propertes, for the gven set of constrants, are obtaned from the unversal equaton (23) by applyng to t approprate dfferental operators. P4. To change natural varables (constrants) so that all the nformaton s preserved, Legendre and Laplace transforms should be performed on TCF and Q respectvely. Postulates 1 and 2 have already been llustrated for the mcrocanoncal ensemble. Consderng, now, the well-known thermodynamc dfferental: S S S ds de dv d (24) E V, V, E, E V wth the denttes: S 1 S p S ; ; (25) E T V T T V, E, E, V the meanng of postulate 3 s obvous: 1 T p T E V V, T E, EV, kln Q E, V, kln Q E, V, kln Q E, V, (26) Let us now apply postulate 3, to change the varable E by T, that s, to move from the mcrocanoncal to the canoncal ensemble. Takng the Legendre transform of S(E,V,)/k n order to E (see Appendx): E V, S/ k AT, V, S S S E Lg E k k E k kt kt, (27) that s, the thermodynamc dmensonless characterstc functon of the canoncal ensemble, where A s Helmholtz s energy. Takng the Laplace transform of Q(E,V,) n order to E (see Appendx), wth parameter β=1/kt : 3 1 QEV (,, )exp EdEh! exp Hr, p dr dp QTV,,, 0 (28) 9

10 that s, the canoncal partton functon. Then, by postulate 3: ATV,, ln QTV,, (29) kt whch s a well-known equaton. As the partton functon s the normalzng factor of the probablty dstrbuton, one obtans for the canoncal ensemble: PTV,, exp H, r p dr dp QTV,, (30) whch s, the probablty of fndng the system n the mcrostate, or phase hypervolume, r dr, p dp. Recognzng that the Hamltonan s the total energy of the system, ether the canoncal partton functon or the probablty dstrbuton can also be wrtten n dscrete (quantzed) equvalent forms: QTV,, exp E (31) PTV,, exp exp E E (32) where the E s are the energy egenvalues of the Hamltonan operator. Dfferentatng A(T,V,) and dentfyng the partal dervatves wth the rght thermodynamc functons (see the Max-Born dagram n the Appendx), as made for the entropy, one obtans, for example: p kt ln Q T, V, V T, S ktln QT, V, (33) T V, kt ln Q T, V, TV, By means of the Legendre-Laplace transform technque, one can move from one ensemble to another dervng, n a smple and elegant way, the correspondng formulae. 3.1 Fluctuatons Once the dfferent ensembles contan all the macroscopc and mcroscopc nformaton about a gven system, the choce of the ensemble to be used s a matter of convenence, generally n accordance wth the partcular expermental stuaton that we want to descrbe. In other words, all ensembles are equvalent. 10

11 There s, however, an ssue that should be clarfed. Let us take the mcrocanoncal and canoncal ensembles. The frst, descrbes an solated system, so the total energy E, the volume V, and the number of molecules are constraned, but other propertes (temperature, pressure, etc) can fluctuate. The second descrbes a system n a heat bath (thermostat) so the temperature T, the volume V, and the number of partcles are constraned, but other propertes (total energy, pressure, etc.) can fluctuate. Suppose we use the mcrocanoncal ensemble. Then, although the temperature fluctuates t has an average value, <T> mc. Suppose now, that we change for the canoncal ensemble usng <T> mc and the same V and. The equvalence between ensembles mples that, although the energy fluctuates n the canoncal ensemble, ts average value <E> can must be: <E> can = E (mcrocanoncal) (34) What s the order of magntude of the energy fluctuatons n the canoncal ensemble? Usng the canoncal probablty dstrbuton, the average nternal energy s: Eexp E Ecan EP ln,, exp E QT V (35) ote that the canoncal energy s also drectly related to the partton functon. The 2 2 averaged square energy s E E P. Then, after some algebra, the averaged square fluctuaton n the canoncal ensemble: E E E E kt C (36) E V where σ E s the standard devaton and C V s the sochorc heat capacty. Snce C V s of the order k and <E> of the order kt: 2 kt C E V 1 O E E, (37) whose value s neglgble for bulk systems (~10-23 ). Ths llustrates the equvalence between the mcrocanoncal and canoncal ensembles. A smlar analyss can be done for the other ensembles. Fnally, t s noteworthy that the sochorc heat capacty s drectly related to the energy fluctuatons n the canoncal ensemble. Ths s a result of the utmost mportance. Indeed, we shall see that such knd of relatons enable us to estmate second order propertes from Molecular Dynamcs or Monte Carlo computer smulatons. 4. Reduced Dstrbuton Functons If the Hamltonan, gven by equaton (5), s ntroduced n the canoncal partton functon (28) or n the probablty dstrbuton functon (30), they can be broken n the knetc (momentum) and confguratonal (excess) contrbutons. Once the knetc energy s, n turn, decomposable nto sngle-molecule terms the respectve contrbuton can be 11

12 further smplfed by a straghtforward analytcal ntegraton. The confguratonal contrbuton, unfortunately, can not be analytcally ntegrated (except for the deal gas or sold, and other relatvely smple models that we shall refer to durng the lectures) snce the potental energy s not, n general, decomposable. Thus, the canoncal partton functon becomes: wth QTV,, QQ P C (38) Q P 2 mkt 2 h 3 /2 (38) and Q C Z VT,, (39)! where Z TV,, expur dr (40) s the confguratonal ntegral. For the deal gas, for example, the confguratonal ntegral s V and the partton functon s readly obtaned. As for the probablty dstrbuton functons, from the above consderatons t s evdent that we also can defne varous momentum and confguratonal dstrbuton functons. Thus, the -partcle confguratonal densty functon s defned as: r r r,,...,! 1 2 r exp U exp U r dr (41) whch expresses the jont probablty of fndng a molecule (any one) at poston r 1, another at r 2,., and another at r, regardless ther momenta. By ntegratng (41) over dfferent postons we can defne a herarchy of reduced densty functons. In general, the reduced n-partcle confguratonal densty functon s: n/ r, r,..., r 1 2! exp U r drn 1... dr n! exp U r dr n (42) whch expresses the jont probablty of fndng a molecule (any one) at poston r 1, another at r 2,., and another at r n, regardless the postons of the remanng -n molecules. When r r j ; 1, j n, the correlatons between molecules approach asymptotcally zero. Then, the molecules become ndependent and we can factorze: n/ 1/ 1/ 1 2 n 1 2 1/ r, r,..., r r r... r (43) Ths suggests the defnton of the normalzed n-dstrbuton functon: n 12

13 Therefore: n n r g n/ r, r,..., r / 1 2 n 1 1/ r n (44) g n/ r 1 when r r j (45) n Consderng that r1 dr 1 (46) 1/ and that for a homogeneous system 1/ r s ndependent of the poston: r (47) V 1/ that s, the number densty, we have for a homogeneous system: For the partcular case n =2: g n/ n n/ n r r (48) n 2/ r, r 2/ 1 2 g r1, r 2 (49) 2 2/ r1, r2 s the jont probablty of fndng a molecule at poston r 1 and another molecule at r 2, regardless the postons of the remanng -2 molecules. It s well-known from probablty theory that the condtonal probablty of the event A gven the event B, denoted by P B A s: P A B PB A f PA 0 (50) P A Thus, the condtonal probablty (for an sotropc flud) that a molecule s found at the dstance r from another molecule located at the orgn of the chosen referental s: 2/ r, r 0, r g0, r gr (51) 2/ 1 2 The functon g(r) s called the radal dstrbuton functon (rdf). It s often referred to as the par correlaton functon or par dstrbuton functon. It s one of the correlaton functons playng a crucal role n the theory of lquds. For example, the average total energy and pressure of a homogeneous system are drectly related to the radal dstrbuton functon: 13

14 3 2 E kt urgr4 r d 2 2 r (52) 0 0 kt du r 3 p gr4 r dr V 6V (53) dr where u(r) s the par potental. We shall return to these equatons durng the lectures. Meanwhle, try to understand them on physcal grounds. Fnally, note that the physcal meanng of (51) s just the average local number densty at r gven that a tagged molecule s at the orgn of the referental. Appendx Max Born-Tsza Dagram It s a very useful mnemonc for thermodynamc functons and ther dervatves. The full detals are gven by Callen [2]. For example, the natural varables of a functon n the mddle of one of the square sdes (E, A, G or H) are the ones n the respectve corners. The partal dervatves of the functon n order to ther natural varables are the functons n the opposte extremes of the dagonals, affected by the mnus sgn f the arrow s pontng to the natural varables. Moreover, all the functons (E, A, G or H) have the number of partcles,, as natural varable. The respectve partal dervatves, n order to, are all equal to the chemcal potental. Legendre Transform A Legendre transform of the functon w(x, y, z), wth respect to x, defnes a new w functon v of the varables: u, y, z, by the relaton vu, y, zwx, y, z ux. x yz, Laplace Transform The Laplace transform of a functon F(x), defned for all postve values of x, wth respect to varable x, s a new functon f(s): exp f s sx F 0 x dx 14

15 where s s called the parameter of the transform. Drac s δ-functon It s a formal functon, defned through the followng relatons: xa 0; x a x a ; x a x a dx 1 ìv f x x a dx f a Consder, for example, the unt step functon: ote that ts dervatve: Recommended readng: x x 1; x 0 0; x 0 d dx [1] Introducton to Modern Statstcal Mechancs, D. Chandler, Oxford Unversty Press, [2] Thermodynamcs and An Introducton to Thermostatstcs, H.B. Callen, J. Wley et Sons, [3] Eght physcal systems of thermodynamcs, statstcal mechancs and computer smulaton, H.W. Graben; J.R.Ray, Mol. Phys., 80 (1993) [4] Molecular Smulaton. The State of the Art, F.M.S.S. Fernandes; P.C.R. Rodrgues, Potugale Electrochmca Acta, 17 (1999) x 15