A New Approach to Classical Statistical Mechanics

Transcription

1 Joural of Moder Physics, 211, 2, doi:1.4236/jp Published Olie Noveber 211 ( A New Approach to Classical Statistical Mechaics Abstract Nijaligappa Uakatha Departet of Physics, Karatak Uiversity, Dharwad, Idia E-ail: uakatha131@yahoo.co.i Received May 2, 211; revised Jue 22, 211; accepted July 6, 211 A ew approach to classical statistical echaics is preseted; this is based o a ew ethod of specifyig the possible states of the systes of a statistical assebly ad o the relative frequecy iterpretatio of probability. This approach is free fro the cocept of eseble, the ergodic hypothesis, ad the assuptio of equal a priori probabilities. Keywords: Statistical Equilibriu, Rado Sequeces, Cetral Liit Theore 1. Itroductio The object of classical statistical echaics is to explai the (statistical) properties of a assebly of a large uber of idetical particles i ters of the (deteriistic) laws of classical echaics. Such a theory has bee developed by Gibbs i the first decade of the last cetury [1]. The Gibbs approach, preseted by Tola [2], has bee accepted as the covetioal approach to classical statistical echaics. The theory preseted by Tola ad the subsequet authors [3,4] is based o: 1) the cocept of state of a ay-particle syste as defied by classical echaics i the sese that the state of a N-particle syste at ay istat of tie ca be specified by a poit i 6N diesioal phase space ad 2) the otio of probability prevalet at the begiig of the last cetury accordig to which probability refers to a ayparticle syste chose at rado fro the eseble of ay-particle systes. I this paper we preset a ew approach to classical statistical echaics based o the sigificat progress ade durig ad after the third decade of the last cetury i the theory of probability (a brach of pure atheatics) ad the ethods of statistical aalysis (a brach of applied atheatics) [5,6]. This ew approach is based o: 1) a ew ethod of specifyig the possible states of a assebly of particles which (ethod) is cosistet with the requireets of statistical aalysis, ad 2) o the relative frequecy iterpretatio of probability. The preset approach is a idepedet approach ad should be viewed as such. For the sake of clarity, the distictive features of the two approaches are also discussed i the text. 2. Preliiary Cosideratios For the sake of clarity we ay just etio the ai features of statistical aalysis ad the relatio betwee statistical aalysis ad probability theory. A large uber of physical etities (such as adult e i a populatio) are said to for a collective, or a populatio, or a statistical assebly [5], if each etity (or eber) of the assebly exists i oe of the ay (at least two) possible states S s (such as the state of parethood of havig uber of childre) ad the states of the ebers collected i ay systeatic aer lead to a rado sequece of these possible states, i which each etry belogs to oe eber. If N is the uber of ties the state S occurs i the rado sequece havig a total uber of N etries, the N /N is the relative frequecy of the state S. As there ca be ay differet systeatic ways of collectig the data, there would be ay rado sequeces of the sae states (relevat to the give assebly). I all these rado sequeces the relative frequecies of the states would have approxiately (i the statistical sese) the sae set of values; such sequeces are said to be statistically equivalet. Oe iportat feature of all statistical properties is that they are idepedet of such details as: 1) which particular eber of the assebly is i which particular state, 2) the regios of space withi which the idividual ebers exist i the assebly (such as the places of residece of e), ad 3) the total uber of ebers of the assebly. Evidetly, the relative frequecies of the possible states i a rado sequece possess these properties. I the theory of probability, statistical properties are dealt with i a ore abstract ad geeral aer by as-

2 1236 N. UMAKANTHA sociatig probabilities with the possible states per se; these probabilities are treated as uspecified costats with the uderstadig that, with referece to ay rado sequece of states (relevat to a statistical assebly), the uerical values of the relative frequecies of the states i the sequece are approxiately equal to the uerical values of the correspodig probabilities. It is extreely iportat to appreciate the poit that (the relative frequecies as well as) the probabilities are associated with the possible states ad ot with the ebers of the assebly. With this backgroud we cosider classical statistical echaics. A physical etity havig fiite o-zero ass boud to a tie-idepedet potetial is said to for a coservative syste; to the extet the iteral structure ad the exteral diesios of the etity do ot play ay role i the pheoeo uder cosideratio, we ca regard the etity as a particle, a poit-ass. We ca attribute ay physical properties to such a syste; the su-total of all the physical quatities which ca be attributed to the syste at ay istat of tie is said to specify fully the istataeous state of the syste at that istat. As tie passes, the syste evolves through a cotiuous sequece of successive istataeous states as govered by the laws of classical echaics; such a sequece ay be called a dyaical state of the syste. Each dyaical state is deteried by oe solutio of Newto s secod law of otio which (solutio) is specified by two costats, r ad p, the positio ad the oetu of the particle at soe iitial istat of tie t. A dyaical state ay be deoted as S(r, p, t ; r, t) which gives the positio r of the particle as a fuctio of tie t ; oce r is give as a fuctio of tie, all the physical quatities attributable to the syste at ay istat t, as well as their variatios with t, ca be derived atheatically fro the fuctio. I the case of a coservative syste i a dyaical state, the eergy E of the syste reais costat so log as the syste is i that dyaical state. There ca be ifiite uber of such possible dyaical states for the give syste, though a syste exists, over a duratio of tie, i oly oe possible dyaical state. I soe special cases, the syste (such as a siple or coical pedulu, or a particle i a closed Kepler orbit) ay go repeatedly through the sae sequece of successive istataeous states; such a dyaical state ay be called a cyclic dyaical state. Our object of study i classical statistical echaics is a statistical assebly cosistig of a large uber of idetical idepedet coservative systes which obey the laws of classical echaics. Such systes ay be: free particles (heliu atos), rigid rotators (diatoic olecules), haroic oscillators (atos i a crystal lattice), etc. I all these cases, first we have to idetify the possible states of the systes relevat to the statistical properties of the give assebly ad the use the laws of probability to deterie the probabilities associated with these states. As our object is to preset the ew approach, we cosider oly a assebly of free particles. 3. A Assebly of Free Particles i Statistical Equilibriu For the sake of defiiteess let us cosider a assebly of a large uber N of idetical particles cofied (by what is orally referred to as the walls of a cotaier) withi a fixed volue of field-free space of acroscopic diesios. Evidetly, over a duratio of tie each particle would be i a dyaical state characterized by a pair of costats such as r, p, with the uderstadig that all such pairs of costats specifyig all the possible dyaical states (of all the particles) correspod to the sae iitial istat of tie t. Though we have referred to the as particles, they are ideed physical etities havig o-zero spatial extesio ad hece they would collide with oe aother exchagig eergy ad oetu. As a result, a particle would travel alog a seget of a straight lie belogig to a particular dyaical state, collide with aother particle, exchage eergy ad oetu, ake a trasitio to aother particular dyaical state, travel alog a seget of a straight lie belogig to the ew dyaical state to collide agai, ad so o; every particle would go through such a process icessatly. It is evisaged that as a result of such repeated trasitios (fro oe dyaical state to aother) ade by all the particles over a sufficietly log iitial duratio of tie, a state of dyaic equilibriu would be reached i the sese that the fractios of the total uber of particles of the assebly i differet possible dyaical states would reai alost costat over subsequet duratios of tie. Such a assebly is said to be i statistical equilibriu. Our iterest is oly i the equilibriu state ad ot i how equilibriu is reached fro a iitial o-equilibriu state [7]. As etioed at the outset, first we have to specify the possible states of the particles i a aer that is cosistet with the laws of classical echaics as well as with the ethods of statistical aalysis. Accordig to classical echaics, the dyaical state of a (free) particle betwee two successive collisios is specified by the oetu of the particle ad by the coordiates of the particle at the two poits of collisio. Accordig to the ethods of statistical aalysis the positios of the particles withi the volue of space of the assebly have o relevace to the statistical properties of the assebly. This eas that, so far as the statistical properties of the assebly are cocered, we eed specify each possible

3 N. UMAKANTHA 1237 state by oetu oly. We refer to the as oetu states. Evidetly, the possible values of oetu p have a cotiuous rage (both i agitude ad directio). We associate probability P(p) dp with the states correspodig to oetu lyig betwee p ad p + dp; here dp is a eleet of costat agitude dp x dp y dp z. Evidetly, P(p) correspods to uit iterval of p values. Our object ow is to fid this probability distributio usig the laws of probability ad the ethods of statistical aalysis. We treat (to begi with) P(p) as a uspecified fuctio of p ad the derive a geeral expressio for it by akig use of the properties of the equilibriu state of the assebly ad the results of probability theory. If we cosider at ay oe istat of tie t 1, the states of differet particles (which exist at differet poits withi the volue of the assebly) oe after the other i ay systeatic aer, we get a rado sequece of the possible oetu states p s. This rado sequece has N uber of eleets correspodig to N uber of particles i the assebly. If the states correspodig to oetu lyig betwee p ad p + dp occur N 1 (p) uber of ties i the sequece, the relative frequecy w 1 (p) = N 1 (p)/n of these states i the sequece would be approxiately (i the statistical sese) equal to the probability P(p) dp. If we cosider the states at aother istat of tie t 2 (after a iterval of tie log eough for particles to udergo trasitios) we get aother rado sequece of the sae states i which also the relative frequecy w 2 (p) is approxiately equal to the probability P(p) dp. Such rado sequeces are statistically equivalet. Thus because of the dyaic ature of equilibriu, we get a large uber of statistically equivalet rado sequeces of the sae states p s, the rado sequeces beig relevat to the state of the assebly at the istats of tie t 1, t 2,. Let us cosider the rado sequece at the istat of tie t 1. Now we defie (what is referred to i statistical aalysis [5,6] as) a bioial rado variable R(p) which assues the value 1 if a state i the rado sequece belogs to oetu betwee p ad p + dp ad assues the value if it does ot. This leads to a (derived) rado sequece of 1 s ad s. Evidetly, R(p) is the total uber N 1 (p) of the particles havig oetu betwee p ad p + dp at the istat t 1. Correspodig to the rado sequece at t 2, we get aother value N 2 (p). Thus correspodig to rado sequeces at differet istats of tie t 1, t 2,, we get differet values of N(p). Sice a well defied (tie-idepedet) probability P(p) dp is associated with the states betwee p ad p + dp, these values of N(p) would have a well defied distributio characterized by the probability P(p) dp. Accordig to the cetral liit theore of the probability theory [5,6], the probability P{N(p)} that the quatity R(p) has (at the coceptual istat of tie) the value N(p) is give by P N 2 2 N p N p N exp 2 p, (1) 2 N 2 where N(p) is exactly equal to P(p) dp N ad (p) = P(p) dp {1 P(p) dp}. Here N(p) is the ost probable value ad, the distributio beig syetric, N(p) is also the ea value of N(p) take over this distributio (which, i effect, is over a duratio of tie); σ is the stadard deviatio. This shows that because of the dyaic ature of the equilibriu distributio, the uber N(p) of particles havig oetu betwee p ad p + dp fluctuates leadig to a tie-idepedet Gaussia distributio of values with the ea value N(p) ad the stadard deviatio σ. Evidetly, this is true of each possible value of p. It is iterestig to cosider the state of a particular sigle particle of the assebly at differet istats of tie. Because of collisios, the particle would be i differet oetu states at differet (well separated) istats of tie, leadig to a rado sequece of oetu states; this sequece would have the sae uber of eleets as the uber of istats of tie at which the state of the particle is cosidered. Sice the probability distributio P(p) dp is associated with the possible states per se (ad ot with the particles) this rado sequece would be statistically equivalet to the rado sequece of states of the particles of the assebly at oe istat of tie. So, the relative frequecy distributio w(p) i ay log seget of this sequece also would be approxiately equal to the probability distributio P(p) dp. We ay regard this sequece as beig ade up of a large uber of successive segets each seget havig uber of eleets. With referece to the first seget, R(p) is the total uber (p) 1 of istats of tie (out of uber of istats) at which this particle has oetu betwee p ad p + dp; with referece to the secod seget we get aother uber (p) 2 ; ad so o. All these ubers are approxiately equal to P(p) dp. These ubers lead to a Gaussia distributio siilar to (1). Agai, the ea uber of istats of tie (p) (out of a total uber of istats of tie ) at which the particular particle has oetu betwee p ad p + dp is exactly equal to P(p) dp. This is true of every possible oetu p of this particular particle uder cosideratio. All this is true also of every other particle i the assebly. Sigificace of this result is that the ea value of a physical quatity take over the states of a large uber of particles of the assebly at ay oe istat of tie is approxiately equal to that take over

4 1238 N. UMAKANTHA the states of ay oe particular particle at equally large uber of istats of tie. A particle akes a trasitio to aother oetu state, p' say, as a result of its collisio with aother particle ad this process is idepedet of the oetu states of all the other particles i the assebly. So the fluctuatios i the uber N(p') of particles havig oetu betwee p' ad p' + dp would be idepedet of the fluctuatios i the ubers relevat to all other oetu values (except for the weak coditio that the su of these ubers should reai costat at N ). So, the probability P{N(p')}, give by (1), that N(p') uber of particles have oetu betwee p' ad p' + dp is idepedet of the probability P{N(p'')} that N(p'') uber of particles have oetu betwee p'' ad p'' + dp for ay two distict values of oetu p' ad p''. The probability P{N(p)}for oe value of p beig idepedet of that for aother value, the probability P(N 1, N 2,, N, ) that N 1 uber of particles have oetu betwee p 1 ad p 1 + dp, ad N 2 betwee p 2 ad p 2 + dp,, N betwee p ad p + dp,, is give by PN P N N N P N P N P N 1, 2,, (2) This is a ulti-diesioal Gaussia distributio ad its axiu (which correspods to each N beig equal to N, the ost probable value) ay be specified by the coditio PN1, N2, N, PN1PN2PN PN1 PN2PN (3) PN1PN2 PN. Here each ter has ot oly the coditio for the probability relevat to oe value of oetu p beig axiu, but also the probabilities relevat to all the other values as well. Thus the coditio (3) is ot cosistet with the fact that the probabilities P(N ) s are all idepedet of oe aother. So we cosider, istead, the coditio l PN1, N2, N,. We have l P N, N, N, l P N l P N l P N, where each ter refers to oe value of oetu oly, cosistet with P(N ) s beig idepedet of oe aother. Thus though both the coditios (3) ad (4) look atheatically equivalet, oly the coditio (4) is physically appropriate. Iportace of this result caot be overephasized. (4) All that has bee said so far is ere explicatio, i ters of the laws of probability, of what we should ea by (dyaic) statistical equilibriu, oce we assue that tie-idepedet probabilities associated with the possible states of the particles exist; i fact, o law physics is ivolved. A little reflectio would show that the above reasoig is so geeral that it is applicable ot oly to oetu p but also to eergy E. The properties of a statistical assebly are better uderstood i ters of eergy (rather tha oetu) of the particles. So we develop the theory by treatig eergy as the idepedet variable. If we associate probability P(E ) de with the states correspodig to the eergy lyig betwee E ad E + de, the the probability P( 1, 2,,, ) that 1 uber of particles have eergy betwee E 1 ad E 1 + de, 2 betwee E 2 ad E 2 + de,, betwee E ad E + de,, is give by P1, 2,, P1P2P (5) P, which is a ulti-diesioal Gaussia distributio ad its axiu correspods to each beig equal to, the ost probable value (which is also the ea value). Agai, the appropriate coditio for this probability beig axiu is specified by l P,,, (6) l P l P l P. Whe we cosider the rado sequeces of states of the systes at a large uber of istats of tie, the rado sequece correspodig to each istat would be characterized by oe set of values. Sice s are the ost probable ubers, the uber of rado sequeces havig the set of values should be larger tha the uber of those havig ay other set of possible values. Now we estiate the uber of such sequeces (havig the set of values). Let S 1, S 2,, S N be the sequece of particles of the assebly i soe order. Each distributio of N uber of eergy states i this sequece of N uber of particles leads to oe sequece of eergy states of the particles; there are N! uber of such sequeces. But all of the are ot distict because the sae states occur ay ties. For istace, i ay sequece the eergy states betwee E 1 ad E 1 + de would occur 1 uber of ties ad perutatio of these states aog theselves does ot lead to a ew sequece of states; this is so with respect to each of the other states as well. So the total uber of distict sequeces of states with 1 uber of particles i the eergy states betwee E 1 ad E 1 + de, 2 betwee E 2 ad E 2 + de,, betwee E ad E + de, is

5 N. UMAKANTHA 1239 give by N N,,,!!!!. (7) Sice s are the ost probable ubers of particles i the relevat states, the uber N is larger tha the uber N correspodig to ay other set of values; the subscript deotes this. So this uber N should be proportioal to P 1, 2,, which is the ost probable value of the probability (5). Sice it is appropriate to express the equilibriu coditio i ters of l P (rather tha P ), let us cosider l N. We have fro (7) l N l N! l!. (8) Usig the Sterlig approxiatio for the factorial of a large uber, give by l! = l, (8) ay be put as l N N l N l. (9) Thus we see that each of the followig three coditios characterize the (sae) equilibriu state of the statistical assebly. Whe each assues its ost probable value : 1) the probability P( 1, 2,,, ) has its axiu value P( 1, 2,, ), 2) the uber N has its largest value N, ad 3) the quatity l has its iiu value l. I statistical physics we are iterested i properties which ca be attributed to a assebly over a tie scale which is large copared to the tie scale relevat to trasitios i the states of the costituet systes. Such properties deped o (the physical properties relevat to) the possible states of the systes ad o the probabilities associated with these states. As etioed before the states are deteried (for the systes of the give assebly) by the laws of physics, ad the probabilities (or equivaletly the ost probable ubers of systes i differet states) are to be deteried by usig the laws of probability (cosistet, of course, with the basic properties of the assebly uder cosideratio). We recogize that the two basic properties of the assebly of particles uder cosideratio are that the total uber N ad the total eergy E of the particles reai costat at all istats of tie. Sice the ost probable ubers s are also oe set of possible ubers s, N (1a) ad E E. (1b) Thus s should satisfy the three coditios give i (9) ad (1). Now N beig the largest uber, ad N ad E beig costats, we have l N l 1 ; (11) ad E E, (12) where s are sall deviatios fro the equilibriu values s. These three coditios should be satisfied siultaeously. Usig Lagrage s ethod of udeteried ultipliers, we ay express the as a sigle equatio. We have l E, (13) where ad are costats. Sice (13) is to be satisfied for ay arbitrary set of s, we should have l E, (14) which eas that 1 exp E. (15) Here is the ost probable, as well as the ea, uber of particles i the eergy rage betwee E ad E + de. Usig the well-kow arguets we ay idetify the costat as 1/kT, where k is the Boltza costat ad T the therodyaic teperature. By virtue of (1a), ca be expressed i ters of other quatities as Z = N exp = exp ( E /kt); Z is called the partitio fuctio. Thus we arrive at the Maxwell-Boltza distributio law for eergy. 4. The Covetioal Approach ad the New Approach The developet of statistical physics has bee reviewed by Lebowitz [8]. Here we copare the distictive features of the covetioal approach (CA) ad the ew approach (NA). I CA [2] the object of our study is N uber of idetical particles cofied withi a acroscopic volue of space; we refer to it as a ay-particle syste. The state of the ay-particle syste at ay istat of tie is specified by the positio ad oetu of the particles at that istat ad chages cotiuously as a fuctio of tie as govered by (deteriistic) laws of classical echaics; the ay-particle syste per se is ot regarded as beig i statistical equilibriu. Because of the difficulties i deteriig the exact state of this ay-particle syste (due to our icoplete kowledge of the syste), statistical approach is adopted. This covetioal approach is based o three basic assuptios. 1) First we select (rather etally) a large uber of idetical ay-particle systes (which have the sae structure as the oe of actual iterest ad are selected i such a aer as to agree with our partial kowledge as to the precise state of the actual syste of iterest, ad) which exist i all the differet

6 124 N. UMAKANTHA accessible states; these are said to for a represetative eseble of ay-particle systes. The cocept of eseble is the ost distictive feature of CA. [9]. 2) Next, the tie-averaged values of physical quatities relevat to the actual ay-particle syste of our iterest are assued to be the sae as the respective values averaged, at oe istat of tie, over the states of the ay-particle systes of the eseble. This is kow as the ergodic hypothesis. 3) I takig the average over the accessible states of the ay-particle systes of the eseble, the sae weight is give to all the states; this is the assuptio of equal a priori probabilities. All the physical reasoig ad atheatical derivatios refer to the represetative eseble of ay-particle systes. I this approach probability refers to a (ay-particle) syste selected at rado fro aog a eseble of (ay-particle) systes; this is cosistet with the otio of probability prevalet at the begiig of the last cetury. I the ew approach also our object of study is N uber of idetical particles cofied withi a acroscopic volue of space; we refer to it as a assebly of ay particles. The ai features of the preset approach are: 1) We do accept (followig classical echaics) that the state of a assebly of particles is specified by the positio ad oetu of the particles, but i recogitio of the statistical properties beig idepedet of where the particles exist withi the assebly, we specify the state by oetu of the particles oly. 2) At the outset we recogize that collisios iduce repeated trasitios i the dyaical states of the particles ad the assebly per se is idetified as beig i statistical equilibriu. All the physical reasoig ad atheatical derivatios refer to this assebly (oly). 3) Though collisio betwee two particles is strictly govered by deteriistic laws of classical echaics, whe we cosider the oetu states of the particles of the assebly (at ay istat of tie) i ay systeatic spatial order, they lead to a rado sequece of these possible states. This justifies itroductio of the cocepts of radoess ad probability ito the theory (withi the coceptual fraework of classical deteriis). 4) Probability of a state is idetified as the relative frequecy of the state i a rado sequece of possible states. 5) The ea uber of particles i the assebly havig eergy betwee E ad E + de, is show to be the sae as the ost probable uber of particles havig eergy i that rage. The values of physical quatities relevat to a statistical assebly i equilibriu deped o the (tie-averaged tie-idepedet) ea ubers of particles i the various eergy states, whereas the coditio for statistical equilibriu is specified i ters of the ost probable ubers of particles i these eergy states. If the theory is to be regarded as beig logically cosistet, these two sets of ubers should be show to be equal. I the preset approach this has bee achieved by usig the cetral liit theore of the probability theory. 6) The ea value of a physical quatity take over the states of the particles i the assebly at ay oe istat of tie, is show to be approxiately the sae as the ea value take over the differet states of ay sigle particle of the assebly at equally large uber of istats of tie. 7) The reaso for axiizig l N ( 1, 2,, ) is justified, ad ot just accepted as a atter of coveiece. I fact, oly as a result of axiizig l N (istead of axiizig N ) do we get the expoetial fuctio i (15). A little reflectio would show that axiizig l N as a atter of coveiece is tataout to assuig what is to be derived. Ad 8) it is show that the iiu possible value of the quatity l give by l (also) specifies the equilibriu coditio of the assebly. This is Boltza s faous H-theore which, accordig to Tola, ay be regarded as aog the greatest achieveets of physical sciece. 5. Cocludig Rearks I coclusio we ay state that by akig use of the oder theory of probability ad statistics, it is possible to develop a ew approach which is free fro the cocept of eseble, the ergodic hypothesis, ad the assuptio of equal a priori probabilities (of the covetioal approach). However, this is oly a alterative approach which leads to the sae fial results as the well established covetioal approach. 6. Refereces [1] J. W. Gibbs Eleetary Priciples of Statistical Mechaics, Dever, New York, 196 (Reprit of 192 Editio). [2] R. C. Tola, The Priciples of Statistical Mechaics, Oxford Uiversity Press, Oxford, [3] D. Ter Haar, Foudatios of Statistical Mechaics, Reviews of Moder Physics, Vol. 27, No. 3, 1955, pp doi:1.113/revmodphys [4] D. Chadler, Itroductio to Moder Statistical Mechaics, Oxford Uiversity Press, Oxford, [5] R. vo Mises, Matheatical Theory of Probability ad Statistics, Acadeic Press, Waltha, [6] M. R. Spigel, Theory ad Probles of Probability ad Statistics, McGraw-Hill, New York, [7] R. Frigg, Typicality ad the Approach to Equilibriu i Boltza Statistical Mechaics, Philosophy of Sciece,

7 N. UMAKANTHA 1241 Vol. 76, No. 5, Deceber 29, pp [8] J. L. Lebowitz, Statistical Mechaics, I: H. Stroke, Ed., The First Hudred Years, The Physical Review, AJP Publicatio, New York, 1995, pp [9] H. O. Georgii, The Equivalece of Esebles for Classical Systes of Particles, Joural of Statistical Physics, Vol. 8, No. 5-6, 1995, pp doi:1.17/bf