Generalized Ideal Gas Equations for Structureful Universe

Transcription

1 Etropy, 2006, 8, Etropy ISSN Full paper Geeralzed Ideal Gas Equatos for Structureful Uverse Shahd N. Afrd 1 ad Khald Kha 2 Departmet of Physcs, Quad--Azam Uversty, Islamabad 45320, Paksta. 1 E-mal: safrd@phys.qau.edu.pk 2 E-mal: mkk@qau.edu.pk Receved: 19 May 2006/ Accepted: 4 September 2006 / Publshed: 4 September 2006 Abstract: We have derved geeralzed deal gas equatos for a structureful uverse cosstg of all forms of matters. We have assumed a uverse that cotas superclusters. Superclusters are the made of clusters. Each cluster ca be further dvded to smaller oes ad so o. We have derved a expresso for the etropy of such a uverse. Our model s rather depedet of the geometry of the termedate clusters. Our calculatos are vald for a o-teractg uverse wth o-relatvstc lmts. We suggest that structure formato ca reduce the expaso rate of the uverse. Keywords: set theory; cluster; temperature; pressure; etropy; thermal eergy; cosmology.

2 Etropy, 2006, 8, Itroducto Ideal gas equatos ca be appled to o-teractg dlute gases. Molecules ad atoms gas phase are so far away from each other, t makes a lttle dfferece f we gore the molecular teracto. These molecules ad atoms are further subject to Heseberg ucertaty prcple [1]. Whe dealg wth the uverse as a whole, we see that the molecules, ucleos ad other subatomc partcles are ot uformly dstrbuted over the space. There exsts varous scales the uverse. The ucleos are partcles a star ad at a larger scale the stars ca be treated as partcles of a galaxy ad so o. I order to corporate all scales oe s requred to geeralze the deal gas equato. I ths paper, we have derved geeralzed gas equatos for such a uverse. I a earler paper, based o cofgurato space, we addressed future oretato of tme wth the growth of etropy [2]. I the preset paper we have exteded the same model to phase space. I ths model we assume that the uverse s structureful whch cossts of clusters. Each cluster cotas subclusters. These subclusters ca further be dvded to eve smaller oes ad so o. Such a descrpto s called the uversal self-smlarty [2]. It should be oted that, for stace, cluster of galaxes are boud, vralzed, hgh over desty system, held together by the cluster self gravty [3]. I ths perspectve we ca say that our model trscally corporates gravty. 2 Model We ca assume that o large scale the uverse s homogeeous, sotropc [4] ad uque [2, 5]. We ca further assume that all forms of the matter the uverse are cotaed clusters. Each cluster cotas sub-clusters ad so o. We treat these clusters, at ther correspodg scales, as partcles of deal gas. We cofe ourself to orelatvstc regme. We ca represet the uverse by a set G [2] G = {G 1 1,G 1 2,..., G 1 N }, where G 1 represets the th cluster ad N s the umber of clusters the G. We ca regard that N s the cardal umber of set G. Smlarly we ca wrte G 1 as G 1 = {G 2 1,G 2 2,..., G 2 N 1 }, We ca cotue ths to sub-atomc level ad fally we get G 1 j = {G 0 1,G 0 2,..., G 0 N 1 }, where G 0 k s correspod to costtuet partcles whch we assume, have o further substructure.

3 Etropy, 2006, 8, The spatal dstrbuto of accessble states for ths system ca be wrtte as [1, 2, 6] Γ N =, (1) V 1 s the spatal volume of G ad V 1 s the volume. Weassumethat V 1 s ot arbtrarly small. It s much smaller as compared,tsmuchlargertha V 2. Therefore where stads for cofguratos space. V occuped by G 1 to V at the scale of G whereasatthescaleofg 1 the correspodg accessble states at the scale of G 1 V Γ N 1 1 = 1, (2) V 2 where V 2 s the volume occuped by Gj 2. Puttg V 1 from eq.(2) to eq.(1) ad o re-arragg we get V =(Γ V ) 1/N Γ 1/N 1 1 (3) 2 We ca geeralze t as follow As we kow that V V 0 = (Γ ) 1/N (4) =1 s = k B l Γ, (5) where s s the etropy ad k B s the Boltzma s costat. Usg eq. (4) ad eq. (5), we get l s = (6) k =1 B N where s s the spatal etropy ad s the spread or ucertaty volume of the costtuet partcles. It s mportat to ote that a relato descrbg average value of physcal quattes such as etropy, mass or thermal eergy of a elemet of G wth a elemet of G 1 ca be wrtte as x = N x 1 (7) where x ca be the average value of etropy, mass or thermal eergy etc. Usg eq.(7) for etropy to eq.(6) ad after terato we get l = N s 0 (8) k B

4 Etropy, 2006, 8, where s 0 s the average of spatal etropy of the costtuet partcle ad 1 N =1+N 1 + N 1 N N (9) It s worth metog the last term the seres of N s much large as compared to other terms the seres. It follows from eq. (7) X x = =1 N x 0 = N t x 0 (10) =1 where X correspods to the average value of etropy, mass, thermal eergy or ay other addtve physcal quatty of the uverse G. N t stads for total umber of massve partcles the uverse. I the above equato, we have further assumed that each G have equal umber of G 1 elemets. If we treat X as etropy eq. (10) ad usg ths equato ad eq.(8), we ca obta the spatal etropy of the uverse whch ca be wrtte as, S = k BN t V N l (11) Next we cosder mometum space. The mometum dstrbuto of accessble states for G ca be wrtte as [1] Γ β = V β V β 1 (12) Here [1, 7] V β = (2πm 1u ) 3N /2 (3N /2)! where superscrpt β stads for mometum space. m 1 s the average mass of G 1 average ketc eergy of G. Smlarly we ca wrte Γ β 1 = V β 1 V β 2 (13) ad u s the (14) Followg the same procedure as for eq. (4), we ca wrte Γ β = V β, (15) N 1! V β 0 =1

5 Etropy, 2006, 8, where V β 0 =( p x ) 3N 1 ad p x s the ucertaty mometum of the costtuet partcle oe dmeso. As we reach smaller ad smaller G s, the we are ultmately quatum regme (.e. ucleos a star). A factor of N 1! appears the deomator of r.h.s of eq. (15) order to avod over coutg of N 1 mometa of detcal ucleos stars [1]. After dog some straght forward calculato we ca wrte a expresso for etropy due to mometum dstrbuto as S β = N tk B N + N t {l V β N 1 l N 1 + N 1 3N 1 l p x } (16) For large N [7] l V β 3N 2πe 2 l.2m 1 u (17) 3N where e = e. base of atural logarthm. Usg above approxmato we ca rewrte eq. (16) as S β = N tk B N + N t 3N 2 l 4πem 1 u N 1 l N 1 + N 1 3N 1 l p x 3N Now usg eqs. (11) ad (18), we ca wrte the combed etropy of our uverse phase space as S = S + S β = N tk B N l + N tk B 3N N + N t 2 l 4πem 1 u 3N (18) N 1 l N 1 + N 1 3N 1 l p x (19) As from eqs. (7) ad (10), we ca wrte m 1 N = m M ad u U, wherem s the mass of the uverse, U stads for the total thermal eergy of the uverse. We ca fally wrte etropy of the uverse as S = N tk B V N l + N tk B 3N 4πeMU V 0 N + N t 2 l N 3N 2 1 l N 1 + N 1 3N 1 l p x, (20) Here N s the umber of top most clusters (say the umber of superclusters the uverse) whch makes the uverse ad N 1 s the average umber of partcles the bottom most cluster (saytheumberofucleosatypcalstar). Itheaboveequatowehavealsodroppedthe dces ad over the spatal volume of the uverse for brevty. 3 Implcato of the Model We ca ow fd the thermal eergy ad the equato of state by usg the followg thermodyamcs relatos. U T = (21) S V,N

6 Etropy, 2006, 8, ad P S T = V U,N (22) where P s pressure ad T s temperature. From eqs. (20) ad (21), we get U = 3N 2. N t N + N t k B T (23) Ths equato gves us the thermal eergy of our structureful uverse. From eqs. (20) ad (22) PV = N tk B T N, (24) Eqs. (23) ad (24) ca be treated as geeralzed equatos for a deal gas for the structureful uverse. To verfy valdty of our system of equatos, the deal gas equatos for molecules/partcles must be deduced from eqs. (23) ad (24). These equatos ca be obtaed, f we take =1as a specal case. I ths case N = N t,wheren t s the umber of partcles a gas. Further we fd that for such a system N =1, whch ca be eglected as compared to N t the deomator of r.h.s. of eq. (23). We fally get ad U = 3 2 N tk B T (25) PV = N t k B T (26) The last two equatos gve us the thermal eergy ad the equato of state for a deal gas respectvely. We ca get terestg results from these equatos. Let us deote the thermal eergy (gve eq. (23) of the structureful uverse by U A ad the pressure by P A (gve eq. (24), ad deotg the thermal eergy ad pressure for structureless uverse (gve eqs. (25) ad (26) by U B ad P B respectvely. For same temperature ad volume U A U B = N N + N T (27) P A P B = 1 N As we ca see that U A U B,adP A P B. It suggests that expaso of structureful uverse ca be much slower as compared to a structureless uverse. (28)

7 Etropy, 2006, 8, Cocluso We have cosdered dfferet scales expaso of uverse startg from sub-atomc scale to super-clusters ad beyod. We have obtaed geeralzed deal gas equatos that are applcable to our structureful dyamcal uverse. It ca be appled to stars, galaxes, cluster, supercluster ad of course to the uverse as a whole. Our model s rather depedet of the shape ad sze of the termedate clusters. I these dervatos we were cofed to a o-teractg ad orelatvstc system. Usg these results ad astrophyscal data oe ca quattatvely compute etropy ad other thermodyamcal observables of the structureful uverse. 5 Ackowledgmets We have beefted from useful commets by Arel Catcha. Oe of us (SNA) would lke to ackowledge the ICSC- World Laboratory for provdg partal facal support. Refereces [1] J. C. Lee, Thermal Physcs: Etropy ad Free Eerges, World Scetfc, Sgapore, [2] S.N. Afrd ad M.K. Kha, Cocepts of Physcs, 3, 71, 2006; physcs/ [3] N.A. Bahcall, Clusters ad Superclusters of Galaxes, Rep. No. Prceto Observatory Preprt 692; astro-ph/ [4] C.W. Mser, K.S. Thore ad J.A. Wheeler, Gravtato, W.H. Freema ad Compay, New York, [5] M. Castago, L. Lara ad O. Lombard, Class. Quat. Grav. 20, 369, 2003; quatph/ [6] R. Bowley ad M. Sachez, Itroductory Statstcal Mechacs, Claredo Press, Oxford,1996. [7] F.J. Vesely, Web Tutoral o Statstcal Mechacs; c 2006 by MDPI ( Reproducto for ocommercal purposes permtted.