MB DISTRIBUTION AND ITS APPLICATION USING MAXIMUM ENTROPY APPROACH

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1 Yugoslav Joural of Opratos Rsarch 6 (06), Numbr, DOI: 0.98/YJOR405906B MB DISTRIBUTION AND ITS APPLICATION USING MAXIMUM ENTROPY APPROACH Suma BHADRA Rsarch Scholar Dpartmt of Mathmatcs IIEST, Shbpur (Formrly BESU, Shbpur) sb.math@gmal.com Saat K. MAJUMDER Profssor, Dpartmt of Mathmatcs IIEST, Shbpur (Formrly BESU, Shbpur) majumdr_sk@yahoo.co. Rcvd: May 04 / Accptd: May 05 Abstract: Maxwll Boltzma dstrbuto wth mum tropy approach has b usd to study th varato of poltcal tmpratur ad hat a localty. W hav obsrvd that th poltcal tmpratur rss wthout gratg ay poltcal hat wh poltcal parts cras thr attractvss by ts publcty, but votrs do ot shft thr loyalts. It has also b show that poltcal hat s gratd ad poltcal tropy crass wth poltcal tmpratur rmag costat wh parts do ot chag thr attractvss, but votrs shft thr loyalts (to mor attractv parts). Kywords: Etropy, Shao Etropy, MB Dstrbuto, Maxmum Etropy Approach. MSC: 94A7, 8B0... Etropy thrmodyamcs. INTRODUCTION Th frst law of thrmodyamcs gvs a gralzato of th law of cosrvato of rgy cludg hat rgy accordg to whch th total rgy of th uvrs rmas costat. Evry systm lft to tslf chags spotaously at a fal stat of qulbrum. As a wll kow xampl s th passag of hat from a hottr rgo to a coldr utl both rgos bcom of th sam tmpratur, but f w thk ths procss to occur rvrs ordr, t caot happ spotaously. Hr coms th cocpt of th scod law of thrmodyamcs, addto to th frst law of thrmodyamcs that dtrms th drcto whch a procss ca tak plac a solatd systm, a systm whch s cogstd wth puts of both mattr ad rgy. Two followg statmts form th bass of th scod law of thrmodyamcs []. Klv Statmt: Thr xsts o thrmodyamc trasformato (A Thrmodyamc trasformato s a chag of stat) whos sol ffct s to xtract a quatty of hat from a gv hat rsrvor ad to covrt t trly to work. Clausus Statmt: Thr xsts o thrmodyamc trasformato whos sol ffct s to xtract a quatty of hat from a coldr rsrvor ad to dlvr t to a hottr rsrvor.

2 90 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato Th scod law of thrmodyamcs abls us to df a stat fucto S, tropy. Th trm tropy was troducd thrmodyamcs by Clausus tth-ctury [], ad s th thm of th scod law of thrmodyamcs, whch stats that a solatd thrmodyamc systm, tropy caot dcras but morovr, t wll stay stabl or augmt towards ts mum. Ths s xplad by Clausus Thorm. Clausus Thorm: I ay cyclc trasformato through whch th tmpratur s dfd, th followg qualty holds: dq 0 T Th tgral xtds ovr o cycl of trasformato. Th qualty holds for rvrsbl cyclc trasformato. I a solatd systm, th systm wll stadly bcom mor ad mor chaotc, utl t rachs mum tropy. Ths mas that sc o w hat rgy ca b addd, th systm ca vr bcom hottr, but ca oly cotu to hav th sam tmpratur or to bcom coldr. As t loss hat ovr tm, ts tropy crass, awatg fally to rach ts mum. Ths stat of mum tropy s calld thrmodyamc qulbrum whch prvals wh th thrmodyamc stat of th systm dos ot chag wth tm. Such thrmodyamc systms ar rrvrsbl trasformatos whr hat caot flow from coldr to hottr parts of th systm, but oly from hottr to coldr aras... Etropy Iformato Thory Iformato thory s a w brach of probablty thory wth xtsv pottal applcatos to commucato systms. It was orgatd by sctsts, whl studyg th statstcal structur of lctrcal commucato qupmt. Iformato Thory has ts org th twtth ctury tslf, wh Hartly trd to dvlop a quattatv masur of formato to assss th capablts of th tlcommucato systms. It s oly durg th last thr dcads or so, that ths masur of formato has b dvlopd ad ts cocpt has foud wdsprad us outsd th tlcommucato grg. Mathmatcal thory of commucato was prcpally orgatd by Claud Shao 948 [3]. It plays a mportat rol modr commucato thors, whr a commucato systm s formulatd as a stochastc or radom procss. Th mum tropy prcpl of Jays [4] has b frqutly usd to drv th dstrbuto of statstcal mchacs by mzg th tropy of th systm subjct to som gv costrats..3. Etropy MB dstrbuto I thrmodyamcs, a systm of dtcal ad dstgushabl partcls of ay sp oby th MB Dstrbuto law ad w gt formato how a total fxd amout of rgy s dstrbutd amog th varous mmbrs of th aformtod systm th most probabl dstrbuto. I Iformato Thory, Jays drvd th dstrbuto a mar totally dffrt from th classcal drvatos. Th ctral da of th dstrbuto s to prdct th dstrbuto of th mcrostats whch ar th partcls of th systm o th bass of th kowldg of som macroscopc data. Th macroscopc data аr spcfd th form of smpl momt costrat. O dstrbuto dffrs from aothr th way whch th costrats ar spcfd, ad Maxwll-Boltzma dstrbuto s obtad wh thr s oly o costrat o th systm that prscrbs th xpctd rgy pr partcl of th systm [5], usg Shao tropy masur. Bos-Est (B.E.) dstrbuto, Frm-Drac (F.D.) dstrbuto, ad Itrmdat statstcs (I.S.) dstrbutos ar obtad by mzg th tropy subjct to two costrats, (s Fort ad Smp [6], J. N. Kapur [5, 7], ad Kapur ad Ksava [8,9]). Though ths dstrbutos aros th frst stac statstcal mchacs (Amrtasu Ray, S. K.

3 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato 9 Majumdr [0]), thy ar wdly applcabl urba ad rgoal plag, trasportato studs (S.K.Mazumdr []), fac, bakg, ad coomcs (Kapur [5], Kapur ad Ksava [8,9], W. Xmg[]). Th papr s orgasd as follows: I Scto, w dscrb our modl; Scto 3 w drv th MB dstrbuto fucto, followd by som rsults. I Scto 4, w gv th applcato of ths MB Dstrbuto poltcal tropy.. MODEL I th prst study w hav cosdrd a localty, whr poltcal parts ar thr wth dffrt attractvss. W fd Maxmum Etropy Probablty Dstrbuto (MEPD) for th proporto of votrs votg for dffrt poltcal parts ths localty by usg th cocpt of MB Dstrbuto. Hr, th put formato of MB Dstrbuto s th xpctd avrag attractvss of th poltcal parts lu of xpctd rgy. Th total proporto of votrs s, whch s th costrat of th dstrbuto. Usg ths modl, w wll provd th cocpt of poltcal tropy, poltcal hat, ad poltcal tmpratur ths localty, whch wll srv as a solatd systm wthout ay trasformato of votrs ad poltcal parts from a xtral sourc. 3. RESULT AND DISCUSSION 3.. Th Maxwll Boltzma (MB) dstrbuto: Lt p, p,..., p b th probablts that a partcl a systm has rgs, rspctvly. Suppos that th oly formato about th systm s that,,..., th xpctd rgy of th partcls of th systm s ˆ,.. w ar gv th formato p ˆ p, p 0,,,..., Now, applyg Jays Maxmum Etropy prcpl w choos that probablty dstrbuto that mzs s p l p, subjct to th abov costrats, th Lagraga L p l p ( )( p ) ( p ˆ ) Sttg th drvatv of th Lagraga wth rspct to p, p,..., p qual to zro, w gt l p p. ( ) 0,,..., p l p 0 l p l p p () Applyg th atural costrat, w gt, ()

4 9 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato (3) Now from quato (), w gt, p,,..., Th dscrt varat dstrbuto s calld th Maxwll Boltzma dstrbuto of statstcal mchacs. Th Lagrag Multplr s dtrmd from, f ( ) [ ] ˆ 0 (4) ( ) ( ) [( )( ) ( ) ] f =( ) [( ) ( )( )] f ( ) 0 by Cauchy Schwarz s qualty, qualty holds f ad oly f.., f ad oly f a a a... a a a ( ) ( ) ( ).... W shall assum that th rgy lvls ar all dffrt so that f ( ) 0. So, f ( ) s a strctly dcrasg fucto of. Wthout loss of gralty, w ca assum that... So that a a a3... a wh 0 a a a3... a wh 0 Wh, a a, a,..., a So that f ( ) ˆ Smlarly, f ( ) ˆ f (0) (... ) ˆ = ˆ If ˆ, th f ( ) wll b postv throughout, ad f ˆ, th f ( ) wll b gatv throughout. I thr cas, soluto of f ( ) dos ot xst; f ( ) wll hav a soluto oly wh.

5 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato Som rsults basd o MB dstrbuto Rsult : f ( ) ˆ s rato of two covx fuctos of. From quato (), w gt f ( ) ˆ f( ) = f ( ) Now, w shall show that f ( ) ad f ( ) ar two covx fuctos of. f ( ) f( ) ( ) f( ) ( ) ( ) 3 ( ) 0 f f So, f ( ) s a covx fucto of. f ( ) f( ) ( ) 0 f So, f ( ) s a covx fucto of. Rsult : f ( ) 0, f ( ) 0, f (0) ( ) From quato (), w gt f ( ) ˆ ( ) ( ) [( )( ) ( ) ] f =( ) [( ) ( )( )]

6 94 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato ( ). f (0) = ( ) (5) Rsult 3: If f ( ) vashs at =0, t vashs vrywhr. Usg quato (3) ad makg t qual to 0, w gt ( ) 0 ( ) , f ( ) =0 vrywhr. Rsult 4: For MB Dstrbuto 0 l( ) ad, th 0 s a covx fucto of. As 0 l( ) So, d0 d d d d d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) { } 0 ( ) ( Hc, 0 s a covx fucto of. ( ) ) 0

7 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato 95 Rsult 5: ds d ˆ d ad that 0, d ˆ ad S s a cocav fucto of ˆ. S p l p = p ( ) Usg quato () = ds d d ˆ. (6) d ˆ d ˆ d ˆ Usg quato (3), w gt d d ˆ = ˆ ( ) d d ˆ Usg quato (6), w gt ds d ˆ ds d ˆ So, d d ˆ ( ) [( )( ) ( ) ] =( ) [( ) ( )( )] 0 S s a cocav fucto of ˆ. 4. APPLICATION OF MB DISTRIBUTION IN POLITICAL ENTROPY Lt p, p,..., p b th proportos of votrs votg towards th poltcal parts havg attractv dcs a, a,..., a, rspctvly. Suppos that th oly formato about th systm s that th avrag attractvss s â,.. w ar gv th formato p a aˆ p, p 0,,,..., Now, applyg Jay s Maxmum Etropy prcpl, w choos that probablty dstrbuto that mzs s p l p, subjct to th abov costrats, th Lagraga L p l p ( )( p ) ( p a aˆ )

8 96 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato Sttg th drvatv of th Lagraga wth rspct to p, p,..., p qual to zro, w gt l p p. ( ) a 0,,..., p l p a 0 l p a l p a a p (8) Applyg th atural costrat, w gt a (9) Now from quato (7), w gt a p,,..., a Th dstrbuto s th sam as MB Dstrbuto. Th Lagrag Multplr s dtrmd from (7) f a a ( ) [ ] aˆ 0 a (0) So that daˆ d a a a a ( ) [( )( a ) ( a ) ] a a a a =( ) [( a ) ( )( a )] daˆ By usg Cauchy Schwarz s qualty, w fd that 0 d, ulss a a... a.. all th poltcal parts hav th sam attractvss, whch s ot th cas. So, â s a mootocally dcrasg fucto of. Usg th costrats, w gt Now, daˆ p da a dp ()

9 S.Bhadra, S.K.Majumdr / MB Dstrbuto ad Its Applcato 97 S p l p a = p ( a l ) = a+l ˆ Poltcal Tmpratur: a () W tak whr T may b dfd as Poltcal Tmpratur, ad T s a kt mootocally crasg fucto of â,.. avrag attractvss. So, th gratr th valus of avrag attractvss, th hghr th poltcal tmpratur s. Cas. If th poltcal parts cras thr attractvss by ts publcty, but votrs do ot shft thr loyalts: I that cas, dp bcoms zro but da s crasg, so attractvss wll cras, so poltcal tmpratur wll also cras. From quato (0), w gt a a ds ˆ ˆ d. a. da d a = d. aˆ. daˆ aˆ. d =. daˆ da ˆ wll cras,.., avrag So, ths cas poltcal tmpratur wll rs but poltcal hat wll ot b gratd. Cas. If parts do ot chag thr attractvss, but votrs shft thr loyalts to mor attractv parts: Usg quato (9), w gt ds ( p da a dp ) (3) kt Hr da bcoms zro. From quato (), w gt ds kt H = kt a dp I that cas poltcal tmpratur wll rs ad poltcal hat s also gratd.

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