
maiks well-boltsman sthithivivarakkanakkukal

stattistikkal mechanicsil, maxwell-boltsman statistiks taapa santhulithaavasthayile vividha oorjja avasthakalil classical metteeriyal kanangalude vitharanathe vivarikkunnu. thaapanila mathiyaya uyarnnatho allengil kwaandam efakttukal nissaaramaakkunnathin kanika saandrata kuravo aayirikkumbol ithu baadhakamaan.
maiks well-boltsman sthithivivarakkanakkukalka oorjja soothravaakyam _ 1 ulla kanangalude prathekshikkunna ennam
evide
thulyamaayi, kanangalude ennam chilappol ingane prakadippikkunnu
ivide "i" soochika ippol oorjja soothravaakyam _ 1, soothravaakyam _ 12 ennivayulla ellah avasthakaludeyum koottathekkaal oru prathyeka avastha vyakthamaakkunnu.
charithram.
maxwell-boltsman sthithivivarakkanakkukal maxwell-boltsman vitharanathil ninnaan valarnnathu, mikkavarum adisthaana saankethikavidyayude vaattiyedukkal aayirikkam. 1860ll huristic adisthaanathil maiks well aanu eee vitharanam aadyamaayi nediyathu. pinneedu, 1870-kalil, boltsman eee vitharanathinte bhauthika uthbhavathekkurich kaaryamaaya anweshanangal nadatthi. sisttathinte entropy paramavadhi vardhippikkunnathinte adisthaanathilaanu vitharanam saadhyamaakunnath.
prayogayogyatha.
oru anuyojyamaaya vaathakathinte maiks well-boltsman vitharanam labhikkunnathin maiks well-boltsman sthithivivarakkanakkukal upayogikkunnu. ennirunnaalum, aapekshika kanikakal (maiks well-jutener vitharanathinu kaaranamaakunnu) polulla vyathyastha oorjja-momentam bandhamullah kanangalilekkum trimana edangalozhikeyulla mattu kanangalilekkum aaa vitharanam vyaapippikkaanum ithu upayogikkaam.
maiks well-boltsman sthithivivarakkanakkukale palappozhum "verthirichariyaan kazhiyunna" classical kanangalude sthithivivarakkanakukalaayi visheshippikkunnu. mattoru vidhathil paranjaal, 1-aamm avasthayilulla kanika "a"-nteyum 2-aamm avasthayilulla kanika "b"-nteyum krameekaranam kanika "b" 1-aamm avasthayilum kanika "a" 2-aamm avasthayilum ullathil ninnu vyathyasthamaan. eee anumaanam oorjjaavasthakalilulla kanangalude sariyaaya (boltsman) sthithivivarakkanakkukali nayikkunnu, ennaal gibbs vairudhyathil ulcherthathupole entropic bhouthikamallaatha phalangal nalkunnu.
athesamayam, maxwell-boltsman sthithivivarakkanakkukalka aavashyamaaya savisheshathakalulla yadhaartha kanangalonnumilla. theerchayaayum, oru prathyeka tharatthilulla ellah kanangaleyum (udha.jii.electronukal, protonukal muthalaayava.) pradhaanamaayum verthirichariyaan kazhiyaathavidham. eee anumaanam nadatthiyaal kanika sthithivivarakkanakkukal maarunnu. mixing udaaharanathinte entropyile entropyile maattam rund tharam kanangalude verthirikkal falamaayundaakunna vipulamallaatha entropy udaaharanamaayi kaanavunnathaanu.
kwaandam kanikakal onnukil bosonukalo (pakaram booas-ainsteen statistiks pinthudarunnu) allengil fermionukaloo (pakaram fermi-diarac statistiks pinthudarunna poolii ozhivakkal thathwathinu vidheyamaan) aanu. eee rund kwaandam sthithivivarakkanakkukalum uyarnna thapanilayudeyum kuranja kanika saandrathayudeyum paridhiyil maxwell-boltsman sthithivivarakkanakkukale sameepikkunnu.
derivations.
vividha stattistikkal mechanical thermodynamic ensembilukalil ninnu maiks well-boltsman sthithivivarakkanakkukal uruthirinjedukkaam
oro saahacharyathilum kanikakal parasparapravartthikkunnum onniladhikam kanikakalkku ore avasthayil vyaapikkaanum swathanthramaayi pravartthikkaanum kazhiyumennum anumaanikkendathu aavasyamaanu.
microcanonical koottatthil ninnulla vyulppannam.
ore bhauthika swabhaavasavisheshathakalulla (pindam, charge muthalaayava) valare cheria kanangalude oru valia ennam ulla oru container namukkundennu karuthuka.). namukk ithine "sisttam" ennu vilikkam. kanangalkk ore gunangalundenkilum avaa verthirichariyaan kazhiyumennu karuthuka. udaaharanathinu, oro kanikayum avayude sanjaarapathangal nirantharam nireekshicho allengil oronninum oru adayaalam vecho namukk thirichariyaan kazhiyum.jii.lottari panthukal pole oronninum vyathyastha nambar varaykkuka.
kanikakal aaa kandeynarinullil ellah dishakalilekkum valia vegathayil neengunnu. kanikakal vegathayil chuttunnathinaal avaykku kurachu oorjjamundu. kandeynarile ethra kanikakalkku oru nischitha oorjjamundennu vivarikkunna oru ganithashaasthra pravarthanamaanu maiks well-boltsman vitharanam. kooduthal krithyamaayi paranjaal, maiks well-boltsman vitharanam oru prathyeka oorjjavumaayi bandhapetta avastha kaivashamullathayi normalise cheyyaatha probability (athaayath probabilittikal 1 vare koottunnilla) nalkunnu.
pothuve, ore alavilulla oorjja soothravaakyamulla niravadhi kanangal undakam. ore oorjja soothravaakyam _ 14 ulla kanangalude ennam soothravaakyam _ 15 ayirikkatte, mattoru oorjja soothravaakyam _ 16 ulla kanangalude ennam soothravaakyam _ 17 ayirikkatte, angane eee sahacharyam vivarikkan saadhyamaaya ellah oorjja soothravaakyam _ 18-num, ellah thozhil samkhyakalum soothravaakyam _ 21 ariyaamenkil soothravaakyam _ 19 ennathu "oorjja nila" soothravaakyam _ 20-nte "thozhil samkhya" aanennu nammal parayunnu, appol sisttathinte motham oorjjam namukkariyam. ennirunnaalum, oro oorjja thalatthilum "eathu" kanikakal ulkkollunnu ennu namukk verthirichariyaan kazhiyunnathinaal, acupation nambar formula _ 22 sisttathinte avasthaye poornnamaayum vivarikkunnilla. sisttathinte avasthaye allengil "microstettu" yeei poornnamaayum vivarikkunnathin, oro oorjja thalatthilum eathu kanangalaan ullathennu naam krithyamaayi vyakthamaakkanam. athinaal, eee samvidaanathinte saadhyamaaya avasthakalude ennam kanakkaakkumbol, oro microstettum naam kanakkaakkanam, saadhyamaaya thozhil samkhyakalude koottam mathramalla.
thudakkathil, oorjja nilawara soothravaakyathil 19 kanikakal ulppeduthaan oreyoru maargamundennu karuthuka (degeneracy illa). kanangalude sambharani krithyamaayi vivarikkunnathil kaaryamaaya bandhamillaattha oru koottaaya chinthayaanu aduthathaayi varunnath. udaaharanathinu, formula 26 ennu label cheythirikkunna motham formula 25 boxukal undennu parayaam. kompineshan enna aashayathode, formula 27 panthukal athath "l"-th boxylake krameekarikkanulla ethra vazhikal namukk kanakkakkaam, athil formula 28 panthukal oordar illaathe undaakum. thudakkathil, motham 27 panthukalil ninnu formula 29 panthukal thiranjedukkukayum avaa boxes formula 31 ll vaykkukayum backie panthukalil ninnu oru panthu polum purathu avasheshikkunnathuvare thiranjedukkunnathu thudarukayum cheiyunnu. panthukal krameekarikkaan kazhiyunna motham vazhikalude ennam
oru panthu polum boxukalkku purathu avasheshippikkathathinaal (ellah panthukalum boxukalil idanam), ithu soochippikkunnathu formula _ 33 enna padangal kondu nirmmicha thuka formula _ 27 nuu thulyamayirikkanam; athinaal mukalilulla bandhatthile formula _ 35 enna padam 0 aayi vilayiruthunnu! (0! = 1), nammal bandham lalithamaakkunnu
ithu multinomial coffificient maathramaan, "n" enangale "k" boxukalaayi krameekarikkunnathinulla vazhikalude ennam, "l"-th boxes "nl" enangal kaivasham vaykkunnu, oro boxileyum enangalude kramappedutthal avaganikkunnu.
ippol, boxes formula 37 kanikakal sthaapikkaan onniladhikam vazhikalulla cases pariganikkuka.iii. degeneracy prashnam kanakkiledukkunnu). formula _ 38-aam boxil formula _ 6 nte "degeneracy" undengil, athaayath, athinu formula _ 6 "sub boxukal" (formula _ 42 boxukal athe oorjja formula _ 43 ullava. ore oorjjamulla eee avasthakal/boxukale degenerate avasthakal ennu vilikkunnu.), upa boxukalile nambar mattunna formula 38-aam boxes poorippikkunnathinul eathu maarggavum boxes poorippikkunnathinul oru vyathyastha maargamaan, thudarnnu "i"-aam boxes poorippikkunnathinul vazhikalude ennam formula _ 6 "sub boxukalile formula _ 19 objectukal vitharanam cheiyunnathinulla vazhikalude ennam vardhippikkanam. formula _ 19 verthirichariyaan kazhiyunna vasthukkale formula _ 6 "sub boxukalil" sthaapikkunnathinulla vazhikalude ennam formula _ 49 aanu (aadya vasthu formula _ 6 boxukalilekkenkilum pookaam, randaamathe vasthu formula _ 6 boxukalilekkum pookaam, angane). angane motham 27 kanangalude formula avayude oorjjamanusarichu oorjja nilakalaayi tharamthirikkaavunna formula 52-nte ennam, athesamayam formula 6 ulla oro level formula 24-num "i"-th level formula 19 kanangale ulkkollunna tharatthilulla vyathyastha avasthakal undu
boltsman aadyamaayi uruthirinja "dablyu" ennathinte roopamaanithu. boltsmante adisthaana samavaakya soothravaakyam _ 58 thermodynamic entroppikale microstettukalude ennavumaayi bandhappeduthunnu, ivide "k" boltsman sthiraankamaanu. ennirunnaalum, "dablyu" ennathinu melpparanja padaprayogam vipulamaaya entropy nalkunnillennum athinaal thettaanennum gibbs choondikkaanichu. eee prashnathe gibbs paradox ennu vilikkunnu. melpparanja samavaakyam pariganikkunna kanikakal verthirichariyaan kazhiyaathavayaan ennathaanu prashnam. mattoru vidhathil paranjaal, rund oorjja upavibhaagangalile rund kanangalkk ("eiii", "bii") [eiii, bii] prathinidheekarikkunna janasamkhya janasamkhyayil ninnu vyathyasthamayi kanakkaakkappedunnu [bii, eiii] athesamayam verthirichariyaan kazhiyaatha kanangalkk avaa allah. verthirichariyaan kazhiyaatha kanikakalkkulla vaadam naam nadathukayaanenkil, nammal "w" ennathinulla booas-ainsteen padaprayogatthilekku nayikkappedunnu
maiks well-boltsman vitharanam eee booas-ainsteen vitharanathil ninnu poornna poojyathinu mukalilulla thapanilayil pinthudarunnu, ithu formula _ 60 soochippikkunnu. maiks well-boltsman vitharanathinum kuranja saandrata aavasyamaanu, ithu formula _ 61 soochippikkunnu. eee saahacharyangalil, factorialinu nammal stirilinginte ekadesha kanakk upayogichekkaam
ezhuthaan
formula _ 61-nuu formula _ 64 enna vastutha upayogichu namukk veendum stirilinginte ekadesha kanakk upayogichu ezhuthaam
ithu adisthaanaparamaayi "n" kondu vibhajikkappettathaan! dablyu" ennathinulla boltsmante yadhaartha padaprayogatthe eee thiruthal ennu vilikkunnu.
kandeynaril oru nischitha ennam kanangalude formula 69 umm oru nischitha oorjja formula 70 umm undenna niyanthranam kanakkiledukkumbol, function formula _ 52 paramavadhi vardhippikkunna formula _ 19 kandethaan njangal aagrahikkunnu. formula _ 52, formula _ 72 ennivayude paramavadhi formula _ 19-nte athe moolyangalaal nedunnu, ganithashaasthraparamaayi poortthiyaakkunnathu eluppamaayathinaal, pakaram nammal randaamathethu paramavadhi cheyyum. function roopappeduthunna lagranch multiplier upayogichu nammal nammude parihaarathe niyanthrikkunnu
oduvil
mukalilulla padaprayogam paramavadhi vardhippikkunnathinu nammal fermet sidhaantham (nischala pooyintukal) prayogikkunnu, athinanusarichu praadeshika theevratha, nilavilundenkil, nirnaayaka pooyintukalil aayirikkanam (bhagika derivateevukal aprathyakshamaakum):
mukalilulla samavaakyangal (soothravaakyam _ 78) pariharikkunnathiloode nammal soothravaakyam _ 19-nte oru padaprayogatthilekku athunnu
formula _ 19-nuu eee padaprayogam formula _ 82-nte samavaakyathilekku maattisthaapikkukayum formula _ 83 nalkunnuvennu karuthukayum cheyyuka
allengil, punakrameekarikkuka
ithu thermodynamicsinte euler-integrated adisthaana samavaakyathinte oru aavishkaaram maathramaanennu boltsman manassilakki. iii" yeei aantharika oorjjamaayi thirichariyunna euler-integrated adisthaana samavaakyam iprakaaram prasthaavikkunnu
ivide "t" ennathu thapanilayum "p" ennathu mardhavum "v" ennathu vooliyam aanu, koodaathe μ ennathu raasa pottanshyalumaan. boltsmante prashasthamaaya samavaakya soothravaakyam _ 87 ennathu entropy formula _ 82 nuu aanupaathikamaanennum aanupaathikathayude sthiraangam boltsman sthiraankamaanennum manassilakkunnu. anuyojyamaaya vathaka samavaakyam upayogichu ("pv" = "nkt"), athu udanadi formula _ 89 umm formula _ 90 umm pinthudarunnu, angane janasamkhya ippol ezhuthaam
melpparanja soothravaakyam chilappol ezhuthappettirikkunnu ennathu shradhikkuka
ivide formula _ 93 ennathu kevala pravarthanamaanu.
pakaram, namukk eee vastutha upayogikkaam
janasamkhyayude ennam eee reethiyil nedunnathinu
ivide "z" ennathu thazhepparayunnavayaal nirvachikkapetta partition functionaanu
εi" oru thudarchayaaya variable kanakkaakkappedunna oru ekadesharekhayil, thomas-fermi ekadesharekha formula _ 97 nuu aanupaathikamaayi thudarchayaaya degeneracy g nalkunnu, angane
ithu oorjjathinte maiks well-boltsman vitharanam maathramaan.
canonical ensemblil ninnulla vyulppannam.
melpparanja charchayil, oru sisttathinte gunithangal nerittu vishakalanam cheythaanu boltsman vitharana pravartthanam labhichathu. pakaramaayi, oraalkk canonical ensemb upayogikkaam. oru canonical koottatthil, oru samvidaanam oru reservoirumaayi taapa sanbarkkathilaanu. samvidaanathinum reservoirinum idayil oorjjam ozhukaan swathanthramaanengilum, samyojitha samvidaanathinu sthiramaaya thaapanila nilanirthunnathinu reservoirinu ananthamaaya valia taapa sheshi undennu karuthappedunnu.
ippozhathe saahacharyathil, nammude systethinu degenerasikal formula 6 ulla oorjja nilakalude formula 99 undennu anumaanikkappedunnu. munbatheppole, nammude systethinu oorjja formula 1 ullathinte saadhyatha kanakkakkaan nammal aagrahikkunnu.
nammude sisttam stettu formula _ 102-laanengil, reservoirinu anubandhamaaya ennam microstettukal labhyamakum. eee samkhyayude soothravaakyam _ 103 ennu vilikkuka. anumaanichaal, samyojitha samvidaanam (namukk thaalpparyamulla samvidaanavum reservoir) ottappettathaan, athinaal ellah microstettukalum thulya saadhyathayullavayaanu. athinaal, udaaharanathinu, formula 104 aanenkil, nammude sisttam formula 106 nekkaal stettu formula 102 ll aayirikkan eratti saadhyathayundennu namukk nigamanam cheyyaan kazhiyum. pothuve, formula 107 ennathu nammude sisttam stettu formula 108 ll aayirikkanulla saadhyathayaanu,
reservoir formula _ 110-nte entropy aayathinal, mukalil paranjava maarunnu.
aduthathaayi nammal thermodynamic aidantitti (thermodynamicsinte aadya niyamathil ninnu) oorkkunnu
oru canonical koottatthil, kanangalude kaimaattam illa, athinaal formula _ 113 padam poojyamaanu. athupole, formula _ 114 ithu nalkunnu
ivide formula 116 umm formula 117 umm yathaakramam reservoirudeyum formula 118 leeii systethinteyum oorjjathe soochippikkunnu. randaamathe samathvathinaayi naam oorjja samrakshanam upayogichu. 119 enna soothravaakyavumaayi bandhapetta aadya samavaakyathilekku mattisthaapikkunnu
ithu soochippikkunnathu, samvidaanathinte eathenkilum samsthaanathinu
motham probability 1 acan "z" uchithamayi thiranjeduttha "sthiraangam" aanu. (thaapanila "t" inverentaanenkil "z" sthiramaanu.)
sisttathinte ellah microstettukaliloode soochika "kall" kadannupokunnu. "z" neey chilappol avasthakalude boltsman thuka (allengil yadhaartha jarmman bhashayil "sustaandsum") ennu vilikkunnu. saadhyamaaya ellah avasthakalkkum pakaram oorjja eejan moolyangal vazhiyaan nammal sangraham soochippikkukayaanenkil, degeneracy kanakkiledukkanam. nammude systethinu oorjja soothravaakyam _ 99 ullathinte saadhyatha ellah anubandha microstettukaludeyum saadhyathakalude aakethukayaanu
evide, vyakthamaaya parishkkaranatthode,
ithu munbathe athe falamaanu.
eee uthbhavathekkurichulla abhipraayangal

