eka peteet gass

quantum yantrikimadhye, eka chaukatitil quantum kananche parinam eka chaukatitil quantum aadarsh vayusathi samtol paristhiti pahanyasathi vaparale jaau shakatat, jyamadhye mothya sankhyene renu asatat jeey parasparanshi parasparkriya karat naheet, keval tatkal aushnik takkar vagalata. yaa sadhya pratimanacha vapar shastriy aadarsh vayoo tasech aadarsh vishal fermi vayoo, aadarsh vishal boses vayoo tasech black boddy kiranotsarg (photon vayoo) yasarkhya vividh quantum aadarsh vayunche varnan karnyasathi keela jaau shakato, jyala vastumanahin boses vayoo manale jaau shakate, jyamadhye samanyatah samatulya vastumanasaha photonchya parasparsanvadamule aushnikikaran sulabh hote asey manale jaate.
maxwell-boltzmann sankhyiki, boses-einstein sankhyiki kinwa fermi-diarac sankhyikichya parinamancha vapar karoon aanee khoop mothya chaukatichi maryada vicharat gheuun, thomas-fermi andaje (enrico fermi aanee levelin thomas yanchya navavarun) urja avasthanchi adhogati vibhedak mhanoon vyakt karnyasathi aanee avasthanvaril yogasutra samakalit mhanoon vyakt karnyasathi vaparale jaate. yamule vayuchya ushmagatikiy gunadharmanchi ganana vibhajan karya kinwa bhavya vibhajan karya vaparoon karata yete. heey parinam mothya aanee vastumanahin asha donhi kananvar lagoo kele jaatil. adhik sampoorn ganana vegalya lekhanvar sodali jaeel, parantu kahee sopee udaharane yaa lekhat dilee jaatil.
avasthanchya adhogatisathi thomas-fermi andaj.
eka chaukatitil prachand aanee vastumanahin asha donhi kanansathi, kananchya avastha asha aahet -
quantum sankhyanchya sanchadvare ganana kelee jaate. gatichi tivrata dilee jaate
jethe "etch" haah plankcha sthirank aahe aanee "ell" hii petichya eka bajuchi lambi aahe. kananchya pratyek sambhavya sthiticha vichar sakaratmak purnankanchya trimiteey gridveril bindu mhanoon keela jaau shakato. moodpasoon konatyahi binduparyantche antar asey asel
samaja, quantum sankhyancha pratyek sanch 'f' sthitee nirdisht karato, jethe 'f' hii kananchya antargat degree auph friedamchi sankhya aahe, jee takkarane badalali jaau shakate. udaharanarth, firki kanamadhye "f" = 2 asel, pratyek firki sthitisathi eka. n" chya mothya mulyansathi, vareel samikaranatun "p" peksha qamee kinwa tyachya barobarichya gatimanachya pariman asalelya avasthanchi sankhya andaje aahe.
jeey trijya asalelya "aine" chya golakachya ghanmanachya fakt "eff" putt aathane bhagile jaate kaaran keval sakaratmak "nee" asalela ashtak manla jaato. satatya andaje vaparoon, "pii" aanee "pii" + "dii. pii" darmyan gatichya parimanasah avasthanchi sankhya mhanoon aahe
jethe "v" = "l" 3 haah chaukaticha aakarman aahe. lakshat ghya qee hii satatyiy andaje, jyala thomas-fermi andaje asehi mhanatat, tee vaparnyat qamee-urja avasthanche vaishishtyikaran karanyachi kshamata nasht hote, jyat jaminichya sthiticha samavesh aahe jithe "nee" = 1. bahutek prakarananmadhye hii samasya nasel, parantu boses-einstein ghanikaran vicharat ghetana, jyamadhye vayucha motha bhag jaminichya sthiteet kinwa tyachya jawal aahe, qamee urja avasthanna samore jaanyachi kshamata mahattvachi tharate.
konatahi andaj naan vaparta, urja asalelya kananchi sankhya "ε" i dware dilee jaate
jethe sootra _ 6 heey 'i' sthitiche apghatan aahe aanee sootra _ 7 jyamadhye 'β' = '1/k' b 't', boltzmancha sthirank 'k' b, tapman 't' aanee rasayanik kshamata 'μ' aahe. (maxwell-boltzmann sankhyiki, boses-einstein sankhyiki aanee fermi-diarac sankhyiki pahaa.)
thomas-fermi andaje vaparoon, 'eae' aanee 'eae' + 'dii' darmyan urja asalelya 'dii. aine. eae.' kananchi sankhya ashee aaheh
jethe sootra _ 9 mhanje "eae" aanee "eae" + "dii" darmyan urja asalelya avasthanchi sankhya.
urja vitaran.
yaa lekhachya mageel vibhaganmadhun midavalelya parinamancha vapar karoon, eka petitil vayusathi kahee vitaran aata nishchit kele jaau shakate. kananchya pranalisathi, chull sootra _ 11 sathee vitaran sootra _ 10 heey sootra _ 12 dware paribhashit kele jaate, jeey sootra _ 11 aanee sootra _ 15 darmyan sootra _ 11 sathee moolye asalelya kanancha apurnank aahe.
kuthe? 
teey pudhilpramane aaheh
gatee vitaran sootra _ 29 sathee, sootra _ 30 aanee sootra _ 31 darmyan gatichya parimanasah kanancha apurnank aaheh
aanee urja vitaran sutrasathi _ 33, sootra _ 34 aanee sootra _ 35 darmyan urja asalelya kanancha apurnank aaheh
eka chaukatitil kanasathi (aanee mukt kanasathi dekhil), urja sootra 34 aanee gatee sootra 30 yanchyatil sambandh mothya aanee vastumanahin kanansathi bhinn asato. mothya kanansathi,
tarr vastumanahin kanansathi,
jethe sootra _ 41 heey kananche vastuman aahe aanee sootra _ 42 haah prakashacha veg aahe.
yaa natesambandhancha vapar karoon,
vishisht udaharane.
khaleel vibhag kahee vishisht prakaranansathi parinamanche udaharan detat.
maxwell-boltzmanche prachand kann.
yaa prakaranasathih
urja vitaran karya ektrit karane aanee 'aine' sathee nirakaran karane dete
mool urja vitaran karyat badal kelyane milate
jeey maxwell-boltzmann vitaranasathi shastriyadrushtya prapt kelele samaan parinam aahet. pudhil parinam aadarsh vayuvareel lekhachya shastriy vibhagat aadhalu shakatat.
mothya aakarache boses-einstein kann.
yaa prakaranasathih
jethe sootra _ 52
urja vitaran karya ektrit karane aanee 'aine' sathee nirakaran karane kann sankhya dete.
jethe lii "ess" ("zed") heey bahupadganitache karya aahe. bahupad ganana hii sandnya nehmich sakaratmak aanee vastavik asali pahije, jyacha arth 'zed' 0 pasoon 1 paryant jaat asatana tyache moolya 0 pasoon 'ζ' (3/2) paryant jaeel. tapman shoonyachya dishene ghasarat jaate tase teey mothe aanee mothe hott jaeel, joparyant shevati 'zed' = 1 aanee 'zed' = 1 aanee 'zed' = 1 aanee 'zed' = 1 paryant nirnayak mulyaparyant pohochat nahee.
jethe sootra _ 55 heey reeman zeeta karya darshavate. jyaa tapmanavar nirnayak tapman asate. yaa gambhir tapmanachya khalee asalelya tapmanasathi, kann sankhyesathi vareel samikaranala konatehi uttar nahee. gambhir tapman mhanje jyaa tapmanavar boses-einstein condensate tayar honyas suruvat hote. varr namood kelyapramane samasya ashee aahe qee satatya andajat jaminivaril sthitikade durlaksh kele gele aahe. tathapi, asey disun yete qee kann sankhyesathi vareel samikaran uttejit avasthanmadhye bosonchi sankhya changlya prakare vyakt karate aanee asha prakareh 
jethe jodaleli sandnya mhanje jaminivaril avasthetil kananchi sankhya. bhoo-avasthechya urjekade durlaksh kele gele aahe. heey samikaran shunyaparyant tapmanat raheel. pudhil parinam aadarsh boses vayuvareel lekhat aadhalu shakatat.
vastumanahin boses-einstein kann (udaa.jee. black boddy radiation).
vastumanahin kananchya babatit, vastumanahin urja vitaran karya vaparale jaane aavashyak aahe. yaa karyala varanvarta vitaran karyat rupantarit karane soyiche aaheh
vastumanahin kanansathi aushnik taranglambi kuthe aahe? varnakrameey urja ghanata (urja prati ekak khand prati ekak varanvarta) nantar aahe
itar thermodynamic mapdand heey mothya kananchya babatit samanpane kadhale jaau shakatat. udaharanarth, frequency distribution functionche ektrikaran karane aanee 'aine' sathee nirakaran karane heey kananchi sankhya deteh
sarvat samanya vastumanahin boses vayoo haah black bodymadhil photon vayoo aahe. boxes' laa black boddy pokali mhanoon ghetalyane, photon satat shoshale jaat aahet aanee bhintindware punha utsarjit kele jaat aahet. jevha asey asate, tevha photonchi sankhya sanrakshit kelee jaat nahee. boses-einstein sankhyikichya vyutpattimadhye, jevha kananchya sankhyevarcha pratibandh kadhoon takala jaato, tevha heey rasayanik kshamata ("maay") shunyavar seet karnyapramanech prabhavipane asate. shivay, photonchya done firki avastha asalyane, 'f' chey moolya 2 aahe. tyanantar varnakrameey urja ghanata aahe
jeey plankchya black boddy radiationchya niyamache fakt varnakrameey urja ghanata aahe. hii prakriya vastumanahin maxwell-boltzmann kanansathi kelee gelee tarr, jee uchch tapman kinwa qamee ghanatesathi plankchya vitaranacha andaj lavte, tee vien vitaran punarprapt hote heey lakshat ghya.
kahee vishisht paristhitinmadhye, photoncha samavesh asalelya pratikriyanmule photonchi sankhya sanrakshit hoil (udaa.jee. prakash utsarjak diode, diode pokalya). yaa prakarananmadhye, photon vitaran karyat shunyetar rasayanik kshamata samavisht asel. (herman 2005)
ushma kshamatesathi debby modeldware aanakhi eka vastumanahin boses vayoo deela jaato. heey model boxmadheel phononchya vayucha vichar karate aanee photonchya vikasapeksha vegale aahe kaaran phononcha veg prakashachya vegapeksha qamee aahe aanee boxchya pratyek akshasathi jastit jaast anumat taranglambi aahe. yacha arth assa qee fages spaceveril integration anantaparyant kele jaau shakat nahee aanee parinam bahupadamadhye vyakt karnyaivaji, teey sambandhit db functionsmadhye vyakt kele jaatat.
prachand fermi-diarac kann (udaa.jee. dhatumadhil electron).
yaa prakaranasathih
urja vitaran karya ektrikaran karane dete
jethe punha, lii "ess" ("zed") heey bahupad karya aahe aanee thermal dii brogley taranglambi aahe. pudhil parinam aadarsh fermi vayuvareel lekhat aadhalu shakatat. fermi vayuche upayog mukt electron modelmadhye, modelmadhye butkyanchya siddhantat aanee sarvasadharanpane ksheen padarthat aadhaltat.

