Zero Point Energy: Thermodynamic Equilibrium and Planck Radiation Law

Transcription

1 Gaug Institut Journa Vo. No 4, Novmbr 005, Zro Point Enrgy: Thrmodynamic Equiibrium and Panck Radiation Law Novmbr, 005 Abstract: In a rcnt papr, w provd that Panck s radiation aw with zro point nrgy is quivant to th combind assumptions of zro point nrgy hypothsis, th quantum aw and th approximatd Boson statistics. Hr, w appy th princip of maxima ntropy to show that Panck s radiation aw with zro point nrgy rprsnts a stat of thrmodynamic quiibrium. Introduction In a rcnt papr [], w provd that Panck s radiation aw with zro point nrgy [] is quivant to th combind assumptions of th zro point nrgy hypothsis, th quantum aw and th approximatd Boson statistics. Panck appid th princip of maxima ntropy to a voum in spac, to obtain th stat of thrmodynamic quiibrium for ida mono-atomic gass []. Bos, in a 94 papr, transatd and pubishd by Einstin [4], appid th princip of maxima ntropy to a voum in phas spac to obtain Panck s radiation aw at a stat of thrmodynamic quiibrium.

2 Gaug Institut Journa Vo. No 4, Novmbr 005, W appy th princip of maxima ntropy in phas spac to show that Panck s radiation aw with zro point nrgy is obtaind at a stat of thrmodynamic quiibrium. Hr, th maxima ntropy princip is ony a too, that dos not rpac any of th thr combind assumptions that w provd quivant to th radiation aw with zro point nrgy. Undr th thr assumptions, th princip of maxima ntropy impis that th radiation aw with zro point nrgy is obtaind at a stat of thrmodynamic quiibrium. Maxima Entropy and ZPE radiation aw: Foowing Bos approach in [4], aboratd by Einstin in [5], th momntum of a radiation quantum is h p p. c That is, radiation quanta with th sam momntum ar on th surfac of th sphr h px py pz. c Thn, for ach of its two poarizations, th momntum of th radiation pr voum mnt, in th sphrica sh btwn, and d, is h h d 4p dp 4 d 4 h. c c c

3 Gaug Institut Journa Vo. No 4, Novmbr 005, Thrfor, for both poarizations, thr ar cs of siz h, and d. For th possib frquncis d 8 c momntum pr voum mnt in th sphrica sh btwn, 0,,,... thr ar d n 8 c momntum cs of siz h, pr voum mnt, in th sphrica sh btwn, and d. Thn, (0) () ()... j 0 n n n n n, () whr (0) n () n () n is th numbr of cs that hav no quanta, is th numbr of cs that hav on quanta, is th numbr of cs that hav two quanta, Th n momntum cs of siz h, pr voum mnt, in th sphrica sh btwn and d, can b arrangd in n! n! n! n!... n! n! (0) () () ( j) j 0 ways. Thus, a th momntum cs of siz can b arrangd in h, pr voum mnt,

4 Gaug Institut Journa Vo. No 4, Novmbr 005, w n! n! ( j) 0 j0 ways. Th ntropy pr voum mnt is ( j) Bog B og(!) B og(!) 0 0 j0, s k w k n k n whr k B is Botzman constant. By String s formua n M! M n M M, ( j) ( j) ( j) ( og ) ( og ) B B 0 0 j0 s k n n n k n n n B og B og 0 0 j 0. k n n k n n Th numbr of radiation quanta h btwn () () () ( j)... j 0 n n n jn., and d is Foowing Panck s assumption of Zro Point Enrgy [], w assum zro point nrgy of h in ach of th cs of siz h, pr voum mnt, btwn n momntum and d Thrfor, th radiation nrgy pr voum mnt is 0 j0 0 0 j0 u h jn h n h ( jn n),. () At thrma quiibrium, th ntropy has a maximum undr th constraints () 0,,,,..., on th numbr of momntum cs, and th constraint (), on th radiation nrgy. 4

5 Gaug Institut Journa Vo. No 4, Novmbr 005, To find that maximum, w appy th Lagrang mutipir mthod with mutipirs, 0,,,,..., for th constraints () 0,,,,..., and 0, for th constraint (), to th auxiiary function F( n,, ) n ogn n ogn 0 0 j0 n n u h j n 0 j 0 0 j0. ( ) ( ( ) ) Th critica points ar at F 0 ogn ( j ) h n F 0 n n, (4) j 0, () From (), or, F 0 u h ( j ) n ( j). (5) 0 j0 og n h, (6) h j h n. (7) By (5), and (6), Thrfor, h h ( j) h j h h h j0 j0. (8) n n Th quotint og n og h h. (9) 5

6 Gaug Institut Journa Vo. No 4, Novmbr 005, n n j 0 h h is th Bos-Einstin distribution function. By (8), d ( ) ( ) h j h h j h jn j n j0 j0 j d( h ). Th sris j h j convrgs uniformy to h, and can b diffrntiatd trm by trm with rspct to h. Thus, Thrfor, d d d j h j ( ). j d( h ) dj 0 d ( ) j0 h h n jn n ( ) ( h ) h. (0) By (5), and (4), and (0), By (6), u h n nh ( ) h 0 j 0 0. () ( j) B B 0 0 j0. s k n og n k n ( ( j ) h ) k n( og n) k h ( j ) n. B B 0 0 j0 Substituting (9), and (0), and (4), Using (), h s( ) k n og k nh( ) h B h B 0 0 6

7 Gaug Institut Journa Vo. No 4, Novmbr 005, h s( ) k n og k u. () B h B 0 Hnc, That is, and max is positiv as rquird. s k B. T u max, kbt Thn, is th maxima ntropy. s( ) k n og h / kbt max B h / kbt 0 u T () To confirm that th ntropy is maxima, w rca that th Lagrang mutipir mthod producs ithr a maximum or a minimum for th function in qustion, which hr is th ntropy. Thus, to confirm a maximum, it is nough to chck th vau of th ntropy at any othr point. For instanc, tak Thn, and for kbt,., h / h / kb T h h / k BT h / h kbt u B B B h / max h kbt 0 0 T. s() k n og k u k n og s( ) 7

8 Gaug Institut Journa Vo. No 4, Novmbr 005, Thrfor, at max, th ntropy is maxima, and at th attaind thrmodynamic quiibrium Thn, u( ) n h ( ) n max h / kbt 0. 0 h ( ) h / k B T is Panck s radiation aw with zro point nrgy. REFERENCES. Dannon, H. Vic, Zro Point Enrgy: Panck Radiation Law, Gaug Institut Journa of Math and Physics Vo. No, August Panck, M., {Annan dr physik 7 (9):p.64}.. Panck, M. Th Thory of Hat Radiation, Dovr 959. Pag Bos, S., {Zitschrift fur Physik 6, 78 (94)} transatd into Panck s Law and th Light quantum Hypothsis in Paui and th Spin-Statistics Thorm by Ian Duck, and E C G Sudarshan, Word Scintific 997, pag Einstin, A. {S. B. d. Pruss. Akad. Wiss. Br., 6 (94)} transatd into Quantum Thory of Mono-Atomic Ida Gas in Paui and th Spin- Statistics Thorm by Ian Duck, and E C G Sudarshan, Word Scintific 997, pag 8. 8