lctical fild gatd by a cagd aic scillat at tdyaic uilibiu STFANO GIODANO Dpatt f Bipysical ad lctic giig Uivsity f Ga Via Opa Pia A 65 Gva ITALY Abstact: - I tis pap w aalys t lctagtic fild gatd by a cagd paticl i Bwia ti. Tis ti is dscibd as kw by t Lagvi uatis []. t fllwig ctibuts a tak it accut: a ficti fc a additiv wit gaussia is fc ad a dtiistic csvativ fc divd by a gic pttial. T Fkk-Plack [] uati is usful t dscib t ti vluti f t pbability dsity i t pas spac. it t pbability dsity w ca fid t avag valu f t distibuti cag ad t dsity cut du t t paticl ti. Tf w dti t a lctagtic fild by as f t tadd pttials. T tal cuplig wit t bat lads t syst t t uilibiu situati: t Fkk-Plack uati adits t Bltza distibuti as asypttic sluti. cput t lctical fild i tis situati f tal uilibiu. I paticula w csid a aic scillat cupld t t tal bat at tdyaic uilibiu. Tis calculati is ad wit classical ad uatu caics: i bt cas w btai sults wic galis t Culb law. I t liit f lw tpatu t classical fula f t lctical pttial bc bviusly cicidt wit t Culb law: t uatal sults i tis liit giv paticula latis wic a itisically dpdig t uatu caics. Oly w t Plack cstat is csidd gligibl t uatal sults bca siila t t classical. Ky-ds: -Pbabilistic tds Stcastic pcsss lctagtics Quatu caics. Itducti T classical ti f a paticl i a tal bat ud t acti f a csvativ fc fild is dscibd by t wll kw Lagvi uatis []. T ffcts f t bat a dlld wit a ficti t ad a is t. T cspdig Fkk-Plack uati [] dscibs t ti vluti f t pbability dsity i t pas spac ad at uilibiu it lads t t Bltza distibuti law. If t paticl i ti is cagd it gats a lctagtic fild wic assus t caact f stcastic pcss. csid a paticl f ass [Kg] ad cag [Cb] dscibd by psiti [] ad vlcity v [.sc - ]. If t fc fild is du t pttial gy U() [J] t Lagvi dyaic syst f uatis is t fllwig: d v dt dv dt U β v D () β [z] is t Lagvi s cllisi fucy ficti cfficit; D[ sc - ] is t diffusi cfficit ad ccs t ffct f t additiv wit gaussia is [sc -/ ]. I t fllwig w always us gaussia is wit a clati T fucti ( ( τ ) I δ t τ ad a a valu ( { } (T as taspsiti as avag valu I is t idtity ati f d t δ ( is t Diac dlta fucti). T
cspdig Fkk-Plack uati f t v t is: dsity ( v t U v β ( v ) D () v v p T cag distibuti ad cut dsity f t paticl accdig t stadad us f t Diac t-disial dlta fucti a: J δ ( ( ) v( δ ( ( ) () ca cput t avag valu f t cag distibuti ad cut dsity usig t pbability dsity i t pas spac: J δ ( ~ ) ( ~ ~ v ~ v ~ v ( δ ( ~ ) ( ~ ~ v ~ v dv ~ v~ dv~ ddv ~ ~ () d~ dv~ T avag f t lctagtic pttials ca b calculatd by as f t tadd pttials btaiig: A ( µ v t v v t c c dvd (5) dvd Tus t a filds a divd by t pttials: B A A t (6) Tf t pbability dsity sluti f t Fkk-Plack uati dscibig t ti f a cagd paticl is t fudatal tl t aalys t avagd lctagtic fild gatd by t paticl itslf. I paticula w aalys t fild gatd by a paticl i tdyaic uilibiu wit a tal bat. It is t difficult t vify tat t fllwig Bltza distibuti is t asypttic sluti ( t vyw) f t Fkk-Plack uati (): vv U (7) ( v ) w is t classical patiti fucti: vv p dv U p d U p d ad w av dfid t tpatu T [ K] by as f t isti lati D β ; K is t Bltza cstat. Tis is t asypttic cct sluti ly if t ipp itgal is cvgt. I t fllwig w always f by yptsis t pttial gis wit cvgt patiti fucti. c at uilibiu t lctagtic pttials bc: ( A U p d U p d (8)
ad t w av a uivalt lctstatic situati wit cag distibuti giv by: U p (9) U p d Q F f p () giv t fial pssi f t a lctical fild gatd by a classical aic scillat at tdyaic uilibiu: Classical aic scillat F w w csid t pttial gy f t stadad aic scillat: U () Tf t cag distibuti at uilibiu is: () p T uivalt lctstatic pbl as spical syty ad t w ca us t Gauss law i stadad f: Q ds S w S is a spical sufac f adius ad Q is t ttal cag ctaid. T siplify t calculati w lt : F ad () t w cput t ttal cag Q usig spical cdiats: f p () T cspdig lctical pttial is t fllwig as ca asily vify: f (5) b tat t a valu f t agtic fild is z. Fut w t tat t pssis () ad (5) duc t t Culb law w t tpatu is at abslut z. Idd t a valu f t gy f t classical scillat is ad if T w fid. O ca fid t alizd gapics f t lctical pttial ad fild i Fig. ad w: w ad y w..8.6. Q Fp siϑ d dϑ d. 5 5 aft s aipulati w btai itgatig by pats: Fig.. lctical pttial.
Fig.. lctical fild. As a csuc f t pssis () ad (5) as t fllwig sults f : d d A plig cag is attactd by t scillat if <. If is cls ug t t f gais a fucy Ω f scillati giv by: Ω Quatu aic scillat csid a disial uatu aic scillat; t gy igvalus ad igfuctis a t fllwig: wit! At uilibiu t dsity ati fllws t Bltza distibuti: w δ T dsity pat agig ad is asily btaid by: * ' ' Lttig w calculat t pbability dsity f t psiti at tal uilibiu: f Tis sis ca b cputd [5]btaiig: w f ct If w csid a istpic t-disial aic scillat t t cdiats a statistically idpdt ad t t avag f t cag distibuti is giv by t latisip: tg tg p (6) Lik i t classical cas w lt: tg F ct t fula () is agai cct givig f t a fild:
f tg tg p tg T lctical pttial is: (7) d tg f tg (8) T fucy f scillati f a plig cag wit ass a t t scillatig cag is: If is gligibl t fulas (7) ad (8) bc t classical latis () ad (5). b t a valu f t gy f a istpic t disial aic scillat i tdyaic uilibiu at a tpatu T : at abslut z t gy bcs stis calld z pit gy ad t scillat is i t gud stat. F T w btai t pttial ad fild: f p (9) tg fcs: [] G..Ulbck L.S.Osti"O t ty f t Bwia ti" Pys.v.6()8-8 (9). [] Mig C ag G..Ulbck"O t ty f Bwia ti II" v.md.pys.7(&)-(95). [] S.Cadaska"Stcastic pbls i Pysics ad Asty" v.md.pys.5()-89( 9). [].isk T Fkk-Plack uati (Spig-VlagBli989). [5] L.Ladau Quatu Mcaics( MI Mscw 976). f () I tis cas w ls t aalgy wit t Culb law bcaus t z pit gy is difft t z. As i t classical cas w pt t pptis at : 5