Relativistic Langevin and Fokker-Planck equations

Transcription

1 elatst Langen and Fkker-Plank equatns STEFANO GIODANO Deartent f Bhysal and Eletrn Engneerng Unersty f Gena Va Oera Pa A, 64 Gena ITALY Abstrat: -The lassal tn f a artle n a theral bath under the atn f a nserate fre feld s desrbed by the renwned Langen equatns [-].The effets f the bath are delled wth a frtn ter and a nse ter. The rresndng Fkker-Plank equatn [4] desrbes the te elutn f the rbablty densty n the hase sae and t leads t the lassal Bltzann dstrbutn law at equlbru. In ths aer we ntdue and analyse the relatst Langen and Fkker-Plank equatns. In artular we fnd the equlbru slutn. e nlude by awng a arsn between lassal and relatst results. In lterature [] we an fnd a relatst analyss f the deal relatst gas whh leads t the sae results btaned here by eans f the Fkker-Plank ethdlgst. Anyway, the Fkker-Plank arah s useful t erfr a theretal systesatn f the t and t buld u a del fr the relatst statstal ehans ut f equlbru. ey-rds: - Prbablst ethds; Statstal ehans; Seal relatty; Therdynas Intrdutn: the lassal del The study f the tn f a artle n a theral bath has been a subet f nterest n the lterature fr a lng te. Indeed, the lassal treatent f ths rble [-] s rather knwn. In ths setn we suarze the an results nernng the lassal thery. T begn, we nsder a artle f ass desrbed by stn r and entu. If the fre feld s due t the tental energy U(r) the Langen dyna syste f equatns s the fllwng: d U r () s the Langen s llsn frequeny r frtn effent; D s the dffusn effent and nerns the effet f the ade whte gaussan nse n (lun etr). Subsequently we always use gaussan nse wth a rrelatn funtn T E{ n( n( τ ) } I δ ( t τ ) and an aerage alue E{ n ( } (T eans transstn, E eans aerage alue, I s the dentty atrx f rder three, δ ( s the Dra delta funtn). The rbablty densty ( r sae ( ),, n the hase r, eles n te wth the Fkker-Plank equatn (see aendx t fnd the general there used t dere any Fkker-Plank equatn entned n ths aer):, t ) t U r ( ) D r () Instead f usng the stn-entu hase sae we an utlze the stn-elty hase sae and nsequently the Langen equatn bees:

2 d U r and the Fkker-Plank equatn fr ( ), t ( ) D U r r,, t s: r () (4) In the lassal analyss f the rble there are n nsuus dfferenes between the tw arahes (elty and entu are erely rrtnal) but the dfferenes are substantal n the relatst ne. e rert bth frulas t ake a arsn wth the relatst del afterwards. It s nt dffult t erfy that the fllwng exressns are asytt slutns ( t eerywhere) f the Fkker-Plank equatns: U e,, T e T ( r ) where s the arttn funtn : and: where: ex T d U ( T ) π ex T U ex T, ), U r e T e T d r d r () (6) T d U ex ex T T π U ex T d r d r e hae defned the teerature by eans f the Ensten relatn D T where s the Bltzann nstant. These are asytt rret slutns nly f the rer ntegral s nergent. In the fllwng we always refer t the tental energes wth nergent arttn funtn. The abe asytt rbablty denstes are the s alled Bltzann dstrbutns. elatst del Nw we want t generalze the abe well knwn results t take nt nsderatn the relatst effets n the dynas f the syste. e study the relatst tn f a artle wth the standard relatn: d F where S, we an ge the Langen equatn n the stn-entu hase sae: d U r (7) The effets f the theral bath are delled wth the sae ters as n lassal ehans (sus frtn and nse). Mreer a nserate fre feld s well ntelated. e ake use f the fllwng tn equatn (n nral fr wth reset t the elty) :

3 d ( F) F t buld the Langen dyna equatn n the stn-elty hase sae: d U U r r ( n ) r nsequently n nents: (8) d a, ( ) (8-b) U a, ( ) D a, ( ) n ( r where we hae let: a δ ( ),, Equatns (7) and (8)-(8-b) are the relatst unterart f lassal equatn () and (); n the lt these equatns bee dental. Obusly, n ths relatst ase elty and entu are nt rrtnal, therefre there are nt analges between Langen equatns nt the tw arahes. Een the Fkker-Plank equatn hanges ts fr; n artular, when we nsder the stn-elty hase sae nt f ew, we ahee a sthast dfferental Langen equatn wth ultlate nse; hene the nstrutn f the Fkker-Plank equatn s re dffult. It s ssble t fnd general ethds t buld the Fkker-Plank equatn nt lterature [4]. The fundaental there used n ths wrk t buld these equatns has been rerted n the aendx, fr nenene. Hweer the tw arahes lead t the fllwng results:, t ) t U r r (9) D t, ( a, ka, k ) (, ) r a a U r D a a, k,,,, k D,,, k () Equatns (9) and () are the relatst unterart f lassal equatns () and (4) and they desrbe the te elutn f the rbablty densty. Therefre they are equatns desrbng the relatst statstal ehans ut f equlbru. These equatns hae a sngle slutn when the ntal denstes are fxed. Here, we want t analyse the equlbru behaur f the equatns (9) and (). Frst f all, n rresndene wth the lassal ase, t s easy t fnd the asytt slutn f (9) n the fr: ) e T U r, e T 4 () where s the arttn funtn, btaned after se straght frward alulatns: 4π ex T T T 4 d U ex T U ex T here n (.) are the dfed Bessel funtn f the send knd f rder n. T fnd the asytt rbablty densty f stn and elty we an sly utlze the rerty f transfratn:

4 and then exltly:, ) (, r,( ), ( ), ex T f < f > det U T () Many ntegrals an be sled nt lsed fr usng the Bessel funtns; the st nterestng results are rerted here: E T T () T { } T E{ } 8 T 8T π π T T E{ } E{ } (4) where s the arttn funtn : 4π ex T U ex T T T U ex T d It s ssble t re by substtutn that () s the asytt slutn f (). e d nt ge here the detals f ths lng alulatn whh s sly based n the utatn f se derates. The relatst dstrbutns () and () they rresnd t the lassal nes () and (6). They are the relatst Bltzann dstrbutns. Carsn between the tw dels e an establsh a arsn between equlbru behaur n lassal and relatst ase utng se aerage alues f se arables by eans f the usual frulas : { ( )} ( ) (, ), E f r, f r, r, d { ( )} ( ) ( ) E g r, g r, r, d r d T T E{ } { } E T () T E{ ( ) } T T E T (6) The rat /T s alled relatst ldness: f t s uh larger than, we hae a nn-relatst syste; f t s uh saller than, we hae an ultrarelatst syste. hen the teerature nreases, n the relatst ase, the aerage alue f elty (4) f the artle nreases but t s lted by the lght elty ; edently, nt the lassal del there s nt ths lt and the aerage alue nreases ndefntely. e learn fr the fundaentals f relatst ehans that the ass ares wth elty and nly f the elty s zer. The teerature, at theral equlbru, s an ndatn f the rs leular agtatn and then the relatst ass deends f the teerature. The relatnsh () desrbes ths relatst deendene; ths effet, f urse, t des nt exst n lassal ehans. 4

5 One fnd se f these results (btaned by eans f dfferent ethds) n a faus wrk f Juttner []. 4 Cnlusn Ths wrk shws that the del f a theral bath wth sus frtn and gaussan nse s nt nly useful nt lassal ehans but t handles nterestng results als nt relatst ne. The Fkker-Plank equatn bult under these nns s a werful and flexble tl whh desrbes the relatst nn-equlbru therdynas. Furtherre ths ethdlgy leads t asytt results that are n ardane wth thse btaned by Juttner []. eferenes: [] G.E.Uhlenbek and L.S.Ornsten, Phys.e.6 (9) 8. [] Mng Chen ang and G.E.Uhlenbek, e.md.phys.7 (94). [] S.Chanasekhar, e.md.phys.( 94). [4] H.sken, The Fkker-Plank equatn (Srnger-Verlag, Berln, 989). [] n Ferenz Juttner, Annalen der Physk. IV.Flge.4(9) 86. Aendx There [4]: we nsder the fllwng syste f dfferental equatn: d x t h t g t n t ( ) ( x, ) ( x, ) ( ) x( t ) x.. n where the sthast resses n ( are Gaussan and: { ( )} { } E n t t E n( t ) n ( t ) δ δ ( t t ) t, t, x s a sthast etr ndeendent f n( and dstrbuted wth densty f ;the te elutn f the rbablty densty f the ress x s exressed by the fllwng Fkker-Plank equatn: n n ( x, g( x, h ( x, gk( x, ( x, t x k xk n n gk ( x, g k ( x, ( x, xx k ( x, f( x)