Du Jiulin. Department of Physics, School of Science, Tianjin University, Tianjin ,China

Transcription

1 arxv:.3498 Trasto state theory: a geeralzato to oequlbrum systems wth ower-law dstrbutos Du Jul Deartmet of Physcs, School of Scece, Ta Uversty, Ta 37,Cha Trasto state theory (TST) s geeralzed for the oequlbrum system wth ower-law dstrbutos. The stochastc dyamcs that gves rse to the ower-law dstrbutos for the reacto coordate ad mometum s modeled by the Lagev equatos ad corresodg Fokker-Plack equatos. It s assumed that the system far away from equlbrum has ot to relax to a thermal equlbrum state wth Boltzma-Gbbs dstrbuto, but asymtotcally aroaches to a oequlbrum statoary-state wth ower-law dstrbutos. Thus, we obta a geeralzato of TST rates to oequlbrum systems wth ower-law dstrbutos. Furthermore, we derve the geeralzed TST rate costats for oe-dmeso ad -dmeso Hamltoa systems away from equlbrum, ad receve a geeralzed rrheus rate for the system wth ower-law dstrbutos. PCS umber(s): 8..Db; 5..-y; 8..Uv; 5.4.-a I. INTRODUCTION Trasto state theory (TST) made t ossble to obta quck estmates for reacto rates of a broad varety of rocesses hyscs, chemstry, bology, ad egeerg. It has bee a corerstoe or a core of reacto rate theory ad rofoudly flueced the develomet of theory of chemcal dyamcs. However, oe key assumto to TST s that thermodyamc equlbrum must reval throughout the etre system studed for all degrees of freedom; all effects that result from a devato from thermal equlbrum dstrbuto, such as Boltzma-Gbbs (B-G) dstrbuto wth the exoetal law, are eglected [. I the reacto rate theory, we are terested the rocesses of evoluto from oe metastable state to aother

2 eghborg state of metastable equlbrum; the assumto thus would be qute farfetched. oe roblem, whch s beg vestgated tesvely, s reacto rate theory away from equlbrum. TST s o loger vald ad caot eve serve as a cocetual gude for uderstadg the crtcal factors that determe rates away from equlbrum [. Thereby TST must be geeralzed f t s emloyed to descrbe rates of reactos the oequlbrum systems wth o-exoetal or ower-law dstrbutos. Theoretcally, f oe suoses to geeralze TST calculatg the rates of reactos comlex systems far away from thermal equlbrum, a aroach may be to fd statoary oequlbrum dstrbuto. But, because most systems away from equlbrum the asymtotc robablty dstrbuto s ot kow due to the lack of detaled balace symmetry, t seems to be a qute formdable task to determe the robablty desty of multdmesoal statoary oequlbrum systems [. Ofte, umercal smulatos [3 of the stochastc dyamcs are the oly ossble aroach. It s worth to otce that all the attemts so far eded u havg a form wth the exoetal law B-G dstrbuto. The eed of geeralzg TST rates such a way as to have a o-exoetal or ower-law exresso s due to the codto away from equlbrum sce ths codto creates geue ower-law dstrbutos that have bee oted revaletly, for examle, glasses [4,5, dsordered meda [6-8, foldg of rotes [9, sgle-molecule coformatoal dyamcs [,, traed o reactos [, chemcal ketcs, ad bologcal ad ecologcal oulato dyamcs [3,4, reacto-dffuso rocesses [5, chemcal reactos [6, combusto rocesses [7, gee exressos [8, cell reroductos [9, comlex cellular etworks [, ad small orgac molecules [ etc. I these stuatos, TST s vald ad the reacto rate equatos have ot exsted yet. O the other had, a tye of statstcal mechacal theory of ower-law dstrbutos has bee costructed va geeralzg Gbbsa theory for systems away from thermal equlbrum [. Esecally worth to meto that recet years, oextesve statstcal mechacs based o the q-etroy roosed by Tsalls has receved great atteto ad t has made very wde alcatos for a varety of

3 B terestg roblems o those systems wth ower law dstrbutos [3. It s studed as a reasoable geeralzato of B-G statstcal mechacs able to erform statstcal descrto of a oequlbrum statoary-state the teractg systems [4-6, so becomg a very useful tool to aroach comlex systems whose roertes go beyod the realm govered by B-G statstcal mechacs. These develomets ow aturally gve rse to a ossblty for us to geeralze TST to the oequlbrum systems wth ower-law dstrbutos, whch s ust our urose ths work. The aer s orgazed as follows. I secto II, we deal wth a tye of Lagev equatos for the reacto coordate ad mometum ad the corresodg Fokker-Plack equatos, ad we study the codtos uder whch those equatos wll gve rse to the ower-law dstrbutos. I secto III, we study the TST for the oequlbrum Hamltoa systems wth ower-law dstrbutos, cludg a geeralzato of TST rate costats to a oe-dmeso oequlbrum system, to a -dmeso oequlbrum system, ad fally we derve the geeralzed rrheus rate for the ower-law dstrbutos. I secto IV, we gve the cocluso. II. STOCHSTIC DYNMICS UNDERLYING POWER-LW DISTRIBUTIONS Let us cosder the followg reacto rocesses a oe system far away from equlbrum, k B B k B B, () where k B s a rate costat of the reacto rocess chagg the reactat B to the roduct, ad k B s that of the verse reacto. If N ( ad N B ( deote the cocetrato of ad B at tme t, resectvely, the dyamcal rate equato goverg evoluto of the stataeous cocetrato s wrtte as dn ( kbnb( kbn (, () dt dn B ( k BN ( kbn B (. (3) dt These reacto rocesses wth ther teractos wth the evromet costtute a oequlbrum dyamcal reacto system. s we kow that the stochastc moto of the reacto coordate x( s a combed effect duced by coulg amog a 3

4 multtude of evrometal degrees of freedom. Covetoally reacto rate theory, oe assumes the reacto coordate to descrbe dyamcs of the escae rocess or the trasto state, ad ts equato of moto to be modeled by the Lagev equato, where the system studed s regarded as ergodc [7 ad the statoary macroscoc dyamcs must have recourse to B-G statstcal dstrbuto [. However, for a comlex system far away from equlbrum, the dyamcs s ot ergodc geeral, ad the statoary robablty has ot to be B-G dstrbuto, but may stuatos shows ower-law dstrbutos. To vestgate the stochastc dyamcs uderlyg the ower-law dstrbutos a system away from equlbrum, we start wth the followg Lagev equato for the dyamcs of stochastc varables x, wth dx K η x (, (4) dt η (, η ( η ( t' ) D( x) δ ( t t' ), (5) x x x where K(x) s a gve fucto of the coordate x, It s requred that the ose η x ( s Gaussa dstrbuto, wth zero mea ad the delta-correlated. The quattes D(x) geeral ca be a fucto of x. Eq.(4) s qute a geeral Lagev equato wth arbtrary fucto K(x), very smlar to a strog frcto or a overdamed moto of teractg artcles f K(x) s assocated wth a otetal V(x) by K(x) dv(x)/dx [. Corresodg to the dyamcs govered by above Lagev equato (4), the Fokker-Plack (F-P) equato for the ose-averaged robablty dstrbuto fucto ρ ( x, [8 ca be wrtte by ρ( x, t x [ K( x) ρ( x, D ρ ( x,, x x. (8) I ths sese, K(x) s usually the drft term ad D(x) the dffuso coeffcet. It s easy to fd that there exsts a geeral statoary-state soluto for the F-P equato (8), whch ca be exressed a exoetal form, ( dx[ K( x) / D( ) ρ s ~ ex x), (9) 4

5 ad ρ whe x ±. Usually, the smlest case, f we take D(x) as a s costat, e.g. D / β, ad K(x) as K(x) dv / dx terms of a otetal fucto V(x), the statoary soluto s Boltzma-Gbbs dstrbuto, ρs ~ ex[, where β /kt. However, for a gve otetal fucto V(x), t s readly foud that f the followg codto s satsfed for the two ay fuctos K(x) ad D(x), ad a gve arameterκ, K( x) β[ dv / dx, () D( x) κ the statoary-state soluto of F-P equato (8) ca be wrtte the form of ower-law dstrbuto by ρ / κ κ [ κ κ x where [y y for y>, ad zero otherwse; determed by x, () κ x s a ormalzato costat that s / κ κ dx [ κ. () s κ, all above recover to the forms wth B-G dstrbutos. ctually, t s clear that there s a set of stochastc dyamcs the Lagev equato wth a whole set of K(x) ad D(x) that gve rse to the ower-law dstrbutos as the oequlbrum statoary-state solutos of the corresodg F-P equato (8). s a secfc examle, f let ρ (x) deote the statoary-state soluto of F-P equato (8), ad take K(x) dv / dx ad D(x) [ wth a arameter, we have D x ρ d dv d ρ Dx [ ρ ( ) x. (3) dx dx dx If the above K(x) ad D(x) are substtuted to E.(), ad the arameter κ s relaced by ( ), the the soluto of the F-P equatos (3) s drectly wrtte as R [ x, (4) 5

6 where the ormalzato costat s gaed by D β. s, all above recover to the stadard forms covetoal theory. It s ovel to otce that ths tye of macroscoc state-deedet ose for reresets a kd of statstcal feedback the system away from equlbrum, ad ths case a aomalous dffuso takes lace, lkely orgatg from the fluece of evromet o the mcroscoc dyamcal rocesses. The other tye mcroscoc dyamcs govered by the Lagev equato ably gvg rse to macroscoc ower-law dstrbutos s for the mometum, as a stochastc varable, wth d dt γ η (, (5) η (, η ( η ( t' ) D( ) δ ( t t' ), (6) where γ s a frcto coeffcet, ad the ose η x ( s Gaussa dstrbuto, wth zero mea ad the delta-correlated. D() ca be a fucto of geeral. Eq.(5) s a lear Lagev equato whose Gaussa statoary soluto s show to be chaged by the multlcatve ose costtuted due to the fluctuato the frcto coeffcet γ, leadg to the ower-law form of Tsalls q-dstrbuto [9. But here we are to touch uo the other cases. The F-P equato corresodg to the above Lagev equato (5) ca be x wrtte for the ose-averaged robablty dstrbuto ρ(, as ρ(, [ γ ρ(, t D( ) ρ (,. (7) I the same way, smlar to the codto (), t s ready to fd that f the followg codto s satsfed for the fucto D() ad a gve arameterκ, mγ D( ) [ κ β / m, (8) β the statoary-state soluto of F-P equato (7) s foud to be the followg ower-law form: x 6

7 ρ [ / κ κ ( ) κ β / m κ where the ormalzato costat κ reads, (9) / κ κ d [ κ β / m. () There s also a ovel examle that f let ρ () be the statoary-state soluto of F-P equato (7), ad take D() d d D [ ρ ( ) wth a arameter, we get d [ γ ρ ( ) D [ ρ ( ). () d fter the above D() s substtuted to E.(8), ad the arameter κ s relaced by ( ), the soluto of F-P equato () s wrtte drectly wth the ower-law form, ρ ( ) [ β / m where the ormalzato costat s gaed by, () D β / mγ. There are a lot of ower-law behavors that have bee observed ad studed varous felds of scece ad techology, whch more comlcated stochastc dyamcal orgs stll eed to be exlored. I our stuatos where the equato of moto for the coordate s modeled by the Lagev equato (4) ad (5), ad for the mometum by the Lagev equato (5) ad (6), the comlcated or aomalous dffusos coordate ad mometum sace, resectvely, may lay mortat roles to gve rse to the ower-law dstrbutos. We meto that the F-P equatos (3) ad () have bee the obect of dverse recet ad revous studes [3-33 ad works elsewhere. If the arameter κ the exressos () ad (9) s relaced by ( q), ad also the arameter the exressos (4) ad () by ( q), t s cked u that these ower-law dstrbutos exactly become Tsalls q-dstrbutos [3. III. THE TST RTES FOR THE SYSTEMS WITH POWER-LW DISTRIBUTIONS We take the reacto rocesses () as a geeral examle of the model to vestgate the TST rate for ower-law dstrbutos. The rate equatos, Eq.() ad 7

8 Eq.(3), for the reacto rocesses are take to accout. Let us follow the stadard le [8 to geeralze TST to the oequlbrum system wth the ower-law statoary dstrbutos. We cosder a system wth degrees of freedom,,,, wth coordates {x } ad mometa { }. The volume elemet of hase sace s deoted by dω dxdx dx d d d. -dmesoal reacto system may be dvded to oe reacto coordate ad o-reactve coordates. The reacto coordate s deoted by x. Let us take a dvdg surface to searate hase sace to two regos B ad, covetoally referred to as the reactat ad roduct states. t the dvdg surface, we take the reacto coordate x, ad x > f x ad x < f x B. Thus the dcator fucto of ca be exressed as the ste fucto θ (x), defed by, f x > θ (3), otherwse Smlarly, the dcator fucto of B s θ ( x). If the hase sace dstrbuto fucto at tme t s ρ ({ x },{ },, the stataeous cocetrato of, N (, ca be defed as the esemble average of θ (x), amely, N ( dxddω θ ρ({ x },{ },, (4) Ω B θ ( x) ad N B ( of B as the esemble average of. d t s determed that dn ( ˆ ( ) ({ },{ }, ) dt dxddω [ Lθ x ρ x t Ω dxddω δ ρ({ x},{ },, (5) Ω m where Lˆ s the Louvlle oerator, ad δ (x) s a delta fucto. The, the stataeous rate of N ( by trastos from to B s exressed, by sertg the dcator fucto θ ( ) Eq.(5), as dn ( dt B dxddω δ θ ( ) ρ({ x},{ },. (6) Ω m Corresodgly, the rate from B to s 8

9 dn ( dt B dxddω δ θ ( ) ρ({ x},{ },. (7) Ω m It s frequetly observed that the relaxato dyamcs of oequlbrum systems has ot to be govered always by a exoetal law, but may stuatos t s by the o-exoetal law or the ower-law [4-. Thermal equlbrum s of emet mortace for the uderstadg of may rocesses hyscs ad chemstry, but may other cases of comlex systems far away from equlbrum, such as for examle lvg matter, fluxes of eergy, matter ad formato revet a system from aroachg a thermal equlbrum state. Cosequetly, temerature s o loger uform but may be wth a dstrbuto. Tme ad sace deedet structure may the ersst the asymtotc log-tme behavor of such a oequlbrum system. Power-law dstrbutos become a tye of metastable states whe the system reaches at a asymtotcally statoary oequlbrum. Due to the ersstet exchages of eergy, matter ad formato betwee the system ad ts evromet, as well as trasformato amog these hyscal quattes by chemcal reactos takg lace the system, each artcle s ot free ad t s always feelg the flueces from the teractos wth other artcles the system ad the evromet. Hece both etroy ad eergy the rocesses are oextesve or seudoaddtve. If the teractos mght be modeled by a otetal, e.g. ϕ (x), the oextesvty would be characterzed terms of a gve arameter dfferet from uty, e.g., referred to as measurg the degree of oextesvty, by the relato [4-6: dt dϕ( x) ~, (8) dx dx where T s temerature. It s clear that the arameter s dfferet from uty f ad oly f the temerature gradet s ot equal to zero; hece t reresets a oequlbrum statoary-state of the system away from thermal equlbrum. Now we retur to our dscussos of the rate theory. For a autoomous oequlbrum system, a statoary robablty dstrbuto wll be aroached asymtotcally at log tmes. To vestgate the trasto rates betwee the regos ad B of the system uder the codto away from thermal equlbrum, we assume 9

10 that the system asymtotcally reaches the metastable states wth statoary ower-law dstrbutos, ad the ower-law dstrbutos ca be exressed the homologous forms to Eq.(4) ad Eq.(), whch have bee vestgated the geeralzed statstcal mechacs [, 3. Let us cosder a geeral may-body system wth the Hamltoa, H V ( x, x,..., x ), (9) m where the otetal fucto s exressed as two-ot teractos, V ( x, x,..., x ) u ( x x ). (3) < The the full asymtotc statoary-state dstrbuto of the oequlbrum system s defed by the ower-law -dstrbuto as ρ ({ x },{ }), (3) [ βh where ca be terchaged by κ ad q, thus s cosstecy wth the formulares revous secto ad some geeralzed statstcal mechacs. The Hamltoa quatty determed by H s -seudoaddtve. The ormalzato costat (3) s Ω dω B [ βh, (3) wth the cotrbutos from the dvdual regos ad B, resectvely, ad dω θ [ βh, (33) Ω B dω θ ( x) [ βh. (34) Ω I a lvg matter ad a chemcal reacto system far away from equlbrum, both etroy ad eergy are oextesve or seudoaddtve due to the resece of ersstet exchages of eergy, matter ad formato betwee the oequlbrum system ad ts evromets as well as by the teractos amog artcles sde the system. Ths kd of seudoaddtve characterstcs has bee studed some geeralzed statstcs made for ower-law dstrbutos, e.g. oextesve statstcal

11 mechacs [3, 34. The relato betwee H ad H [35, 36 s exressed as βh β [ x x ), (35) m < so that the dstrbuto fucto (3) ca be wrtte as ρ ({ x },{ }) β [ x x ). (36) m < The statoary-state cocetrato rego s N /, ad rego B, N B. The local oequlbrum dstrbuto ca be cosdered as the full B / statoary-state dstrbuto weghted o ether sde by the actual amout of ad B that are reset at tme t. So the local oequlbrum dstrbuto rego s wrtte as N ( ρ ({ x},{ }, ρ ({ x },{ }). (37) N ccordgly, by substtutg Eq.(37) to Eq.(6), we obta dn ( dt B ( dxddω δ θ ( ) ρ ({ x},{ }), (38) Ω m N N whch ca be rewrtte as the rate equato such as Eq.(), amely dn ( dt B k N (. (39) B Thereuo we fd the reacto rate costat, gve by k B dxdd ( ) ({ x},{ }) N Ω δ θ ρ Ω m dxddω δ θ ( ) [ βh. (4) Ω m Or, equvaletly, t ca be wrtte as k dω δ υθ ( υ) Ω [ βh Ω B dω θ [ βh, (4) where υ s the velocty. I the same way, oe has the rate equato from B to,

12 dn ( dt B ad the we fd the rate costat, k B Ω dω δ υθ ( υ) Ω dω θ ( x) kb N B (, (4) [ βh [ βh. (43) s exected, the TST rate costats for the thermal equlbrum assumto wth B-G dstrbuto are recovered the lmt Eqs.(4) ad (43). Thus we obta a geeralzato of the TST rate costats to a oequlbrum Hamltoa system wth the ower-law dstrbuto. I order to derve more secfc exressos of the rate costats for the ower-law dstrbuto, we here reset the followg three cases for further dscussos.. The rate costats for a oe-dmeso Hamltoa system For oe dmeso system, the Hamltoa reads H / m V, the from E.(35) ad Eq.(36) oe has the relato, H V, (44) m m ad the statoary-state dstrbuto fucto, [ β / m [ ρ ( x, ). (45) So the rate costat (4) becomes k B dxdδ θ ( ) ρ ( x,,) m /( ) [ () d [ β / m x m, (46) where the ormalzato costats have to be calculated the two cases as follows, amely < < ad >, resectvely. Partcularly, for >, the cutoff codtos must be take to accout. For < <, we have

13 ad x dx[ /( ) [ d β / m, (47). (48) d for >, due to the cutoff codto V /( ) β, there s a value, x x a, that should be determed by V(x a ) max V(x) /( ) β, ad also there s a maxmum of, max, that s m / β, so that ad x x a max max max dx[ [, (49) d β / m. (5) fter comletg the tegrals of Eqs. (48) ad (5), we obta (, ), (, ), / m B >, (5) / β ( ) B < <, where B(a, b) s the beta fucto. The average velocty Eq.(46) also eeds to be calculated the two cases, but the results the both cases are the same: </m> /β. Substtutg the above results to Eq.(46), we fd the rate costats ad /( [ () dx[ ( ) / mβ k B, for < <, (5) /( ) B(, ) k B / mβ B(, ) x a /( [ () dx[ ) ), for >. (53) It s clear that the TST rate exresso wth B-G dstrbutos s recovered by the lmt above Eqs.(5) ad (53), where the calculato the lmt for the beta fucto s erformed by trasformg t to gamma fuctos ad the usg α the asymtotc formula for gamma fuctos [37: lm [ z Γ( z α) / Γ( z) z (the calculatos hereafter for the beta ad gamma fuctos are made the same way). Thus, we receve a geeralzato of the TST rate costat to a oe-dmeso oequlbrum system wth the ower-law dstrbutos. 3

14 B. The rate costats for a -dmeso Hamltoa system Whe we calculate the TST rate costats for a -dmeso system, we meto aga that a -dmesoal reacto system has oe reacto coordate ad ( ) o-reactve coordates. Hereby, the umerator of Eq.(4) for the rate costat k B s calculated by Ω dω δ υθ ( υ) d [ β / m [ β / m [ x x ) d β m m [ βu ( ) [ x x ) x < < Corresodgly, the deomator of Eq.(4) for k B s Ω dω θ dx. (54) [ β / m [ x x ) d [ β / m dx[ x ) d < x [ β / m dx [ x x ) <. (55) So the rate costat k B, Eq.(4), reads k B /( d ( / m) [ β / m [ x ) d[ β / m dx[ x x ) ). (56) The calculatos of the refactors Eq.(56) also have to be erformed the two cases, amely < < ad >, resectvely. d, for >, the cutoff codtos should be take to accout. Fally, we derve the TST rate costats exressed wth the ower-law forms: /( [ x ) ( ) / mβ k B B(, ) dx[ x x ) ), for < <, (57) 4

15 ad /( [ x ) / mβ k B B(, ) xa dx[ x x ) ), for >, (58) where due to the cutoff codtos, there s a value, x ( ) β, that should be determed byu ( x x ) max u x x, ad at the same tme u x ) also should a x a ( be cut by the costrat u ( ) β. It s also clear that the TST rate costat x for a -dmeso system wth the B-G dstrbuto s recovered by takg the lmt Eqs.(57) ad (58). Thus, we obta a geeralzato of the TST rate costats to the -dmeso oequlbrum system wth the ower-law dstrbutos. C. geeralzed rrheus rate for ower-law dstrbutos We take a tycal alcato as a examle to derve a geeralzed rrheus rate. I the chemcal reacto system uder cosderato, the two regos ad B are assocated wth mma of the otetal eergy ad are dvded by a hgh barrer where the otetal eergy has a saddle ot. ssume that the eghborhood of the bottom of rego, located at {x a }, the otetal eergy s dagoal ad exaded as a harmoc fucto, V({x }) V a mω ( x x a ) m ( ) ω x xa (59) where at the bottom, the otetal has a mmum V a. I the eghborhood of the saddle ot, located at {x }, the otetal eergy s exaded as V({x }) V s mω x m ω x (6) where the otetal eergy has a maxmum V s. Substtute Eq.(59) ad Eq.(6) to the exresso of k B, Eq.(4), the we fd that the umerator of Eq.(4) s [ s ( )[ ( ) dxδ x β mω x d m [ β / m d β m 5

16 [ dx βm ω x /, (6) ad corresodgly, the deomator of Eq.(4) s [ a [ ( ) ( ) dx β mω x xa d m [ β / m [ β m ω ( x x ) a d β m dx. (6) So the geeralzed TST rate costat, Eq.(4), becomes k B d( / d where the tegral m) [ β / m /( [ β / m ) [ [ a dx[ β mω ( x xa ) s dx [ β mω ( x xa ) [ ( ) dy y βmω x a βmω /, (63). (64) The robablty dstrbuto fucto the tegral must be zero at y ±. I the case of lower temerature, β s suffcetly large so that the lower lmt of the tegral, βmω /, may be regarded as fte ths tegral. Cosequetly, t x a s foud that the geeralzed TST rat costats are ad k B k B Γ Γ Γ Γ ( ) ω [ s ( ) π [ ( 3 ) ω [ s ( ) π [ a a, for < <, (65), for >. (66) s exected, whe takg the lmt, they recover the TST rate the famlar form wth B-G exoetal law, k B ω ex[ β ( V π s V ), (67) a 6

17 whch cotas the famlar rrheus actvato eergy V V ) ad the frequecy ( s a factor ω/π. Thus we have receved a geeralzed rrheus rate for a oequlbrum system wth the ower-law dstrbuto. IV. CONCLUSIONS I cocluso, we vestgate the trasto state theory for a system away from thermal equlbrum whe the system asymtotcally reaches a oequlbrum statoary-state wth the ower-law dstrbutos. We frstly deal wth the stochastc dyamcs for the reacto coordate ad mometum, whch s modeled by the Lagev equatos ad the corresodg Fokker-Plack equatos, ad the we study uder what codtos the stochastc dyamcs wll gve rse to the statoary ower-law dstrbutos. It s derved that f the codto Eq.() for the two fuctos K(x) ad D(x) s satsfed for a gve otetal V(x) ad a ay arameter κ, the Lagev equato (4) for the coordate x( gves rse to the ower-law dstrbutos wth the form such as Eq.(). Eq.(4) s qute a geeral Lagev equato wth arbtrary fucto K(x), very smlar to a strog frcto or a overdamed moto of teractg artcles a otetal V(x) f K(x) s assocated wth the otetal by K(x) dv(x)/dx. s a secfc case, t s show that a aomalous dffuso,.e. D(x) D x [ wth a ay gve arameter, the stochastc dyamcs lays a mortat role to gve rse to the statoary ower-law dstrbuto. I the homologous way, t s obtaed that f the ρ codto Eq.(8) for the fucto D() s satsfed for a ay gve arameter κ, the Lagev equato (5) for the mometum ( gves rse to the ower-law dstrbutos wth the form such as Eq.(9). It s also show that as a secfc case of the codto (8), a aomalous dffuso the mometum sace,.e. D() [ ( ) wth a gve dex, the stochastc dyamcs lays a D ρ mortat role to gve rse to the statoary ower-law dstrbuto. I the vestgato of the geeralzed TST for a oequlbrum system wth the ower-law dstrbutos, we cosder a geeral may-body Hamltoa system. It s 7

18 assumed that the system far away from equlbrum has ot to relax to a thermal equlbrum state wth the B-G dstrbuto, but asymtotcally aroaches to a oequlbrum statoary-state wth the ower-law dstrbutos. Thus, followg the stadard le of TST we obta a geeralzato of TST rates made sutable for a oequlbrum system wth the ower-law dstrbuto, gve by Eq.(4) ad Eq.(43) for ostve ad reverse reacto rocesses. Furthermore, we derve more secfc exressos of the geeralzed TST rate costats for a oe-dmeso ad a -dmeso cases for a oequlbrum Hamltoa system, gve by Eqs.(5)-(53) ad Eqs.(57)-(58), resectvely. Fally, we cosder the otetal fucto as a harmoc aroxmato at the saddle ot of the hgh barrer ad the mmum of the otetal bottom, resectvely, thus we receve a geeralzed rrheus rate, gve by Eqs.(65)-(66), for the oequlbrum system wth the ower-law dstrbuto. Refereces [ P. Hagl, P. Talker ad M. Borkovec, Rev. Mod. Phys. 6, 5(99). [ E. Pollak ad P. Talker, Chaos 5, 66(5). [3P. E. Kloede ad E. Plate, Numercal Soluto of Stochastc Dfferetal Equato (Srger, Berl, 999) [4 V. Lubcheko ad P. G. Wolyes,. Rev. Phys. Chem. 58, 35(7). [5S. Sastry, P. G. Debeedett, ad F. H. Stllger, Nature 393, 554(998). [6 K. Kelley ad M. D. Kost, J. Chem. Phys. 85, 738(986). [7. K. Raagoal, K. L. Nga, R. W. Redell ad S. Teler, Physca 49, 358 (988). [8 J. W. Haus ad K. W. Kehr, Phys. Re. 5, 63(987). [9 H. Frauefelder, S. G. Slgar, ad P. G. Wolyes, Scece 54, 598(99). [ H. P. Lu, L. Xu, X. S. Xe, Scece 8, 877(998). [ W. M, G. Luo, B. J. Cherayl, S. C. Kou ad X. S. Xe, cc. Chem. Res. 38, 93(5). [ R. G. DeVoe, Phys. Rev. Lett., 63(9). [3 V. qulat, K. C. Mudm, M. Elago, S. Kle ad T. Kasa, Chem. Phys. Lett. 498, 9(). [4 R. K. Nve, Chem. Eg. Sc. 6, 3785(6). [5 L. K. Gallos ad P. rgyraks, Phys. Rev. E 7, 7(5). [6 J. R. Claycomb, D. Nawaratha, V. Varala, J. H. Mller, J. Chem. Phys., 48(4). [7 V. N. Skokov,. V. Reshetkov, V. P. Koverda ad. V. Vogradov, Physca 93, (). [8 C. Furusawa ad K. Kaeko, Phys. Rev. Lett. 9, 88(3). 8

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