Non-Markovian Effects on the Brownian Motion of a Free Particle

Transcription

1 Non-Markovian Effecs on he Brownian Moion of a Free Paricle A. O. Bolivar Insiuo Mário Schönberg de Física-Maemáica-Filosofia, Ceilândia, Caixa Posal 7316, DF , Brazil bolivar@cbpf.br Absrac Non-Markovian effecs upon he Brownian movemen of a free paricle in he presence as well as in he absence of inerial force are invesigaed wihin he framework of Fokker Planck equaions (Rayleigh and Smoluchowski equaions). More specifically, i is prediced ha non-markovian feaures can enhance he values of he mean square displacemen and momenum, hereby assuring he mahemaical propery of differeniabiliy of he hese physically observable quaniies. Keywords: Brownian Moion; Smoluchowski equaion; Ornsein Uhlenbeck process; Fokker Planck equaion; non-markov effecs 1

2 1 Inroducion The Brownian moion of a paricle wih mass m and posiion X = X() may be described by he Langevin equaion [1] m d2 X d 2 = dv(x) dx dx + L, (1) d where he inerial force md 2 X/d 2 offses a se of hree sors of forces: a conservaive force derived from an exernal poenial V(X); a linearly velociydependen dissipaive force, dx/d, accouns for sopping he paricle s moion; and a sochasic force L, dubbed Langevin s force, responsible for acivaing he paricle s movemen. The damping consan exhibis dimensions of mass per ime. As far as a free paricle is concerned, he Langevin equaion (1) may be wrien in erms of he linear momenum P = P = mdx/d as dp d = P + L. (2) m On he oher hand, in he absence of inerial force, i.e., md 2 X/d 2 = 0, he free Brownian paricle is described by he sochasic differenial equaion dx d = 1 L. (3) In he Gaussian approximaion i is assumed ha he Langevin force L presen in (1 3) has he following saisical properies [2] L() = 0, (4) L L( ) = Dδ, (5) where D is a consan and he average value is evaluaed from he probabiliy disribuion funcion associaed wih he environmenal random funcion L. Equaion (4) characerizes he irregulariy feaure of he Langevin force and Eq. (5) means ha he ineracion beween he agged paricle and a generic environmen is deemed o be Markovian in he sense ha he auocorrelaion funcion (5) is dela-correlaed (whie noise). For hermal open sysems characerized by he emperaure T and he Bolzmann consan k B, he Ornsein-Uhlenbeck process (2), along wih (4) and (5), leads o he mean square momenum [3] P = mk B T 1 e 2 m, (6) 2

3 whereas he Fokker Planck equaion (he so-called diffusion equaion) generaed by he sochasic differenial equaion (3) wih he saisical properies (4) and (5) yields he Einsein s mean square displacemen [4] X = 2k B T. (7) Boh Markovian resuls (6) and (7) are non-differeniable funcions a = 0. Hence, i has been claimed ha physically here is no force applied o a free Brownian paricle in he presence of inerial forces, as well as no velociy in he absence of inerial forces [5,6]. This fac suggess ha he concep of rajecory of a free Brownian paricle seems o be an elusive feaure in he Markovian regime. The purpose of he presen paper is o show how non-markovian effecs upon a free Brownian paricle are responsible for he differeniabiliy propery of he mean square displacemen (6) and momenum (7), hereby predicing an increase of he values of hese physically observable quaniies. Our paper discusses he Brownian moion of a free paricle reckoning wih wo physical siuaions described by Fokker Planck equaions: In Sec. 2 he case of a Brownian paricle in he presence of inerial forces (Ornsein Uhlenbeck sochasic process) described by he non-markovian Rayleigh equaion is examined, whereas in Sec. 3 we solve our non-markovian Smoluchowki equaion describing a free Brownian paricle in he absence of inerial effecs. Concluding remarks are presened in Sec. 4. In addiion, wo appendices are included. 3

4 2 Brownian Moion in he Presence of Inerial Force A paricle wih mass m, posiion X = X() immersed in a generic environmen (e.g., a non-hermal fluid) undergoes a jiering movemen dubbed Brownian moion. The moion of his agged paricle may be mahemaically described by he sochasic differenial equaion (1) (he so-called Langevin equaion) in he form m d2 X d 2 = dv(x) dx dx + bψ, (8) d wih he Langevin force given by L = bψ, where he parameer b conrols he environmenal influence (flucuaions) upon he Brownian paricle. Because he erm bψ in (8) has dimensions of velociy, i.e., [leng ime 1 ], we can readily check ha b may be expressed in dimensions of [leng ime 1/2 ] and he funcion Ψ in dimensions of [ime 1/2 ]. From he mahemaical viewpoin he quaniies X = X and Ψ = Ψ in he Langevin equaion (8) are inerpreed as random variables belonging o he Kolmogorov probabiliy space [7] in he sense ha here exiss a probabiliy disribuion funcion, F XΨ x, ψ,, associaed wih he sochasic sysem {X, Ψ}, expressed in erms of he possible values x = {x i } and ψ = {ψ i }, wih i 1, disribued abou he sharp values q and φ of X and Ψ, respecively. In addiion, he average values of X and Ψ are expressed as + X = xf XΨ x, ψ, dxdψ, (9) + Ψ = ψf XΨ x, ψ, dxdψ. (10) For a free Brownian paricle he Langevin equaion (8) may be expressed in erms of he concep of linear momenum P = P = mdx/d as Noicing ha dp d = P + bψ. (11) m lim ε 0 +ε Ψ d = 0, (12) wih 4

5 Ψ() = + ψf Ψ ψ, dψ, he Fokker Planck equaion generaed by (11) in he Gaussian approximaion reads (see Appendix A) g p, = p p b Ψ m g p, + D p 2 g p, p 2 (13) in erms of he probabiliy disribuion funcion g p,, he mean value of Ψ and he diffusion coefficien 1 Ψ = lim ε 0 ε +ε Ψ d, (14) D p = E (15) where he funcion E, given by E = b2 2 b2 I() = 2 lim 1 ε 0 ε +ε Ψ Ψ d d, (16) has dimensions of energy, i.e., [mass leng 2 ime 2 ]. Hence we call E he diffusion energy responsible for he Brownian moion of he paricle dipped in a generic environmen. An ousanding feaure underlying he diffusion energy concep (16), which fulfils he validiy condiion 0 < E <, (17) is ha i conveys in andem flucuaion and dissipaion phenomena hrough he funcion I() and he fricion consan, respecively. We can hen sae ha (16) ses up a general flucuaion dissipaion relaionship underlying all open sysems described by he Langevin equaion (11) and is corresponding Fokker Planck equaion (13). Boh cases E = 0 and E = are no concerned, for hey may violae he validiy condiion of he flucuaion dissipaion relaion (17). The former case may lead o dissipaion wihou flucuaion, whils he laer one may give rise o flucuaion wihou dissipaion. 5

6 From he physical poin of view he pivoal issue inheren in heory of Brownian moion is o deermine he ranspor coefficien, ha is, he imedependen diffusion coefficien (15). A = 0 he answer o his quesion seems o be fairly sraighforward, for he diffusion energy is null, i.e., E 0 = D 0 = 0, meaning ha here is no diffusive moion, D 0 = 0, associaed wih he iniial condiion g p, = 0 = δ p o our Fokker Planck equaion (13). On he oher hand, a he long-ime regime,, we assume ha he seady diffusion energy can be idenified wih he hermal energy, k B T, a hermodynamic equilibrium, i.e., b2 lim E = 2 = k BT, (18) wih he dimensionless funcion I in (15) having he asympoic behavior lim I = 1. (19) The physical significance of condiion (19) has o do wih he fac ha environmenal flucuaions decay o Markovian correlaions, i.e., I displays a local (shor) range behavior in he seady regime. By conras, non-markovian effecs show up a he range 0 < <. The correlaional funcion I in (16) fulfilling condiion (19) can be buil up as (see Appendix B) I = 1 e c, (20) where he correlaion ime c accouns for non-markovian effecs upon he Brownian paricle. I is worh underscoring ha he ideniy (18) expresses he principle of energy conservaion, for he diffusion energy E = b 2 /2 comes from he Markovian Brownian dynamics (Langevin and Fokker Planck equaions a he seady regime), whereas he hermal energy k B T is a physical quaniy semming from he environmen a hermodynamic equilibrium (equaion of sae for perfec gases: PV = Nk B T, where N is he number of paricles wihin he volume V under he pressure P and emperaure T; he consan k B denoes he Bolzmann consan and displays dimensions of energy per emperaure). As a consequence of he conservaion of energy (18), from (15) a he hermal equilibrium we can readily derive he Ornsein Uhlenbeck diffusion consan as D p = k B T, (21) and he parameer b urns ou o be of hermal naure, i.e., b = ± 2k B T. (22) 6

7 Using (22), (20), (16), and (15) our Fokker Planck equaion (13) reads g p, = p p m 2k B T Ψ g p, + k B T 1 e c 2 g p, p 2. (23) or g p, = m p p g p, + k B T 1 e c 2 g p, p 2, (24) in erms of he hermal posiion p = p m 2k B T Ψ. (25) According o (23) a paricle immersed in a hea bah undergoes a Brownian moion owing o he diffusion energy semming from he hermal reservoir, k B T, muliplied by he correlaional effecs presen in he funcion I = 1 e / c. In general, he non-equilibrium hermal energy E = k B T 1 e / c is differen from he equilibrium hermal energy, k B T. They equal only a he seady regime (18). In his hermal equilibrium sae he sochasic process is called normal diffusion, whereas in he non-equilibrium hermal regime characerized by 0 < I < 1 he Brownian moion is said o be subdiffusive. I is worh poining ou ha he diffusion consan (21) has been derived wihou specifying ab iniio he form of he auocorrelaion funcion of he Langevin force, L( )L( ) = 2k B T Ψ Ψ = 2D p Ψ Ψ, as well as is average value, i.e., L() = ± 2k B T Ψ() = ± 2D p Ψ(). The crucial poin is he long ime behavior (19). This means ha he arbirariness as o he form of he saisical properies of he Langevin force L() leaves room o incorporae non-markovian and averaging effecs ino he sudy of Brownian moion hrough he funcions I and Ψ in he Fokker Planck equaion (23). So, saring from he iniial condiion g p, = 0 = δ(p ) a non-equilibrium soluion o (23) reads g p, = 1 4πℲ e 4Ⅎ, (26) p 2 where 7

8 Ⅎ = mk BT 2 1 e 2 m 2 c 2 c m e c. (27) The probabiliy disribuion funcion (26), which is expressed in erms of he evoluion ime, he relaxaion ime r = m/, and he correlaion ime c, gives rise o P = 0, (28) P 2 = 2Ⅎ = mk B T 1 e 2 m 2 c 2 c m e c. (29) Making use of (29) he mean mechanical energy associaed wih he free Brownian paricle is given by which reduces o E() = P 2 2m = k BT 2 1 e 2 m 2 c 2 c m e c (30) E() = k BT 2 1 e 2 m (31) in he Markovian limi. The mean square momenum, P () = P 2 P 2, reads P () = P = mk B T 1 e 2 m 2 c 2 c m e c, (32) meaning ha he average value of Ψ in Eq. (25) has no influence upon he physically observable quaniy (32). I is readily o check ha he quaniy d P d =0 = 22 k B T c 2 c m m 2 c 2mk B T c (33) does no diverge provided ha 0 < c < m/2, hereby implying ha non- Markovian effecs are responsible for enhancing he mean square momenum (32). Ye in he Markovian limi we find from (32) he resul P = mk B T 1 e 2 m (34) ha is non-differeniable a = 0. Non-Markovian effecs in (32) herefore accoun for he exisence of he concep of force acing upon a free Brownian paricle in he presence of inerial force, since 0 < c < m/2. 8

9 A he equilibrium regime, from (30) he energy equipariion is readily obained as E( ) = k BT 2, (35) while from he non-equilibrium soluion (26) we derive he Maxwell Bolzmann probabiliy disribuion funcion a hermal equilibrium g p = 1 2πmk B T e 2mk B T (36) p 2 for he hermal momenum (25). 3 Brownian Moion in he Absence of Inerial Force Now we sar from he Langevin equaion (8) in he Smoluchowski limi (he large fricion case) P m + bψ dp d, (37) such ha inerial effecs in (8) can be negligible, i.e., md 2 X/d 2 = 0. So he Brownian moion is approximaed by he sochasic differenial equaion dx d = 1 dv(x) + bψ, (38) dx which gives rise o he following Fokker Planck equaion on configuraion space in he Gaussian approximaion (see Appendix A) f x, = x 1 dv x dx + b Ψ f x, + D x() 2 f x, x 2, (39) where he ime-dependen diffusion coefficien is given by D x = E, (40) wih he diffusion energy E being given by (16). Averaging effecs of he random velociy, b Ψ, upon he drif coefficien bring abou he ime-dependen effecive velociy v eff x, = 1 dv x dx + b Ψ. (41) For hermal sysems a hermodynamic equilibrium in which (18) is valid, he diffusion consan (40) becomes he Einsein relaion 9

10 D x = k BT. (42) Accordingly, he non-markovian Fokker Planck equaion (39) in he presence of hermal flucuaions reads f x, = x 1 dv x dx 2k B T Ψ f x, + k BT I 2 f x, x 2. (43) For Ψ = 0 and I = 1, he Fokker Planck equaion (43) yields he equaion of moion f x, = 1 x dv(x) dx f x, + k BT 2 f x, x 2, (44) early found ou by Smoluchowski [8] and widely invesigaed in he lieraure [9 14]. The Smoluchowski equaion (44) is defined for Markovian correlaions and no averaging effecs in conras o our Fokker Planck equaion (43) which akes ino accoun non-markovian effecs in view of he ime-dependen funcion I as well as averaging effecs presen in he drif coefficien. Again, our accoun of he Einsein Langevin sochasic approach presened above has poined ou ha he main upsho of Einsein he Founding Faher of Brownian moion heory ress on he fac ha his diffusion coefficien (42) is a consequence of he conservaion of energy (43) for hermal open sysems a hermodynamic equilibrium. I is worh recalling ha he Einsein relaion (42) has been aained wihou specifying ab iniio he form of he auocorrelaion funcion of he Langevin force, L()L( ) = 2 2 k BT Ψ Ψ = 2 2 D x Ψ Ψ, as well as is average value, i.e., L() = ± 2 k BT Ψ() = ± 2D x Ψ(). 10

11 3.1 Non-Markovian Brownian Moion of a Free Paricle As far as he non-inerial Brownian moion of a free paricle is concerned, our non- Markovian Smoluchowski equaion (43) wih (20) a poin x reads f x, = 2k B T Ψ f x, x + k BT 1 e c 2 f x, x 2. (45) Saring from he deerminisic iniial condiion f x, = 0 = δ x, (46) characerized by E 0 = 0 and Ψ 0 = 0, we obain he following ime-soluion o (45) f x, = 4πA e 4A, (47) x 2 in erms of he hermal posiion and he funcion x = x 2k B T Ψ d, (48) A = k B T I d = k B T + c e c. (49) Besides reducing o (46) a = 0 (along wih c = 0), soluion (47) yields and X = 0, (50) X 2 = 2k BT + c e c, (51) where he sochasic posiion X = X corresponding o (48) is given by X = X 2k B T Ψ d. Accordingly, he mean square displacemen, X = X 2 X 2, reads X = X = 2k B T + c e c. (52) 11

12 We noice ha averaging effecs are unobservable, for hey do have no influence upon he physically measurable quaniy (52), albei hey can bring abou a shif in he posiion (48). By conras, non-markovian effecs accoun for enhancing he mean square displacemen (52) of a free Brownian paricle. by For very shor imes c, quaniy (52) reduces o a consan value given X = 2k B T c (53) which is differeniable, i.e., d X/d = 0. On he oher hand, in he Markovian limi, c 0, quaniy (52 reduces o he Einsein s renowned upsho X = 2k B T. (54) ha is non-differeniable a = 0, i.e., d X() d =0 = k B T 2 =0. (55) Hence, i is claimed ha here is no concep of velociy of a Brownian paricle in he srong fricion regime [5,6]. In fac, he physically embarrassing oucome (55) is a consequence of neglecing non-markovian feaures in (52). 12

13 4 Concluding Remarks In his paper we have examined he Brownian moion of a free paricle immersed in a hermal reservoir reckoning wih non-markovian effecs. In Sec. 2 we have shown ha he mean square momenum (32) is a differeniable funcion in he presence of inerial force provided ha non-markovian effecs increase he value of (32). In he absence of inerial force in Sec. 3 i has been prediced ha non- Markovian effecs enhance he mean square displacemen (52), hereby assuring he mahemaical propery of differeniabiliy of his physically observable quaniy. I is worh poin ou ha our chief resuls have been obained wihou making use of he generalized Langevin equaion m d2 X d 2 = dv(x) dx β( ) 0 dx( ) d + L d pu forward by Mori [15] on he basis of a close relaionship beween memory effecs ingrained in he fricion kernel β and non-markovian effecs (colored noise) showing up in he auocorrelaion funcion L L( ), given by L L( ) = D β. By conras, our approach has prediced ha non-markovian effecs independen of memory effecs can be physically measured for a free Brownian paricle immersed in a hea bah by means of he mean square displacemen in he absence of inerial force as well as he mean square momenum in he presence of he inerial force. Surprisingly, his feaure sanding ou in he Einsein Langevin framework has been overlooked in he cenenary lieraure abou Brownian moion [1 20]. 13

14 Appendix A: Generalized Fokker Planck Equaions Le us consider he sochasic differenial equaion dz d = 1 dv(z) + aψ. (A1) dz For Z = P, V(P) = 2 P 2 /2m, and a = b Eq. (A1) reduces o he Langevin sochasic equaion (11), whereas for Z = X, V = V(X), and a = b we obain he Langevin equaion (44). Equaion (A1) gives rise o he Kolmogorov equaion [7,11,17,18] f z, = Kf z,, A2 where he Kolmogorovian operaor K acs upon he probabiliy disribuion funcion f z, according o Kf z, = k=1 1 k k! k z k A k z, f z,, A3 he coefficiens A k z, being given by A k z, = lim ε 0 Z k ε, (A4) where he average values, Z k, are o be calculaed abou he sharp values z from he probabiliy disribuion funcion F ZΨ z, ψ, = δ z z F Ψ ψ,. (A5) According o he Pawula heorem [18], if he coefficiens A k z, in (A4) are finie for every k and if A k z, = 0 for some even k, hen A k z, = 0 for all k 3, hereby assuring he posiiviy of he probabiliy densiy funcion f(z, ). So, if z 2 z 3 1 ~0 such ha z 2 z 4 1 = 0, i follows hen ha A k z, = 0, k 3, where z 2 = z + ε and z 1 = z. Accordingly, he Kolmogorov equaion (A2) can be approximaed by he Fokker Planck equaion f z, 2 = z A 1 z, f z, z 2 A 2 z, f z,, A6 where he drif coefficien is given by A 1 z, = lim ε 0 Z ε = 1 dv + a Ψ() dz (A7) and he diffusion coefficien reads 14

15 Z 2 A 2 z, = lim ε 0 ε = a 2 1 lim ε 0 ε +ε Ψ Ψ d d, (A8) wih lim ε 0 1 ε +ε Ψ( ) d = Ψ() (A9) and lim ε 0 +ε Ψ d = 0. (A10) To evaluae he coefficiens (A7) and (A8) we have used (A1) in he form Z = Z + ε Z = ε dv dz + a +ε Ψ d. (A11) For z= p, V(p) = 2 p 2 /2m, and a = b he Fokker Planck equaion (A6) reduces o he Rayleigh equaion (13), whereas for z= x, V = V(X), and a = b (A6) leads o he Smoluchowski equaion (45). 15

16 Appendix B: The Time-dependen Diffusion Energy The ime-dependen diffusion energy E = E I (B1) in he Fokker Planck equaions (13) and (45) presens he correlaional funcion 1 I = lim ε 0 ε +ε Ψ Ψ d d. (B2) given by On he condiion ha he auocorrelaion funcion, Ψ( )Ψ( ), can be Ψ( )Ψ( ) = 1 e + 2 c δ, (B3) where c is he correlaion ime of Ψ() a imes and, i follows ha (B2) becomes I = 1 e c, (B4) reducing o I = 1 in he seady regime. The auocorrelaion funcion (B3) defines a colored noise which in he Markovian limi, c 0, changes ino he so-called whie noise Ψ( )Ψ( ) = δ, (B6) meaning ha he sochasic funcion Ψ() is dela-correlaed [16,19]. I has been argued ha he Markov propery (B6) is a highly idealized feaure [19], because he physical ineracion beween he Brownian paricle and he hermal bah always comes abou for a finie correlaion ime. By he same oken, a decade ago van Kampen has laconically saed: Non-Markov is he rule, Markov is he excepion 20]. Hence, our auo-correlaion funcion (B3) seems o be a more realisic feaure of he Langevin force. 16

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