Langevin-Vladimirsky approach to Brownian motion with memory

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1 The Open-Access Journal for he asic Principles of Diffusion Theory, Experimen and Applicaion Langevin-Vladimirsky approach o rownian moion wih memory Jana Tohova, Vladimir Lisy Deparmen of Physics, Technical Universiy of Kosice, Slovakia Corresponding auhor: Vladimir Lisy, Dep. of Physics, Technical Universiy of Kosice, Park Komenskeho, Kosice, Slovakia, vladimir.lisy@uke.sk (received 14 July 011, acceped 5 July 011) Absrac A number of random processes in various fields of science is described by phenomenological equaions of moion conaining sochasic forces, he bes known example being he Langevin equaion (LE) for he rownian moion (M) of paricles. Long ago Vladimirsky (194) in a lile known paper proposed a simple mehod for solving such equaions. The mehod, based on he classical Gibbs saisics, consiss in convering he sochasic LE ino a deerminisic equaion for he mean square displacemen of he paricle, and is applicable o linear equaions wih any kind of memory in he dynamics of he sysem. This approach can be effecively used in solving many of he problems currenly considered in he lieraure. We apply i o he descripion of he M when he noise driving he paricle is exponenially correlaed in ime. The problem of he hydrodynamic M of a charged paricle in an exernal magneic field is also solved. Keywords Generalized Langevin equaion, colored sochasic force, hydrodynamic rownian moion 1. Inroducion During he pas decades, a coninuously increasing ineres in various fields of science and echnology is observed o diffusion processes in sysems exhibiing differen kinds of memory [1]. A variey of physical, chemical, biological and oher naural phenomena can be adequaely described by random processes ha a long imes show a behavior, which is characerized by a nonlinear dependence of he variance of hese processes on ime. Then he famous Einsein formula [] for he long-ime behavior of mean square displacemen (MSD) of he rownian paricle (P) changes o [r() - r(0)] wih he consan ha can differ from 1. Mahemaically, such anomalous diffusion is described by fracional differenial equaions or Volerra-ype inegro-differenial equaions wih colored sochasic forces driving he paricle [3, 4]. These equaions generalize he well-known Langevin equaion (LE) [5] designed for he descripion of memory-less rownian moion (M). In some cases hey a 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

2 lead o normal diffusion wih = 1 bu due o he memory in he sysem here are imporan correcions o he Einsein formula a finie imes. In he presen work we use a very simple way of soluion of he generalized LE (GLE) for he M wih memory. The mehod, proposed long ago by Vladimirsky in a hardly accessible and lile known work [6], can be used for he descripion of he M in linear classical sysems wih any kind of memory. Using i, he general properies of anomalous diffusion can be easily derived. We consider a special kind of he memory in he sysem, which is associaed wih he driving sochasic force represened by he Ornsein-Uhlenbeck process. The exac soluion of he GLE is given for his case. The effeciveness of he mehod is shown also on he example of he hydrodynamic M of a charged paricle moving in an incompressible liquid along and across an exernal magneic field.. Efficien rule for solving he Langevin equaion In Ref. [6], Vladimirsky considered a mehod for he evaluaion of he averaged producs of wo hermally flucuaing quaniies relaed o differen momens of ime. The difficul ask of he calculaion of such means using he Gibbs mehod was reduced o a much simpler soluion of equaions of moion. The obained rules of finding he ime correlaion funcions are hus consisen wih he general principles of saisical mechanics and simulaneously wih he linear phenomenological equaions of moion for he sudied quaniies. For he purposes of he presen paper, le us consider he LE for he coordinae x of he P. If we are ineresed in finding he MSD of he paricle, () = x () = [x() - x(0)], according o he Vladimirsky rule we have merely o subsiue x() in he LE by () and replace he sochasic force driving he paricle wih he consan force f = k T, where k is he olzmann consan and T he emperaure. Of course, having he dimension of energy, f is no a rue force; i only plays his role in he equaion for he paricle posiion (). This ficiious force begins o ac on he paricle a he ime = 0. Up o his momen he paricle is nonmoving so ha he equaion of moion mus be solved wih he iniial condiions (0) V (0) 0, where we have inroduced he velociy V ( ) d ( ) / d. I is imporan ha a he iniial momen he whole sudied sysem is assumed o be in he hermodynamically equilibrium sae. From he poin of view of he macroscopic equaions of moion such iniial sae corresponds o he long-lasing res of he sysem. Thus he values of all he parameers enering he equaions of moion are specified for any 0. This allows one o sudy no only equaions for x() in he form of differenial equaions (ha require only he specificaions of he iniial values of he quaniies) bu also any Volerra-ype inegral equaions. Such equaions can describe sysems for which he whole hisory of is moion is imporan. The discussed mehod is no limied o a concree law of he afereffec. Is advanage is ha i is applicable o any kind of he memory in he sudied (linear) sysem and he only requiremen is he lineariy and exisence of he unique soluion a he given iniial condiions. Le us illusrae he described rule by a simple example of a damped harmonic oscillaor under he influence of he hermal Langevin force f(). The moion of he oscillaor is described by he equaion for he posiion x(), mx x m0 x f () (he parameers m, and can be regarded as he mass of a paricle, he fricion consan, and he oscillaor frequency, respecively). y he same way as in he original Langevin paper [5] (see also [4]), assuming in he averaging ha x, x and f are saisically independen, and wih he help of he equipariion heorem m x = k T, he original equaion can be for saionary random processes easily rewrien in he form m m0 k T, wih he eviden iniial condiions (0) (0) 0. Analogously, more complicaed phenomenological (bu linear) sochasic equaions can be convered o deerminisic ones, e.g., in he case when he fricion 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp 1-10

3 force - x is replaced by a convoluion of he ime dependen memory kernel () and he paricle velociy x ( ). This corresponds o he GLE considered in he nex secion. 3. Solving he generalized Langevin equaion So, already in 194 [6] an effecive approach o various problems ha are now exensively sudied in he lieraure was available. Many non-markovian processes are described by he GLE [1, 3, 4, 7, 8] 0 d M, (1) where, for he M of paricles, M could be he mass of he P wih he velociy ( ) x ( ). The memory in he sysem is described by he kernel (), and () is a sochasic force wih zero mean. According o he flucuaion-dissipaion heorem he condiion (0)() = k T() a 0 mus be saisfied [3, 7-9]. One is usually ineresed in finding he MSD of he paricle, () V( )d, is ime-dependen diffusion coefficien D() = V()/, or he 0 velociy auocorrelaion funcion (VAF) ( ) D ( ) [10]. The iniial condiions for () and V() have been discussed above. Obviously, also he condiion V (0) kt / M mus hold. I follows from he equaion ha concreizes he Vladimirsky rule for his special case, 0 MV V dk T. () Taking he Laplace ransformaion of his equaion wih he use of he iniial condiions, i can be easily solved for he specific memory kernels considered in he lieraure. Also he known general properies of he MSD and oher ime correlaion funcions [11, 1] can be obained in his way, which proves he Vladimirsky rule for equaion (1). The long-ime properies of he MSD can be relaed o he behavior of he Laplace ransform of he kernel, () s = {()} a small s [11, 13]. A shor imes, equaion () describes he ballisic moion, () kt / M as 0, independenly of a concree form of (). As an example, le us specify he memory kernel. We chose () in he Ornsein-Uhlenbeck form () = ( /m)exp(-/m). Then he force () = m u () corresponds o he soluion of he usual LE mu () u() f (). Here u() can be he velociy of a P, is a consan fricion coefficien, and f() he sochasic (whie noise) force. The moion of he Ps wih he mass M can be hus inerpreed as being induced by he force caused by he paricles of mass m. The Laplace ransformaion of () yields 1 kt V () s s s s Ms m m mm (3) The denominaor in his expression can be given he form (s - 1 )(s - ), 1, ( / m)(1 14 m/ M), so ha he soluion is expressed as 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

4 kt V () s 1. (4) M 1 m ss s1 Using he ables of Laplace ransforms [14] we ge kt V () M m + 1 exp 1 exp (5) 1 m m1 The MSD is obained by simple inegraion of his equaion. I can be used ha 1 /( mm ); i is hen seen ha he firs erm in {} is he Einsein long-ime limi k T/. A shor imes we ge he expeced ballisic moion. If we denoe 1 4 m/ M, D() can be given a compac form kt 1 D( ) 1 exp( / m) (1 ) exp( / m) (1 ) exp( / m) 4. (6) If we forge ha he moion of he paricles was inerpreed as induced by paricles wih smaller mass m, an ineresing resul follows from equaion (5) in he case when M 4m, i.e. when he roos are complex. Then he soluion describes damped oscillaions; e.g. for D() we have kt 4m m/ M 1 4m D () 1exp cos 1 sin 1 m m M 4 m/ M 1 m M kt m 1 exp( / m) sin cos M (if M 4m). (7) mm mm The special case of consan memory kernel [15] follows from here when /m 0 ( = /m = cons): kt D () sin M M. (8) Noe ha he resuls (5) and (6) (or he corresponding VAF) can be obained also from he soluion of he Fokker-Planck equaion (FPE), associaed wih he GLE (1). The FPE possesses he disribuion funcions of flucuaing macroscopic variables, which in many cases give a more convenien probabilisic descripion of he sudied processes. As disinc from he used approach ha does no explicily explore he correlaion properies of he random force (), building he FPE requires he knowledge of hese properies [16]. The generalized FPE corresponding o he GLE (1) wih he memory kernel ( - ) and o equaion () (wih he iniial insan of ime 0 = 0) was sudied in Ref. [17]. A more general FPE was obained in he remarkable work [18]. The FPE obained in ha work is associaed wih he sysem of Langevin equaions for mulidimensional non-saionary process wih he memory kernels (, ) and arbirary 0 for Gaussian random forces wih zero mean. As an example, he moion of a spherical paricle in an unbounded viscoelasic fluid wih a paricular relaxaion 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

5 ime was considered. The velociy correlaion funcion found afer edious calculaions from he obained FPE coincides in he case of saionary random process wih he resul ha is easily calculaed using he approach of he presen work. 4. Hydrodynamic rownian moion in he magneic field The descripion of he diffusion-like processes wih he use of equaion (1) is no he mos general one. In some cases differen phenomenological equaions can possess a more appropriae descripion. As discussed in Inroducion, he used mehod is no limied o (1). Here we shall demonsrae he efficiency of he Vladimirsky approach by he soluion of a problem ha naurally arises in he heory of he M, namely, he M wih hydrodynamic memory [19]. This problem can be formulaed using he LE as well. However, he Sokes fricion force -, which is valid only for he seady moion of he paricle (a long imes), should be replaced by a force ha akes ino accoun he hisory of he paricle moion. This force appears as a consequence of he inerial effecs in he moion of a paricle in a liquid, and for incompressible fluids was firs derived by oussinesq [0]. The soluion of his problem for Ps was given in he work [1] where, among ohers, i has been shown ha he classic LE gives he correc resuls only for paricles wih he densiy much larger han he densiy of he surrounding fluid (and simulaneously a shor imes), or a long imes when, however, he Einsein resul is valid. Laer his ask was sudied in a number of papers and exacly solved in Ref. [] (for a review see [3-5]). Wih he approach used in he presen work he soluion can be obained in a simpler way han so far. Here we will consider a more complicaed problem of he movemen of a charged P in he magneic field. For he moion along he field or in is absence he previous resuls [1, ] will be recovered. The hydrodynamic moion of he P across he field was sudied already in he older paper [6]. Our soluion correcs he resuls from Ref. [6] in several poins. The oussinesq force on a spherical paricle of radius R is R dυ R dυ d F υ 9 d, (9) d where he fricion facor = 6R, is he densiy and he viscosiy of he solven. This equaion is valid for he imes R/c (c is he sound velociy), i.e., very shor imes when he compressibiliy effecs play a role are excluded from he consideraion. If Q is he charge of he paricle of mass m and is he consan inducion of he magneic field along he axis z, hen he LE for he P reads mυ () F() Qυ η(). (10) The projecion of his equaion ono he axis z does no conain he magneic force so ha along he field we have he moion of a free P. Equaion () for V z = d z /d wih z being he MSD in he z direcion can be wrien in he form 1 R Vz( ) 1 kt V z() d Vz(), (11) M 0 where M = m + m s / (m s is he mass of he solven displaced by he paricle). The characerisic imes in his equaion are = M/ (he relaxaion ime of he usual P) and R = R / (he voriciy ime). The Laplace ransformaion (11) gives 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

6 1 R kt V s s s s M 1 1 1/ 1 z(). (1) The inverse ransform is found afer expanding he erm (...) -1 in simple fracions (s - 1, ) -1, where 1, R /1 14 / R Then are he roos of equaion s 1 1/ 1 R s = 0. kt V z () s Ms 1 s s, (13) 1 so ha [14] kt Vz () exp( )erfc 1 exp( )erfc 1 M 1 1. (14) 1 1 The VAF is expressed by a similar equaion, if one divides V z () by, in {...} replaces 1/ 1, wih 1,, and in boh square brackes omi -1. This expression exacly corresponds o he soluions found in Refs. [1, ] by differen mehods bu differs from he soluion [6] due o he difference in he roos 1,. The soluion a long imes conains he so called long-ime ails ha became famous afer heir discovery in he compuer experimens [7, 8]. For he VAF a i follows from equaion (14) ha he longes-lived ail is -3/. The MSD is obained by inegraing his equaion, 1/ R z D R 1 1 exp exp1 erfc 3 erfc 3 1, (15) 1 1 z R m R m R D ,, (16) 9 ms 9 ms where D kt / M is he diffusion coefficien of he paricle. Equaion (15) corresponds very well o experimen [9]. For he paricle moion across he magneic field we have from (10) wo equaions for he x and y componens of he velociy [6] and, afer he applicaion of Vladimirsky rule, he equaion for he quaniy V x + V y = d xy /d ha deermine he MSD xy = x + y of he paricle in he plane perpendicular o he field [19]. In he Laplace ransformaion we obain for V () s = {V x ()+ V y ()} xy 4 kt () s V xy () s Ms, (17) () s c 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

7 where (s) = s s and he new characerisic ime c is conneced o he 1 1 R cycloron frequency c = Q/M = 1/ c. The roos 1, of (s) have been deermined afer equaion (1) and for he four roos of he denominaor () s c we have 1, 1 ( 1 ) 4ic and 3,4 1 ( 1 ) 4ic dependen fracion in (18) can be decomposed in simple fracions,. Using hem, he s- ( s) () s c 1 s s, s 4 s 3 afer which he Laplace ransform ables can be used [14]. This yields kt 1 1 Vxy () f, f,, M , (18) 1 where f (, ) exp( )erfc( ). The MSD is readily obained by he inegraion of V xy () from 0 o. Using 1 f(, )d exp erfc( ) 1, (,) = one finds 0 3 kt xy () M ( 1 ) ( 1 ) exp 3 erfc exp 3 1erfc ,. (19) In he absence of he field ( c = 0) 1 = 3 = 1, = 4 =, and xy = z, wih z from (16). To rewrie equaion (19) in erms of he characerisic imes, c, and R, one has o use i shown afer equaion (17) and i from (13). This gives,, 1 i c 3 4 i c 1/ R, ec. For long imes we use he expansion of he funcion erfc(z) for large argumens z [14], which, up o he longes-lived ail, gives 4kT 1 / R c xy ( ) 1...,, (0) M 1 / c 1 / c wih / c MQ / (6 Rm) Q(1 / ) / (6 R). Here and are he densiy of he fluid and he paricle, respecively. Noe ha even he Einsein limi of equaion (0) differs 011, V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

8 from he previous resul [6] (he agreemen is only when M = m (or a c = 0)). A shor imes we have he ballisic moion (independen of he magneic field) as from equaion (), () k T / M, bu wih a differen M, which is now M = m + m s /. This apparen xy conradicion wih he equipariion heorem is jus a consequence of he assumpion of solven incompressibiliy, due o which he limi 0 canno be accomplished. The correc resul ha conains only he mass of he paricle, m, is achieved when he compressibiliy of he solven is aken ino accoun (on he ime scale 10-1 s ha characerizes he collisions of he P wih he surrounding molecules he VAF () decays from he equipariion value k T/m o k T/(m + m s /) hrough he emission of sound waves). This is one more ask ha is sraighforwardly solvable by he presen mehod. Recenly [30], he moion of a charged P in a magneic field was sudied in he case when he afereffec exponenially decreases in he ime. Also his problem can be effecively solved by he proposed mehod [31]. 5. Conclusions In conclusion, we have described an efficien approach o he soluion of he problems of diffusion wih memory ha are modeled using he generalized Langevin equaion. The mehod consiss in convering his sochasic inegro-differenial equaion ino a deerminisic one for he mean square displacemen. I allowed us o solve in a very simple way he specific problem when he rownian paricle is driven by an exponenially correlaed sochasic force. The descripion of diffusion-like processes wih he use of he GLE is no he mos general one. In many cases differen phenomenological equaions could possess a more appropriae model. I is hus imporan ha he used mehod is no limied o he GLE equaions. We have demonsraed i on he problem of he hydrodynamic rownian moion of charged paricles in he magneic field. Oher applicaions ha are currenly of grea ineres concern, for example, he moion of rownian paricles dragged by opical weezers [3, 33], he anomalous moion of colloidal paricles under he influence of various exernal fields [34, 35], he behavior of mesoscale elecric circuis in conac wih he hermal bah [36, 37], he moion of magneic domain walls [38], or he dynamics of polymers [4, 5, 39-41]. Among he menioned problems hose close o he asks considered in he presen work relae o he rownian moion in he presence of magneic field [4, 43]. In Ref. [4], he moion of a charged rownian paricle in a harmonic rap was heoreically described in he case when he rap is dragged wih a consan velociy and when he paricle is subjeced o an ac force. The work disribuion of he paricle has been found analyically and he Jarzynski equaliy [44, 45] was verified. In he recen paper [43], a similar problem was considered for a rapped paricle under he acion of a consan magneic field and a ime-dependen elecric field. The main aim of ha work was o prove he validiy of he saionary sae flucuaion heorem [46] and he ransien flucuaion heorem [47]. Noe ha he equaions of moion for he paricle were chosen in he form of he usual Langevin equaions wih he Sokes fricion and he whie noise force. I would be ineresing o develop hese works o he case of he GLE wih hydrodynamic memory, which is a suiable model if he rownian paricle moves in a liquid. In our opinion, fuure effor should be also oriened on he generalizaion of he proposed mehod o siuaions when quanum effecs are significan [36], and o overcome is resricion o linear models. Acknowledgmens. This work was suppored by he Agency for he Srucural Funds of he EU wihin he projecs NFP and , and by he gran VEGA 1/0300/ , V. Lisy diffusion-fundamenals.org 15 (011) 5, pp

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