On the velocity autocorrelation function of a Brownian particle

Transcription

1 Co. Dept. Che., ulg. Acad. Sci. 4 (1991) [axiv 15.76] On the velocity autocoelation of a ownian paticle Rouen Tsekov and oyan Radoev Depatent of Physical Cheisty, Univesity of Sofia, 1164 Sofia, ulgaia Meoy effect of ownian otion in an incopessible fluid is studied. The easoning is based on the Moi-Zwanzig foalis and a new foulation of the Langevin foce as a esult of collisions between an effective and the ownian paticles. Thus, the stochastic foce autocoelation with finite dispesion and the coesponding ownian paticle velocity autocoelation ae obtained. It has been theoetically accepted by Langevin [1] that the inteaction of a acopaticle with a ediu can be pesented by two foces: a fiction foce, elated to the concept of fluid hydodynaic viscous esistance [], and a ando foce, called usually the Langevin foce, which eflects the stochastic chaacte of collisions. Accoding to this pictue, the oentu balance of the ownian otion has the fo of a genealized Langevin equation [3-5] MdU / dt GU ˆ F (1) whee M and U ae the ass and velocity of the ownian paticle, ĜU is the fiction foce and F is the Langevin foce with a zeo ean value F. The Langevin foce is not coelated with the paticle velocity in pevious oents, F( t ) U( t) fo. Due to the coon olecula-kinetic oigin of the two foces ĜU and F they ae not independent. Thei link, naed by Kubo [3] the second fluctuation-dissipation theoe, is ost geneally foulated by Moi [4] and Zwanzig [6] t k ˆ TGU ( t) ( t s) U( s) ds () whee ( ) F( t ) F( t) is the autocoelation of the Langevin foce and T is tepeatue. The opeato Ĝ is popotional to the fiction coefficient b and thei elationship, expessed by Eq. (), is given by the well-known equation [3, 5, 7] k Tb d (3) ()

2 whee is the unit tenso. At slow tanslation of a spheical paticle in an incopessible fluid (Stokes flow []) the fiction coefficient b is equal to 6 R with being the fluid viscosity and R being the paticle adius. An equation fo the evolution of the ownian paticle velocity autocoelation ( ) U( t ) U( t) can be obtained fo Eqs. (1) and () [5, 8] k TMd / d ( s) ( s) ds Futhe the Laplace tansfoation of tie-dependent s will be eployed and Laplace iages will be denoted by a tilde supescipt. Using the Maxwell expession fo the dispesion () k T / M of the ownian paticle velocity, the equation above acquies the fo ( k TM p ) ( ) kt (4) whee p is the Laplace tansfoation vaiable. Usually the fiction foce ĜU is appoxiated by bu, the Langevin foce autocoelation tends to ktb ( ), and Eqs. (1) and (4) acquies thei classical fos [3, 5, 7] MdU / dt bu F k T /( Mp b) ( k T / M) exp( b / M) In the taditional echanical teatents [5, 7] the tansition ĜU bu is analyzed on a icoscopic level as a of the atio / M between the asses of the fluid and ownian paticles. It is shown that in the ass point appoxiation G ˆ( / M ) U bu. Fo objects with a finite sufficiently lage size and acoscopic inteaction with the suounding ediu the agnitude of the ass should be deteined by the ass of fluid displaced by the ownian paticle, i.e. 3 R [9-11] with being the fluid ass density. This can be shown [1, 11] in the classical esult fo the fequency dependent spectal density of the Langevin foce autocoelation ktb (1 pr / ) (5) The liit, at which ( p ) ( M / b is the elaxation tie of the ownian paticle velocity) tansfos into a white noise, coesponds to the liit R / R / M. 3

3 The ai of the pesent pape is to descibe in oe details the behavio of the Langevin foce autocoelation as copaed to the classical odels, which show soe physical discepancies. Fo instance, the dispesion of the Langevin foce coesponding to the spectal density (5) is an infinite quantity, which is unphysical. The pesent new odel is based on the assuption that the action of the suounding ediu can be teated as an ipact of an effective paticle on the ownian paticle, which allows a useful efoulation of the Langevin foce. The inteaction between a ownian paticle and the suounding ediu can be geneally pesented by an opeato C ˆ( V, U ) acting of the velocity V of an effective paticle and the velocity U of the ownian paticle. Since the inteaction foce should be independent of the whole syste dift velocity, it follows that C ˆ ( V, U) C ˆ ( V U). In accodance with the classical theoy of collisions [7], the opeato Ĉ is a linea one and hence the oentu balance of the ownian paticle acquies the fo MdU / dt CV ˆ CU ˆ This esult is equivalent of Eq. (1) and assuing identity of the opeatos expession fo the Langevin foce Ĉ and Ĝ it follows an t F( t) F() GV ˆ ( t s) V ( s) ds / kt (6) Using the stationay natue of the consideed pocesses, V, Eq. (6) and its obvious consequence k T( df / dt)() () V() yields a link between the autocoelation s of the Langevin foce and of the velocity of the effective paticle VV ( ) V( t ) V( t) ( ) / () VV ( ) ( ) k T d d s s ds which, afte application of the Laplace tansfoation, acquies the fo ( kt) [ () p ] () VV (7) The eployent of this esult equies an adequate expession fo the autocoelation of the velocity V by chaacteistics of the ediu. ecause the velocity field excited by the ownian paticle in an infinite incopessible fluid has the sae shape and size of the

4 acopaticle (the ean depth 4 R / b of the hydodynaic field [] is of ode of R ) the effective paticle will have the sae geoetic paaetes as those of the ownian paticle. Of couse, the ass of the effective paticle will be equal to 3 R. Thus the velocity autocoelation of the effective paticle can be expessed by the ownian paticle velocity autocoelation fo Eq. (4) as follows ( M ) ( k T) ( k T p ) (8) 1 VV It is assued hee that the independence of the Langevin foce on M ; is independent on the ownian paticle ass, which follows fo fo Eq. (5) has the sae behavio. An explicit expession of the Langevin foce autocoelation follows fo the esults above. Substituting VV fo Eq. (8) into Eq. (7) leads to the following expession ktb [ 1 ( p / ) p / ] (9) whee / b is the coelation o eoy tie. The invese Laplace tansfoation of Eq. (9) povides an analytical expession fo the Langevin foce autocoelation k Tb J ( / ) / 1 (1) whee J 1 is the essel of fist kind and fist ode. Accoding to Eq. (1) the odulus 3/ of this autocoelation tends to zeo at lage ties as, thus leading to a known esult [9, 1]. Equations (4) and (9) allow the obtaining of the spectal density of the autocoelation of the ownian paticle velocity in the fo kt / b[ 1 ( p / ) p( / )] (11) whee M / b is the ownian paticle elaxation tie. In soe paticula cases the invese Laplace tansfoation of Eq. (11) is possible to be pefoed and the coesponding solutions ae given below: ( ) ( k T / M) exp( / ) heavy ownian paticle ( ) ( k T / M) J ( / )( / ) neutal, equal densities 1 ( ) ( k T / M) J ( / ) lighte ownian paticle

5 whee the second expession has the sae tie-dependence as fo Eq. (1). It should be noted that such oscillatoy-decaying velocity autocoelation s ae obseved in nueical siulations [8, 1]. Equation (11) shows that diffeent behavio of will be obseved at sall and lage ties, which has been discoveed by nueical siulations [13, 14] as well (the so-called long-tie tails). Also by nueical siulations [1] the dependence of by the atio between the fluid and ownian paticle densities is obtained, which is siila to the pedicted by Eq. (11) dependence of on the coesponding atio / / M. In the faes of the linea hydodynaics the poble of finding of the Langevin foce autocoelation can be solved exactly and the esult is Eq. (5). As noted befoe it diveges at. This inconsistency is inheent fo the acoscopic desciption of diffusive pocesses and its eliination equies eployent of icoscopic concepts. The pesent odeling of the inteaction of the ownian paticle with the suounding ediu as ipacts by an effective paticle is not diectly elated to the linea hydodynaics. Hydodynaic concepts ae used only fo deteination of the paaetes of the effective paticle and the autocoelation. Fo instance, the size of the effective paticle is assued to be popotional the ownian paticle size in analogy with the paticle-fluid viscous inteaction in a Stokes flow. The obtained in Eq. (1) finite dispesion () k Tb / of the Langevin foce possesses a clea physical explanation. The velocity dispesion () / coesponds to the classical Maxwell expession VV k T /., geneated by the Langevin foce, [1] P. Langevin, Copt. Rend. Acad. Sci. (Pais) 146 (198) 53 [] L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pegaon, New Yok, 1959 [3] R. Kubo, Rep. Pog. Phys. 9 (1966) 55 [4] H. Moi, Pog. Theo. Phys. 33 (1965) 43 [5] D. Foste, Hydodynaic Fluctuations, oken Syety and Coelation Functions, enjain, Massachusetts, 1975 [6] R. Zwanzig, Annu. Rev. Phys. Che. 16 (1965) 67 [7] P. Resibois and M. de Leene, Classical Kinetic Theoy of Fluids, Wiley, New Yok, 1977 [8].J. ene and G.D. Hap, Adv. Che. Phys. 17 (197) 63 [9] R. Zwanzig and M. ixon, Phys. Rev. A (197) 5 [1] T.S. Chow and J.J. Heans, J. Che. Phys. 56 (197) 315 [11] D. edeaux and P. Mazu, Physica A 76 (1974) 47 [1].J. Alde, D.M. Gass and T.E. Wainwight, J. Che. Phys. 53 (197) 3813 [13].J. Alde and T.E. Wainwight, Phys. Rev. A 1 (197) 18 [14] G. Subaanian, D.G. Levitt and H.T. Davis, J. Che. Phys. 6 (1974) 591