Non-Markovian Brownian Motion in a Viscoelastic Fluid

Transcription

1 The University of Akron College of Polymer Science an Polymer Engineering Non-Markovian Brownian Motion in a Viscoelastic Flui V. S. Volkov Arkaii I. Leonov University of Akron Main Campus, leonov@uakron.eu Please take a moment to share how this work helps you through this survey. Your feeback will be important as we plan further evelopment of our repository. Follow this an aitional works at: Part of the Polymer Science Commons Recommene Citation Volkov, V. S. an Leonov, Arkaii I., "Non-Markovian Brownian Motion in a Viscoelastic Flui" (1996). College of Polymer Science an Polymer Engineering This Article is brought to you for free an open access by IeaExchange@UAkron, the institutional repository of The University of Akron in Akron, Ohio, USA. It has been accepte for inclusion in College of Polymer Science an Polymer Engineering by an authorize aministrator of IeaExchange@UAkron. For more information, please contact mjon@uakron.eu, uapress@uakron.eu.

2 Non-Markovian Brownian motion in a viscoelastic flui V. S. Volkov a) an A. I. Leonov b) Department of Polymer Engineering, The University of Akron, Akron, Ohio Receive 14 September 1995; accepte 10 January 1996 A theory of non-markovian translational Brownian motion in a Maxwell flui is evelope. A universal kinetic equation for the joint probability istribution of position, velocity, an acceleration of a Brownian particle is erive irectly from the extene ynamic equations for the system. Unlike the extene Fokker Planck equation which correspons to Mori Kubo generalize Langevin equation an provies only with calculations of one-time moments, the universal kinetic equation obtaine gives complete statistical escription of the process. In particular, an exact generalize Fokker Planck equation in the velocity space vali for any time instant is erive for the free non-markovian Brownian motion. It shows that both the master telegraph an the respective kinetic equations, obtaine in the molecular theory of Brownian motion, are type of approximations. The long an short time behavior of velocity an force correlations for a free Brownian particle is investigate in the general case of a nonequilibrium initial value problem. A corresponing iffusion equation in the coorinate space, an the generalize Einstein relation between the iffusion coefficient an the mobility are erive American Institute of Physics. S I. INTRODUCTION Statistical escription of Brownian motion belongs to the most funamental problems in physics with a wie variety of applications. The Brownian motion of small particles suspene in a viscous flui has been stuie since In pioneering papers by Einstein, 1 Smoluchovski, 2 an Langevin 3 this motion has been treate as a Gaussian Markovian ranom process. The classical moel of Brownian motion uses the simple Stokes formula for the hyroynamic force acting on the spherical particle in a viscous incompressible flui. This familiar Stoke s law was erive originally for the steay motion of a sphere. In 1851 Stokes 4 calculate the frequency-epenent friction coefficient for a sphere oscillating in a viscous quiescent flui. The corresponing expression for motion with arbitrary changing velocity was foun by Boussinesq in Thus the Brownian motion of a particle in a viscous, inertial flui has to be consiere as a non-markovian ranom process because the frictional resistance of a particle epens on its history. The non-markovian theory of Brownian motion in viscous flui with ue account for the hyroynamic aftereffect escribe by Stokes Boussinesq s formula was propose by Vlaimirsky an Terletsky 5 in 1945 an evelope later by several authors. 6 9 If the liqui surrouning moving particle is viscoelastic, as all the liquis are in fact, 6 the stochastic motion of a Brownian particle is non-markovian, even if the inertia of the liqui is negligible The interpretation of the Brownian motion as a non-markovian stochastic process is corroborate by the molecular theory which consiers the motion of a heavy particle in a meium consisting of light F i t F iu i. 2.2 Here r an u are the raius vector an velocity of the Brovnian particle of mass m;6a is the friction coeffia Permanent aress: Institute of Petrochemical Synthesis, Russian Acaemy of Sciences, Leninsky Pr., 29, Moscow, Russia. b The author to whom corresponence shoul be aresse. particles Several statistical approaches to the stuy of non-markovian ranom processes have been evelope. They are summarize in reviews This work analyzes the Brownian motion in a viscoelastic liqui with one relaxation time. The analysis is base on increasing the imension of space of ynamical variables. In oing this, we introuce an aitional variable representing the Markovian ranom force, which is consiere as the solution of an aitional stochastic equation with a eltacorrelate ranom force. Thus the problem of Brownian motion in the simplest viscoelastic liqui can be reuce to the statistical escription of an extene ynamical system subjecte to a elta-correlate ranom force. The moel of Brownian motion stuie here is more realistic than the classical one, since it gives a finite variance of the acceleration force of a Brownian particle. II. BROWNIAN DYNAMICS WITH MARKOVIAN RANDOM FORCE The motion of an elastically boun Brownian particle in a quiescent viscoelastic liqui with a single relaxation time is escribe by the stochastic equations of motion t r iu i, m u i t F ir i i with the relaxe friction J. Chem. Phys. 104 (15), 15 April /96/104(15)/5922/10/$ American Institute of Physics

3 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui 5923 cient; a is the raius of the particle; an are the viscosity coefficient an relaxation time of the flui, respectively. A systematic viscoelastic rag F i an a ranom force i, along with an external quasielastic force r i, act on the particle. This escription of the Brownian motion is vali only for time intervals that are not too short. In this case, the force exerte by the surrouning meium on the particle can be ivie into systematic an ranom parts. The solution of the hyroynamic problem of inertialess translational motion of a spherical particle in a viscoelastic flui with a single relaxation time 12,14 leas to the simple expression 2.2 for the viscoelastic friction force F. Inthe particular case of a viscous Newtonian flui 0 it reuces to the well known Stokes law F i 0 u i. Equation 2.2 is easily solve to yiel the following expression for the rag on the sphere in the Maxwell flui in terms of the initial force F 0 i, F i tf 0 i e tt 0 / t Btsui ts. 2.3 t0 Here the friction kernel B(t) is given by Bt et/. 2.4 In the case of the zero initial conition, F i 0 0, Eq. 2.3 is simplifie to F i t t0 t Btsui ts. For tt 0, the rag on the sphere is t 2.5 F i t Btsu i ts. 2.6 Equations 2.3, 2.5, an 2.6 lea to ifferent Langevin equations with memory. We will use here the universal relaxation Eq The ranom force i (t) maintains the thermal motion of the particle. At any time, the average value of this force vanishes, i t0. The ranom force i (t) being affecte by a large number of equally strong inepenent impulses changes irection rapily. Therefore one can assume that it satisfies the conitions of the central limit theorem an has the Gaussian istribution. Accoring to the Callen Welton fluctuation-issipation theorem 22 the spectral ensity of the ranom force is etermine by the relation K ik TZ ik Z ki. 2.7 Here K ik are the Fourier components of the correlation function i (t) k (0); Z ik is the impeance matrix of the system an T is the temperature in energy units. We use the following notations for the two-sie an one-sie Fourier transforms: x xte it t, x 0 xte it t. The impeance matrix of the system uner stuy has the form Z ik B/iim ik. 2.8 Here B is the complex friction coefficient of a Brownian particle in a Maxwellian flui, B 1i. Substituting Eq. 2.8 into Eq. 2.7 yiels the fluctuationissipation relation for the non-markovian stochastic equations of motion 2.1, K ik 2T 1 2 ik. Since the resulting spectral ensity epens on the frequency, the ranom force i (t) is not elta-correlate. The corresponing correlation function has the form i t k 0T et/ ik. 2.9 Thus the ranom force i (t) acting on a Brownian particle in a viscoelastic Maxwellian flui is correlate exponentially. Its statistical properties o not epen on the quantity characterizing the external force. In the limit 0, the ranom force i (t) is represente by the Gaussian white noise, an Eq. 2.5 reuces to the familiar Einstein relation, i t k 02Tt ik. Note that Eq. 2.9 also follows from the fluctuationissipation theorem for non-markovian Langevin equation with memory, obtaine by Mori 22 an Kubo 23 from ifferent approaches. The Markovian ranom force i (t) can then be regare as the solution of the first orer stochastic ifferential equation 12,13 t it i t i t with a elta-correlate ranom force (t), i t k 02Tt ik, an initial conition referre to. Equation 2.10 is a result of the solution for an inverse problem of the classical theory of Brownian motion to obtain the white noise i (t) from a ranom function with given statistical characteristics 2.9. Equations 2.10 an 2.11 with arbitrary initial conition may be consiere as a most general form of fluctuation-issipation theorem for non-markovian Langevin Eq The well known fluctuation-issipation relation 2.9 is the special case for the initial conition referre to

4 5924 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui. Accoring to Eq. 2.10, the starting ranom force i (t) is relate to white noise i (t) by the following integral relation: t t i 0 e tt 0 / 1 t0 t Bsi ss, where i 0 is the initial ranom force. It has a memory over the time equal to the relaxation time of the Maxwell flui. Using the relaxation equations for systematic 2.2 an ranom force 2.10, we can rewrite the set 2.1 of stochastic equations into an equivalent one with a elta-correlate ranom force, r i t u i, u i t u i, m t u imu iu i r i i Thus in the resulting equation of motion 2.12 the set of inepenent variables inclues the first-orer acceleration u. This efines a multiimensional Markovian process r,u,u. It shoul be note however that not every ranom process can be reuce to a Markovian one, even in the most general sense. For instance, this is impossible when significant resiual effects are associate with inertia of the liqui or in the case of a viscoelastic flui with a continuous spectrum of relaxation times. III. KINETIC DESCRIPTION The statistical characteristics of the Brownian motion uner stuy can be foun irectly from the linear stochastic equations In many cases, however, the ifferential equation for the istribution function of the solution of Eqs gives a more convenient probabilistic escription. For this reason, it is esirable to establish a precise corresponence between Eq an the equation for the istribution function, analogous to the Fokker Planck equation. In the erivation we follow the metho of Klyatskin Tatarskii. 25 This metho allows us to erive kinetic equations for various istribution functions irectly from the stochastic equations of motion. We efine the istribution function for the solution of the system of Eqs as follows: f r,u,u,trrtuutu u t. 3.1 Here r(t), u(t), an u (t) are the solution of Eqs corresponing to a certain realization of the ranom force i (t) an the averaging is performe over the set of all realizations. Taking the time erivative of Eq. 3.1 an using Eqs. 2.12, we obtain f t u e f r e u e f r eu e u e m f u e 1 u ef 1 u e m e tr. 3.2 u e Here Rrr(t)uu(t)u u t is a nonlinear functional of the Gaussian stochastic process (t) with zero average. To close Eq. 3.2 we nee to express the average value e R in terms of f. It is one by employing the Furutsu Novikov formula 26,27 which in our case has the form e tr e t n s R s. 3.3 n s Using the formulas, r i t j t 0, u i t j t 0, u it j t 1 m ij which follow immeiately from Eqs. 2.12, we can obtain the explicit expression for the functional erivative R[]/ e (s), R e t 1 R. m u e As a result we fin a close kinetic equation for the one-time istribution function, t f r,u,u,tu e u e u e D u f r e u e u e f. f r e u e u e m f u e 3.4 The iffusion coefficient in the acceleration space is etermine in the form D u T m 1 m 2, where the following notations for the relaxation times m m/, B, B / are introuce. The istribution function f r,u,u,t of position, velocity, an acceleration can be foun from Eq. 3.4 for a given initial istribution f 0 f r,u,u,t 0. Further on, one can obtain the phase-space istribution function f r,u,t f r,u,u,tu. This istribution cannot be establishe from the classical Fokker Planck equation, since the process r(t),u(t) is not Markovian. For free Brownian motion in a Maxwellian flui 0 the kinetic equation

5 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui 5925 t f r,u,u,tu e f r e u e f u e u e m f u e u e u e D u 3.5 u ef follows from Eq Using Eq. 3.4 we can etermine one-time statistical characteristics of the multiimensional Markovian process r(t),u(t),u (t). Since the solution of the system 2.12 is a linear functional of a Gaussian ranom force the joint istribution of the probabilities of r(t), u(t), an u (t) is also Gaussian an it is sufficient to calculate only the first an secon moments of this istribution. For the average values, the following system of linear equations hols: t r iu i, t u iu i, t u iu i m r i u i. The stationary solution of this system has the form r i 0, u i 0, u i0. The evolution of the secon one-time moments, x ik tr i tr k t, y ik tr i tu k t, z ik tu i tu k t, n ik tr i tu kt, m ik tu i tu kt, e ik tu itu kt, is escribe by the set of equations t x iky ik y ki, t y ikz ik n ik, t n ikn ik m ik m x ik y ik, t m ikm ik e ik m y ki z ik, 3.6 t z ikm ik m ki, t e T ik2e ik 2 m ik 2 m m n ik m ik. The stationary solution of the system 3.7 has the form r i u k 0, u i u k0, u i u k T m ik, r i u k T m ik, 3.7 u iu k T m m 1 ik. 3.8 m Equations 3.8 exactly correspon to the law of equipartition of energy over the egrees of freeom. Thus as t, the Brownian particle comes into thermoynamic equilibrium with the surrouning viscoelastic meium an is characterize by the correlation matrix 0 r ir k r iu k r u i r k u i u k i u k u ir k u iu k u iu kt T m T 0 0 m 2ik. T m 0 T m Therefore the multiimensional stochastic process uner stuy has the stationary istribution f s r,u,u C exp mu2 m m u 22 m ru 1/ B r 2 2T. 3.9 Here C is a constant etermine by the normalization conition fruu 1. Equation 3.9 represents a generalization of the well known Maxwell Boltzmann istribution. A new feature of the istribution 3.9 is statistical epenence of coorinates an accelerations. It is easy to verify that this istribution is the stationary solution of Eq The equilibrium istribution 3.9 epens on the iniviual properties of the Brownian particle an on the external parameters an, characterizing the viscoelastic properties of the surrouning meium. Integrating it over accelerations results in the Maxwell Boltzmann istribution f s r,uexp mu2 r 2 2T. Therefore the stationary istribution functions in the phasespace for the Brownian motion in viscous an viscoelastic liquis are ientical. IV. ONE-TIME STATISTICAL CHARACTERISTICS In orer to etermine how a Brownian particle attains the stationary state, starting from arbitrary initial conition, let us now turn our attention to the analysis of one-time correlations. As mentione, the asymptotic state of Brownian particle at t may be interprete as a thermal equilibrium with the surrouning flui at temperature T. We now will stuy the particular form of Eqs. 2.12, m 2 u i t 2 m u i t u i i t, 4.1 escribing the free Brownian motion in a Maxwell flui, i.e., in the absence of an external force fiel. The statistical properties of a Gaussian elta-correlate ranom force i (t) is efine by Eq Accoring to Eq. 4.1, the velocity of Brownian particle u is a non-markovian stochastic process, wich can be consiere as a projection of the Markovian process u(t),u (t). It is interesting that Eq. 4.1 is mathematically equivalent to the secon-orer stochastic ifferential equation in position space, that escribes Brownian motion of a simple harmonic oscillator in viscous flui. 28 From the non-markovian stochastic ifferential Eq. 4.1 with a

6 5926 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui elta-correlate ranom force in the right-han sie one can erive the following evolution equations for the average values: u i u i, t m t u i m u iu i 0 with initial conition y 0 i u 0 i, z 0 i u 0 i an the secon-orer moments z ik m t ik m ki, m 2 z ik t 2 e ik t z ik m 2 t m e ik 2z ik, 2e ik 2 T m ik 1 z ik m m t The erivation Eqs. 4.3 is base on the Furutsu Novikov formula 3.3. The averages appearing in the moment equations by using one are easily etermine i tu k t0, i tu T kt ik. m We consier the nonequilibrium initial value problem with the initial secon-orer moments z 0 ik u 0 i u 0 k, m 0 ik u 0 i u k 0, e 0 ik u i 0 u k 0. The initial velocity u 0 i an the initial acceleration u i0 of Brownian particle are assume to be Gaussian istribute. One of the main problems in the theory of Brownian motion is the calculation of velocity moment functions. From Eqs. 4.2 an 4.3 it is a simple matter to obtain the close equations for the average velocity y i (t)u i an one-time velocity z ik (t) an acceleration e ik (t) correlations, m 2 y i t 2 m y i t y i0, m 2 3 z ik t 3 4 T m ik, m 2 3 e ik t 3 3 m 2 z ik t 2 3 m 2 e ik t 2 22 m z ik 4z t ik 22 m e ik 4e t ik T 4 m ik. 4.4 m Solving Eqs. 4.2 an 4.3, we fin the expressions for the one-time moments, y i u i ty i 0 tz i 0, z i u i ty i 0 tz i 0, z ik u i tu k t T m ikz 0 ik z e ik t 0 m ik te 0 ik e e ik 2 2t, e ik u itu kt T m ik z ik m 0 z ik e t 0 m ik te 0 ik e e ik 2 2t, m ik u i tu kt 1 2 z 0 ik z e ik tm 0 ik t e 0 ik e e ik 2 t, 4.5 where (t) 2 (t) an (t) 2(t). Here we have efine the relaxation times by 1 2 m m 2 4 m. The function 4.6 t 1 e t/ e t/ 4.7 may be interprete as the response function, associate with system 4.1, to the external force i (t). Another useful way of expression of (t) is te t/2 Cosh t 2 Sinh t 2, 4.7a where 1/ 1/. The convenience of formula 4.7a is that it gives for the moments 4.5 finite an real expressions even in aperioic 0 an unerampe i 1 cases. Equation 4.5 shows that the short-time behavior of the stochastic process u,u epens on an initial conition. However, after the transient time, which is etermine by the relaxation time of viscoelastic flui, the solution asymptotically approaches a unique, equilibrium state etermine by the statistical characteristics z e ik T m ik, m e ik 0, e ik e T m ik. m 4.8 If the initial conition is etermine by the accelerations equilibrium istribution an z 0 ik u 2 0 ik, we obtain u i tu k t T m u 2 0 m T 2 t ik. 4.9 Equation 4.9 shows how the equipartition value is reache. This result is similar to that obtaine originally in the classic papers 28 for the Brownian motion in viscous flui, where, however, a ifferent nonexponential function (t) was foun.

7 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui 5927 For the one-time cumulants, u i t,u k tu i u i u k u k, u i t,u ktu i u i u ku k, u it,u ktu iu iu ku k, we have the simple expressions u i t,u k t T m 12 t 2t ik, u it,u kt T 1 2t m m 2 2t ik, u i t,u kt T 2t ik Analysis of Eqs. 4.5 an 4.10 shows that the force an the velocity of Brownian particle suspene in a Maxwell flui are correlate over long time intervals. In this case, velocity an acceleration of Brownian particle are characterize in general by the nonexponential one-time correlations. The steay state is reache after a long time. The physical origin of the slow approach of correlations to the equilibrium values lies in the viscoelasticity of the surrouning flui. We now consier the asymptotic behavior of the expression 4.5 for the one-time correlations of velocities, z ik (t), an accelerations, e ik (t), at t, i.e., when approaching to the equilibrium. Depening on the value of root in Eq. 4.6, there are two cases. 1 m 4. The most interesting situation here is when m. In this case, the viscous behavior of the liqui is ominant over the viscoelastic one, an except for the equilibrium value of e e ik, where the relaxation parameter is essential, one can expect that all the transitional phenomena will epen mostly on the viscosity of liqui an the mass of Brownian particle. Brief calculations using Eq. 4.5, yiel the following asymptotic result: z ik tz ik ik z 0 ik /z exp t/ m, e ik te ik / m ik z 0 ik /z exp t/ m. t, z T/m, e z / m Since m, the correlations of accelerations reach the equilibrium much faster than that of velocities. 2 m 4. In this case, approaching to the equilibrium is accompanie by oscillations, both the correlations z ik (t) an e ik (t) reach the equilibrium almost synchronously, an the relaxation properties of liqui are very essential. This is the case of a behavior appropriate for polymer solutions an melts. V. KINETIC EQUATION IN VELOCITY SPACE In this section we erive the equation for the velocity probability ensity of free Brownian motion in the Maxwell flui, starting irectly from the non-markovian Langevin Eqs The erivation follows the functional metho similar to that propose by Klyatskin Tatarskii. The istribution function f u,t of a Brownian particle in a velocity space may be efine by f u,tuut, 5.1 where u(t) is a solution of Eq. 4.1 for a given realization of ranom force i (t) with initial conition specifie at t0. The velocity of Brownian particle u(t) epens on time t an initial velocity u 0, an is a linear functional of the noise i (t). The averaging is one over the set of all realizations (t) an over the istribution of initial velocities. Equation 5.1 yiels f u,t u etuut. t u e We now take the time erivative of Eq. 5.2 to obtain 2 f u,t t 2 ü u e tuut e u etu ntuut. 5.3 u e u n Taking into account the initial stochastic Eq. 4.1, wefin from Eqs. 5.2 an 5.3 the following equation for the istribution function: 2 f t 2 1 f t 1 u m u e f 1 e m u e tuut e 2 u etu ntuut. 5.4 u e u n To calculate the averages in Eq. 5.4 we use the Furutsu Novikov formula 3.3 an a similar formula for the mean of the prouct of two nonlinear functionals P an R of a Gaussian stochastic process (t) with a zero mean value 11 PRPR P e t 1 P n t 2 e t 1 n t 2 t 1 t Equation 5.5 was erive by employing a functional Taylor-series expansions of P an R an using then the statistical properties of (t). As a result we get for the velocity istribution function the close kinetic equation 2 f f m t 2 m t u e u e m u etu nt u nfu,t. 5.6 Here u e (t)u n(t) is the one-time acceleration correlations efine in Eq The generalize Fokker Planck equa-

8 5928 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui tion in velocity space 5.6 for the non-markovian Brownian motion in a Maxwell flui is important result of this paper. Equation 5.6 can be also expresse as follows: 2 f t 2 f t u s u s D sn t u nfu,t. Here D sn t T m sn z 0 sn z e sn t m sn te 0 sn e e sn 2 2t. This equation is vali for any time an any value of the parameters of the system, given in terms of the response function (t). The kinetic coefficients D sn (t) have ifferent forms epening on the statistical properties of the initial velocity an initial acceleration of Brownian particle. This is in agreement with the physical fact that the evolution of non-markovian processes epens significantly on initial conitions. If the Brownian motion starts from zero initial values of velocity an acceleration, the equation for the velocity istribution function 5.7 still hols but D sn t T m 1 2t 2 2t sn. 5.8 In the case of a Brownian motion in a viscous flui, when m/ an 0, Eqs. 5.7 an 5.8 are reuce to the known classical Fokker Planck equation in velocity space. The generalize Fokker Planck Eq. 5.7 can be represente as a retarte equation f u,t t m sbts t 0 u u e D en s e u nfu,s 5.9 with the same memory kernel 2.4 as in the starting Langevin equation with exponential memory. For a long time interval, when the equilibrium acceleration istribution has alreay reache the asymptota Eq. 4.11, we have u e(t)u n(t)(t/m m ) en. In this particular case, the equation for the velocity istribution function 5.6 takes the form of the telegraph equation, 2 f t 2 f t m u e u e T m u e f u,t 5.10 with constant coefficients. The master telegraph equation analogous to Eq has been also erive in the papers 29,30 for non-markovian processes associate with nonequilibrium phenomena. Note that the telegraph Eq coincies with that erive in the molecular theory of Brownian motion 15,18 f u,t 1 t sbts t m 0 u efu,s u u e T e m 5.11 if the memory kernel B(t) is the exponential function 2.4. In this case, Eq can be rewritten as a ifferential Eq VI. THE DYNAMICS OF TWO-TIME DISTRIBUTION FUNCTION We now stuy the time correlations of the ranom process ar(t),u(t),u (t), etermine by the system of stochastic Eqs All statistical characteristics of the Markovian Gaussian process a(t) can be etermine with the help of the two-time istribution function f 2 r,u,u,t;r,u,u,taataat. 6.1 If f 2 is known, it is possible to establish any n-time istribution function. Therefore f 2 completely characterizes the process uner stuy. Differentiating the expression 6.1 with respect to time an using the ynamic Eqs. 2.12, the causality conition an the Furutsu Novikov formula, we obtain the following equation for f 2 : t f f 2 f 2 2r,u,u,t;r,u,u,tu e u r e e u e r e u e m f 2 u e u e u e D u u e f In contrast to the initial system of stochastic equations, Eq. 6.2 represent the stochastic information in a much more compact form. With the help of the kinetic Eq. 6.2, one can fin a set of equations for the two-time correlations. For efiniteness, let tt. Then Eq. 6.2 yiels t r itr k tu i tr k t, t r itu k tu i tu k t, t u itu k tu itu k t, m t u itr k tmu itr k t r i tr k t u i tr k t, m t u itu k tmu itu k t r i tu k t u i tu k t. 6.3 The initial conitions for this system are expresse in terms of the one-time correlations 3.7. The final state of the system uner stuy is a stationary stochastic process which is

9 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui 5929 etermine by the two-time istribution function f 2s r,u,u,r,u,u ;s. Here stt. Note that in this case, the one-time istribution function 3.9 is completely time inepenent. As follows from Eq. 6.2, the stationary twotime istribution satisfies the equation t f f 2s f 2s 2sr,u,u,r,u,u,su e u r e e u e r e u e m f 2s u e u e u e D u u e f 2s. 6.4 We now turn our attention to etermining the stationary probabilistic characteristics of the Brownian motion uner stuy, which o not epen on the time instant of measurement. In this case we can fin from Eq. 6.3 the equilibrium correlation functions for the velocity an acceleration of a free Brownian particle moving in a Maxwell flui u i tu k 0 T m u itu k0 T m 1 exp t ik, 1 t ik. exp t 1 exp t 1 exp 6.5 The velocity correlation function ik (t)u i (t)u k (0) obeys the equation 2 t 2 ik t t ikt ik t It may be of interest to note that the Gray s statistical moel 31 of transport process in a monatomic liqui, where no assumption of the Brownian motion type was mae about the statistical features of the molecular motion, resulte in a similar secon-orer equation for the velocity correlation function of a liqui molecules. From Eq. 6.6 one can erive the relation between the velocity correlation function an the friction kernel B(t) efine by Eq. 2.4, m t ikt 0 t Btsik ss. 6.7 This result may be interprete as the equation of motion for the velocity correlation function. Note also that using a projection operator technique, Zwanzig 32 erive an equation escribing the time evolution of the autocorrelation function of a ynamical variable in terms of a well-efine memory function. The form of equation obtaine from this approach formalism is ientical to Eq The equation for the istribution function of a Brownian particle in velocity space 5.12 erive in the molecular theory of Brownian motion 15,18 leas to Eq. 6.7, too. From the practical viewpoint, it is important to know the mean square isplacement of the Brownian particle (r i ) 2, where r i r i (t)r i (0), in a given irection i over the time interval t. In this case we have r i 2 2T m t 3 3 exp t 1 exp t At long times when t, the mean-square isplacement of a free Brownian particle in a viscoelastic Maxwell flui is a linear function of time an is etermine by the Einstein s formula, r i 2 2Dt. Here D T m T is the iffusion coefficient. Therefore over long time intervals the Brownian particle forgets its past an the process becomes inertialess. In the particular case 0 an m, corresponing to a viscous liqui surrouning the Brownian particle, Eqs. 6.5 an 6.8 are reuce to the classical results 28 u i tu k 0 T m exp t m ik, r i 2 2T t mexp t m Accoring to Eq. 6.9, the exponential correlation of the velocity iffers from zero only within a time interval of orer m m/. In aition, it iffers qualitatively from generally nonexponential correlation 6.5 for the non-markovian Brownian motion in a Maxwell flui. VII. THE DIFFUSION APPROXIMATION Excluing from the analysis any processes occurring over short time intervals, we can neglect the inertia of Brownian particle. Accoring to Eqs. 2.1 an 2.6 the equation of motion of an inertialess Brownian particle in a Maxwell flui has the form t Btsṙ i ssr i tf i t The friction kernel Bt et/ characterizes the viscoelastic resistance, to which a spherical particle moving in a resting Maxwell flui is subjecte. It is relate to the complex viscosity of the flui by

10 5930 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui B6a, /1i. We analyze the ranom process efine by Eq. 7.1 by the metho of spectral analysis. Using the Fourier expansion for the ranom functions r(t) an f(t) we obtain from Eq. 7.1 an equation relating the Fourier components of the coorinates r i an the generalize ranom force r i ()(1i)f i () of the Brownian particle, r i r i. 7.2 The generalize susceptibility, etermine in the form 1/i 7.3 satisfies all requirements of a spectral characteristic for the one-sie Fourier transform. However, more general expression *1/iB in the relation r i () *[] f i () cannot be use as the generalize susceptibility, since it oes not approach zero as. The statistical properties of the generalize ranom force r i (t) can be etermine with the help of the Callen Welton fluctuation-issipation theorem. The formulation of the theorem given by Lanau an Lifshitz, 33 K r ik it 1 1 ik, 7.4 is now wiely use. Substituting the specific expression 7.3 for we fin from Eq. 7.4 the correlation function of the ranom force i r (t), i r t i r s2tts ik. Hence the resiual ranom force f i (t) in the inertialess ynamical Eq. 7.1, is exponentially correlate f i tf k st exp ts ik. 7.5 In much the same way as in the previous section it is possible to erive the corresponing equation for the istribution function Wr,t from the non-markovian inertialess Langevin Eq In oing so we obtain Wr,t t 1 r e W 2 W D r r 2. e r e 7.6 The generalize Smoluchowski Eq. 7.6 escribes the iffusion of an elastically boun Brownian particle in a viscoelastic flui with one relaxation time. Here the iffusion coefficient is expresse as follows: D r T. 7.7 This result is a generalization of the well-known Einstein relation between the iffusion coefficient an the mobility. Accoring to Eq. 29 the mobility of a Brownian particle in a relaxing flui b1/ epens on the external fiel. In the particular case 0, Eq. 7.6 reuces to the well-known Smoluchowski equation for a Markovian Brownian oscillator. If there are no external forces, i.e., 0, then the generalize an classical iffusion equations are ientical. In the case 0 only their stationary solutions coincie. The mean square isplacement of an elastically boun Brownian particle in a quiescent Maxwell flui in a irection i is etermine by the expression r i 2 2T 1exp t. 7.8 Equation 7.8 is reuce to well known equation for the Brownian motion of simple harmonic oscillator in viscous flui when 0. It emonstrates that the mean-square isplacement increases much more slowly in a viscoelastic flui than in a viscous flui. This is cause by the fact that the mobilities of a Brownian particle in these fluis are ifferent. VIII. CONCLUDING REMARKS The theory of Brownian motion evelope here is phenomenological as oppose to the molecular theory of Brownian motion The later provies a more etaile explanation of the effect of the flui properties on motion of Brownian particle. Averaging microscopic equations of motion has resulte in a generalize Fokker Planck equation with retare kernel However in general, the equation obtaine in molecular approach is only an approximation which can exactly efine only the first-orer one-time moments. 34,35 For instance, in the case of the exponential memory kernel, Eq yiels only a secon-orer ifferential equation for the one-time velocity correlations of Brownian particle. This is in contraiction to our exact result efine by the thir-orer ifferential Eq The present work was originally motivate by an attempt to fin an exact equation for the velocity istribution function corresponing to a non-markovian Langevin Eq As a simple check for the obtaine Eq. 5.6, we can mention that the secon-orer evolution equation for the average velocity an the thir-orer equation for the one-time correlations of velocity of Brownian particle can be obtaine using Eq. 5.6 exactly in the forms of Eqs An important conclusion can be mae about the comparison of the statistical properties of the ranom forces of the inertialess Eq. 7.1 an exact Eq. 2.1 stochastic equations of motion. In a transition from Eq. 2.1 to Eq. 7.1, along with the limit m 0, it is also necessary to change the statistical characteristics of the ranom force. Accoring to Eq. 7.5, they epen on parameter in coorinate space, which characterizes the external forces. Thus the statistical properties of the ranom forces in coorinate space in the absence of external forces are ifferent from those in the presence of external forces. If this is neglecte, one can obtain incorrect results for the equilibrium correlations of the coorinates of the Brownian particle, corresponing to the irect limit m 0 in the expression for the equilibrium istribution function Eq However, the more systematic transition from the stationary istribution function f s r,u,u to the istribution function W s r shoul be mae by integrating the istribution Eq. 3.9 over the velocities an accelerations. The interesting feature of the Brownian motion

11 V. S. Volkov an A. I. Leonov: Brownian motion in a viscoelastic flui 5931 moel stuie here is that, in contrast with classical theory of Brownian motion, these two transitions to the istribution function W s r are not equivalent. Finally, it shoul be mentione that the stochastic equation in velocity space Eq. 4.1 for free Brownian motion in the Maxwell flui is mathematically equivalent to the non- Markovian ifferential equation in position space, escribing the Brownian motion of a harmonic oscillator in viscous flui. 28 In Sec. V we erive the Eq. 5.7 for the istribution function of the solution of secon-orer stochastic ifferential Eq. 4.1 with the arbitrary initial conitions. It is possible to associate Eq. 5.7 with the valiity of the solution for an ol problem of the erivation of exact equation for the istribution function in position space for the Brownian motion in viscous flui an external perioic potential 36,37 for any time instant. An approximate answer to this problem is given by the Smoluchowski iffusion equation 2 vali only for long times an high frictions. This aspect will be treate in more etail elsewhere. ACKNOWLEDGMENTS This work was supporte in part by the U.S. National Acaemy of Science uner the Collaboration in Basic Science an Engineering program an the International Science Founation Grant No. MRE A. Einstein, Ann. Phys. 17, M. von Smoluchowski, Ann. Phys. 21, P. Langevin, C. R. Aca. Sci. Paris 146, G. G. Stokes, Camb. Trans. IX 1851; Mathematical an Physical Papers, Vol. III. 5 V. Vlaimirsky an Ya. Terletsky, Eksp. Theor. Fiz. USSR 15, R. Zwanzig an M. Bixon, Phys. Rev. A 2, A. Wiom, Phys. Rev. A 3, K. M. Case, Phys. Fluis 14, T. S. Chow an J. J. Hermans, J. Chem. Phys. 56, T. S. Chow an J. J. Hermans, J. Chem. Phys. 59, V. S. Volkov an V. N. Pokrovsky, J. Math. Phys. 24, V. S. Volkov an G. V. Vinograov, J. Non-Newt. Flui Mech. 15, V. S. Volkov, Sov. Phys. JETP 71, R. F. Roriquez an E. S. Roriguez, J. Phys. A 21, J. Lebowitz an E. Rubin, Phys. Rev. 131, P. Resibois an H. Davis, Physica 30, J. Lebowitz an P. Resibois, Phys. Rev. 139A, R. M. Mazo, J. Stat. Phys. 1, R. F. Fox, Phys. Rep. 48C, P. Hanggi an H. Thomas, Phys. Rep. 88, M. W. Evans, P. Grigolini, an G. Pastori Parravicini, Memory Function Approaches to Stochastic Problems in Conense Matter Wiley, New York, H. B. Callen an T. A. Welton, Phys. Rev. 83, H. Mori, Prog. Theor. Phys. 33, R. Kubo, Rep. Prog. Phys. 29, V. I. Klyatskin an V. I. Tatarskii, Sov. Phys. Usp. 16, K. Furutsu, J. Res. NBS 667D, E. A. Novikov, Sov. Phys. JETP 20, G. E. Uhlenbeck an L. S. Ornstein, Phys. Rev. 36, ; S. Chanrasekhar, Rev. Mo. Phys. 15, 11943; M. C. Wang an G. E. Uhlenbeck, ibi. 17, E. W. Montroll an G. H. Weiss, J. Math. Phys. 6, V. M. Kenker, E. W. Montroll, an M. F. Shlesinger, J. Stat. Phys. 9, P. Gray, Mol. Phys. 7, R. Zwanzig, Lectures in Theoretical Physics Interscience, New York, 1961, Vol. 3, p L. D. Lanau an E. M. Lifshitz, Electroynamics of Continuous Meia Pergamon, New York, S. A. Aelman, J. Chem. Phys. 64, S. Grossman, Z. Phys. B 47, G. Wilemski, J. Stat. Phys. 14, M. San Miguel an J. M. Sancho, J. Stat. Phys. 22,

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