IImpulse Response Function for Brownian Motion
Nicos Makris ∗ Dept. of Civil and Environmental Engineering, Southern Methodist University, Dallas, Texas, 75276 (Dated: February 4, 2021)Motivated from the central role of the mean-square displacement and its second time-derivative— that is the velocity autocorrelation function (cid:104) v (0) v ( t ) (cid:105) =
12 d (cid:104) ∆ r ( t ) (cid:105) d t in the description of Brow-nian motion, this paper examines what is the physical meaning of the first time-derivative of themean-square displacement of Brownian particles. By employing a rheological analogue for Brownianmotion, we show that the time-derivative of the mean-square displacement d (cid:104) ∆ r ( t ) (cid:105) d t of Brownianmicrospheres with mass m and radius R immersed in any linear, isotropic viscoelastic material isidentical to NK B T πR h ( t ), where h ( t ) is the impulse response function (strain history γ ( t ), due to animpulse stress τ ( t ) = δ ( t − m R = m πR . With the introduction ofthe impulse response fucntion h ( t ), we show that there is a direct analogy between the description ofthe deterministic motion of a free particle with its displacement r ( t ), velocity v ( t ) and acceleration a ( t ); and the description of the random, thermally driven, Brownian motion of a collection of par-ticles with the associated deterministic creep compliance J ( t ) = πRNK B T (cid:10) ∆ r ( t ) (cid:11) , impulse responsefunction h ( t ), and impulse strain–rate response function ψ ( t ) = πRNK B T (cid:104) v (0) v ( t ) (cid:105) , of the viscoelasticmaterial–inerter parallel connection. Keywords: Random process; rheological network; viscoelasticity; inerter; impulse fluidity; time–responsefunctions; determinism
I. INTRODUCTION
Brownian motion is the random and perpetual move-ment of microparticles immersed in a fluid or in any vis-coelastic material which originates from the collisions ofthe molecules of the surrounding material on the Brow-nian particles and is driven by the thermal energy of thematerial. The dynamics of Brownian motion have beentraditionally expressed with the mean-square displace-ment, (cid:10) ∆ r ( t ) (cid:11) = 1 M M (cid:88) j =1 ( r j ( t ) − r j (0)) (1)where M is the number of suspended microparticles;while r j ( t ) and r j (0) are the positions of particle j attime t and the time origin, t = 0; in association with thevelocity autocorrelation function of the Brownian parti-cles (cid:104) v (0) v ( t ) (cid:105) = (cid:104) v ( ξ ) v ( ξ + t ) (cid:105) (2)= lim T →∞ T (cid:90) T v ( ξ ) v ( ξ + t ) d ξ where v ( t ) is the velocity of the Brownian particle [1–5].The Laplace transform of the mean-square displace-ment, L (cid:8)(cid:10) ∆ r ( t ) (cid:11)(cid:9) = (cid:10) ∆ r ( s ) (cid:11) = (cid:90) t (cid:10) ∆ r ( t ) (cid:11) e − st d t is related with the Laplace transform of the velocity ∗ [email protected]; Also at Office of Theoretical and AppliedMechanics, Academy of Athens, 10679, Greece autocorrelation function L {(cid:104) v (0) v ( t ) (cid:105)} = (cid:104) v (0) v ( s ) (cid:105) = (cid:90) t (cid:104) v (0) v ( t ) (cid:105) e − st d t via the identity [4–6] (cid:104) v (0) v ( s ) (cid:105) = s (cid:10) ∆ r ( s ) (cid:11) (3)while, according to the properties of the Laplace trans-form of the derivatives of a function s (cid:10) ∆ r ( s ) (cid:11) = L (cid:40) d (cid:10) ∆ r ( t ) (cid:11) d t (cid:41) (4)+ s (cid:10) ∆ r (0) (cid:11) + d (cid:10) ∆ r (0) (cid:11) d t From the definition of the mean-square displacementgiven by Eq. (1), at the time origin t = 0, (cid:10) ∆ r (0) (cid:11) = 0.Furthermore, the time-derivative of Eq. (1) givesd (cid:10) ∆ r ( t ) (cid:11) d t = 2 M M (cid:88) j =1 ( r j ( t ) − r j (0)) d r j ( t )d t (5)therefore, at t = 0, d (cid:104) ∆ r ( t ) (cid:105) d t = 0. Accordingly, substitu-tion of Eq. (4) into Eq. (3) gives L {(cid:104) v (0) v ( t ) (cid:105)} = 12 L (cid:40) d (cid:10) ∆ r ( t ) (cid:11) d t (cid:41) (6)and inverse Laplace transform of Eq. (6) yields (cid:104) v (0) v ( t ) (cid:105) = 12 d (cid:10) ∆ r ( t ) (cid:11) d t (7) a r X i v : . [ c ond - m a t . s o f t ] F e b which shows that the velocity autocorrelation function ishalf the second time-derivative of the mean-square dis-placement.The phenomenon of Brownian motion was first ex-plained in the 1905 Einstein’s celebrated paper [7] whichexamined the long-term response of Brownian micro-spheres with mass m and radius R suspended in a mem-oryless, Newtonian fluid with viscosity η . Einstein’s the-ory of Brownian motion predicts the long-term expres-sion for the mean-square displacement of the randomlymoving microspheres (diffusive regime) (cid:10) ∆ r ( t ) (cid:11) = 2 N Dt = N K B T πR η t (8)where N ∈ { } is the number of spacial dimensions, K B is Boltzman’s constant, T is the equilibrium tempera-ture of the Newtonian fluid with viscosity η within whichthe Brownian microspheres are immersed and D = K B T πRη is the diffusion coefficient. The time derivative of Eq. (8), d (cid:104) ∆ r ( t ) (cid:105) d t = 2 N D is a constant which is in contradictionwith the result of Eq. (5) at the time origin ( t = 0).At short time-scales [8–10], when t < m πRη = τ , theBrownian motion of suspended particles is influenced bythe inertia of the particle and the surrounding fluid (bal-listic regime); and Einstein’s “long-term” result offeredby Eq. (8) was extended for all time-scales by [2] (cid:10) ∆ r ( t ) (cid:11) = N K B T πR η (cid:104) t − τ (cid:16) − e − t / τ (cid:17)(cid:105) (9)where τ = m πRη is the dissipation time-scale of the per-petual fluctuation–dissipation process. The time deriva-tive of Eq. (9) isd (cid:10) ∆ r ( t ) (cid:11) d t = N K B T πR η (cid:16) − e − t / τ (cid:17) (10)which indicates that at t = 0, d (cid:104) ∆ r ( t ) (cid:105) d t = 0 which is inagreement with the result of Eq. (5). Equation (7) inassociation with the result of Eq. (9) yields the veloc-ity autocorrelation function of Brownian particles withmass m when suspended in a memoryless, Newtonianfluid with viscosity η (cid:104) v (0) v ( t ) (cid:105) = 12 d (cid:10) ∆ r ( t ) (cid:11) d t = N K B Tm e − t / τ (11)which is the classical result derived by [2] after evaluat-ing ensemble averages of the random Brownian process.Equation (11); while valid for all time-scales it does notaccount for the hydrodynamic memory that manifests asthe energized Brownian particle displaces the fluid in itsimmediate vicinity [11–15].The reader recognizes that the exponential term of thevelocity autocorrelation function given by Eq. (11) iswhatever is left after taken the second time derivativeof the mean-square displacement given by Eq. (9) that is valid for all time-scales. Consequently, by accountingfor the “ballistic regime” at short time-scales, Uhlenbeckand Ornstein’s (1930) expression for the mean-square dis-placement given by Eq. (9) is consistent with the identitygiven by Eq. (7) and yields the correct expression for thevelocity autocorrelation function of Brownian particlessuspended in a memoryless Newtonian fluid given by Eq.(11). In contrast, Einstein’s (1905) “long-term” expres-sion for the mean-square displacement given by Eq. (8)(diffusive regime) yields an invariably zero-velocity auto-correlation function.Studies on the behavior of hard-sphere systems haveidentified a / t decay of d (cid:104) ∆ r ( t ) (cid:105) d t with time [16]; whilewith reference to Eq. (8), the time-derivative of themean-square displacement, d (cid:104) ∆ r ( t ) (cid:105) d t , has been inter-preted as a time-dependent diffusion coefficient [17].Given that the mean-square displacement displacementdefined by Eq. (1) and its second time derivative — thatis the velocity autocorrelation function defined by Eq.(2), play such a central role in the description of Brow-nian motion in association with the role of d (cid:104) ∆ r ( t ) (cid:105) d t tointerpret the Brownian motion at various time scales andconfined spacings [18]; the natural question that arises iswhat is the physical meaning of the first time derivativeof the mean-square displacement, d (cid:104) ∆ r ( t ) (cid:105) d t . The aim ofthis paper is to address this question. II. A RHEOLOGICAL ANALOGUE FORBROWNIAN MOTION
In a recent publication, Makris [19] presented aviscous–viscoelastic correspondence principle for Brown-ian motion which reveals that the mean-square displace-ment, (cid:10) ∆ r ( t ) (cid:11) of Brownian microspheres with mass m and radius R when suspended in any linear isotropic,viscoelastic material and subjected to the random forcesfrom the collisions of the molecules of the viscoelasticmaterial is identical to NK B T πR γ ( t ), where γ ( t ) = J ( t ) isthe strain due to a unit-step stress on a rheological net-work that is a parallel connection of the linear viscoelasticmaterial (within which the microspheres are immersed)with an inerter with distributed inertance m R = m πR .Accordingly, (cid:10) ∆ r ( t ) (cid:11) = N K B T πR J ( t ) (12)where J ( t ) is the creep compliance of the rheological net-work shown in Fig. 1 (right). Laplace transform of Eq.(12) gives [19] (cid:10) ∆ r ( s ) (cid:11) = N K B T πR C ( s ) = N K B T πR J ( s ) s (13)where C ( s ) is the complex creep function [20–22] and J ( s ) = G ( s ) is the complex dynamic compliance of the Brownian motion of particles (microspheres) with mass m and radius R in a linear, isotropic viscoelastic material subjected to random forces. M= number ofprobe microspheres.Linear, Isotropic Viscoelastic Material ~= ReferencePosition r j ( t ) Δ r ( t ) = γ ( t ) NK B T π R M | r j ( t ) -r j (0) | = j= Σ M Linear, Isotropic ViscoelasticMaterial γ ( t ) =J ( t ) τ ( t ) t m R = m π R FIG. 1. Statement of the viscous–viscoelastic correspondence principle for Brownian motion [19]. The mean squared displace-ment, (cid:10) ∆ r ( t ) (cid:11) , of Brownian particles (microspheres) with mass m and radius R suspended in any linear, isotropic viscoelasticmaterial when subjected to the random forces from the collisions of the molecules of the viscoelastic material, is identical to NK B T πR γ ( t ), where γ ( t ) = J ( t ) is the strain due to a unit step–stress on a rheological network that is a parallel connection ofthe linear viscoelastic material and an inerter with distributed inertance m R = m πR . rheological network shown in Fig. 1 (right). The com-plex dynamic compliance, J ( s ) of a rheological networkis the inverse of the complex dynamic modulus, G ( s ); andis a transfer function that relates a strain output, γ ( s )to a stress input τ ( s ) [23–25]. In structural mechanics,the equivalent of the complex dynamic compliance at thedisplacement–force level is known as the dynamic flexibil-ity, often expressed with H ( ω ) = K ( ω ) [26], where K ( ω )is the dynamic stiffness of the structure. The Laplacetransform of the time-derivative of the mean-square dis-placement is: L (cid:40) d (cid:10) ∆ r ( t ) (cid:11) d t (cid:41) = s (cid:10) ∆ r ( s ) (cid:11) − (cid:10) ∆ r (0) (cid:11) (14)From Eq. (1), at the time origin, t = 0, (cid:10) ∆ r (0) (cid:11) = 0,and substitution of Eq. (13) into Eq. (14) gives: L (cid:40) d (cid:10) ∆ r ( t ) (cid:11) d t (cid:41) = N K B T πR J ( s ) (15)The inverse Laplace transform of the complex dynamiccompliance, J ( s ), appearing in the right-hand side ofEq. (15) is the impulse fluidity, φ ( t ) = L − {J ( s ) } [27–29], defined as the resulting strain γ ( t ), due to an impul-sive stress input, τ ( t ) = δ ( t − δ ( t − L { δ ( t − ξ ) } = (cid:90) t − δ ( t − ξ ) e − st d t = e − ξs . The equivalentof the impulse fluidity, φ ( t ), at the displacement–forcelevel is the impulse response function often expressed as h ( t ). Given that the term “impulse response function”(rather than the term “impulse fluidity”) is widely knownand used in dynamics, structural mechanics, electricalsignal processing and economics; in this paper we adoptthe term “ impulse response function = h ( t )”, rather thanthe term “impulse fluidity” used narrowly in the vis-coelasticity literature alone. Accordingly, inverse Laplacetransform of Eq. (15) gives:d (cid:10) ∆ r ( t ) (cid:11) d t = N K B T πR L − {J ( s ) } = N K B T πR h ( t ) (16)Equation (16) indicates that the time-derivative of themean-square displacement, d (cid:104) ∆ r ( t ) (cid:105) d t , of Brownian parti-cles suspended in any linear, isotropic viscoelastic ma-terial is proportional to the impulse response function, h ( t ), of the rheological network shown in Fig. 1 (right),defined as the resulting strain history, γ ( t ), of the vis-coelastic material–inerter parallel connection due to animpulsive stress input, τ ( t ) = δ ( t − m andradius R immersed in a memoryless Newtonian fluid withviscosity η is a dashpot–inerter parallel connection withconstitutive law [19] τ ( t ) = η d γ ( t )d t + m R d γ ( t )d t (17)where m R = m πR is the distributed inertance of the in-erter with units [M] [L] − (i.e. Pa s ). The Laplace trans-form of Eq. (17) gives: τ ( s ) = G ( s ) γ ( s ) = ( ηs + m R s ) γ ( s ) (18)where G ( s ) = J ( s ) = ηs + m R s is the complex dynamicmodulus of the dashpot–inerter parallel connection (iner-toviscous fluid, [31]). The complex dynamic compliance, J ( s ), of the inertoviscous fluid expressed by Eq. (17) is: J ( s ) = 1 G ( s ) = 1 ηs + m R s = 1 η (cid:18) s − s + τ (cid:19) (19)where τ = m R η = m πRη is the dissipation time , whichis the time scale needed for the kinetic energy stored inthe inerter with distributed inertance, m R , to be dissi-pated by the dashpot with viscosity, η . Inverse Laplacetransform of Eq. (19) gives: L − {J ( s ) } = h ( t ) = 1 η (1 − e − t / τ ) (20)By substitution of the result of Eq. (20) into Eq. (16) werecover Eq. (10) that was reached by merely taking thetime-derivative of Eq. (9), initially derived by [2] aftercomputing ensemble averages of the random Brownianprecess.The analysis presented in this section, in associationwith the viscous–viscoelastic correspondence principle forBrownian motion [19] concludes that the time-derivativeof the mean-square displacement, d (cid:104) ∆ r ( t ) (cid:105) d t , of Brownianmicrospheres with mass m and radius R suspended inany linear viscoelastic material is identical to NK B T πR h ( t ),where h ( t ) is the impulse response function of a rheolog-ical network that is a parallel connection of the linearviscoelastic material and an inerter with distributed in-ertance m R = m πR . III. IMPULSE RESPONSE FUNCTION FORBROWNIAN MOTION IN A HARMONIC TRAP(KELVIN–VOIGT SOLID)
The Brownian motion of microparticles trapped in aharmonic potential when excited by random forces f R ( t )has been studied by [2] and [3]. The mean-square dis-placement of a Brownian particle in a harmonic trap hasbeen evaluated by [3] after computing the velocity au-tocorrelation function of the random process. For theunderdamped case ( ω τ > ): (cid:10) ∆ r ( t ) (cid:11) = 2 N K B Tmω (21) × (cid:20) − e − t / τ (cid:18) cos( ω D t ) + 12 ω D τ sin( ω D t ) (cid:19)(cid:21) where ω = (cid:113) km is the undamped natural frequency ofthe trapped particle with mass m and radius R , τ = m πRη flywheel pinion rack θ ( t ) τ ( t )= δ ( t- γ ( t )= h ( t ) η Gm R = m6 π R FIG. 2. Inertoviscoelastic solid which is a parallel connectionof an inerter with a distributed inertance m R , a dashpot withviscosity η and a linear spring with elastic shear modulus G . In analogy with the traditional schematic of a dashpotthat is a hydraulic piston, the distributed inerter is depictedschematically with a rack–pinion–flywheel system. is the dissipation time and ω D = ω (cid:114) − (cid:16) ω τ (cid:17) is thedamped angular frequency of the trapped particle [8, 10].The rheological analogue of the Brownian motion ofparticles trapped in a Kelvin–Voigt solid is the inertovis-coelastic solid shown in Fig. 2 which is a parallel connec-tion of a spring with elastic shear modulus G and a dash-pot with shear viscosity η (Kelvin–Voigt soild); togetherwith an inerter with distributed inertance m R = m πR [19]. Given the parallel connection of the three elemen-tary mechanical elements shown in Fig. 2, the constitu-tive law of the combined inertoviscoelastic solid is: τ ( t ) = Gγ ( t ) + η d γ ( t )d t + m R d γ ( t )d t (22)The Laplace transform of Eq. (22) is: τ ( s ) = G ( s ) γ ( s ) = ( G + ηs + m R s ) γ ( s ) (23)where G ( s ) = G + ηs + m R s is the complex dynamicmodulus; while the complex dynamic compliance of theinertoviscoelastic solid is: J ( s ) = 1 G ( s ) = 1 G + ηs + m R s (24)= 1 m R (cid:0) s + τ (cid:1) + ω R − (cid:0) τ (cid:1) where again τ = m R η = m πRη is the dissipation timeand ω R = (cid:113) Gm R is the undamped rotational angular fre-quency of the inertoviscoelastic solid shown in Fig. 2. Forthe inertoviscoelastic solid described by Eq. (22) and aBrownian particle in a harmonic trap to have the sameundamped natural frequency ω R = (cid:113) Gm R = (cid:113) km = ω , critically damped: BallisticRegime ( (
Viscous fluid with Eq. (29):
DiffusiveRegime
FIG. 3. Normalized time-derivative of the mean-square dis-placement of Brownian particles trapped in a harmonic po-tential for the underdamped, ω τ > , critically damped, ω τ = , and overdamped cases which is equal to √ Gm R h ( t )of the inertoviscoelastic solid shown in Fig. 2. For the over-damped cases (weak spring) at early times, the time–responsefunctions of the Brownian particles in a harmonic trap coin-cide with the corresponding time–response functions of Brow-nian particles in a viscous fluid with viscosity η . the shear modulus needs to assume the value G = k πR ,where k is the spring constant of the harmonic trap[3, 10]. Accordingly, by setting ω R = ω , the lasttwo terms in the denominator of Eq. (24) combine to ω R (cid:20) − (cid:16) ω R τ (cid:17) (cid:21) = ω (cid:20) − (cid:16) ω τ (cid:17) (cid:21) = ω D .The inverse Laplace transform of Eq. (24) yields theimpulse response function of the inertoviscoelastic soliddescribed by Eq. (22) [32] L − {J ( s ) } = h ( t ) = 1 m R ω D e − t / τ sin( ω D t ) (25)Substitution of the result of Eq. (25) into Eq. (16), yieldsthat the time derivative of the mean-square displacementof Brownian particles trapped in a Kelvin–Voigt solid is:d (cid:10) ∆ r ( t ) (cid:11) d t = N K B T πR h ( t ) (26)= 2 N K B Tmω (cid:114) − (cid:16) ω τ (cid:17) e − t / τ sin( ω D t )The result of Eq. (26) that was computed herein aftercalculating the impulse response function of the inertovis-coelastic solid in association with Eq. (16) is identical tothe first time derivative of Eq. (21) derived by [3] aftercomputing ensemble averages of the random Brownianprocess.In a dimensionless form Eq. (26) or (25) which is forthe underdamped case (cid:0) ω τ > (cid:1) is expressed as: mω N K B T d (cid:10) ∆ r ( t ) (cid:11) d t = (cid:112) Gm R h ( t ) = 1 (cid:114) − (cid:16) ω τ (cid:17) e − t / τ sin ω τ (cid:115) − (cid:18) ω τ (cid:19) tτ (27)For the overdamped case (cid:0) ω τ < (cid:1) , the normalized im- pulse response function for Brownian motion of particlestrapped in a Kelvin–Voigt solid is: mω N K B T d (cid:10) ∆ r ( t ) (cid:11) d t = (cid:112) Gm R h ( t ) = 1 (cid:114)(cid:16) ω τ (cid:17) − e − t / τ sinh ω τ (cid:115)(cid:18) ω τ (cid:19) − tτ (28)For small values of the dimensionless product ω τ (weakspring), Eq. (28) at early times contracts to the solutionfor Brownian motion of particles in a Newtonian viscousfluid since the inertia and viscous terms dominate over the elastic term mω N K B T d (cid:10) ∆ r ( t ) (cid:11) d t = ω τ (1 − e − t / τ ) (29)Equation (29) is obtained after multiplying both sides ofEq. (10) with ω and replacing η with πRτm . Figure 3 T A B L E I . M e a n - s q u a r e d i s p l a c e m e n tt og e t h e r w i t h i t s fi r s t a nd s e c o nd t i m e d e r i v a t i v e ( a u t o c o rr e l a t i o n f un c t i o n ) o f B r o w n i a n m i c r o s ph e r e s w i t h m a ss m a nd r a d i u s R s u s p e nd e d i n a v i s c o u s N e w t o n i a nflu i d , a K e l v i n – V o i g t s o li d , a M a x w e ll flu i d a nd a s ubd i ff u s i v e S c o tt – B l a i r flu i d . T h e m e c h a n i c a l a n a l og u e s f o r B r o w n i a n m o t i o n i n t h e s e m a t e r i a l s a r e s h o w n i n t h e fi r s t c o l u m n , w h e r e a s t h e d e t e r m i n i s t i c e x p r e ss i o n s o f t h e i r c r ee p c o m p li a n c e s , J ( t ) ,i m pu l s e r e s p o n s e f un c t i o n s , h ( t ) a nd s t r a i n – r a t e r e s p o n s e f un c t i o n s , ψ ( t ) a r e s h o w n i n t h e s ub s e q u e n t c o l u m n s . B r o w n i a n M o t i o n o f m i c r o s ph e r e s w i t h m a ss m a nd r a d i u s R s u s p e nd e d i n a : (cid:10) ∆ r ( t ) (cid:11) = N K B T π R J ( t ) J ( t ) = L − { C ( s ) } = C r ee p C o m p li a n ce C ( s ) = L { J ( t ) } = C o m p l e x C r ee p F un c t i o n d (cid:104) ∆ r ( t ) (cid:105) d t = N K B T π R h ( t ) h ( t ) = L − { J ( s ) } = I m pu l s e R e s p o n s e F un c t i o n J ( s ) = L { h ( t ) } = C o m p l e x D y n a m i c C o m p li a n ce ( D y n a m i c F l e x i b ili t y ) d (cid:104) ∆ r ( t ) (cid:105) d t = (cid:104) v ( ) v ( t ) (cid:105) = li m T →∞ T (cid:90) T v ( ξ ) v ( ξ + t ) d ξ = N K B T π R ψ ( t ) ψ ( t ) = L { φ ( s ) } = I m pu l s e S t r a i n – R a t e R e s p o n s e F un c t i o n φ ( s ) = L − { ψ ( t ) } = C o m p l e x D y n a m i c F l u i d i t y ( A d m i tt a n ce ) N e w t o n i a n V i s c o u s F l u i d w i t h v i s c o s i t y η J ( t ) = η (cid:2) t − τ (cid:0) − e − t / τ (cid:1) (cid:3) h ( t ) = η (cid:0) − e − t / τ (cid:1) ψ ( t ) = m R e − t / τ , τ = m R η = m π R η K e l v i n – V o i g t S o li d w i t h e l a s t i c i t y G a nd v i s c o s i t y η J ( t ) = m R ω (cid:34) − e − t / τ (cid:32) c o s ( ω D t ) + τ ω D s i n ( ω D t ) (cid:33) (cid:35) , ω D τ > h ( t ) = m R ω D e − t / τ s i n ( ω D t ) ω D = ω (cid:114) − (cid:16) ω τ (cid:17) , ω D τ > ψ ( t ) = m R (cid:34) c o s ( ω D t ) − τ ω D s i n ( ω D t ) (cid:35) e − t / τ , ω D τ > M a x w e ll F l u i d w i t h e l a s t i c i t y G a nd v i s c o s i t y η J ( t ) = m R η (cid:40) t τ − β (cid:34) β − − e − t τ β × (cid:32) β − β √ − β s i n (cid:16) t τ β (cid:112) − β (cid:17) + (cid:0) β − (cid:1) × c o s (cid:16) t τ β (cid:112) − β (cid:17) (cid:33) (cid:35) (cid:41) , β = τ ω R < h ( t ) = η (cid:34) U ( t − ) − e − t τ β × (cid:32) β − β √ − β s i n (cid:16) β (cid:112) − β t τ (cid:17) + c o s (cid:16) β (cid:112) − β t τ (cid:17) (cid:33) (cid:35) , β = τ ω R < ψ ( t ) = m R e − t τ β (cid:34) c o s (cid:16) t τ β (cid:112) − β (cid:17) + β √ − β s i n (cid:16) t τ β (cid:112) β − (cid:17) (cid:35) , β = τ ω R < S c o tt - B l a i r s ubd i ff u s i v e flu i d w i t h m a t e r i a l c o n s t a n t µ α J ( t ) = m R t E − α , (cid:16) − µ α m R t − α (cid:17) h ( t ) = m R t E − α , (cid:16) − µ α m R t − α (cid:17) ψ ( t ) = m R E − α , (cid:16) − µ α m R t − α (cid:17) plots the normalized impulse response function given byEqs. (27) and (28) as a function of the dimensionless time tτ for various values of ω τ = √ km πR = √ Gm R η togetherwith the results from Eq. (29) (Newtonian viscous fluid)for values of ω τ = 0.2 and 0.3.The results reached in Sections II and III are summa-rized in Table I which lists the mean-square displacementtogether with its first and second time-derivatives (veloc-ity autocorrelation function) of Brownian microparticlessuspended in a Newtonian fluid, a Kelvin–Voigt solid, aMaxwell fluid and a subdiffusive Scott–Blair fluid. TableI also shows the rheological analogues for the Brownianmotion of microparticles suspended in the above men-tioned viscoelastic materials together with the expres-sions of the corresponding deterministic creep compli-ance J ( t ) = πRNK B T (cid:10) ∆ r ( t ) (cid:11) , impulse response function h ( t ) = πRNK B T d (cid:104) ∆ r ( t ) (cid:105) d t and impulse strain–rate responsefunction ψ ( t ) = πRNK B T d (cid:104) ∆ r ( t ) (cid:105) d t = πRNK B T (cid:104) v (0) v ( t ) (cid:105) ofthe viscoelastic material–inerter parallel connection. IV. IMPULSE RESPONSE FUNCTION FORBROWNIAN MOTION IN A MAXWELL FLUID
According to the viscous–viscoelastic correspondenceprinciple for Brownian motion illustrated in Fig. 1, therheological analogue for Brownian motion of particlessuspended in a Maxwell fluid with a single relaxationtime λ = ηG is the mechanical network shown in Fig. 4which is a parallel connection of a Maxwell element withshear modulus G , and shear viscosity η , with an inerterwith distributed inertance m R = m πR . The mechanicalnetwork shown in Fig. 4 is described by a third-orderconstitutive equation [19] τ ( t ) + ηG d τ ( t )d t = η d γ ( t )d t + m R d γ ( t )d t + η m R G d γ ( t )d t (30)By defining the dissipation time τ = m R η = m πRη and therotational angular frequency ω R = (cid:113) Gm R = (cid:113) πRGm , Eq.(30) assumes the form τ ( t ) + 1 τ ω R d τ ( t )d t (31)= m R (cid:18) τ d γ ( t )d t + d γ ( t )d t + 1 τ ω R d γ ( t )d t (cid:19) The Laplace transform of Eq. (31) gives γ ( s ) = J ( s ) τ ( s )where J ( s ) is the complex dynamic compliance of the mechanical network shown in Fig. 4 J ( s ) = 1 G ( s ) = γ ( s ) τ ( s ) = 1 m R s τω R s (cid:16) τ + s + τω R s (cid:17) (32)In addition to s = 0, the other two poles of the complexdynamic compliance J ( s ) given by Eq. (32) are s = − τ ω R ω R (cid:114)(cid:16) τ ω R (cid:17) − − ω R (cid:16) β − (cid:112) β − (cid:17) (33)and s = − τ ω R − ω R (cid:114)(cid:16) τ ω R (cid:17) − − ω R (cid:16) β + (cid:112) β − (cid:17) (34)where β = τω R = η (cid:113) mG πR is a dimensionless parameterof the mechanical network shown in Fig. 4 and of theBrownian particle–Maxwell fluid system. By virtue ofEqs. (33) and (34), the complex dynamic compliance J ( s ) = (cid:90) ∞ h ( t ) e − st d t given by Eq. (32) is expressed as J ( s ) = 1 η (cid:34) s − (cid:32) β − β (cid:112) β − (cid:33) s − s (35)+ (cid:32) β − β (cid:112) β − − (cid:33) s − s (cid:35) For the case where β = τω R > ω R t = 2 β tτ , the inverse Laplace transform ofEq. (35) gives flywheel pinion rack θ ( t ) η G m R = m π R τ ( t ) = δ ( t -0) γ ( t ) =h ( t ) FIG. 4. Mechanical analogue for Brownian motion in aMaxwell fluid. It consists of the Maxwell fluid with shearmodulus G and viscosity η that is connected in parallel withm an inerter with distributed inertance m R = m πR . h ( t ) = L − {J ( s ) } = 1 η (cid:34) U ( t − − e − tτ β (cid:32) β − β (cid:112) β − (cid:18) β (cid:112) β − tτ (cid:19) + cosh (cid:18) β (cid:112) β − tτ (cid:19) (cid:33)(cid:35) , β > U ( t −
0) is the Heaviside unit-step function [30].For the case where β = τω R < h ( t ) = L − {J ( s ) } = 1 η (cid:34) U ( t − − e − tτ β (cid:32) β − β (cid:112) − β sin (cid:18) β (cid:112) − β tτ (cid:19) + cos (cid:18) β (cid:112) − β tτ (cid:19) (cid:33)(cid:35) , β < πRηN K B T d (cid:10) ∆ r ( t ) (cid:11) d t = ηh ( t ) (38)where the impulse response function, h ( t ), is offered byEq. (36) or (37) depending on the value of β = τω R . Forlarge values of β (stiff spring) the solution contracts tothe solution for Brownian motion of particles suspendedin a Newtonian viscous fluid since the dashpot essentiallyreacts to a non-compliant element. Viscous fluid:
FIG. 5. Normalized time-derivative of the mean-square dis-placement of Brownian particles with mass m and radius R suspended in a Maxwell fluid with shear modulus G and vis-cosity η for various values of the parameter β = η (cid:113) mG πR = η √ m R √ G . For large values of β (stiff spring), the solutioncontracts to the solution for Brownian motion of particles im-mersed in a Newtonian viscous fluid: 1 − e − t / τ . V. IMPULSE RESPONSE FUNCTION FORBROWNIAN MOTION WITHIN ASUBDIFFUSIVE MATERIAL
Several complex materials exhibit a subdiffusive be-havior where from early times and over several tempo-ral decades the mean-square displacement of suspended particles grows with time according to a power law; (cid:10) ∆ r ( t ) (cid:11) ∼ t α , where 0 (cid:54) α (cid:54) γ ( t ) = U ( t − τ ( t ) = G ve ( t ) ∼ t − α , where 0 (cid:54) α (cid:54) G ve ( t ) is the relaxation modulus of the viscoelasticmaterial. Following Nutting’s [35] observations and theearly work of [36, 37] on fractional differentials, Scott-Blair [38, 39] proposed the springpot element, which is amechanical idealization in-between an elastic spring anda viscous dashpot with constitutive law τ ( t ) = µ α d α γ ( t )d t α , 0 (cid:54) α (cid:54) α is a positive real number, 0 (cid:54) α (cid:54) µ α is a phenomenological material parameter with units[M] [L] − [T] α − (i.e. Pa s ), and d α γ ( t )d t α is the fractionalderivative of order α of the strain history, γ ( t ).A definition of the fractional derivative of order α isgiven through the Riemann–Liouville convolution inte-gral [40–43] I α γ ( t ) = 1Γ( α ) (cid:90) t − t − ξ ) − α γ ( ξ ) d ξ , α ∈ R + (40)where R + is the set of positive real numbers and Γ( α ) isthe Gamma function. The integral in Eq. (40) convergesonly for α >
1, or in the case where α is a complex num-ber, the integral converges for R ( α ) >
0. Nevertheless,by a proper analytic continuation across the line R ( α ) =0 and provided that the function γ ( t ) is n -times differ-entiable, it can be shown that the integral given by Eq.(40) exists for n − R ( α ) > α ∈ R + exists and is defined as[40–45]d α γ ( t )d t α = I − α γ ( t ) (41)= 1Γ( − α ) (cid:90) t − t − ξ ) α γ ( ξ ) d ξ , α ∈ R + where the lower limit of integration, 0 − may engage anentire singular function at the time origin such as γ ( t ) = δ ( t −
0) [30].The relaxation modulus (stress history due to a unit-amplitude step-strain, γ ( t ) = U ( t − flywheel pinion rack θ ( t ) τ ( t ) = δ ( t -0) γ ( t ) =h ( t ) μ α with units [M][L] -1 [T] α - m R = m6 π R FIG. 6. A springpot–inerter parallel connection which isthe mechanical analogue for Brownian motion of micro-spheres with mass m and radius R suspended in a Scott-Blair subdiffusive fluid with material constant µ α with units[M][L] − [T] α − (say Pa s α ). element (Scott-Blair fluid) expressed by Eq. (39) is [46–51] G ve ( t ) = µ α − α ) t − α , t > J ve ( t ) = 1 µ α α ) t α , t (cid:62) t α , appearing in Eq. (43) renders the ele-mentary springpot element expressed by Eq. (39) (Scott-Blair fluid), a suitable phenomenological model to studyBrownian motion in subdiffusive materials.The mean-square displacement of Brownian particlessuspended in the fractional Scott-Blair fluid described byEq. (39) was evaluated in [52, 53] after computing thevelocity autocorrelation function of the random motionof the suspended microspheres with m and radius R , (cid:10) ∆ r ( t ) (cid:11) = 2 N K B Tm t E − α , 3 (cid:18) − πRµ α m t − α (cid:19) (44)where E α , β ( z ) is the two-parameter Mittag–Leffler func-tion [54, 55] E α , β ( z ) = ∞ (cid:88) j =0 z j Γ( jα + β ) , α , β > τ ( t ) = µ α d α γ ( t )d t α + m R d γ ( t )d t , α ∈ R + (46)The Laplace transform of Eq. (46) is τ ( s ) = G ( s ) γ ( s ) = ( µ α s α + m R s ) γ ( s ) (47)where G ( s ) = µ α s α + m R s is the complex dynamic mod-ulus of the springpot–inerter parallel connection, whilethe complex dynamic compliance is J ( s ) = 1 G ( s ) = 1 µ α s α + m R s = 1 m R s α (cid:16) s − α + µ α m R (cid:17) (48)The inverse Laplace transform of Eq. (48) is evaluatedwith the convolution integral [56] h ( t ) = L − {J ( s ) } = (cid:90) t f ( t − ξ ) g ( ξ ) d ξ (49)where f ( t ) = L − (cid:26) m R s α (cid:27) = 1 m R α ) 1 t − α , α ∈ R + (50)and g ( t ) = L − (cid:40) s − α + µ α m R (cid:41) (51)= t − α E − α , 2 − α (cid:18) − µ α m R t − α (cid:19) where E − α , 2 − α (cid:16) − µ α m R t − α (cid:17) is the two-parameterMittag–Leffler function defined by Eq. (45). Thefunction g ( t ) expressed by Eq. (51) is also knownin rheology as the Rabotnov function, ε − α ( − λ , t ) = t − α E − α , 2 − α ( − λt − α ) [45, 57]. Upon substitution ofthe results of Eqs. (50) and (51) into the convolutionintegral given by Eq. (49), the impulse response func-tion of the springpot–inerter parallel connection shownin Fig. 6 is merely the fractional integral of order α ofthe Rabotnov function given by Eq. (51). h ( t ) = 1 m R α ) (cid:90) t t − ξ ) − α ξ − α E − α , 2 − α (cid:18) − µ α m R ξ − α (cid:19) d ξ = 1 m R I α (cid:20) ε − α (cid:18) − µ α m R , t (cid:19)(cid:21) = 1 m R tE − α , 2 (cid:18) − µ α m R t − α (cid:19) (52) Substitution of the result of Eq. (52) into Eq. (16) together with m R = m πR , yields the time derivative of0 : Newtonian viscous fluid: : Undamped harmonic trap: sin ω R t BallisticRegime
DiffusiveRegime
FIG. 7. Normalized time-derivative of the mean-square dis-placement of Brownian microspheres suspended in a frac-tional, subdiffusive Scott–Blair fluid with material constant µ α with units [M][L] − [T] α − for various values of the frac-tional exponent 0 (cid:54) α (cid:54) (cid:16) µ α m R t − α (cid:17) − α , where m R = m πR and τ = m πRη . the mean-square displacement of Brownian particles sus-pended in a Scott-Blair subdiffusive fluidd (cid:10) ∆ r ( t ) (cid:11) d t = N K B T πR h ( t ) (53)= 2 N K B Tm tE − α , 2 (cid:18) − πRµ α m t − α (cid:19) The result of Eq. (53) that was computed herein af-ter calculating the impulse response function of thespringpot–inerter parallel connection shown in Fig. 6in association with Eq. (16) is identical to the first timederivative of Eq. (44) derived by [52] and [53] after com-puting ensemble averages of the random Brownian pro-cess.For the limiting case where α = 1, the Scott-Blair fluidbecomes a Newtonian viscous fluid with µ α = µ = η andEq. (53) reduces tod (cid:10) ∆ r ( t ) (cid:11) d t = 2 N K B Tm tE
1, 2 (cid:18) − tτ (cid:19) (54)where τ = m πRη is the dissipation time. By virtue of theidentity E
1, 2 (cid:0) − tτ (cid:1) = τt (cid:0) − e − t / τ (cid:1) , Eq. (54) returns Eq.(10) which was obtained by taking the time derivative ofthe mean-square displacement for Brownian motion in amemoryless Newtonian fluid derived by [2].At the other limiting case where α = 0, the Scott-Blairelement becomes a Hookean elastic solid with µ α = µ = G and Eq. (53) givesd (cid:10) ∆ r ( t ) (cid:11) d t = 2 N K B Tm tE
2, 2 (cid:18) − πRGm t (cid:19) (55)where πRGm = Gm R = ω . By virtue of the identity E
2, 2 (cid:0) − ω t (cid:1) = sinh (i ω t )i ω t = 1 ω t sin ( ω t ) , (56)Eq. (55) givesd (cid:10) ∆ r ( t ) (cid:11) d t = 2 N K B Tmω sin( ω t ) (57)which is the special result of Eq. (26) for Brownian mo-tion in an undamped harmonic trap. Figure 7 plots thenormalized time-derivative of the mean-square displace-ment for Brownian motion in a subdiffusive Scott–Blairfluid (cid:18) πRµ α m (cid:19) − α m N K B T d (cid:10) ∆ r ( t ) (cid:11) d t (58)= (cid:18) µ α m R (cid:19) − α m R h ( t )as a function of the dimensionless time (cid:16) µ α m R t − α (cid:17) − α with m R = m πR for various values of the fractional expo-nent α ∈ R + . VI. SUMMARY
This paper builds upon past theoretical and exper-imental studies on Brownian motion and microrheol-ogy in association with a recently published viscous–viscoelastic correspondence principle for Brownian mo-tion and shows that for all time-scales the time-derivativeof the mean-square displacement, d (cid:104) ∆ r ( t ) (cid:105) d t , of Brown-ian microspheres with mass m and radius R suspendedin any linear, isotropic viscoelastic material is identi-cal to NK B T πR h ( t ), where h ( t ) is the impulse responsefunction (strain history γ ( t ) due to an impulse stress τ ( t ) = δ ( t − m R = m πR .With the introduction of the impulse response function h ( t ) for Brownian motion, the paper uncovers that thereis a direct analogy between the description of the deter-ministic motion of a free particle and the random motionof a collection of Brownian particles immersed in a linearviscoelastic material as illustrated in Fig. 8. This anal-ogy shows that the random process of Brownian motionof a collection of particles can be fully described with thedeterministic time–response functions of the viscoelasticmaterial–inerter parallel connection.1 yzx r ( t ) Deterministic motion of a free particle ( t ) Random, thermally driven Brownian motion of M particles (microspheres) with mass m and radius R suspended in the linear viscoelastic material Linear, Isotropic Viscoelastic Material ~= r j ( t ) m Δ r ( t ) j= M ( r j ( t ) -r j (0)) Σ M = Linear, Isotropic ViscoelasticMaterial γ ( t ) τ ( t ) m R = m π R = = mean square displacement FIG. 8. In a analogous way that in Newtonian mechanics the deterministic motion of a free particle is described by itsdisplacement r ( t ), velocity v ( t ) = d r ( t )d t , and acceleration a ( t ) = d r ( t )d t , the random, thermally driven Brownian motion ofa collection of microspheres with mass m and radius R immersed in a linear, isotropic viscoelastic material is described bythe deterministic creep compliance, J ( t ) = πRNK B T (cid:10) ∆ r ( t ) (cid:11) , impulse response function, h ( t ) = πRNK B T d (cid:104) ∆ r ( t ) (cid:105) d t , and impulsestrain–rate response function, ψ ( t ) = πRNK B T d (cid:104) ∆ r ( t ) (cid:105) d t , of a mechanical network that is a parallel connection of the viscoelasticmaterial (within which the microspheres are immersed) with an inerter with distributed inertance m R = m πR .[1] P. Langevin, Compt. Rendus , 530 (1908).[2] G. E. Uhlenbeck and L. S. Ornstein, Physical Review ,823 (1930).[3] M. C. Wang and G. E. Uhlenbeck, Reviews of ModernPhysics , 323 (1945).[4] P. Attard, Non-equilibrium thermodynamics and statisti-cal mechanics: Foundations and applications (OUP Ox-ford, 2012).[5] Y. P. Kalmykov and W. T. Coffey,
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