Coupled translational and rotational diffusion in liquids

Coupled translational and rotational diffusion in liquids Ulrich Steiger and Ronald F. Fox Citation: J. Math. Phys. 23, 296 (1982); doi: 10.1063/1.525352 View online: http://dx.doi.org/10.1063/1.525352 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v23/i2 Published by the American Institute of Physics.

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Coupled translational and rotational diffusion in liquids Ulrich Steiger and Ronald F. Fox School 0/ Physics, Georgia Institute o/Technology, Atlanta, Georgia 30332

(Received 29 January 1981; accepted for pUblication 11 August 1981) The equations for coupled translational and rotational diffusion of asymmetric molecules immersed in a fluid are obtained. The method used begins with the Kramers-Liouville equation and leads to the generalized Smoluchowski equation for diffusion in the presence of potentials. Both external potentials and intermolecular potentials are considered. The contraction of the description from the Kramers-Liouville equation to the Smoluchowski equation is achieved by using a combination of operator calculus and cumulants. Explicit solutions to these equations are obtained for the two-dimensional case. Comparison of our results with earlier literature is also presented. PACS numbers: 05.40.

+ j, 51.90. + r

I. INTRODUCTION

In this paper we study the translational and rotational motion of molecules immersed in a fluid. The molecules experience translational and rotational Brownian motion as a result of the bombardment by fluid molecules. The description of this essentially stochastic process in terms of the probability-distribution function P(t,x) leads to a diffusion equation

a

a

2

-P(t,x) at

= Ialj(x)--P(t,x) + ax;axj

;J

a

Ib;(x)-P(t,x) ; ax;

AP(t,x)

1

F(t,x',x)P(O,x') dx'

f

Js"

dyc Ic (t,xc> Yc)

(4)

fulfills the first order differential equation in time:

a

-P(t,x c ) = G(t,xc)P(t,xc)' at

(5)

In general nothing is gained, since G (t,x,) might be a very complicated operator. We will use the cumulant expansion 5 ,6 to approximate the operator G (t,xc)' G (t,xc) =

I

G Inl(t,X,).

(6)

n=1

It turns out, that the diffusion operator A is the first nonvanishing term in the expansion (6). Equation (I), where A is now replaced by the second cumulant G 121(t,xc) [G l'l(t,xc) = 0], is a very good approximation of (5). K describes the time evolution of the distribution of the momenta due to random forces. The momentayc(t) can be considered as random variables, which very quickly become independent. Yc (t ) is independent ofYc(t +..1t) if the lapse of time ..1t is large compared with the correlation time 1'k' It can be shown,7 that the nth cumulant is proportional to

(7)

(2)

satisfies Eq. (1). Sx is the space containing all points x. P (O,x) is the initial distribution at time t = O. The concept of a Markov process is an idealization of the underlying physical reality. For a complete dynamical description, it is necessary to consider the distribution functionfc(t,xc> Yc) defined on the phase space Sx, XSy , consisting of all points (xc, Yc) with Xc = (Q"Q2,Q3,cp,e,t/J) and the canonically conjugate momentayc = (p" P2' Po' Pd>' Pe, Pv,). J. Math. Phys. 23(2). February 1982

L is Liouville's operator and K denotes Kramers operator, which describes the effect of all random forces acting on the Brownian particle. IfEq. (3) can be solved for some initial distributionfc (O,x c' y c) then it is possible to find an operator G (t,x c ) such that the averaged distribution P (t,x c ) defined by P (t,x c )==

s,

296

(3)

(1)

for all times t~O and all points x, x = (Q"q2,q3,cp,e,t/J). Q = (Q"Q2,Q3) describe the position and the Euler angles a = (cp,e,t/J) fix the orientation. The differential operator A is a diffusion operator. All eigenvalues of the symmetric matrix [alj(x)] are non-negative. For translational diffusion AT is simply a diffusion constant multiplied by the Laplace operator. Favro 1 derived the diffusion equation for rotational Brownian motion and was able to solve it for axial symmetric molecules using the fact that the diffusion operator AR has the same form as the quantum mechanical Hamilton operator for a rigid body,2 the properties of which are well known. In general the translational and rotational motions are coupled in a complicated way. Already 50 years ago, Kolmogorov showed that under very general conditions a Markov process defined in terms of the transition probability F(t,x,x')dx' of finding a particle initially at point x in the infinitesimal small set dx' after a lapse of time t, leads to a diffusion equation. The probability density P(t,x)

The distribution function Ic (t ,xc, y c) satisfies the KramersLiouville equation 3 ,4

fis a dimensionless quantity. f=1'kI1'. 1'is some typical macroscopic time unit. Intuitively, it is clear that we obtain a Markov process on Sx described by (I) if the correlation time l'k of the momentaY,(t) becomes very small. It is the short correlation time which makes the higher order contributions small. The idea of deriving the diffusion operator A as the lowest order of a cumulant expansion (6) is not new. The actual calculation of the operators A ,G 1.11, ... , is complicated by the

0022-2488/82/020296-14$02.50

® 1982 American Institute of Physics


296

nonlinearity of the equation of motion for a rigid body. The time derivatives of the angular momentum L ' and translational momentum p' expressed in an orthogonal coordinate frame attached to the moving particle are

a - e-IKL e'Kf- L- (t)f-f= (13) at The operator L (t )can be expressed in terms of the differential operators alaq and alap using the identity

£' =L'XI-IL' +N',

(14) (8)

jJ' = p'XI-IL'

+ F'.

N' and F' are the torques and the forces acting on the particle. The prime denotes vectors in the body fixed coordinate frame. I is the tensor of inertia. It is necessary to choose body fixed coordinates for both L ' and p' since otherwise the friction tensor C depends on the orientation [see (70)].8 The purpose of this work is to analyze the rotational and translational diffusion in the most general case using a mathematically transparent method. We will show that (i) The generalized Smoluchowski equation is the lowest order contribution of G (t,x c ). Starting off with a Maxwell distribution at time t = 0 the diffusion tensor is time dependent. For t < 1'k the diffusion tensor depends on the mass and the moments of inertia, and becomes stationary for t).1'k' (ii) The diffusion equation couples the translational and rotational degrees of freedom even in the simplest case. 8 As an illustration, the two dimensional diffusion equation is solved. The solutions are obtained in terms of exponential and Mathieu functions. (Sec. V). (iii) A suspension of N interacting Brownian particles leads to a diffusion equation for the N particle density P(t,x~I),X~2), ... ,X~N)). (Sec. IV). In Sec. II the operator calculus used later is introduced and applied to the translational motion. Section III treats coupled translational and rotational diffusion. II. OPERATOR CALCULUS, TRANSLATIONAL DIFFUSION

The starting point of the theory is the Kramers-Liouville equation. 3 •4

a - f(t,q, p) = B f(t,q, p) = (L + K )f(t,q, pl.

-m-Ip'~f+ au '~f

aq aq ap u denotes the potential. Kramers operator is

Kf=a~.(m-Ip + kT~)f

ap ap It is convenient 3 to work in the "interaction picture" f-e'K]

(10)

(11 )

(12)

J. Math. Phys., Vol. 23, No.2, February 1982

f

+

[K,·j"L [( - t)"lnfJ.

(15)

n~1

The commutators [K,· j"L can be defined by recursion,

[K,.j1L =[K,L ], [K,·fL =[K,[K,L]],

(16)

[K,.]nL =[K,.]([K,.]n -IL). We can calculate all terms in the infinite sum (15). Applying the commutator algebra discussed in Ref. 3 leads to

L(t) = -

e-la/m)I~. aq

(Lm + kT~) ap

+ ela/mll ~ • (aU + ~).

(17) ap aq aq In Sec. III the corresponding expression for translational and rotational motion is derived in great detail. Formally, the solution of(13) can be written

j(t)

E(t 1l0=T exp ('ds L(sllo, ~ Jo

=

(18)

in which T exp is the time ordered exponentiaLS jo is the .~-

initial distribution. The time ordered exponential must be used becauseL(t l) does not commute with L (t 2 ) iftl #t 2 • We would like to derive the time evolution for the averaged distribution P (t,q),

J

P (t,q)= d 'p f(t,q, p) =

J

= d'p e'Kj(t,q, p)

J

d 3pj(t,q,p)=' is omitted. The generalization of Liouville's equation including stochastic forces can be obtained from (71). 5 The result is the Kramers-Liouville equation 301


The following identities hold for arbitrary operators A,B,C: [A,BC] = [A,B]C [AB,C]

+ B [A,C],

= A [B,C] + [A,C]B. U. Steiger and R. F. Fox


(79) 301

In order to find the commutators [K,. ]"Lq we make the ansatz that there exist some vectors Win), Xln), yin), Zln) such that

With (79), (78) becomes

[K,Lo] = Ien,(Dx)rn{Vn[y"Ym)

+

[Vn,Yrn)yJ

n,/,m

+ Xln)*·v + yln)(y® V ® V) + Zln)·(v ® V ® V).

[K,. rLq = Wln).( y ® y ® V)

+

I

Cn,(Dx)m {V n [V"Ym)

+

[Vn,Ym

)vJ.

n.l.m

With (75) and using the fact that the matrix C = CkTis symmetric, 14 leads to

= Dx'Cy + 2Dx ·Cv.

[K,Lo]

(80)

For the higher order commutators one obtains

I

[K,.JnL o = Dx·Cny + 2

The vector X In)*ER is defined X rlo"==~,x 17k. The definition of the nth commutator [(16)] [K,.]nL q = [K,.]([K .. r-ILq) allows us to derive recursion relations for the vectors Win), Xln), yin), zln). Lemma: WIO) = - a, X (0) = 0, y(O) = 0, Z (0) = 0,

Win + II = Wlnl..!'

Dx·CrnC(-ct)'v.

xln+ I) =Xlnlfl + wlnlE

rn+'~n-I

(81)

(89)

This equation can be proved by induction on n. The calculation is similar to the calculation of [K,LoJ. We observe that the matrix CrnC is symmetric for all m>O since

CrnC =CM-ICM-I .. ·CM-ICkT

yin + II = ylnly + Wlnll/l Z In + I) = Z In)cp

..!' = C ® 1 ® 1

Dx·Cny, ICv ,

1/1 = 41

n~O

+ Dx.(e tC -

e- tC) C-ICv,

(83)

and with the definitions of C, C, and Dx [(76)] one obtains Lo(t) = - y.M-IE( - t)Dx +kTV.[E(t)-E(-t)]Dx . The matrix E (t) is the exponential

E (t )==e tCM

(84)

(85)

'.

The corresponding expression used earlier for the translational motion

- P . ~e-Ia/m)t + 2kTsinh(!!.... m aq m

t) ~. ~ ap aq

=- a·(y® y® V).

302


1 ®1 ®

ct,

C® 1,

fl= -1®I®C t . 1 is the 6 X 6 identity matrix. Wrlrn is symmetric in the first two indices Wrlrn = Wlz~ for all n = 0,1,2, .... Proof All these relations follow directly from the definition of xln l, yin), Win), Zln) [(88)] and the definition. of the commutator [K,.]"[(16)]. The following equations are true for arbitrary vectors Xln l, yin), Wln l, zln l with the only restriction that Wln l is symmetric in the first two indices.

W~1m = WIZ~· (1) [v.cy,xln)*.v] = (Xln)fl )*.V,

(86)

(87)

(91)

(2) [V.Cy,Wln).(y® y® V)] = Wln)..!'.(y® y® V), (3) [v.cy,ylnl.(y®V® V)] = ylnly.(y®V®V),

= Zln)cp·(v ® V EB V),

(5)

In the final step we calculate the operator Lq which is quadratic in the momentay. This leads to major complications, but it turns out that the operator Lq(t) contains no higher order derivatives than a third order derivative in the momentaq. We will write Lq as the scalar product of two vectors with 63 = 216 components [(74)]:

Lq

®

(4) [V.Cy,Zln).(v EB V EB V)]

is a special case of (84). It is remarkable that no higher than second order derivatives appear in Lo(t )! The calculation of the operator Lf is similar. One obtains Lf(t) = V.E(t)[DxU(x)).

+ 1 ®C ® 1 -

E=2C®1®1,

[K,.]"Lo[(-ttln!]

= Dx.e- ICy

E, I/I,..!' are defined

(90)

n even n odd.

The final result for the operator Lo(t ) is

f

Cp,

cp = - C t ® 1 ® 1 - 1 ® ct ® 1 - 1 ® 1 ® Ct,

(82)

Lo(t) =

WlnlE

y= C® 1 ® 1 - 1 ® ct ® 1 - 1 ® 1 ® Ct,

C and M are symmetric. Using this property the last term in (81)becomes2~m+'~ n- I Dx·Cm(C)'Cv. The sum vanished for even n. For odd n it is equal to 2Dx·C n- ICv.

+ 2Dx .en-

+

The 216 X 216 matrices y,

= (CmC)t = CC tm .

[K,. JnL o = {Dx .Cny

(88)

6

[v.CV,Wlnl.(y® y® V)] = (Wln)E)*.V

(92)

+ Wln)I/I.(y® V ® V),

(6) [v·Cv,yln).(y® V ® V)] = yln)E.(V EB V ® V),

(7) [v·Cv,Zlnl.(V ® V ® V)] = O. The proof of these equations is mostly straightforward. For instance, the first equation is [v.Cy,xlnl*.V] = =

I

CapX~)· [Va yp,Vy

"

C Xln)·v a (- 0ap ) = Xln)*( - et).v

a,p.l'

]

~aPl'

a,/3,l'

= (Xlnl)*.V.

The fifth equation is different since there are two different U. Steiger and R. F. Fox


302

The first cumulant is according to (25)

types of terms: [v.Cv,Wln).(y@ y@ V)] =

I

u,{3,y,6,.

CapW~l.{(Va[Vp,yy]

+ (Vu,Yy]VpY6V.

+ Y6 V.)

+ YyVu [Vp 'Y6]V.

By assumption W~2. = W~~. and [Ya,V{3] = -8ap , This gives the result [v.Cv,Wln).(y@ y@V)] = (Wln)E)*·V + Wln)I[I.(y@V@V), The proof of the other equations is similar. We define the vector valued function Wit ):R---+R 216 ,

I (( - t )nln!) WIn)

G(I)(t)P(t,x) =

o.

(99)

The second cumulant gives the first nonvanishing contribution,

(93)

and similarly X *(t), Y (t), and Z (t ). The recursion relations (89) for Win), X*ln), yIn), and zln) lead to the differential equations W(O) = - a, X(O) = 0, Y(O) = 0, Z(O) = 0,

LJ

G (2 )(t) =

ds

d 6yL(t)L(s)g(y)

with [(96)] G(2)(t) = - fdS

f

d 6yy.M- IE(-t)D x

Y(t) = - Y(t)r - Wit )1[1,

+ V.E(s)[DxU(x)]

Z(t)= -Z(t)-Y(t)E.

+ X*(s)·V

These differential equations can be integrated and the results are t~),

So' ds exp( - s~ )Eexp((s -

t]D), (95)

Y(t) = a

So' ds exp( - s~)1[1 exp([s -

Z (t) = -

So' dsY(s)E exp([s -

+ y(s)·(y@ V@ V)]g(y).

The remaining terms ofthe productL (t)i (s)vanishafterintegrating by parts. The only term left from the operator L (t ) is - y·M - I E ( - t )Dx • Also the term Z (s)·(V @V @V) vanished after integrating by parts three times. At first we can show that the contribution due to the terms W (sH y ® y ® V), X *(s). V, and Y (s).( y ® V ® V) cancel each other. We will show that the following integral vanishes for k = 1,2, ... ,6 and all times s;;;.O:

(102) + Y(s).(y® V ® V)]g(y).

We recall thatSd 6y Yi Yj g(y) = MijkT. Again integrating by parts (102) becomes

E( - t )]Dx

Jds)

= -

(96)

+ V·E(t)[DxU(x)]

+ W(t).(y@y@V)+X*(t).V + Y(t ).(y@V@V) + Zit )·(V@ V @V). This is the Liouville operator in the interaction picture. The quadratic term Lq caused all the additional terms. Even if they are not explicitly known, we will be able to show that they do not contribute to the first and second cumulants. D. First cumulant

e - y,M - 'y/2kT.

J, Math. Phys., Vol. 23, No.2, February 1982

(97)

I(kTWnmk(S)Mnm n,m

+ Xnnds) -

Ynnds)).

(103) The function Jds) may be written as J k(s) = 'I.;;~ oJ~n)( - sl" In!. For the constants J~n) one obtains, according to (93),

L

(kTW~~kMnm + X~~k - Y~~k)' (104) n,m The recursion relations (89) allow us to define J ~ )in terms of J (I-I) J~) = -

k

We calculate the cumulants under the assumption that initially the distribution in the momenta y is a Maxwell distribution, 1 (21TkT)3(det M)J/2

y@V)

t]O.

(109)

Equation (108) is the generalized Smoluchowski equation for coupled translational and rotational diffusion. Since we started with a Maxwell distribution at t = 0, the diffusion matrix A (t) is time dependent. Equation (108) includes as a special case the translational diffusion and the rotational diffusion discussed in Ref. 1. The operator Dx depends on the orientation a = (f/J,e,¢). (110)

dom is still coupled even if the potential U vanishes, since D q" dependsona. In Sec. V we will solve (114) in two dimensions for U (q,a) = O. In Refs. 10 and II, different expressions for the operators corresponding to Dq , and Da , which are wrong in our opinion, are used. Instead of D a the operator J = - iqx(J/Jq) was used. J is, up to a constant factor, the quantum mechanical angular momentum operator for a rotatingpoint particle. Both operators Do and Jhave the same commutator algebra since they are both infinitesimal generators of a representation ofSO(3). Do and J correspond to two different representations; see (136). A connection between J = - iqx(J/Jq) and the three Euler angles (f/J,e,¢) also used in Refs. 10 and II is not obvious. For axially symmetric particles one can factorize the angular dependence of P (t,q,f/J,e,¢) in ¢. The operator D;; is in general not equal to..:1 Ir ~ I , the Laplace operator in spherical coordinates on the unit sphere. This is only true if we set J /J¢ = O. Ifwe consider only axial symmetric molecules and do not distinguish between two orientations which differ only by a rotation about the axis of symmetry, then we may use D;; I¢ =..:1 Ir ~ I ; see (136). Reference 10 obtained wrong results by setting J 2 = ..:1. It is important to keep in mind that the operator D q' depends on the orientation. Dq , is the gradient along the body fixed coordinate axis. If Dq, is replaced by Dq = J/Jq one obtains wrong results. 10,11 The coupling of translational and rotational diffusion of the two dimensional model discussed in Sec. V is a consequence of the a dependence of Dq' , only. These claims will be justified in detail in Sec, V.

(Ill)

IV. N PARTICLE DIFFUSION

J + sm '.1. I J e'sm ¢J'I' - - - - - cot Je sin e Jf/J J¢ . J I J J -sm ¢ - + cos ¢ - - - - -cot ecos¢Je sin e Jf/J J¢ cos

Da =

.1. 'I' -

J J¢

(112) The rotation R (f/J,e,¢) is defined in (33) and (34). The expression for Da follows from (59) and (63). Usually the friction tensor C is split into four 3 X 3 matrices.

_ (C

C-

TT

CTR ) .

&

kT

= kTC TTl (1 AR = kTC RRI (1 At

ee-

(115)

with X=(XiI),xI21, ... ,XINI), The volume element dll x is the product measure (114)

N

dll x =

II dqVldq~ldq~ldf/J Iii sin e lilde lild¢lil.

(116)

i= 1

kIT [Da U(q,a)]) }p(t,q.a),

. tCrm tCRRI

), '),

t>O,

The diffusion of translational and rotational degrees of free304

f

EO (t)== dllxP(t,X)O (x)

= {Dq,.AT(D q, + _I_[Dq, U(q,a)])

+ Da.Ar(Da + with

For an arbitrary observable 0 (xiI),Xi21, ... ,xINI) depending on the position and orientation of the particles 1",.,N the expectation value is defined

(113)

CRT CRR For axialsymmetric molecules it is easy to show that C TR = CRT = 0. 8 In this case the diffusion equation is

!..-P(t,q,a)

We consider N particles moving in a fluid interacting via arbitrary forces. In general the N particle density P (t,x(\),x(2i, ... ,x(NI) is not the product of the distributions P (t,xlil), where xiii denotes the six coordinates of the ith particle Xiii = (qlI),a lil ), The N particles are correlated. The interaction energy is U (Xi I),x(2), ... ,x IN I),

J. Math. Phys .• Vol. 23. No.2. February 1982

The objectives of this section is to derive the evolution equation for the N particle density P (t,x) based on the Kramers-Liouville equation for the N particle motion. For the complete description of the N particle dynamics all positions xiii and all momenta ylil are required, U. Steiger and R. F. Fox


304

(117)

z(t )=(ZOI(t ),zmllC iTlll and to>IIC RRI] II, at

kT

b )(Da)~ ]P(t,q,a)

for t> to.

(135)

This equation is a special case oft 114) where we used expression (134) for the friction tensor. We also used A (t )~ k TC - I for f>t o. 306

a

Llq is the Laplace operator in Cartesian coordinates. We define the new density P (t,q,¢;,e), P(t,q,¢;,e) =

f

dl/; P(t,q,¢;,e,I/;).

(137)

Integrating Eq. (135) on both sides with respect to I/; leads to a -a P (t,q,¢;,e) = kT [ aLl q + (a 3 - a) - a . B (¢;,e)~

~

+ bL1 Ir=

I


~

k(t,q,¢;,e).

(138)

The matrix B (a) defined in Eq. (136) does not depend on 1/;. B (¢;,e)

=

(:~:: ~:::) (:~:: ~:::). ®

(139)

cos e cos e The contributions of Eq. (135) which contain a deriva.tive with respect to I/; vanish after integrating by parts. Therefore the operator D; reduces to Ll Ir = , , the Laplace operator in spherical coordinates on the unit sphere. (140) We assume that the initial condition factorizes. For t > to the solution of (138) has the form P(to,q,¢;,e) = POt (q)POR (¢;,e), P(t,q,¢;,8)=PT(t,[PR ])PR(t,¢;,8),

[aD~. + (a 3 - a)(Dq')~

+ bD; + (b 3 -

ae '

2

(134)

We assume that we know the distribution at time t = to, where to is large compared with the translational and rotational relaxation time of the momenta.

~ P(t,q,a) =

cote~

(Dab = aZI/;'

V. CORRELATIONS BETWEEN THE VARIABLES q AND a

We consider the one particle diffusion equation (114). In general the positions and orientations are correlated. The correlations are not only caused by the potential U = U (x), x = (q,a) or by non vanishing elements of the matrix CTR = CkT' We will show that, if the positions q and the orientations a are uncorrelated at t = to there are in general correlations for t> to even if the potential vanishes and also C TR = O.

+

(136)

t>to'

(141)

The function PT(t) is also a functional of the distribution PR(t ). P T(t ) and PR(t ) are probability densities, Sd 3q P(t,q, [PR ]) = 1 and Sd¢;d8 sin 8PR(t,¢;,e) = 1 for all times t> to' The boundary conditions are: PT(t,q, [PR ]) = 0 ifqi = 00 for some i = 1,2,3. Substituting(141)intoEq. (138) and integrating with respect to ¢; and (using the weight sin 8) leads to Eq. (142). Similarly one obtains (143) by inte-

e

U. Steiger and R. F. Fox


306

Without solving (144) explicitly it is already possible to make some statements about the lowest moments of ql' q2, and,p. One obtains the following differential equations for the expectation values ( ... ), = fdq, dq2 d,p ...P(t,ql,q2',p):

grating with respect to q.

~PT(t,q,[PR]) = kT{aLl q + (a 3 -

a)

at

xL fd,p sin edeBij(,p,e)pR(t,,p,e) ~~} iJ aqi aqj xPT(t,q,[PR ]),

~ PR(t,,p,e ) = at

I

kTM

PR(t,,p,e )

r~

for

d

dt (ql), (142)

t'~to·

= 0,

!!. (q~), =

2kT8

!!. (q Iq2),

= - 2kT€(sin 2,p )"

dt

I

(143)

The second equation describes the "Brownian motion on the unit sphere." The eigenfunction of Ll Ir~ I are the spherical harmonics Y lm (e,,p ). Substituting a solution PR(t,,p,e ) of (143) into Eq. (142) one obtains an expression which is formally a diffusion equation with time-dependent diffusion coefficients. The off diagonal elements of the diffusion matrix vanish if the distribution P R(t,,p,e ) is uniform. Similarly, one can show that for arbitrary molecules with CTR = oa solution of the form (141) (including tP) exists, if the positions and orientations are uncorrelated at time t = to and if U = o.

B. Diffusion in two dimensions

In two dimensions the diffusion equation without external potential can be solved for arbitrary initial conditions. Equation (108) reduces to

dt

!!.(cos2,p), dt

+ 2kT€(cos 2,p )" (149)

= -4ykT(cos2,p)"

!!. (sin 2,p ), =

4ykT (sin 2,p ) I '

-

dt

This leads to (cos 2,p ) I = e - 4yk Tt (cos 2,p ) 10 '

(qi>, = 2kT8t

;y (1 - e- 4ykTt)(cos 2,p )'0

+

+ (qi>,o' (150)

(qlq2), =

~y€

(1 - e-

4ykTt

)(sin 2¢> )'0

+ (qlq2),o'

The calculation of arbitrary expectation values (0) I ' 0= 0(ql,q2',p) can be reduced to the problem of finding the eigenvectors and eigenvalues of the diffusion operator A in Eq. (144).

a

(151)

- P(t,ql,q2',p) = AP(t,ql,q2',p), at

For the symmetric case a = f3 the solutions of (151) are (144)

A

./,' A.) 'f'k,k,t (q I' q2,'f'

(~')'A (ai') + ~:, (,p)

=

A (,p) = kT

(152)

kTy

aq2

~ (ql,q2,'f' A.) = ----u2 1 - 1 e~ ,q, e~ ,q, cos(l,p ). 'f'k,k,/

aq2

(a( f3cos- a),p sin+ f3,psincos ,p,p 2

2

= _1_..!.. eik,q, eik,q, sin(lrl) 1TI/2 L 'f' ,

L

1T

( f3 - a) sin ,p cos ,p ) a sin 2 ,p + f3 cos 2 ,p . (145)

We choose a box oflength L and assume periodic boundary conditions,

kTa, kTf3, and kTy are the diffusion constants corresponding to the degrees of freedom ql' q2, and,p. We assume that a> f3. We use the following identities to simplify the matrix A (,p): a cos 2 ,p a sin 2 ,p

+ f3 sin 2 ,p = + f3 cos 2 ,p =

a a

+ f3 + 2

+ f3

a - f3 cos 2,p,

The possible values for k

2

"

_ a - f3 cos 2¢>,

2

(146)

2

-

- sin 2,p) . - cos 2,p (147)

8 is the average translational diffusion constant and measure for the asymmetry of the particle.

+ f3, 2

307

€=

± T'

k 2 , and I are

k2=

2m1T

± T'

n,mEN

(154)

0) + kT€ ( co~sm2,p2,p 8==. a

2n1T

k, =

2 sin ,p cos,p = sin 2,p,

1

(153)

a - f3 . 2


€

/=0,1,2, ....

In the general case a > f3 we make the ansatz that the eigenfunctions can be written

is a

tPk,k,/(ql,q2',p) = (148)

)12 ~ eik,q, eik,q'gk,k,/(,p ).

(155)

One obtains the following differential equation for the unknown functiongk,k,M) [(144), (147), (151)]: U. Steiger and R. F. Fox


307

[ - o(k i

+ k ~) -

E(k i

+ 2Eklk2 sin2cp

- k ~) cos 2cp

+r

a~2

2

-

J

dq l dq2 d CPtPt,k,dQI,q2,CP)0(ql,q2'CP)

Ok,k,1 =

AZ';I ] gk,k,M ) = O. (156)

We define the complex wavenumber k',

k '=:=k , + ik2' (157)

=(tPk,k,!'O),

o k,k,1 =

(163)

(tPk,k,/'O),

Pk,k,1

= (P (to),tPk,k,l)'

P k,k,1

=

(P (to),tPk,k,I)'

As an illustration we consider the following two observables:

OS(q\,cp )=sin(k,q\)se\(r,cp), (164)

OC(ql'cp )=sin(k,qtlcedr,cp), withk , = 21TILandr = ~(a - P )/rL 2. Weassumethatthe asymmetry is small. In this case r( 1 and the Mathieu functions se I and ce I are approximately )~cos(cp)

cedr,cp The exponentials can be combined to cos(2[cp + tP)). Equation (158) is equivalent to Mathieu's equation. 16 d - Y/(Z)

dz

r=

+ (al(r) -

(165)

se\(r,cp )~sin(cp) - ~ sin(3cp). 8

2r COS2zlY/(Z)

= 0,

(159)

The corresponding eigenvalues are al(r)~

+ k~)(a -p) ------(k~

1 + r,

(166)

4r

b,(r)~

(160)

1 - r.

The eigenvalues A ±

A±k,O\

A '±k,OI The eigenvalues al(r) of Mathieu's equation are negative for certain values of r and 1,16 but the eigenvalues Ak k I are always less or equal to zero for all klk2' and I. ', Equation (159) has a complete set of orthogonal solutions cel(r,z) and sel(r,z) with the corresponding eigenvalues denoted by al(r) and bl (r).16 The eigenfunctions of (151) are 1

1

'k

'k

tPk,k,/(ql,q2'CP) = 1T1/2 Le' ,q'e' ,q'sel (r,cp+arctan(k 2Iktl) tPk,k,/(ql,q2'CP)

=

1 1 'k 'k 1T1/2 Le' ,q'e' ,q'cel (r,cp

+ arctan(k2Iktl)· (161)

since {tPk,k,l,tPk,k,/} is a complete set of orthogonal eigenfunctions of the diffusion operator (151), the expectation value (0), can be found by

(0),

=

=

J

dq 1 dq2 d cpP(t,q\,q2,CP)0(q\,q2'CP)

'" L

r - - cos(3cp), 8

e,t"" ,. Pk,k,1 0 k,k,1

(162)

~~

-

k,OI

and A '± k,O\ are

t2 + t2

kT(r+

kT(r

(3a +P)). (167) (a

+ 3P)).

and for the expectation values of 0 sand 0' one obtains (0'), ~ce- kTll'+ (rr'IL'il3a +{3))', (168) (0 "), ~c'e - kTll' + (rr'IL 'lia + WII'. The constants c and c' can be written c = (0 c,P (to)) and c' = (0 ",P (to)). The state 0 C decays faster since we assumed a > p. a corresponds to the diffusion along the e; axis of the molecule. In the state 0 C the molecule axis e; is mainly parallel to the e I direction of the laboratory frame; in the state 0 se; is mainly parallel to the e 2 axis. The average speed of the molecules in state 0 is bigger in the direction e I; e I is also the direction of the spatial inhomogenity. Therefore 0 C decays faster than 0". This example is typical for the type of coupling of q \,q2' and cp, which occurs in the translational and rotational diffusion if the otential Uvanishes and also C TR = O. C

VI. CONCLUDING REMARKS

k,k,!

I e ',kI P k,k,1 0 k,k,[' k,k,1 The coefficients Ok,k,I' 0 k,k,I' Pk,k,I' P k,k,1 are obtained from o (ql,q2'CP) and the initial distribution P (to,ql,q2'CP ). +

308

,t'


We have shown that a "contraction of the description" is achieved when a Kramers-Liouville process is averaged with respect to its momenta variables. The second cumulant of an ordered time evolution cumulant expansion yields the generalized Smoluchowski equation as the contracted deU. Steiger and R. F. Fox


308

scription. We have examined the details of the dynamical operator algera generated by the contraction procedure for translational and rotational degrees of freedom, and for as many as N distinct particles. A more thorough description of the higher order cumulants, shown to be small here, will appear in a forthcoming paper. ACKNOWLEDGMENT

NSF Grant 7921541 for theoretical physics has supported this research. This work is from the dissertation ofU. Steiger, Georgia Tech, 1981. 'L. D. Favro, Phys. Rev. 119, 53 (1966). 2R. Jost, Quantenmechanik I (Yerein der Mathernatiker und Physiker,

309


ETH, Ziirich, 1969). 'J. L. Davis and R. F. Fox, J. Stat. Phys. 22, 627 (1980). 's. Chandrasekhar, Rev. Mod. Phys. 15, 1(1943). ~R. F. Fox, Phys. Rep. 48, 179 (1978). "R. F. Fox, J. Math. Phys. 20, 2467 (1979). 'R. F. Fox, J. Math. Phys. 17, 1148 (1976). "D. W. Condiffand J. S. Dahler, J. Chern. Phys. 44,3988 (1966). 9K. Yosida, Functional Analysis (Springer, Berlin, 1974). It'G. T. Evans, J. Chern. Phys. 65, 3030 (1978). "L.-P. Hwang and J. H. Freed, J. Chern. Phys. 63,118 (1975). '2H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959), Chaps. 4 and 5. '"'V. r. Arnold, Mathematical Methods o/Classical Mechanics (Springer, Berlin, 1978). '4E. H. Hauge and A. Martin-Lof, J. Stat. Phys. 7, 259 (1973). "J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (prentice Hall, Englewood Cliffs, N. J., 1965). '"r. S. Gradshteyn and I. M. Ryzhik, Table 0/ Integrals and Series and Products (Academic, New York, 1965), Sec. 8.6.

U. Steiger and R. F. Fox


309