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CHEMICAL PHYSICS LETTERS 3 March 1984

EIGENVALUES OF THE FOKKER-PLANCK AND BCK OPERATORSFOR A DOUBLE-WELL POTENTIALK VOIGTLAENDER and H. RlSKENAbreikng fir l k o r e t i s ch e Phys i k de r Un i ver J i t ~ t U i m , D -7 900 U lm . Fede r a l Repub l i c of e rma n yReceived 1 December 1983; in final form 16 January 1984

Eigenvalues of the Fokker-Planck and BGK operators for a d 2 x 2 / 2 + d a 4 / 4 double-well potential are ulculated by thematrix continued-fraction method. A dependence of the eigenvalues on t h e friction constant or coupbng strength is shownfor the lowest non-zero real eigenvalue and for some higher, generally complex eigenvalues.

1. IntroductionIn chemical physics, physics, electrical engineering

and other fields, b&able operations appear quiteoften_ Some of these systems can be modelled to theonedimensional motion of particles in a bistable po-tential with the inclusion of some rrreversiile terms.One of the simplest expressions for a b&able poten-tial (per mass) ti of the formf(x)= d2x2/2+d4x4/4, d,O- I)

The rreversible terms may be described by addinga damping term and a Langevin force of the equa-tion of motion, leading to a Fokker-Planck equation[eqs. 3) and 4) below] for determining the distri-bution function [l-lo] _Another simple equationfor de termimn g the distribution function is theBoltzmann equation with a Bhatnagar


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Volume 105. number S CHEMICAL PHYSICS LETTERS 3 0 arch 19842. Fokker-Planck and BGK operators

Ihe Fokker-Phurek equation and the Boltzmannequation with a BGK collision operator for particlesmoving in a potential Cper mass) offix) is [I ,2]aru/ar=Lw , (3)where L i s either the Fokker-Planck operatorL FP = -$u+grxT)+T$(+.$-) (4)or the EGK operator [l l--13]

+y g(u)) ..du-1 .( )-_ (5)In (4) and (5), y is the friction constant or the cou-

pling constant. respectrvely, 0 = k T / n t is a smtabletemperature scale anci gM(u) is the Maxwell distribu-tion,gM(u) = (2n0)-1/2exp (-G/20) _ (6)The stationary solution of (3) (4) and (5), is given bythe Boltzmann distributionfVi.&,u) =Nexp{--ax)+&?]/@) _ (7)

For large d~p~g, the operator L of the corre-sponding Smoluchowski equation can be brought tohermitian form by multiplying from the left with theinverse of the square of the stationary distributionfunction and from the right by the square root itself,i.e.E = (Wst)-1 f2L( W 2 _ (8)

For intermediate or smali d~pmg constants,Lisnot a hermitian operator for either the Fokker-Planck or the BGK case. The dynamics of eq. (3)follows from the eigenvalues and eigenfunctions of Lor L, i.eEJ/(x, u) =-X$(x, tJ) _ (9)

In this letter, we are mainly concerned with theeigenvalues (the eigenfunctions can also be obtainedin a manner similar to that in ref. [19])_ For the po-tential (l), the eigenvahres X depend on the parame-ters d,, d, , y and Qr

h=h(d2.d4,y,0). (10)By using proper scaling factors for the variables x, tand u, rt is easy to show that in (10) two of the fourparameters may be normahzed to unity. Some usefulnormalizations used later are--h = X(dz, 7) = X(& , 1, 7, 1)and

(1 la)

x=:(~,~)=h(--l. l,T,G)_ (1 lb)The eigenvalue h can then be expressed in either ofthe two forms in the following wayh(d2, d 4, y, 0) = (O d4) 1/45;(d 2/(Od 4y2, ~/ (@d ~)

= (-d 12 r ;(y/(-d.,) I 2 , (%4/d:) _ (12)Note that the parameter AEjkT appears in both

expressions:d2 = d , J (~d~) l12 = -2(AE/kT)1~2 ,6 = Odq fd $ = kT /4AE , 113)where .AJZ= mdz/4dq is the energy differencebetween the hump in the middle of the potential (1)and its mimma, multiplied by the mass m.

3. Matrix continued fraction

Ihe expressions in this section stmplify greatly Ifwe use the normalization in (I la), i.e. we put 0 =da = 1. We first expand the ergenfunctions 9(x, u)into nonnaked Hermite functions [20]#&i) =H,(7_-t~2U)exp(-~i79)/~~~ 7n(3~)1~2]1~2

(14)and &_T): Lp m

(Wwhere cr is the proper scaling factor. Inserting (15)into (9) and using the orthonormahty relations andthe recursion expression for the Hermite pol~omi~sand their denvatives 1201, w e obtain

507


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Volume 105. number 5 CHEMICAL PHYSICS LETTERS 30 March 1984

+Fw c BPqcgl =o _ 116)4Here, I, are the eigenvalues of the operators con-

taining y in (4) and (5) and are given by [12]1 , = lFPn =n ,

I , = l J . JGKn I 1 - 6 ,($ (17)For the fW and J%q, only the follower elementscan be non-zero-WP-3 =(T)(1/2a3)Ip@ - I)@ - 2)] I3 )Dpap-l = (-ia lIZa 3p/2&)p1/* ,IN-1.P =(&y(T) l/2&) 3p/2a3)p12 ,m-3yp =(T)(l/Zd)[p@ - 1)(-p - 2)ff/2 . 08)The signs in the parentheses have to be used to obtain@+r_ Truncating the expansion (15) at an mdexp =Q and using the vectors and matrices

D = (l.FJ) d= @m) , (1

the recurrence relation (16) may be cast into the tri-diagonal vector recurrence relation(n + l)~2Dcn+l +(x - i;ln)cn +&2Lkr,_1 = 0 _

(20)As explained elsewhere [14,15,21], all cn with n > 1can be eliminated by repeated use of (20) leadingfmahy to[xl + K,(-r;)]co = 0, (21)where i s the unit matrix and Ku(s) is given by theiufmite matrix continued fractionKo(s) = D [(s -j- Tfl) i

-2D[(s+~y12)i -...]-ti]-fti. (22)A non-trivial solution of (21) can occur only if the

corresponding deternunant vanishes:D(x) =det[Xl+ K,-,(-h)] ~0. (23)This equation determines the eigen~a~ue x. The trun-cation mdex Q and the truncation indexAl of the in-finite matrix continued fraction (22) has to be chosenin such a way that a further increase of Q and M doesnot alter the final result beyond a given accuracy. Bychoosing a proper scaling factor Q, Q and N can beminimized. The eigenvalues are determined from 23)by an ordinary regula falsi method (real eigenvahres)or by a regula falsi method for two variables (complexeigenv~ues) Because of the s~et~ of the prob-lem, the eigenfunctions are either symmetric or anti-symmetric. For symmetric eigenfimctions, only ~$2and c$$i can be different from zero. Therefore, thedimension of the vectors cn and the matrices can bereduced by a factor of 2 by using this symmetry

4. Results and comparison with analytic expressionsIn fig. 1, some of the non-zero eigenvalues of the

Fokker-Planck operator are shown as a function ofthe damping constant for a barrier height AE = k l r .Whereas for large y the eigenvalues are real, theeigenvalues become complex at those points wherethe curve of the real eigenvalue has a vertical tangent.If complex eigenvalues occur, the complex conjugatesare also eigenvalues Some of these eigenvahres remaincomplex down to 7 = 0; others become real again forcertain values of 7 (see the bubble in the upper partof fig. la). The eigenvalues of the BGK operator havefeatures similar to those for the Fokker-Planck oper-ator. For small friction constants the real eigenvahresare proportional to y, whereas the real parts of thecomplex ergenvalues have a more complicated stmc-ture as explained in ref. [IS], eq. (3.27). In fig. 2, thelowest real non-zero eigenvalues for the Fokker-Planck and BGK operators are shown for moderateand large barrier heights. For the normalization (1 lb),the following asymptotic form of the transition rateI = A/2 for high barrier herghts was derived for theFokker-Planck case [ 1,22,23] :2 = (2112/zr)[fi2/4 + 1)lj2 - 71 exp(-1/4G). (24)

As seen in fig. 2, this expression agrees quite wellwith the exact one for finite friction constants 7.

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Volume 105. number 5 CHEMICAL PHYSICS LETTERS 30 hlarch 1984

0 0 1 2 3 LY

xl 1 . . ...i..... :\@A \ b _. 21 13\ \ \1 \ 1

5/ l\\ \_ . \ \ b. 1 \\ \ .\ I m \ I\ L .

a- 1 )I ;


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Volume 105. number 5 CHEMICAL PHYSICS LETIERS 30 March 1984

References [13] D.K Garrity and J .L. Skinner, Chem. Phys. Letters 95(1983) 46_

[1] H A. Kramers. Physica 7 (1940) 284. 1141 H. Risken. H.D. Vollmer and H. Denk.Phys Letters[2] S. Chandrasekhar. Rev_ Mod. Phys. 15 (1943) 68. 78~ (1980) 22.[3] H C. Brinkman,Physica 22 (1956) 29. [15] H. Risken. H.D. Vollmer and M. M6rsch. 2. Physik B40[4] R. Landauer and J .A Swanson, Phys. Rev. 121 (1961) (1981) 343.

1668 [16] H. Risken and H.D. Vollmer. Z_ Physik El33 (1979) 297.[5] P.B. V her. Phys Rev. I314 (1976) 347. [17] H.D. Vollmer and H. Rkken. 2. Physik B34 1979) 313.[6] C. Blomberg. Physica 86A (1977) 49_ [lS] H.D. Vollmer and H Risken, Physica 1lOA 1982) 106.

0. Edbolm and 0. Leimar. Physica 98A (1979) 313. [19] H. Risken and H D. Vollmer. Mol. Phys 46 (1982) 555J L Skinner and P-G. Wolynes, J . Chem. Phys. 69 [20] W. Magnus, F. Oberhettinger and R.P. Soni. rormulas(1978) 2143. and theorems for the special functions of mathematicalK. Schulten, 2. Schulten and A. Szabo. I. Chem. Phys. physics (Springer, Berlin, 1966).74 (1981) 4426. [21] H. Risken and HD. Vollmer, Z. Physik B39 (1980) 339.M. Mangel, J . Chem Phys 72 (1980) 6606. 1221 BJ . Matkowski. Z. Schuss and E. Ben-J acob. SIAM J .P-L. Bhatnagar. E.P. Gross and M. Krook. Phys. Rev. Appl. Math. 42 (1982) 835.9At19CAI ill [23] C-W. Gardiner, J . Stat. Phys. 30 (1983) 157.

I71181PI

IlO11111I121

_ . . ___ . _ __J -L. Skinner and P G Wolynes, J . Chem. Phys. 72(1980) 4913.

[24] H_ Risken and K. Voigtlaender. in preparation.[25] J -T. Hyncs, Chcm. Phys. Letters 79 (1981) 344.

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