Relaxation time for nonlinear response of a Brownian particle subject to a step external force: Analytical solutions for one-dimensional models

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1 PHYSICAL REVIEW E VOLUME 55, NUMBER 3 MARCH 1997 Relaation time for nonlinear response of a Brownian particle subject to a step eternal force: Analytical solutions for one-dimensional models Yu. P. Kalmykov Centre d Etudes Fondamentales, Université de Perpignan, 52 Avenue de Villeneuve, 6686 Perpignan Cede, France and Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryaino, Moscow Region, 14112, Russian Federation J. L. Déjardin * Centre d Etudes Fondamentales, Université de Perpignan, 52 Avenue de Villeneuve, 6686 Perpignan Cede, France W. T. Coffey Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland Received 1 November 1996 The nonlinear response to a step eternal force of a system with relaational dynamics governed by a one-dimensional Fokker-Planck equation is considered. An eact analytical epression for the step response nonlinear relaation time is derived in terms of an integral which can be evaluated numerically. Applications to nonlinear problems concerning the dynamic Kerr effect, dielectric relaation of liquid dielectrics, and magnetic relaation of systems of single domain ferromagnetic particles are given. The results are compared with solutions previously obtained. S X PACS numbers: 5.4.j, 76.2.q, Gm, 78.2.Jq I. INTRODUCTION A system initially in an equilibrium stationary state and suddenly disturbed by an eternal stimulus e.g., by applying a step eternal field will evolve into another equilibrium stationary state. Presently a satisfactory theory is available for linear response only the energy of the system arising from the eternal stimulus is much lower than the thermal energy 1,2. Here we need only linear in the eternal stimulus deviations of the epectation value of the dynamical variable of interest in the stationary state in order to evaluate the generalied susceptibility and/or response functions in terms of the appropriate equilibrium stationary correlation function. Linear response theory is widely used for an interpretation of nonequilibrium phenomena such as dielectric and magnetic relaation, conductivity problems, etc. Here we wish to study relaation following a steplike stimulus in systems described by one-dimensional Fokker- Planck equations for the distribution function W(,t) of a variable 2, t WL FPW. 1.1 We shall therefore first summarie the principal results of linear response theory 2, Chap. 7 for systems the dynamics obey a diffusion equation like Eq Thus let us consider the Fokker-Planck operator L FP of a system subject to a small perturbing force F(t). On account of this, L FP may be represented as with L FP evbft D 2 e VBFt L FP L et Ft, L FP D 2 e V ev, L et D2 B, L FP W, 1.2 B d d B, L FP () is the Fokker-Planck operator in the absence of the perturbation, W is the equilibrium stationary distribution function, V is called a generalied effective potential 2, D (2) () is the diffusion coefficient, and B() denotes a dynamical quantity. The step-off and step-on relaation functions when, on the one hand, a small constant force F 1 is suddenly switched off and, on the other hand, switched on at time t, respectively, statistical equilibrium having been achieved prior to the imposition of the stimulus in both instances for a dynamic variable A() are then A off ta F 1 C AB t, A on ta F 1 C AB C AB t t, 1.3 *Author to whom correspondence should be addressed. the quantity X/97/553/2597/$ The American Physical Society

2 251 YU. P. KALMYKOV, J. L. DÉJARDIN, AND W. T. COFFEY 55 C AB ta B t A B 2AA e L FP t BB W d 1.4 is the equilibrium stationary correlation function, and the symbols and designate the statistical averages over W and W, respectively, with defined in the range 1 2. Furthermore, the spectrum of A(t) ac response is A F C ABi CAB te it dt, 1.5 A and F are the Fourier components of A(t) and F(t), respectively. We remark that Eqs. 1.3 and 1.5 are particular eamples of Kubo s linear response theory 1. Moreover, an eact integral formula eists for the correlation time A defined as the area under the curve of the normalied autocorrelation function C AA (t) as is apparent from Eq. 1.4 for AB. This is see Eq. S9.14 in Ref. 2 A 1 C AA CAA tdt 1 C AA 2 1 D 2 W 2AA W d2d. 1.6 In contrast, nonlinear response theory has been much less well developed by reason of its inherent mathematicalphysical compleity see, e.g., 3,4. The calculation of the nonlinear response even for systems described by a single coordinate is a difficult task as there is no longer any connection between the step-on and step-off responses and the ac response because the response now depends on the precise nature of the stimulus as no unique response function valid for all stimuli unlike linear response eists. Such results as have been obtained have mainly emerged either by perturbation theory or by numerical simulations. However, a few eact analytical solutions of particular nonlinear step response problems eist e.g., 5 7. We shall now demonstrate that it is possible to derive an eact general equation in terms of an integral similar to Eq. 1.6 for the nonlinear step response relaation time of a system governed by onedimensional Fokker-Planck equation 1.1 just as in linear response 2. II. ANALYTIC EQUATION FOR THE NONLINEAR RESPONSE RELAXATION TIME We consider the one-dimensional Brownian movement of a particle subject to a potential V(), and we assume that the relaational dynamics of the particle obeys the Fokker- Planck equation 1.1. Let us suppose that at time t the value of the generalied potential V is suddenly changed from V I to V II e.g., by applying a strong eternal field or by a change in some parameter characteriing the system. We are interested in the relaation of the system starting from an equilibrium stationary state I with a distribution function W I () which evolves under the action of the stimulus to another equilibrium stationary state II with the distribution function W II (). Our goal is to evaluate the relaation time A of a typical dynamical variable A. This problem is intrinsically nonlinear, because we assume that changes in the magnitude of the potential are now significant. Thus the concept of relaation functions and relaation times must be used rather than correlation functions and correlation times. We define the normalied relaation function f A (t) ofa dynamical variable A by t AtA II, t f A A I A II 2.1 1, t, A I and A II are equilibrium stationary averages defined as A I 2AWI d, A II 2AWII d, 2.2 and A(t) is the time-dependent average, At 2AW,td. 2.3 The relaation time A defined as the area under the curve of f A (t) att is then given by A f A tdtlim s e st f A tdtf A, 2.4 f A(s) is the Laplace transform of f A (t). On interchanging the orders of integration over and t in Eq. 2.4, we have and 1 A A I A II 2AAII W,d, W, limw,s s W,s W,te st dt The quantity W (,) can be calculated analytically by quadratures as follows. On using the final value theorem of Laplace transformation, vi. lim sw,slim W,tW II, s t and on taking into account Eq. 1.2, from Eq. 1.1 at t we obtain

3 55 RELAXATION TIME FOR NONLINEAR RESPONSE OF A W II W I d d D d 2 d W,W d, d V II. The solution of Eq. 2.8 is W,W II ydy D 2 yw II y, y y W II W I d. Thus from Eqs. 2.5 and 2.9 we obtain 1 A A I A II 2AAII W II so that, on integration by parts, 1 A A II A I y D 2 dy d, yw II y 2 D 2 W II d, AyA II W II ydy Equation 2.11 is an eact equation for the nonlinear step response relaation time, which is analogous to Eq. 1.6 for the linear response. III. EXAMPLES As a first eample, we consider the one-dimensional noninertial translational Brownian motion of a particle in a potential V(), the variable specifies the position of the particle. The relevant Fokker-Planck Smoluchowski equation for the distribution function W of the position is given by 2 t W W II V W t, 3.1 with the initial conditions W(,)W I (), is the friction coefficient and 1/kT. For potentials, the probability current S in the stationary state is not equal to ero e.g., for the tilted periodic potential V()V cos a, the relaation time is given by Eq. 2.11, with the stationary distribution functions W I () and W II () defined as 2 W i N i e V i S i e V i e V i y D 2 y dy ii,ii, D 2 / and the constants N i and S i must be determined from the boundary and the normaliation conditions 2. On the other hand, for potentials, the probability current S at the stationary state e.g., for the bistable potential V()V (b 2 4 ), W I () and W II () are the Mawell-Boltmann distribution functions, vi. W i e V i /Z i ii,ii. Here Z I and Z II are the partition functions, and 1 and 2. The relaation time is therefore given by A A II A I e V II d, Z 1 II e V II y Z 1 I e V I y dy, AA II e V II d Similar results are obtained for the longitudinal relaation arising from the noninertial rotational Brownian motion of a dipolar particle in an eternal uniaial potential V. The relevant Fokker-Planck equation for the distribution function W of the orientations of the particle is 8,9 2 D t W sin sin W II V 1 sin sin W t, 3.5 is the polar angle which specifies the orientation of the particle or introducing a variable cos 2 D t W 1 2 WW t, V II 3.6 D is a characteristic Debye relaation time. In this case W I () and W II () are the Mawell-Boltmann distribution functions W I e V I /Z I, W II e V II /Z II, , 2 1, and Thus Eq yields D D. A 2 1 D e A II A I V II d 1 1 2, 3.8

4 2512 YU. P. KALMYKOV, J. L. DÉJARDIN, AND W. T. COFFEY 55 Z 1 II e V II y Z 1 I e V I y dy, Z i 1 2 1/2 i e i h 2 i erf i1hi i erf i1h i i 4.6 AA II e V II d IV. APPLICATION TO NONLINEAR DYNAMIC KERR EFFECT, DIELECTRIC AND MAGNETIC RELAXATION Equation 3.8 can be used to calculate the relaation time of the nonlinear dielectric and dynamic Kerr effect step responses of systems consisting of permanently polar and polariable molecules. It may also be applied to the calculation of the nonlinear relaation time of nematic liquid crystals as well as single domain ferromagnetic particles the amplitude of a strong dc field F electric or magnetic as appropriate is suddenly changed at t. The longitudinal relaation of all these systems ecluding interparticle interactions is governed by the Fokker-Planck equation 3.5, with a uniaial potential V given by Vcos cos 2 cos is the partition function, and erf i 2 e t 2 dt 4.7 is the error function of imaginary argument. From Eqs. 2.11, 4.5, and 4.6, we have 2 1 D n P n II P n I n e II 2 II d 1 1 2, 1 n1 and 2, W II W I d 1/2 e II h 2 II 2 1/2 II Z II 4.8 or V 2 2h cos, 4.1 K, F and h/2 4.2 erf ih II II erf i1h II II 1/2 e I h 2 I 2 1/2 erf ih I Z I I I erf i1h I I, 4.9 are the appropriate dimensionless anisotropy and eternal field parameters and their ratio, respectively, K is an anisotropy constant, and is the electric or magnetic dipole moment. The appropriate interpretation of the parameters D,, and in each case is given, e.g., in Refs The potential Eq. 4.1 is symmetrical when and has a barrier at /2, the potential has a maimum the height relative to the minima at, and is equal to. The potential becomes asymmetrical for and the double well structure disappears at hh s 1. The quantities of greatest interest in the nonlinear response of these systems are the relaation times n n1 and 2 of the relaation functions f 1 (t) and f 2 (t) of the first and second Legendre polynomials, vi. and f 1 tp 1 cos tp 1 cos II f 2 tp 2 cos tp 2 cos II Equation 4.3 governs the dielectric and magnetic relaation, and Eq. 4.4 governs the dynamic Kerr effect. The distribution functions in the equilibrium states I and II are given by 1 1 P 1 P 1 II e II 2 2h II d 1 2 II e II 2 2h II e II 12h II 2 1 e II 1h II 2 1/2 sinh2 II h II 3/2 erf ih Z II II II 2 II erf i1h II II, P 2 P 2 II e II 2 2h II d 3 4 II e II 2 2h II h II e II 12h II 1h II e II 1h II /2 cosh2 II h II h II sinh2 II h II 1/2 II Z II erf ih II II erf i1h II II, 4.11 W i e i 2h i 2 /Z i ii,ii, 4.5 P 1 i e i sinh2 i h i i Z i h i, 4.12

5 55 RELAXATION TIME FOR NONLINEAR RESPONSE OF A P 2 i 3e icosh2 i h i h i sinh2 i h i 2 i Z i 3h 2 i i 2 ii,ii First, we shall compare the foregoing results with those previously given. We have obtained in Refs. 6,7 by solving the infinite hierarchy of differential-recurrence relations for the averaged spherical harmonics, the eact analytical solution for the relaation time 2 for the nonlinear rise transient of the dynamic Kerr effect for nonpolar polariable molecules, which is in the notation of Ref D 2 64M 1 2, 3 2,M 3 2, 7 2, n1 2 2n2 4n12nM 2 n 1 2,2n 3 2, n1 2 2n 3 2 Here () is the gamma function 17, and M(a,b,) is the confluent hypergeometric Kummer function defined as 17 Ma,b,1 a b 1! aa1 bb1 2 2! aa1a2 3 bb1b2 3!, F, 2kT and 1 and 2 are the components of the electric polariability parallel and perpendicular to the ais of symmetry of the molecule. We remark that all the confluent hypergeometric functions appearing in Eq may be epressed in terms of the more familiar error function of imaginary argument erf i(). In particular 18, pp. 58 and 581, M 1 2, 3 2, 1 2 1/2 erf i, M 3 2, 7 2, e /2 erf i. Equations for the other M functions occurring in Eq may be obtained from Table of Ref. 18, and the recurrence relations for the confluent hypergeometric function. In our notation the model considered in Refs. 6,7 corresponds to Eq. 4.8 for n2 with h I, I, h II, II, 4.15 according to Eqs and 4.15, erf i 2 erf i 2, e2 e erf i i, erf FIG. 1. Nonlinear dielectric relaation time in the form ln 1 / D calculated from Eq. 4.8 as a function of and for a suddenly reversed dc field: II I, and II I. P 2 3 2e 4 erf i The result of calculations of the correlation time 2 from Eq is in complete agreement with that predicted by Eqs. 4.8 and e.g., for 5 we obtained 2 / D in both representations. In Ref. 7 we also evaluated the nonlinear dielectric and birefringence rise transients for a more general model of polar and polariable molecules the solution was obtained in terms of matri continued fractions. The model corresponds to h I, I, h II h, II Here we have also complete agreement between the relaation times yielded by both solutions. Equation 4.8 can also be applied to other nonlinear problems of the dynamic birefringence and dielectric relaation considered by Morita and Watanabe 4,5. In particular, Eq. 4.8 for h I h II or I II, II I yields the nonlinear dielectric relaation time for a transient process a homogeneous electric field F applied to a system of polar and polariable particles for a time sufficient to allow the system to reach the equilibrium state for t, is suddenly reversed at t Fig. 1. Furthermore, Eq. 4.8 for h I h II, II I gives the eact solution for the nonlinear birefringence and dielectric relaation times when a strong homogeneous electric field F is suddenly applied to the system a Mawell-Boltmann distribution of the orientations of particles has been established by another homogeneous electric field F 1. Let us now suppose that the amplitude of the strong dc field F is suddenly altered at time t and that, unlike in Eq. 4.19, the value of the anisotropy parameter remains unaffected, i.e., we assume h I h II, II I. 4.2

6 2514 YU. P. KALMYKOV, J. L. DÉJARDIN, AND W. T. COFFEY 55 FIG. 2. ln 1 / D calculated from Eqs. 4.8 and 4.19 as a function of for h I.1 h II.15 curve 1, h I.1 h II.25 curve 2, h I.1 h II.5, curve 3, and h I.1 h II 1 curve 4. This form of the model is appropriate to the nonlinear dielectric response of nematic liquid crystals and to the nonlinear magnetic response of an assembly of single domain ferromagnetic particles, is mainly determined by the crystalline or shape and magnetic anisotropy rather than the eternal field, in contrast to nonlinear dielectric and Kerr effect relaation of an assembly of permanently polar and polariable molecules. The linear response of this model has been evaluated else 13,16 this corresponds to an infinitesimal change in amplitude of F. The step-on, step-off, and ac linear responses are now entirely determined by the equilibrium dipole autocorrelation function C 1 (t) cos cos (t) cos 2 13, * *e cos2 cos sin d. e cos 2 cos sin d FIG. 3. ln 1 / D calculated from Eqs. 4.8, 4.19, and 4.2 as a function of for linear response h II h I.1 curve 1, nonlinear decay transient h I.25 h II.1 curve 2, nonlinear rise transient h I.1 h II.25 curve 3, and linear response h II h I.25 curve 4. bias parameter h is shown in Fig. 2, and the linear and nonlinear relaation times for the rise and reverse decay transients are shown in Figs. 3 and 4. It is apparent from Figs. 2 4 that the relaation process obeys an activation Arrhenius law behavior i.e., an eponential increase of 1 with increasing barrier height in a restricted range of the parameters and only. This may be eplained as follows. The relaation dynamics in the potential given by Eq. 4.1 which has in general two potential wells is determined by two relaation processes. One relaation activation process governs the crossing of the potential barrier between two positions of equilibrium by a current of particles. Another process describes relaation inside the wells. Potential 4.1 becomes more and more asymmetrical with increasing, and the activation process is suppressed due to the depletion of the upper well 16. This depletion is achieved at values of the constant electric field F which are considerably smaller than the value of a critical field at which the double well In particular, the relaation time 1 lin of the linear response step-off relaation function f 1 lin (t)c 1 (t) is16,21, cf. Eq. 1.6, lin 2 D 1 cos 2 2 cos cos e 2 d2d, e 2 cos P 1 and cos 2 2P 2 1/3 may be readily etracted from Eqs and 4.13, respectively. Equation 4.2 may be obtained from Eq. 4.8 by writing I II, and proceeding to the limit. Alternative methods of derivation of Eq. 4.2 are given, for eample, in Refs The results of our calculations of the nonlinear step response relaation time 1 and comparison with those of the linear response are given in Figs. 2 4: the relaation time of the rise transient as a function of for various values of the FIG. 4. ln 1 / D calculated from Eqs. 4.8, 4.19, and 4.2 as a function of for a further selection of values of h: linear response h II h I.1 curve 1, nonlinear decay transient h I.5 h II.1 curve 2, nonlinear rise transient h I.1 h II.5 curve 3, and linear response h II h I.5 curve 4.

7 55 RELAXATION TIME FOR NONLINEAR RESPONSE OF A structure of the potential disappears this critical field is given by /21. In the case of the strong bias field, when the potential 4.1 transforms to the single well potential cos, we observe intrawell relaation modes only. Similar results have been obtained for the linear response in magnetic relaation of single domain ferromagnetic particles with high anisotropy barriers in the presence of a strong constant magnetic field following an infinitesimal change in that field 13,16,21. It is also apparent from Figs. 3 and 4 that nonlinear relaation times for the step-on h I h II and the reverse step-off h II h I responses may differ considerably in marked contrast to the linear response step-on and step-off solutions according to Eq. 1.3 the relaation behavior is characteried by the same relaation time. In other words, in linear response the rise and decay transients are mirror images of each other. One can also see in Figs. 3 and 4 that the nonlinear response relaation times for the step-on h I h II and for the reverse step-off h II h I responses are always less than the linear response relaation time for an infinitesimal change in h I, and are always higher than those for an infinitesimal change in h II for h I h II. This is due to the greater contribution of intrawell relaation modes to the nonlinear relaation time than to the linear one. ACKNOWLEDGMENTS One of us Yu. P. K. thanks the French Ministry of High Education and Research for a financial assistance high level Grant PECO-CEI. W.T.C. acknowledges the financial support of the Forbairt Research Collaboration Fund and the French Foreign Office. The partial support of this work by the Russian Foundation for Basic Research Grant a is also gratefully acknowledged. 1 R. Kubo, J. Phys. Soc. Jpn. 12, H. Risken, The Fokker-Planck Equation, 2nd ed. Springer, Berlin, A. Morita, Phys. Rev. A 34, A. Morita and H. Watanabe, Phys. Rev. A 35, H. Watanabe and A. Morita, Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice Wiley, New York, 1984, Vol. 56, pp J.-L. Déjardin, P. Blaise, and W. T. Coffey, Phys. Rev. E 54, W. T. Coffey, J. L. Déjardin, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 54, J.-L. Déjardin, Dynamic Kerr Effect World Scientific, Singapore, W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation World Scientific, Singapore, A. J. Martin, G. Meier, and A. Saupe, Symp. Faraday Soc. 5, W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Physica A 213, W. F. Brown, Jr., Phys. Rev. 13, W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B 51, Yu. L. Raikher and M. I. Shliomis, Advances in Chemical Physics, edited by W. T. Coffey Wiley, New York, 1994, Vol. 87, p W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, E. S. Massawe, and J. T. Waldron, Phys. Rev. E 49, D. A. Garanin, Phys. Rev. E 54, Handbook of Mathematical Functions, edited by M. Abramowit and I. Stegun Dover, New York, A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Additional Chapters Science, Moscow, A. Sabo, J. Chem. Phys. 72, G. Moro and P. L. Nordio, Mol. Phys. 56, W. T. Coffey and D. S. F. Crothers, Phys. Rev. E 54,