Low-Resistant Band-Passing Noise and Its Dynamical Effects

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1 Commun. Theor. Phys. (Beijing, China) 48 (7) pp c International Academic Publishers Vol. 48, No. 1, July 15, 7 Low-Resistant Band-Passing Noise and Its Dynamical Effects BAI Zhan-Wu Department of Applied Physics, North China Electric Power University, Baoding 713, China (Received July 14, 6; Revised November 13, 6) Abstract We propose an n-order noise, which is realized by driving an n-order linear differential equation with a Gaussian white noise. The time-derivative noise is a low-resistant band-passing noise. If the derivative noise is regarded as a thermal one, the system has a vanishing effective friction and it should induce ballistic diffusion of a free particle at long times. The simulation method for the generalized Langevin equation driven by the n-order noise is discussed systematically. The features of three-order derivative noises are presented when they are applied to a ratchet system. PACS numbers: 5.4.-a Key words: n-order noise, ballistic diffusion, Langevin simulation, ratchet system 1 Introduction Much work has been carried out on the processes that take place in disordered media and systems, which show anomalous diffusive behaviors. [1] As a limit of anomalous diffusion, ballistic diffusion has been studied theoretically. [ 6] The experimental results were reported recently. [7] The noise corresponding to anomalous diffusion is a colored one. The common noises, Ornstein Uhlenbeck noise (OUN) and harmonic noise (HN), have been applied widely in various physical situations, either as an internal noise or an external noise. [8 1] On the other hand, the harmonic velocity noise (HVN) proposed in Ref. [13] can induce some novel phenomena, such as ballistic diffusion anon-unique stationary state. The HVN has the feature that the power spectrum is equal to zero in the zero frequency limit and the effective friction ˆγ() = dtγ(t) vanishes. The noises mentioned above can be realized by the solution of a one- or two-order linear differential equation driven by a Gaussian white noise. As an extension, we propose an arbitrary n-order noise, which is generated by driving an n-order linear differential equation with a Gaussian white noise. It can be expected to find applications in physical situations. They will bring different results in nonequilibrium systems. The time derivative noises have properties similar to HVN. We shall focus on the derivative noises in the present paper, and discuss the numerical simulation of generalized Langevin equation (GLE) driven by the n-order noise. The three-order noise, especially the derivative noise, is studied in detail as an example of the n-order noise; the characteristics of derivative noises are also shown when they are applied to a ratchet system. Generation of n-order Noise and Its Dynamical Effects Let y(t) denote a random force, which satisfies the following linear differential equation: dt n y(t) + a 1 n 1 dt n 1 y(t) + + a y(t) = ξ(t) (1) with ξ(t)ξ(s) = Dδ(t s). () We demonstrate firstly that y(t) is a stable process and the spectrum of autocorrelation function y(t)y(s) is a rational one. Taking the Fourier transform to Eq. (1), we obtain ỹ(ω) = ξ(ω) χ n (ω), (3) where χ n (ω) is a polynomial function of ω, χ n (ω) = (iω) n + 1 (iω) n a, (4) and ξ(ω) is the Fourier transform of ξ(t). The solution of Eq. (1) reads y(t) = 1 dω ξ(ω) exp(iωt), (5) χ n (ω) then y(t)y(s) = 1 () ξ(ω)ξ(ω ) = dωdω ξ(ω)ξ(ω ) χ n (ω)χ n (ω ) exp(iωt + iω s), (6) = D = D So equation (6) becomes y(t)y(s) = 1 dtds ξ(t)ξ(s) exp(iωt + iω s) dtdsδ(t s) exp(iωt + iω s) dt exp(i(ω + ω )t) = D()δ(ω + ω ). (7) D dω exp(iω(t s)), (8) i.e., the process y(t) is stable and the spectrum of y(t)y(s) is rational. Multiplying y(s) (s < t) to the

2 No. 1 Low-Resistant Band-Passing Noise and Its Dynamical Effects 113 two sides of Eq. (1) and taking the ensemble average, we have dt n y(s)y(t) + a 1 n 1 y(s)y(t) + dtn 1 + a y(s)y(t) =. (9) Here we have used y(s)ξ(t) = for s < t, which is the demand of the Fluctuation-Dissipation Theorem (FDT). By using the FDT, the memory kernel function γ(t s) satisfies Eq. (9), which is the corresponding homogeneous equation of Eq. (1), dt n γ(t s) + a 1 n 1 γ(t s) + dtn 1 + a γ(t s) =. (1) If y(t), the solution of Eq. (1) in the coordinate space, is used as a thermal coloreoise to drive GLE, we have ẋ = v, v = dt γ(t t )v(t ) + 1 m y(t) + 1 m f(x) = w f(x), (11) m where m is the mass of the particle, f(x) = U/ x is the potential force and w 1 (t) = dt γ(t t )v(t )+y(t)/m. By further introducing variables w,..., w n in a similar way, we give a set of Markovian Langevin equations, where dx dt = AX + B, (1) X = x v w 1 w n 1 w n f(x)/m, B =, A = ξ(t)/ 1 1 γ() γ (n ) () 1 γ (n 1) () a For the derivative thermal noise y (m) (t) (1 m n), the GLE is written as ẋ = v, v = ηδ mn v dt γ m (t t )v(t ) + 1 m y(m) (t) + 1 m f(x) 1 = ηδ mn v + w m y(m) (t) + 1 f(x), (13) m In Eq. (13), we have introduced a variable w 1 = dt γ m (t t )v(t ). By introducing variables w,..., w n similarly, we have another Markovian Langevin equations, where X = x v w 1 w n 1 w n, B = y (m) (t)/m + f(x)/m dx dt. = AX + B, (14), A = Here the initial values γ (i) n () should be regarded as the δ-function term in γ n (t) has been removed. It is worth while to notice that the derivative thermal noises can induce the ballistic diffusion of a free particle. [3] Applying / t s to both sides of Eq. (8), we can obtain ẏ(t)ẏ(s) = 1 dω Dω exp(iω(t s)). (15) 1 ηδ mn 1 γ m () γ (n ) m () 1 γ m (n 1) () a Similarly, y (m) (t)y (m) (s) = 1 1 dω Dωm. exp(iω(t s)). (16) For Markovian anormal non-markovian processes, the viscosity strength η is defined as η = γ(t)dt; however, this quantity is equal to zero in the case of bal- listic diffusion. The noise y n (m) ( m n) can re-

3 114 BAI Zhan-Wu Vol. 48 duce to a white noise by taking limits a m, namely y n (m) (t)y n (m) (s) = mηk B T δ(t s). So the noise intensity square D of y n (m) should be taken as D a m, where D = mηk B T. The power spectrum of the noise yn m (t) is given by γ (ω) = mηk BT a m. (17) The memory kernel function is given by γ(t) = 1 dω ηa m exp(iω(t s)). (18) The initial values γ m (i) () in Eqs. (1) and (14) can be calculated as follows. Expanding ˆγ(p) (term ˆγ(p )δ mn in ˆγ(p) has been removed) into a Laurant series, C j+1 ˆγ(p) =, (19) pj+1 j= we have (the possible δ-function term has been removed) C i+1 γ(t) = t i, () i! γ (i) i= m () = ( 1) m a m γ (m+i) () = ( 1) m a m a C m+i+1. (1) When the potential is a quadratic form, we can obtain the analytical solution of Eqs. (1) and (14) by matrix theory, a X(t) = exp(ta)x() + exp(ta) ds exp( sa)b(s). () Note that the second row of A has been replaced by (f 1,, 1,,..., ), the second row of B has been replaced by f, where f 1 and f are constants. There are several methods to calculate the matrix function f(a) = exp(ta), for example, the method of undetermined coefficient. The matrix elements f ij (t) can be expressed as f ij (t) = R n+1 k=1 m,l= C k mlt m exp(lλ k t), (3) where λ k (k = 1,,..., R) are roots of characteristic polynomial det(λi A), R is the number of roots, an the degree of the linear differential equation. y (m) (t) can be calculated by ordinary method or matrix theory. Based on the Markovian typical Eqs. (1), (14), and (1), an accurate and fast approach for numerically solving GLE with n-order noise is developed. The algorithm combines the closed integration for both damping anoise terms with the Runge Kutta method for nonlinear force in a set of Markovian Langevin equations transferred from the original equation. Simulating GLE driven by noise y n (m), we can obtain the equal-time dynamical quantities of a force-free particle v (t) and ( x(t)). If n is fixed, the process driven by y n (m) tends to its asymptotic value faster as m increases. When m is fixed, the process departs from its initial state slower as n increases. These behaviors can be understood by the spectra of them. The spectra of three kinds corresponding to γ m (t) and the memory kernel functions are given in Figs. 1 and. The instantaneous values γ m (t) can take positive or negative values for m = to 3. The spectra of derivative thermal noises vanish at zero frequency. Fig. 1 The power spectrum of three-order noise. a = 1, a 1 = 1.5, a = 1, a 3 = 1, m =, 1,, 3. Fig. The memory kernel function of three-order noise, a = 1, a 1 = 1.5, a = 1, a 3 = 1, m =, 1,, 3 (the δ(t) function term in γ 3(t) has been removed). 3 Apply Three-Order Noise to a Ratchet System Only the second-order coloreoise was used in the previous study. For a diffusive ratchet model with a green noise, the non-markovian Langevin equation describing the motion of the particle was transferred into a set of Markovian

4 No. 1 Low-Resistant Band-Passing Noise and Its Dynamical Effects 115 Langevin equations, the average velocity of the particle was numerically evaluated by Langevin simulation. [14] The simulation of Langevin equation with a broad-band coloreoise was proposed in Ref. [15]. Here we apply three-order noise to the correlation ratchet. The dynamical equation reads where U(x) is a ratchet potential and chosen to be ẋ = U (x) + Dξ 1 (t) + ɛ(t), (4) U(x) = 1 [sin(x) +.5 sin(4πx)]. (5) The external noise ɛ(t) is a three-order noise with the intensity Q, ξ 1 (t) is a Gaussian white noise with strength D and its correlation reads ξ 1 (t)ξ 1 (s) = δ(t s), which is independent of the white noise ξ(t) in ɛ(t). Fig. 3 The current J as a function of color τ, a 3 = 1, a =, a = 15, a 1 = 1.5 to (133.3 for m = ), η = 1, Q = 1, T =.1, T =, m =, 1,, 3. Fig. 4 The current J m (m =, 1,, 3) as a function of frequency ω, a = 1, a 1 = 1.5, a = 1, a 3 = 1, η = 1, T = 1, T =, A =.5. In Fig. 3, we calculateumerically the current of the particle as a function of parameter τ, where τ is the correlation time of y 3 (t), given by τ = dt y 3 ()y 3 (t) / y3(). [1] The parameters used here are T =.1, x =.19, T =, Q = 1, a 3 = 1, a =, a = 15, and a 1 varying from 1.5 to (133.3 for m = ). As τ increases, the maxima

5 116 BAI Zhan-Wu Vol. 48 ω max of power spectra for m = 1,, 3 increase, but ω max decreases for m =. The parameters allow the noise used a transition from a reoise to a green one, the current occurs reversal when τ increases continuously (the current for three-order derivative noise remains negative). In contrast, the current for m = noise remains positive. This is because the derivative noises show more greenness, but the noise with m = shows more redness. [16] The current induced by the noise with m = occurs reversal at a larger τ (corresponding to a larger ω max ), since the difference of the power spectrum in high and low frequencies is smaller. Because of a large Gaussian white noise component, the current induced by m = 3 noise is small (not shown here). We now discuss another rocking ratchet system. The dynamical equation of a particle reads mẍ + m where U(x, t) is the rocking periodic potential and is taken in the form ds γ(t s)ẋ(s) = U (x, t) + ɛ(t), (6) U(x, t) = 1 [sin(x) +.5 sin(4πx)] A cos(ωt)x, (7) γ(t) is the friction memory kernel and ɛ(t) is a thermal three-order noise that has zero mean and obeys the FDT: ɛ(t)ɛ(s) = k B T γ(t s). The dependence of the current J m on driving signal frequency ω is shown in Fig. 4 with parameters x =.19, T = 1, T =, A =.5. The magnitude of the current for time-derivative noises is much larger than y 3 (t) noise. This can be qualitatively understood from the generalized Einstein relation: x(t) f = x (t) free f /k B T, owing to the particle diffuses very fast for the derivative noises. The most remarkable feature of Fig. 4 is the oscillatory reversal of the current as a function of the external frequency. Besides roughly single-hump, there exists roughly two-humps in the same direction (J ). The current can be larger in positive direction (J, J ) or negative direction (J 1, J 3 ). 4 Conclusions We have propose-order noise, which is the solution of an n-order linear stochastic differential equation driven by a Gaussian white noise. It can result in different dynamical effects in nonequilibrium systems. The noise might be applied to various problems in the future. The GLE driven by the n-order noise can be transformed into a set of Markovian Langevin equations, so a fast and accurate simulation method has been developed. The thermal derivative noises can induce ballistic diffusion of a force-free particle. In particular, the three-order time derivative noise has been discussed in detail. Some characteristics are shown when it is applied to a ratchet system. References [1] R. Metzler and J. Klafter, Phys. Rep. 339 () 1. [] J.D. Bao and Y.Z. Zhuo, Phys. Rev. Lett. 91 (3) [3] U. Weiss, Quantum Dissipative System, World Scientific, Singapore (1999). [4] H. Grabert, P. Schramm, and G.L. Ingold, Phys. Rev. Lett. 58 (1987) 185; Phys. Rep. 168 (1988) 115. [5] E. Barkai and R.J. Silbey, J. Phys. Chem. B 14 () [6] R. Morgado, F.A. Oliveira, G.G. Batrouni, and A. Hansen, Phys. Rev. Lett. 89 () 161. [7] X.L. Wu and A. Libchaber, Phys. Rev. Lett. 84 () 317. [8] J. Iwanniszewski, I.K. Kaufman, P.V.E. McClintock, and A.J. Mckane, Phys. Rev. E 61 () 117. [9] Wu. Shun-Guang, Ren Wei, Kai Fen, and Huang Zu-Qia, Phys. Lett. A 79 (1) 347. [1] J.D. Bao, Phys. Lett. A 4 (1995) 1. [11] M. Millonas and M.I. Dykman, Phys. Lett. A 185 (1994) 65. [1] R. Bartussek, P. Hänggi, B. Lindner, and L. Schimansky- Geier, Physica D 19 (1997) 17. [13] J.D. Bao, Y.J. Song, Q. Ji, and Y.Z. Zhuo, Phys. Rev. E 7 (5) [14] J.D. Bao, H.Y. Wang, and Y.L. Song, Commun. Theor. Phys. (Beijing, China) 4 (4) 77. [15] J.D. Bao and S.J. Liu, Phys. Rev. E 6 (1999) 757. [16] J.D. Bao and Y.Z. Zhuo, Phys. Lett. A 39 (1999) 8.