Physica A 392 (2013) 5283–5288

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Physica A

journal homepage: www.elsevier.com/locate/physa

Stochastic resonance in periodic potentials drivenby colored noiseKaihe Liu, Yanfei Jin ∗

Department of Mechanics, Beijing Institute of Technology, 100081, Beijing, PR China

h i g h l i g h t s

• Stochastic resonance exists in the underdamped periodic potential system.• The colored noise delays the transitions between these two dynamical states.• Noise correlation time can improve the stochastic resonance.

a r t i c l e i n f o

Article history:Received 25 April 2013Received in revised form 15 June 2013Available online 5 July 2013

Keywords:Stochastic resonancePeriodic potentialColored noiseStochastic energetics

a b s t r a c t

Westudied themotion of anunderdampedBrownianparticle in a periodic potential subjectto a harmonic excitation and a colored noise. The average input energy per period andthe phase lag are calculated to quantify the phenomenon of stochastic resonance (SR). Thenumerical results show that most of the out-of-phase trajectories make a transition to thein-phase state as the temperature increases. And the colored noise delays the transitionsbetween these two dynamical states. The each curve of the average input energy per periodand the phase lag versus the temperature exist a mono peak and SR appears in this system.Moreover, the optimal temperature where the SR occurs becomes larger and the region ofSR grows wider as the correlation time of colored noise increases.

© 2013 The Authors. Published by Elsevier B.V.

1. Introduction

Stochastic resonance proposed by Benzi et al. [1] and Nicolis and Nicolis [2] is a phenomenon where the response of adynamical system subject to an input is enhanced by the addition of an optimal amount of noise. Since SR was proposed,it has been extensively investigated both theoretically and experimentally in many fields [3–10]. In order to detect thisphenomenon, there are several observables to quantify SR. For example, Benzi et al. [1] used the intensity of a peak in thepower spectrum to quantify SR. Zhou et al. [11] employed the residence-time distribution to explain SR as a resonancesynchronization phenomenon. Stochastic energetics were first proposed by Sekimoto [12], which enable us to analyze theenergetics of both thermodynamical processes [13–15] and non-equilibrium processes [16,17] described by the Langevinequations and the Fokker–Planck equations as their equivalent counterparts. Stochastic energetics mainly explore someconcerning energies of the system like the work done by some external agent. When the resonant motion is enhanced, thework is expected to become larger. Toshiya Iwai [18] numerically analyzed SR by themethod of the stochastic energetic andthe results were quite satisfactory.

Periodic potential is quite common in some physical and biological systems like Feynman’s ratchet and Josephsonjunction. However, most of these studies considered the bistable systems; only a few publications dealt with SR in periodic

∗ Corresponding author.E-mail address: jinyf@bit.edu.cn (Y. Jin).

0378-4371© 2013 The Authors. Published by Elsevier B.V.http://dx.doi.org/10.1016/j.physa.2013.06.050

Open access under CC BY-NC-ND license.

Open access under CC BY-NC-ND license.

5284 K. Liu, Y. Jin / Physica A 392 (2013) 5283–5288

potential systems [19–21]. Among them, Kim and Sung [19] found the resonance behavior in the frequency-dependentmobility in a periodic potential by using a linear-response theory. Saikia and co-workers [20] explored the possibility of theoccurrence of SR in a periodic sinusoidal potential with a Gaussian white noise by using input energy and hysteresis looparea as quantifiers. Their numerical work shows that the periodic potential system indeed exhibits SR in the high-frequencyregime. Reenbohn and co-workers [21] employed the stochastic energetics to study the motion of an underdampedBrownian particle in a bistable periodic potential and a washboard potential subjected to a sinusoidal external field. Theyfound that SR did occur.

Generally, Gaussian white noises without any time correlation are used to describe the environment fluctuations forsimplicity. However, in most situations the correlation times are long enough not to be ignored and colored noises arenecessarily considered. Previous investigations show that the colored noise plays an important role for SR [22–26]. Kisset al. [22] reported that the colored noise could be used as additive noise for SR in a physical system. Zhang et al. [23]investigated a single-mode laser cubic model driven by multiplicative colored noise and discussed the effects of cross-correlation between real and imaginary parts of quantum noise on first-passage time. Ghosh [24] studied the diffusion ratefor an overdamped Brownian particle in a cosine potential with colored noise and found that the rate could be enhancedby the finite correlation time. Jin et al. [25,26] studied the effects of two different kinds of colored noises on the mean first-passage time and SR. To our knowledge, the SR of an underdamped Brownian particle in a periodic potential subject to aharmonic excitation and a colored noise has not been reported yet. In this article, we show the effects of colored noise onthe motion of an underdamped Brownian particle in a periodic potential driven by Gaussian colored noise. In Section 2, wenumerically calculate the input energy and the phase lag, which can be used to explore the possibility of the occurrence ofSR. In Section 3, we present the summary of our results and discussions.

2. The model and some numerical results

2.1. The periodic potential model

Consider an underdamped Brownian particle in a periodic potential subject to a harmonic signal and a colored noise, theoutput of which is described by the following Langevin equation:

d2x(t)dt2

= −γ0dx(t)dt

+ cos x(t) + η(t) + F(t), (1)

where γ0 is the friction coefficient, and F(t) = F0 cos(ωt) is the harmonic signal. The noise term η(t) represents the inherentrandom fluctuation in the system and is colored noise, which satisfies the following statistical properties:

⟨η(t)⟩ = 0,η(t)η(t ′)

=

√γ0Tτ

exp

−

t − t ′

τ

; (2)

here T is the environment temperature and is in the units of the Boltzmann constant kB. T represents the noise intensity. Inthe following, the values of system parameters are fixed as γ0 = 0.12, F0 = 0.2, ω = π/4, τ = 0.6, and some numericalresults can be given by using the stochastic energetics.

Using the stochastic energetics formulation proposed by Sekimoto [12], the input energy, or work done by the field onthe system W in a period τω =

2πω, is calculated as

W (t0, t0 + τω) =

t0+τω

t0

∂U(x(t), t)∂t

dt, (3)

where the potential U(x(t), t) = − sin x − F0x cos (ωt). The average input energy per period over an entire trajectoryW is

W =1N

Nn=0

W (nτ , (n + 1) τ ) . (4)

In the following numerical simulation, the number of periods is chosen as N = 2500 and the initial conditions are taken asv(0) = 0 and x(0) at N1 = 100 equispaced intervals, xi = −π/2+ i∗2π/100, i = 1, 2, . . . , 100. Numerical calculations ofthe model are performed for various values of T by the Heunmethod. The average input energy per period

Wis calculated

by averagingW over all the trajectories.

2.2. The input energy

Fig. 1 shows the plot of input energyW averaged over a trajectory with initial position x(0) at different temperatures T .From Fig. 1(a), it is seen that W is confined to two narrow bands around 0.062 and 1.217 corresponding to x(0) values atvery low temperature. That is, the trajectories exist for two dynamical states and are quite stable at the low temperatureT = 0.001. In Fig. 1(b), the trajectories are given for the temperature T = 0.021. It is observed that there are only afew transitions from the higher value (out-of-phase state) to the lower value (in-phase state). And in Fig. 1(c) and (d), the

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(a) T = 0.001. (b) T = 0.021.

(c) T = 0.059. (d) T = 0.101.

Fig. 1. Plot of W̄ with x(0) for different temperatures: (a) T = 0.001; (b) T = 0.021; (c) T = 0.059; (d) T = 0.101.

Fig. 2. Plot of x(t) with T = 0.001 for different initial conditions.

trajectories are presented for the case of T = 0.059 and T = 0.101. Obviously, most of the out-of-phase trajectories make atransition to the in-phase state as the temperature increases. It is found that the critical temperature of transition is much

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Fig. 3. Plot of x(t) with T = 0.5.

Fig. 4. Plot ofWas a function of T .

larger than that presented in Ref. [18] for Gaussian white noise. So the colored noise delays the transitions between thesetwo dynamical states.

The two dynamical states of trajectories are distinctly identified by the phase difference ϕ between the harmonic signalF = F0 cos(ωt) and the output signal x(t) = |A(ω)| cos(ωt + ϕ), one having a fixed phase lag ϕ = ϕ1 and the otherϕ = ϕ2. Fig. 2 shows the output signal x(t) for the temperature T = 0.001 with two different initial positions x(0) = 0 andx(0) = −π/2+ π/50. According to Fig. 1(a), the input energiesW = 0.062 correspond to in-phase trajectories with phaselag ϕ1 = −0.013π , and those with W = 1.217 correspond to out-of-phase trajectories with phase lag ϕ2 = −0.5π . Fig. 3is a plot of x(t) for T = 0.5 with x(0) = 0. It is found that there are numerous interwell transitions between the in-phaseand out-of-phase states as the temperature increases. These interwell transitions assist intrawell transitions. The inset is anamplified figure to show these two dynamical states.

2.3. Stochastic resonance

In this section, we explore the possibilities of the occurrence of SR by studying the T dependence of the average inputenergy and the phase lag. The average input energy

Was a function of temperature T with different correlation times

K. Liu, Y. Jin / Physica A 392 (2013) 5283–5288 5287

Fig. 5. Plot ofWas a function of T for T < 0.1.

Fig. 6. Plot of φ as a function of T .

τ is shown in Fig. 4. It is seen that the curve ofWversus T exhibits a minimum first and a maximum later. This typical

phenomenon is a signature of stochastic resonance. The optimal temperature where the SR occurs becomes larger and theresonance region grows wider as the correlation time increases. Fig. 5 shows the curves of

Wfor the case of T < 0.1. It is

seen that the positions of the higher peaks and the lower peaks are both shifted to right with the increase of τ . In Fig. 6, thecurves of the phase lag φ are plotted as a function of temperature T with different correlation times τ . It can be found thatthe φ demonstrates a resonance-like structure with an increase of T . The position of the peak shifts right with an increase incorrelation time. Moreover, comparing Figs. 5 and 6, we find that at the temperature where the phase lag has a maximum,the average input energy has a minimum.

3. Conclusion

The SR of an underdamped Brownian particle in a periodic potential driven by Gaussian colored noise is numericallyanalyzed by the stochastic energetics. Compared with the Gaussian white noise system, the particles in our system are alsoshown to be effectively in two and only two dynamical states of their trajectories with distinct amplitudes and phase lagto the external signal. These dynamical states are stable with fixed energies at low temperatures, but transitions betweenthem take place as the temperature increases. Furthermore, the colored noise can delay the transitions between these two

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dynamical states, thus influences the process of SR. The average input energy per period of the drive signal and the phaselag shows SR as a function of temperature. It is found that as the noise correlation time increases, the optimal temperaturewhere the SR occurs becomes larger and the region of SR becomes wider.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No. 11272051.

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