We have analyzed the interplay between an externally added noise and the
intrinsic noise of systems that relax fast towards a stationary state, and
found that increasing the intensity of the external noise can reduce the total
noise of the system. We have established a general criterion for the appearance
of this phenomenon and discussed two examples in detail.Comment: 4 pages, 4 figure

A. van der Ziel

B. McNamara

F. Chapeau-Blondeau

F. Jülicher

F. Moss

F. Sagues

J. M. G. Vilar

J. M. Rubí

K. Wiesenfeld

L. Gammaitoni

L. J. DeFelice

M. San Miguel

P. C. Gailey

P. Jung

R. Benzi

S. Fauve

S. M. Bezrukov

W. Horsthemke

Z. Gingl

Physical Review Letters

English

arXiv

Crossref

Noise Suppression by Noise

We have analyzed the interplay between an externally added noise and the intrinsic noise of systems that relax fast towards a stationary state, and found that increasing the intensity of the external noise can reduce the total noise of the system. We have established a general criterion for the appearance of this phenomenon and discussed two examples in detail

Vilar, J. M. G. (José M. G.), 1972-

Rubí Capaceti, José Miguel

Diposit Digital de la Universitat de Barcelona

VOLUME 86, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 5 FEBRUARY 20019Noise Suppression by NoiseJ. M. G. Vilar1 and J. M. Rubí21Departments of Physics and Molecular Biology, Princeton University, Princeton, New Jersey 085442Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain(Received 22 May 2000)We have analyzed the interplay between an externally added noise and the intrinsic noise of systemsthat relax fast towards a stationary state, and found that increasing the intensity of the external noise canreduce the total noise of the system. We have established a general criterion for the appearance of thisphenomenon and discussed two examples in detail.DOI: 10.1103/PhysRevLett.86.950 PACS numbers: 05.40.Ca, 05.40.–aFor a long time, noise was considered to be only a sourceof disorder, a nuisance to be avoided. Recently, this viewhas been changing due to several phenomena that showconstructive facets of noise. Among them, the most widelystudied is the phenomenon of stochastic resonance, wherethe addition of noise to a system enhances its response to aperiodic force [1,2]. This counterintuitive aspect of noisehas been found under a wide variety of situations, includ-ing bistable [3] and monostable [4] systems, nondynamicalelements with [5] and without [6] threshold, and patternforming systems [7]. Similar constructive outcomes arealso found in other remarkable phenomena, such as noiseinduced transitions [8] and noise induced transport [9]. Tosome extent, the presence of noise is an unavoidable fea-ture, and as one moves from macroscopic to microscopicscales that presence becomes more and more prominent.To withdraw the noise, it is customary to reduce as muchas possible all of the external noise sources that affect thesystem since it still seems paradoxical that adding noisemight result in a less noisy system.In this Letter we show that the intrinsic noise displayedby some systems can substantially be reduced through itsnonlinear interplay with externally added noise and we es-tablish sufficient conditions for this phenomenon to oc-cur. The systems we consider are those relaxing fast toa stationary state where their properties are completelydetermined by some parameters: the state of the system,denoted by It,V, is a function of some input parameters,denoted by the set V . These systems are usually callednondynamical systems. For the sake of simplicity, we con-sider the case of a single input parameter, i.e., V  V . Thetemporal dependence of the state of the system takes intoaccount the intrinsic fluctuations. This stochastic behaviorcan be described by the mean valueIt,V   HV  (1)and the correlation functionIt,V It 1 t,V  2 It,V 2  G˜V , t , (2)where . indicates average over the noise. For practicalpurposes, it is convenient to write IV , t asIt,V   HV  1 jt,V  , (3)50 0031-90070186(6)950(4)$15.00where jt,V , with jt,V jt 1 t,V   G˜V , t, rep-resents the intrinsic noise. When the characteristic timescales of the fluctuations are smaller than any other enter-ing the system, G˜V , t  GV dt, and the power spec-trum of the fluctuations is flat for the range of frequenciesof interest. The function HV  corresponds to the deter-ministic response and GV  corresponds to the intensity ofthe intrinsic fluctuations.We now analyze how adding external noise affects theoutput of the system. We consider that a random quan-tity z t is added to a constant input V0, i.e., V t  V0 1z t. The random term is assumed to be Gaussian noisewith zero mean and correlation function z tz t 1 t s2 exp2ttF, where s2 defines the noise level and tFis the correlation time. If the correlation time tF is not toosmall, the system adapts to each value of V t and can bedescribed by an input-output relationship where the inputis V0. The external noise, however, changes the character-istics of the system which is then described byIt,V0  H0V0 1 j0t,V0 , (4)where the mean value and the total fluctuations of theoutput are taken into account by the terms H0V0 andj0t,V0, respectively. For times higher than tF , thenoise term can again be approximated by Gaussian whitenoise with zero mean, but now with correlation functionj0t,V0j0t 1 t,V0  G0V0dt.For small s2, H0V0  It,V t and G0V0 R`2`	It,V tIt 1 t,V t 1 t2 It,V t2 dt canreadily be obtained by expanding HV  and GV  inpower series of V around V0. Now the average . mustbe carried out over the two noises. By averaging first overthe internal noise, we obtainH0V0  HV0 112H 00V0 z t2 , (5)G0V0 Z `2`Ω∑GV0 112G00V0 z t2∏dt1 H 0V02z tz t 1 tædt , (6)© 2001 The American Physical SocietyVOLUME 86, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 5 FEBRUARY 2001where (0) indicates the derivative of the function withrespect to its argument. The second average leads toH0V0  HV0 112H 00V0s2, (7)G0V0  GV0 1∑2tFH0V02 112G00V0∏s2.(8)Therefore, the output noise may be decreased by theaddition of external noise when G00V0 is negative andthe correlation time of the external noise is sufficientlysmall.It may also be interesting to decrease the output noiseintensity for a fixed mean value of the output. For thispurpose, the input V0 must be tuned to a new value Vcso thatH0Vc  HV0 , (9)G0Vc  GV0 1 DGV0 . (10)If the noise level is small, the new value of the input isVc  V0 2H 00V02H 0V0s2, (11)and the variation of the output noise intensity DGV0 isgiven byDGV0s2 2tFH0V02 112G00V02 G0V0H 00V02H 0V0 , (12)which can take negative values as well.As a first example illustrating the applicability of ourresults, we will analyze a model for electrical conductionwhich displays saturation. In this model, It,V  corre-sponds to the current intensity and V corresponds to aninput voltage. To be explicit, we considerHV  VR1 1 V 212, (13)GV  Q1 1 V 212, (14)where R and Q are constants. A well-known example ofsystems exhibiting this nonlinear behavior are semicon-ductor systems that display hot electron effects [10,11].According to our previous analysis, the output noise for aninput with mean value V0 is given byGV0  G0V014tF 2 QR21 2 2V 20  1 1 V20 122R21 1 V 20 3s2.(15)This indicates that, for small correlation times and smallmean voltages, the output noise of the device may be de-creased by the addition of noise. Similar results are ob-tained when the input V0 is changed to Vc in order forthe mean value of the output to have the same value as inabsence of noise. For the new value of the input,Vc  V0 13V02 1 2V 20s2, (16)the output noise changes toGVc  G0V0 14tF 2 QR21 1 V 20 322R21 1 V 20 3s2. (17)As in the previous case, the output noise can be decreasedfor small values of the correlation time of the input noise,but now it is not required for the applied voltage to besmall. These features are illustrated in Fig. 1 for differentvalues of the correlation time tF .To extend our results to higher input noise levels, wehave computed H0V0 and G0V0 by numerically averag-ing Eqs. (13) and (14) over the fluctuating input. In Fig. 20 1 2 3V0−0.5−0.3−0.10.10.30.5[G(V0)−G0(V0)] / σ20 1 2 3V0−0.5−0.3−0.10.10.30.5[G(Vc)−G0(V0)] / σ2FIG. 1. Increase of the output noise [(a) constant input and(b) fixed mean value of the output] over the input noise for thenonlinear electric conduction model [Eqs. (13) and (14)] withQ  1 and R  1. The different lines correspond to differentinput noise correlation times (from bottom to top): tF  0.0,0.1, 0.2, 0.3, 0.4, and 0.5. All quantities and parameters aregiven in arbitrary units.951VOLUME 86, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 5 FEBRUARY 20010 1 2 3V00.20.40.60.81.0G0σ2=0σ2=1σ2=30 1 2 3V00.00.20.40.60.81.0H0σ2=0σ2=1σ2=3FIG. 2. Change of (a) the output mean and (b) the output fluc-tuations for the nonlinear electric conduction model [Eqs. (13)and (14)] with tF  0.001, Q  1, and R  1. All quantitiesand parameters are given in arbitrary units.we have shown the results for different intensities of thenoise. This figure indicates that qualitatively the same phe-nomenon appears for higher noise intensity.The second example we consider is an ionic channelmodel [12]. The characteristics of the current intensityIt,V  as a function of the input voltage are given byHV  V1 1 eDW2V , (18)GV  V 2eDW2V 1 1 eDW2V 2, (19)where W and D are constants. By using Eqs. (8) and(12), one can easily see that the input noise can reduce theoutput noise for a fixed average of either the input or theoutput. Figure 3 shows the results obtained by numericallyaveraging Eqs. (18) and (19). As in the previous example,there is a range of values of V0 where the input noise canreduce the output noise.9520 1 2V00.00.10.20.3G0σ2=0σ2=0.3σ2=0.60 1 2V00.00.40.81.21.62.0H0σ2=0σ2=0.3σ2=0.6FIG. 3. Change of (a) the output mean and (b) the output fluc-tuations for the ionic channel model [Eqs. (18) and (19)] withtF  0.001, D  10, and W  1. All quantities and parame-ters are given in arbitrary units.The way noise affects the output of the two systemsconsidered previously is depicted in Fig. 4. This figureshows the output noise intensity as a function of the meanvalue of the output. For the values of the parameters usedin Fig. 4(a), the system can have the same average outputwith less fluctuations by just adding noise. Similarly, inFig. 4(b), externally added noise greatly reduces the out-put noise around H0  0.5, but, for higher values of H0,the situation is just the opposite: noise plays its usual detri-mental role.Our analysis indicates that addition of noise to systemsthat have intrinsic noise can change their properties to theextent that they may display less noise. Other studieshave shown that intrinsic noise can be responsible for theappearance of stochastic resonance [5,13] and aperiodicstochastic resonance [14] in nondynamical systems. Inthose studies, external noise enhances the response of thesystem to a periodic or an aperiodic signal. In our case,noise does not need to cooperate with an external signal toplay a constructive role. A line of investigation for futureVOLUME 86, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 5 FEBRUARY 20010.0 0.5 1.0H00.40.60.81.0G0σ2=0σ2=1σ2=30 1 2H00.00.10.20.3G0σ2=0σ2=0.3σ2=0.6FIG. 4. Output noise intensity as a function of the mean out-put for (a) the nonlinear conduction model [Eqs. (13) and (14)with tF  0.001, Q  1, and R  1] and (b) the ionic chan-nel model [Eqs. (18) and (19) with tF  0.001, D  10, andW  1]. All quantities and parameters are given in arbitraryunits.work would be to extend our results to dynamical systems.This would need the development of new methodologies,in a similar way as has been done for nonlinear noisesources in stochastic differential equations [8,15]. Ourwork then offers new perspectives on what concerns theconstructive effects of noise in general nonlinear systems.This work was partially supported by DGICYT (Spain)Grant No. PB98-1258. J. M. G. V. thanks MEC (Spain) forfinancial support.[1] R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14,L453 (1981); S. Fauve and F. Heslot, Phys. Lett. 97A, 5(1983); B. McNamara, K. Wiesenfeld, and R. Roy, Phys.Rev. Lett. 60, 2626 (1988); L. Gammaitoni, F. Marchesoni,E. Menichella-Saetta, and S. Santuchi, Phys. Rev. Lett. 62,349 (1989).[2] F. Moss, in Contemporary Problems in Statistical Physics,edited by G. H. Weiss (Society for Industrial and Ap-plied Mathematics, Philadelphia, 1994); K. Wiesenfeld andF. Moss, Nature (London) 373, 33 (1995); L. Gammaitoni,P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys.70, 223 (1998).[3] B. McNamara and K. Wiesenfeld, Phys. Rev. A 39, 4854(1989).[4] J. M. G. Vilar and J. M. Rubí, Phys. Rev. Lett. 77, 2863(1996).[5] Z. Gingl, L. B. Kiss, and F. Moss, Europhys. Lett. 29,191 (1995); F. Chapeau-Blondeau and X. Godivier, Phys.Rev. E 55, 1478 (1997).[6] S. M. Bezrukov and I. Vodyanoy, Nature (London) 385,319 (1997).[7] P. Jung and G. Mayer-Kress, Phys. Rev. Lett. 74, 2130(1995); J. M. G. Vilar and J. M. Rubí, Phys. Rev. Lett. 78,2886 (1997).[8] W. Horsthemke and R. Lefever, Noise-Induced Transitions(Springer, Berlin, 1984).[9] F. Jülicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. 69,1269 (1997).[10] Hot Electron Transport in Semiconductors, edited byL. Reggiani (Springer-Verlag, Berlin, 1985).[11] A. van der Ziel, Noise in Solid State Devices and Circuits(Wiley, New York, 1986).[12] L. J. DeFelice, Introduction to Membrane Noise (PlenumPress, New York, 1981).[13] J. M. G. Vilar, G. Gomila, and J. M. Rubí, Phys. Rev. Lett.81, 14 (1998).[14] P. C. Gailey, A. Neiman, J. J. Collins, and F. Moss, Phys.Rev. Lett. 79, 4701 (1997).[15] M. San Miguel and J. M. Sancho, Z. Phys. B 43, 361(1981); F. Sagues, M. San Miguel, and J. M. Sancho,Z. Phys. B 55, 269 (1984).953

Noise suppression by noise

http://dx.doi.org/10.1103/PhysRevLett.86.950

Noise suppression by noise

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