Work Fluctuations and Stochastic Resonance

0.19 MB·English

by  Shantu Saikia

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WorkFluctuations andStochasticResonance
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Shantu Saikia1,RatnadeepRoy2,A.M. Jayannavar3∗
]
h 1St.Anthony’sCollege, Shillong-793003, India
c 2Women’s College, Shillong-793003, India
e 3Institute of Physics, Sachivalaya Marg, Bhubaneswar-751005, India
m
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t
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t Abstract
s
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t We study Brownian particle motion in a double-well potential driven by an ac force. This system exhibits the
a
m phenomenonofstochasticresonance.Distributionofworkdoneonthesystemoveradriveperiodinthetimeasymptotic
regime havebeencalculated. Weshow thatfluctuationsintheinputenergy orwork donedominatethemean value.
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d Themeanvalueofworkdoneoveraperiodasafunctionofnoisestrengthcanalsobeusedtocharacterisestochastic
n resonance in thesystem. Wealso discussthevalidityof steady statefluctuation theoremsin thisparticular system.
o
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[ Keywords: Stochastic Resonance, Fluctuation Theorems
PACS: 05.40.−a;05.40.Jc;05.60.Cd;05.40.Ca
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0 1. Introduction: theresidencetimedistribution[2]haveturnedoutto
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be goodquantifierscharacterizingSRasabonafide
1
0 Stochasticresonance(SR)referstotheenhanced resonance.Recentlyithasbeenshownthatthe dif-
7 response of a system to a small deterministic peri- ferent quantifiers of SR are mathematically related
0 odic forcing in the presence of an optimal amount to each other[6]. Earlier studies on the work done
t/ ofnoise[1].Itoftenoccursinbistableandexcitable in a periodically driven bistable system by an ex-
a systemswithsubthresholdinputs.Noiseplaysacon- ternal agent has established that the average work
m
structiveroleinthisphenomenon.Thisisduetoaco- peaks around the resonance, as expected [4,5]. In
d- operativeinterplaybetweennonlinearityofthesys- ourpresentwork,westudythefluctuationsandna-
n tem,inputsignalandnoise.Thusthepowerfromthe ture of the probability distributions of work done
o wholespectrumispumpedintoasinglemodethatis over a period in a driven double-well system in the
c coherentwiththeexternaldrivingforce.Becauseof time asymptotic regime (where all averaged quan-
:
v itsgenericnature,thisphenomenon,boastsapplica- tities become periodic in time with a period of the
Xi tions in almost all areas of natural sciences. Differ- externaldrive).
ent quantifiers of SR have been discussed[3,4,5,6,7] Motivation for this study also comes from re-
r
a as regards to its validity as a bonafide resonance, cent interest in the so called fluctuation theorems
i.e.,theresonancephenomenaasafunctionofnoise and transient violation of second law in systems
strengthaswellasthedrivingfrequency.Hysteresis driven to a nonequilibrium state[8,9]. Fluctuation
loop area[3], input energy or the work done on the theorems describe properties of the distribution of
systempercycle[4,5]andareaunderthefirstpeakin various nonequilibrium quantities, such as work,
heat, entropy, etc. [8,9,10]. It is important to note
that these theorems are rigorous and applicable
∗ CorrespondingAuthor to systems driven arbitrarily far away from equi-
Email address: [email protected] (A.M.Jayannavar3).
Preprintsubmitted toElsevier 6February2008
librium. These theorems can be extremely useful 1.2 5
in analysing the role of fluctuations on the perfor-
mancecharacteristicsofenginesatnanoscales(e.g., 4
1
molecular motors). In these tiny systems interac- Vr3
tions with the environmentare dominated by ther-
mal fluctuations[8]. Moreover fluctuations in phys- 2
0.8
ical quantities are more than the mean values and
1
large fluctuations may occur which can lead to un- 0 0.1 0.2 0.3 0.4
p D
expected consequences. It has been shown recently W0.6
thatoverdampeddrivenharmonicoscillator(linear
system)satisfiesasteadystatefluctuationtheorem
(SSFT)fortheworkdoneoverasingleperiod(W ) 0.4
p
in the time asymptotic regime[11]. SSFT implies
that the probability distribution P(W ) of work
p
0.2
W satisfies the following relation[10,11,12,13,14],
p
namely,
P(Wp) =e(βWp). (1) 0 0.05 0.1 0.15 0.2 0.25 0.3
P(−Wp) D
Weshowthatthisisanartifactofthelinearityof
Fig.1.FigureshowsthevariationofaverageworkWp done
the model and in general does not hold for driven
perperiodwithtemperatureD.Insetshowsthecorrespond-
nonlinearsystems.However,weshowthatSSFTfor ingplotofrelativevarianceVr.Theparameters areA=0.1,
theworkisindeedsatisfied,ifoneinsteadconsiders and ω=0.1.
the work done over a large number of cycles[14].
Our study of input energy fluctuations also reveals a barrier height ∆V = 0.25 between the two sym-
that the relative variance is larger than one across metrical wells (minima) located at distances xm =
SR indicating that fluctuations dominate the mean ±1. We consider the case of weak forcing A|xm| <
value. ∆V.All the physical quantities are taken in dimen-
sionlessunits as prescribedinreferences[4,5].
2. The Model:
Theworkdonebytheexternaldriveonthesystem
2π
orthe input energy τ = isdefined as[4,5]
ω
ω
(cid:18) (cid:19)
The overdampedLangevinequationof a particle
t0+τω
in a double-well potential in the presence of a time ∂U(x,t)
W = dt,
periodic forceis givenby[12] p ∂t
Z
t0
dx ∂U(x)
=− +ξ(t), (2) t0+τω
dt ∂x
=−Aω x(t)cosωtdt (5)
where Z
t0
x4 x2
U(x)= − −Axsinωt. (3) This follows from the stochastic energetics formal-
4 2
ismdevelopedbySekimoto[16].
We have set friction coefficient (γ) to unity. The Numerical simulation of this model was carried
randomforcefieldξ(t)isazeromeanGaussianwhite out by using Huen’s method [17]. To calculate the
noise,i.e, work done we first evolve the system and neglect
initial transients and then work done overa period
′ ′
hξ(t)ξ(t )i=D δ(t−t ). (4)
0 (W ) is calculated (eqn.(5)). The values of W are
p p
whereD =2γk T isthestrengthofthenoise.The differentfordifferentcycles.Togetbetterstatistics
0 B
x4 x2 we have calculated Wp for 100000 different (some-
static double-well potential V(x) = − , has times evenmore)cycles.
4 2
2
70 5
D=0.02 D=0.12
60 D=0.04 D=0.16
D=0.06 4 D=0.50
50
3
p) 40 p)
W W
( (
P 30 P
2
20
1
10
0 0
-0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.4 -0.2 0 0.2 0.4
Wp Wp
(a) (b)
Fig.2. Probabilitydistributionforvarious values ofD, other parameters beingthe same as inFig.1.
3. Resultsand Discussion: imum. The nature of this maximum is a function
of other physical parameters as discussed in detail
InFig.1weplottheaverageworkdone(W )over in[4,5].IntheinsetofFig.1wehaveplottedtherel-
p
asingleperiod(inthetimeasymptoticregime)asa hW 2i−hW i2
p p
function of noise strength D(= kT = β1, in dimen- ativevarianceVr =s hWpi2 ofworkdone
sionlessunits)forlowamplitudedrivingofstrength W overaperiodasa functionofnoisestrength.
p
A = 0.1 and for frequency ω = 0.1. We clearly see We see that V > 1 in the parameter range we
r
that input energy exhibits a peak as a function of havechosen.Thisshowsthatfluctuationsdominate
D. This can be attributed to the synchronized es- the mean value. V exhibits a minimum near the
r
cape from the potential well with the external pe- valueofDatwhichSRoccurs.Itmaybenotedthat
riodic drive. We reproduce exactly the same figure whenV becomesmuchlargerthanonetheaverage
r
as in reference[5], which has been obtained using a valueceasestobeagoodphysicalvariable(nonself-
differentnumericalmethod. averaging quantity) and hence to characterise the
For the case where noise is small the particle behavior of work one has to resort to the study of
remains in one of the potential wells for a longer theentiredistributionfunction.Inthetimeperiodic
time(intrawell motion dominates) with an occas- state the average work done is dissipated into the
sional random jump to another well (Kramers’ es- systemasheat.ThusonecanidentifyhW iasahys-
p
cape over the barrier) as a function of time. When teresislossinthemedium.However,itmaybenoted
the noise is very strong a large number of interwell that fluctuations of work done cannot be identified
random switches occur for each period of the si- with heatfluctuations [11,12].
nusoid and the systems response is again random Nowwewillturntotheunderstandingofthena-
(interwell motion dominates). In between these tureofprobabilitydistributionP(W )ofW forvar-
p p
two conditions, surprisingly, there exists an opti- ious values ofnoise strengthD acrossSR peak.For
mal value of the noise that cooperatively concurs this we have plotted P(W ) as a function of W in
p p
with the periodic forcing to make almost exactly Figs.2aand2bforvariousvaluesofD.Forlowvalues
one switch per period. Hence interwell motion will of D (∼ 0.02) particle dynamics is mostly confined
be in synchrony with the input signal. Quantita- to a small amplitude intrawell oscillatory motion.
tivelythis conditionis determinedbythe matching HencethedistributionisclosertotheGaussian.For
of the two time scales [1,5], namely, the period of slightly higer values ofD(≃ 0.04and 0.06)the par-
the input signal and the Kramers’ escape rate rk, ticle makes occassionalrandom excursions into the
1 = τω → ω =rk,withrk = 1 e(0.D25)) . At other well as a function of time. This is clearly re-
(cid:18)rk 2 π 2π (cid:19) flectedasasmallasymmetryandahump(interwell
the optimal value of noise the input energyis max- motion) in the plot of P(W ). It is interesting to
p
3
55
4
4
2
70
P(Wp) )} 3 0
60 P(-Wp)exp(βWp) np2
W -2
Wp) 50 (- 1 -4
β P
p)exp( 40 np)/00 -4 -2 0 2 4
P(Wp), P(-W 2300 ln {P(W---231
10
-4
0
-0.04 -0.02 0 0.02 0.04 --55
--55 -4 -3 -2 -1 00 1 2 3 4 55
Wp βWnp
(a)
Fig. 4. Plot of ln{P(Wnp)/P(−Wnp)} vs. βWnp (over 10
3 periods).InsetshowsthecorrespondingplotofWnp (over5
P(Wp) periods). The plots areforD=0.16,ω=0.1 andA=0.1
2.5 P(-Wp)exp(βWp)
p)
W
βp( 2 note that there is substantial weight towards nega-
p)ex tive values ofWp.The negativevalues ofWp corre-
W1.5 sponds to transient second law violating hysteresis
P(- loops.AsweincreaseDfurthertheinterwelldynam-
Wp), 1 ics starts playing a prominent role. It is interesting
P( tonotethatpositionofthefirstandthesecondpeak
0.5
does not shift much, however, its width increases.
0 The weight of the probability distribution towards
-0.4 -0.2 0 0.2 0.4
Wp higher negative values of W increases. On further
p
(b) increasingD(Fig.2b)finallyasinglepeakstructure
appears. The shape of the single peak structure is
4
closertoaGaussianathighD(∼0.5).Forsuchhigh
P(Wp)
P(-Wp)exp(βWp) values of D particle makes several random excur-
Wp) 3 sionsbetweenthetwowellsduringasingletimepe-
β
p( riod of an external drive. Hence total work can be
x
e
p) treatedasanadditionofindependentincrementsof
W 2
P(- work.Thenthecentrallimittheoremleadsustoex-
p), pect that the distribution of work will be approxi-
W
P( 1 mately Gaussian. This is indeed the case. The pa-
rameterrangeforwhichwehaveexploredthedistri-
butions are broad and the variance of fluctuations
0
-0.4 -0.2 0 0.2 0.4 dominateovertheaveragevalues(Fig.1,inset).This
Wp
is oneofourmainobservations.
(c)
Finally we discuss the applicability of SSFT for
theworkdoneoverasingleperiod.Forthiswehave
plotted in Fig.3,P(W ) and P(−W )exp(βW ) for
p p p
Fig.3. three different values of D. For small D (≃ 0.02),
Plot of P(Wp) (line with circles), Fig.3a and for large D (≃ 0.5), Fig.3c P(Wp) and
P(−Wp)exp(βWp)(line with crosses) vs. Wp. Plots P(−Wp)exp(βWp)arealmostequal,thussatisfying
are for D=0.02(a), D=0.16(b) and D=0.5(c) re- SSFT for work distribution. For the intermediate
spectively.Forallplots ω =0.1andA=0.1. value of D (≃0.16), Fig.3b we see that SSFT does
not hold[18]. This clearly indicates that prediction
ofvalidityofSSFTforworkdoneoverasingleperiod
4
is restrictedonlyto adrivenoverdampedharmonic
oscillator (linear problem)[11]. This will not be the
case in the presence of nonlinearity. We consider a References
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5. Acknowledgement:
SS and RR gratefully acknowledges Institute
of Physics, Bhubaneswar for hospitality. RR also
wishes to thank DST for financial assistance
(SR/FTP/PS-33/2004).
5