Stochastic Time

0.1 MB·

by  Toru Ohira

Stochastic Time
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0 Toru Ohira
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SonyComputerScienceLaboratories,Inc.,Tokyo,Japan141-0022

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J Abstract. Wepresentasimpledynamicalmodeltoaddressthequestionofintroducingastochastic

7 nature in a time variable. This model includes noise in the time variable but not in the “space"
variable,whichisoppositetothenormaldescriptionofstochasticdynamics.Thenotablefeatureis

] thatthesemodelscaninducea“resonance"withvaryingnoisestrengthinthetimevariable.Thus,
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theyprovideadifferentmechanismforstochasticresonance,whichhasbeendiscussedwithinthe
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normalcontextofstochasticdynamics.
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Keywords: StochasticResonance,Time,Delay
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PACS: 05.40.-a,02.50.-r,01.55.+b
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. “Time" is a concept that has gained a lot of attention from thinkers in virtually all
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c disciplines[1]. In particular, our ordinary perception of time is not the same as that of
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s space,andthisdifferencehasbeenappearinginavarietyofcontemplationsaboutnature.
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h It appears to be the main reason for the theory of relativity, which has conceptually
p brought space and time closer to receiving equal treatment, continues to fascinate and
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attract discussion in diverse fields. Also, issues such as “directions" or ”arrows" of

2 time are current interests of research[2]. Another manifestation of this difference is the
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treatmentofnoiseorfluctuationsindealingwithdynamicalsystems.Whenweconsider
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0 dynamicalsystems,whetherclassical,quantum,orrelativistic,timeiscommonlyviewed
2 as not having stochastic characteristics. In stochastic dynamical theories, we associate
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noiseandfluctuationswithonly“space”variables,suchasthepositionofaparticle,but
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6 notwiththetimevariables.Inquantummechanics,theconceptoftimefluctuationiswell
0 accepted through the time-energy uncertainty principle. However, time is not treated as
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s adynamicalquantumobservable,and aclearer understandinghas been explored[3].
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i Against this background, our main theme of this paper is to consider fluctuations
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y of time in classical dynamics through the presentation of a simple model. There are
h varietyofwaystobringingstochasticitytosometemporalaspectsofdynamicalsystems.
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The model which we present is one way, it is an extension of delayed dynamical
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models[4, 5, 6, 7, 8]. With stochastic time, we have found that the model exhibits
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X behaviors similar to those investigated in the topic of stochastic resonance[9, 10, 11],
r which are studied in a variety of fields[12, 13, 14, 15, 16]. The difference is that the
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phenomenaareinduced bynoisein timeratherthanby noisein space.
The general differential equation of the class of delayed dynamics with stochastic
timeis
dx(t¯)
= f(x(t¯),x(t¯−t )). (1)
dt¯
Here, x is the dynamical variable, and f is the “dynamical function” governing the
dynamics. t is the delay. The difference from the normal delayed dynamical equation
appearsin“time”t¯,whichcontainsstochasticcharacteristics.Wecandefinet¯inavariety
of ways as well as the function f. To avoid ambiguity and for simplicity, we focus on
the followingdynamical map system incorporating the basic ideas of the abovegeneral
definition.
xn = f(xn ,xn −t ),
k+1 k k
n = n +x (2)
k+1 k k
Here, x is the stochastic variable which can take either +1 or −1 with certain prob-
k
abilities. We associate “time” with an integral variable n. The dynamics progress by
incrementing integer k, and n occasionally “goes back a unit” with the occurrence of
x =−1.Lettheprobabilityofx =−1be pforallk,andwesetn =0.Thennaturally,
k 0
with p=0, this map reduces to a normal delayed map with n =k. We updatethe vari-
k
able x with the larger k. Hence, x in the “past” could be “re-written” as n decreases
n n
withtheprobability p.
Qualitatively,wecanmakeananalogyofthismodelwithatele–typewriteroratape–
recorder,whichoccasionallymovesbackonatape.AschematicviewisshowninFigure
1A. The recording head writes on the tape the values of x at a step, and “time” is
associated with positions on the tape. When there is no fluctuation (p = 0), the head
moves only in one direction on the tape and it records values of x for a normal delayed
dynamics.Withprobability0< p, it movesback a unitof“time”to overwritethevalue
ofx.Thequestionishowtherecordedpatternsofxonthetapeareaffectedaswechange
p.
Thedynamicalfunctionischosentobeanegativefeedback function(Figure1B)and
theconcrete mapmodelthatwewillstudyisgivenas follows.
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x =x +dd (−a x − +1), (3)
nk+1 nk nk 1+e−b xnk−t
with a , b and dd as parameters. With both a and b positive and no stochasticity
in time, this map has the origin as a stable fixed point with no delay. We consider
the case of a = 0.5, b = 6, and dd = 0.1. By a linear stability analysis, the critical
delay t , at which the stability of the fixed point is lost, is given as t ∼ 6. The
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larger delay gives an oscillatory dynamical path. We have found, through computer
simulations, that an interesting behavior arises when delay is smaller than this critical
delay. The tuned noise in the time flow gives the system a tendency for oscillatory
behavior. In other words, adjusting the value of p controlling x induces an oscillatory
dynamical path. Some examples are shown in Figure 1C. With increasing probability
for the time flow to reverse, i.e., with p increasing, we observe oscillatory behavior
both in the sample dynamical path as well as in the corresponding power spectrum.
However, when p reaches beyond an optimal value, the oscillatory behavior begins to
deteriorate. The change in the peak heights is shown in Figure 1D. This phenomenon
resembles stochastic resonance. A resonance with delay and noise, called “delayed
stochasticresonance”[20],hasbeenproposedforanadditivenoisein“space”.Analytical
understanding of the mechanism is yet to be explored for our model. However, this
mechanismofstochastictimeflow isclearly ofadifferenttypeand new.
We would like to now discuss a couple of points with respect to our model. First,
wecan viewthis modelas adynamicalmodelwith non-localityand fluctuation ontime
A B D
f[x]
1.5 25
xn-τ xn-2 xn-1xn xn+1 1 20
f 0.5 S/N 15
x
-3 -2 -1 1 2 3 10
p 1-p -0.5 5
-1
-1.5 0.1 0.2 p 0.3 0.4
C
(a) 2 7 500
1.5 6
1 5 400
xn0.5 n P(λ) 4 n 300
-0.5 100 200 300 400 500 3 200
-1 2
-1.5 1 100
-2 10 20 30λ 40 50 60 70 100 200 k 300 400 500
(b)
2 7
1.5 6 400
xn0.51 n P(λ) 45 n 300
100 200 300 400 500 3 200
-0.5
-1 2 100
-1.5 1
-2 10 20 30λ 40 50 60 70 100 200 k 300 400 500
(c) 2 7
1.5 6 300
1 5 250
xn-00.5.5 100 200 300 400 500 n P(λ) 34 n 125000
-1 2 100
-1.5 1 50
-2
10 20 30 λ40 50 60 70 100 200 k300 400 500
(d)
2 7
1.5 6 200
1 5
xn0.5 n P(λ) 4 n150
-0.5 100 200 300 400 500 3 100
2
-1.5-1 1 50
-2 10 20 30 λ40 50 60 70 100 200 k300 400 500
(e)
2 7
1.5 6 150
1 5 125
xn0.5 100 200 300 400 500 n P(λ) 34 n10705
-0.5-1 2 50
-1.5 1 25
-2 10 20 30 40 50 60 70 100 200 300 400 500
λ k
FIGURE 1. A: A schematic view of the model. B: Dynamical functions f(x) with parameters as
examples of simulations presented in this paper. Negative feedback function with parameters b = 6.
Straightlinehasslopeofa =0.5.C:Dynamics(left)andpowerspectrum(middle)ofdelayeddynamical
modelwithstochastictimeflow(right).Thisisanexampleofthedynamicsandassociatedpowerspectrum
throughthesimulationofthemodelgiveninEq.(3)withtheprobabilitypofstochastictimeflowvaried.
Theparametersaresetasa =0.5,b =6,dd =0.1,t =5andthestochastictimeflowparameterpareset
to(a) p=0,(b) p=0.1,(c) p=0.25,(d) p=0.35,and(e) p=0.45.Weusedtheinitialconditionthat
x =0.5(n≤0),andn =0.Thesimulationisperformeduptok=5000steps.Atthatpointthevalues
n 0
ofx for0≤n≤LwithL=512arerecorded.The50averagesaretakenforthepowerspectrumonthis
n
recordedx .Theunitoffrequencyl issetas 1,andthepowerP(l )isinarbitraryunits.D:Thesignalto
n L
noiseratio S atthepeakheightasafunctionoftheprobability pofstochastictimeflow.Theparameter
N
settingsarethesameasinC.
axis. Both factors are familiar in “space”, but not on time. We may extend this model
to include non-locality and fluctuations in the space variable x. Proceeding in this way,
we have a picture of dynamical systems with non-locality and fluctuations on both the
time and space axes. The analytical framework and tools for such a description need to
bedeveloped,alongwithasearch forappropriateapplications.
Anotherwaymightbetoextendthepathintegralformalism.Thequestionofwhether
this extension bridges to quantum mechanics and/or leads to an alternative understand-
ing of such properties as time-energy uncertainty relations also requires further investi-
gation. Also, there is a theory of elementary particles with a fluctuation of space–time,
wherethenoisetermisaddedtothemetric[21].Ifwecanconnectourideasheretosuch
atheoryremainsto beseen.
Finally,ifthesemodelsdocapturesomeaspectsofreality,particularlywithrespectto
stochasticityin thetimeflow, this resonance may be used as an experimentalindication
forprobing fluctuations orstochasticityin time.We have previouslyproposed “delayed
stochasticresonance”[20],aresonancethatresultsfromtheinterplayofnoiseanddelay.
It was theoretically extended[22], and recently, the effect was experimentally observed
in a solid-satelaser system with a feedback loop[23]. It is left for the future to see if an
analogousexperimentaltestcould bedevelopedwithrespect tostochastictime.
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