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First passage times distribution dependence on noise statistics and colour in a simple dynamical system
Volume
128, number
6,7
PHYSICS
LETTERS
A
I I Aprd 1988
FIRST PASSAGE TIMES DISTRIBUTION DEPENDENCE AND COLOUR IN A SIMPLE DYNAMICAL SYSTEM Vincenzo PALLESCHI Istliuro di Fisica .4towca c Molecolare
de1 CNR,
Received 9 November 1987; revised manuscript Communicated by A.R. Bishop
Cia d~l Glardino
received
ON NOISE
STATISTICS
No. 3. 56100 Plsa. Ital?
27 January
1988: accepted
for publication
5 February
1988
The effect of changing noise statistics on first passage times (FPT) distribution 1s studied by means of d&al simulation m a simple dynamical system. The first experimental determination of the FPT distribution is given in three different situations. involving “shot”, dichotomic and gaussian noise statistics. In the first two cases, the experimental results are compared with an exact theory recently developed by Lindenberg et al., whilst the results obtained with gaussian noise statistics are compared with the theoretical prevision of Doering et al. A remarkably good agreement between theory and experiment is found. Important effects on the FPT distribution are found by increasing the correlation time 5,. of the noise.
1. Introduction
x
There has recently been a great deal of interest in the study of systems driven by stochastic fluctuations with a finite correlation time To. In the last few years a number of theoretical techniques has been developed to understand the equilibrium and dynamical properties of such systems [ l-6 1. The main purpose of this paper is the study of the dynamical properties of a system defined by the stochastic differential equation i=((t)
.
(1)
The “free process” described by eq. ( 1) can be considered as the simplest limit case of the general Langevin equation i-=4x)
+&x)<(t)
9
(2)
with v(x) =O and g(x) = 1. The fluctuating term r(t) is a zero-centered stochastic quantity of prescribed statistics. The two-times correlation function of the noise t(t) is written as (<(t)T(t’)>=Q(t-t’).
(3)
The intensity D of the noise and its correlation time ~~ can be defined in terms of the two-times correlation function Q( t- t’ ), 318
(4) XI
q.=;
s
tQ(t)dt.
(5)
0
The time that a physical process described by the general Langevin equation ( 2 ) takes to reach a given value x starting from the initial state x0 is called the first passage time (FPT). In the particular case of the system described by eq. ( 1 ), we shall define the FPT as the time needed to cross either the level +Z or the level --x. This quantity in this case thus coincides with the escape time of the system out of the interval t--X, +Z]. The first passage time, as well as its probability distribution T( x0 (X), depends greatly on the statistics and on the correlation properties of the variable c(t) [ 2,3 1. The main reason for comparing noises with the same intensity D and correlation time TV, but governed by different statistics, lies in the fact that even in the simple process described by eq. ( 1) no explicit exact solution for the first passage time distribution is known, when the coloured noise obeys gaussian statistics. As clearly pointed out in a recent paper by Lindenberg et al. [ 11, the main the-
Volume
128, number
PHYSICS
6,7
oretical difficulties arise from the fact that the FPT problem for non-markovian processes cannot generally be formulated as a boundary value problem, as usual in a Markov process (a promising alternative approach to these arguments has been recently developed by Doering et al. [ 6 ] ) . The difficulties in dealing with gaussian coloured noise have motivated studies of models of non-gaussian coloured noises. In particular, exact theories have been developed for the mean first passage time by Lindenberg et al. [ 1,4], for a system described by eq. ( 1) and driven by “shot” or dichotomic noise. The same results have been independently obtained by Hanggi and Talkner [ 5 1, in the case of dichotomic noise. However, the effects due to a change of noise statistics can often be dramatic, as will be shown in the following section by using the results of a digital simulation experiment. These effects become more and more pronounced by increasing the noise correlation time 5,. A similar study has been performed in a previous paper by Faetti et al. [ 71, for what concerns the equilibrium static properties of a different system described by a Langevin equation of the general form of eq. (2). The authors in that case studied the effect on the equilibrium probability distribution p(x) of varying the correlation time 7c of the noise, a) Shot noise
+X
xll -F
LETTERS
11April 1988
A
in the case of gaussian tomic) noise statistics.
or non-gaussian
(dicho-
2. Experiment The numerical simulation of eq. ( 1) has been performed by using the algorithm proposed by Sancho et al. [ 8 1. The system is prepared at the time t = 0 in the state x=0, k=O. Because of the absence of deterministic forces, there is no “natural” length scale for the variable x. Therefore, we can calculate the FPT distribution r(x,,l_?) by arbitrarily fixing the lower and upper boundaries at the values x0 = 0 and X= 1, respectively. The first passage time probability distribution T( 0 ) 1) between the two states x0 = 0 and X= 1 has been experimentally obtained by histogramming the results of 5000 independent realizations. Some typical trajectories x( t) are shown in fig. . 1.
The first moment of the FPT distribution first passage time) is defined by the relation
(mean
This quantity, together with the second moment the distribution
b) Dichotomic
noise
c) Gaussian
of
noise
iEJFr:~:~ FPT
-
.Y
-Y _
\ FPT
-
0
Fig. 1. Some typical realization of the escape from the region [ -2, +X], under the effect of shot, dichotomic lower part of the figure are reported the corresponding forms of the noise t(t).
and gaussian
noise. In the
319
Volume 128,
number 6,7
PHYSICS
LETTERS
ourselves to the case of positive a,, assuming the variables a, and t, to be random uncorrelated variables with distribution functions
is obtained by numerical integration directly from the experimental distribution T(xo IX). The numerical simulation of eq. ( 1) has been performed in three different cases, characterized by the different statistics which the noise t(t) obeys.
n(4)=
a random
process c(t) defined
.
r(t) = c G(t-r,) I
kexp(-4/a),
(9)
Wt,)=yexp(-X).
(10)
The parameters a and y represent the average jump size and the inverse of the average time between two successive shots, respectively. The particular choice of the distribution function Y( t,) given in eq. ( 10) always yields a delta-correlated noise r(t), i.e. Q(t-t’)=s(t-t’). Therefore, in this case the FPT distribution depends on the characteristic time 1/;J of the noise only via a simple time scale factor. Nevertheless, the comparison between experimental results and theoretical prediction can be considered a valid test of the reliability of the simulation programme. It must be noticed that, as a consequence of the particular choice of amplitude probability dis-
2. I. Case A: “shot” noise Let us consider
11 April 1988
A
as (8)
The noise consists of random impulses of amplitude a, at the times t,, as shown in fig. 1a. This fluctuating process is a kind of “shot” noise; it has a physical correspondence in the noise that often affects some electronic circuits, producing random amplitude spikes casually distributed in time. Following the theoretical tractation of ref. [ 11, we shall restrict
C
2
-Ii”\. I*
I-
.
a00
0
0
”
FPT
loo
”
150
So FPT loo
Shot
A) az0.05 B) az0.1 C) az0.2
0
5o
loo FPT
Fig. 2. First passage times distribution at the value 1 /y= 3.
320
0 150
as a function
5o
100
.
1 150
5o
E
D
,
0
FPT
loo
Noise
D) a=03 E) az0.4
150
FPT
of the shot mean amplitude
a. The characteristic
time 1 /y of the noise is kept fixed
Volume
128, number
PHYSICS
6,7
tribution ii( only the escape through the +X boundary is, in this case, allowed. The stochastic trajectory analysis technique (STAT) developed by the authors of ref. [ 1 ] leads to the following relation, for the mean first passage time between the two limits x0 and X,
(11) The results of the digital simulation of the system described by eq. (1 ), subjected to the shot noise characterized by eqs. (8)-( lo), are shown in figs. 2-4. In fig. 2 is reported the experimental dependence of the FPT distribution T(0) 1) on the shot mean amplitude a for a given value of the characteristic time 1/y of the noise. This is the first expera)
LETTERS
11April 1988
A
imental determination of the FPT distribution for a system described by a Langevin equation of the kind of eq. (2 ). In figs. 3a and 3b the mean first passage time ( T( 0 I 1) ) is reported as a function of the shot mean amplitude a and of the characteristic time of the noise. A comparison of these experimental results with the theoretical predictions of ref. [ 1 ] shows a close agreement in the whole range of parameters chosen for the experiment. In figs. 4a and 4b the standard deviation
~=J(~2(011))-(~T(OI~))2
(12)
is plotted as a function of the noise mean amplitude and characteristic time. 2.2. Case B: dichotomic noise In the case of a dichotomous stochastic process the fluctuating quantity C(t) alternately takes the values
8ow
a)
20 •I
0
B’O
0 3 0
a
’ ‘Y Fig. 3. (a) Mean first passage time as a function of the shot average amplitude a. The characteristic time ofthe noise is 1/y= 3. (b) The same quantity reported as a function of the noise characteristic time. The average amplitude ofthe shots is a~0.3. The continuous lines correspond to the predictions of the theory developed in ref. [ 11,
Fig. 4. (a) Standard deviation of the first passage times distribution as a function of shot average amplitude. The characteristic time of the noise is 1 /y= 3. (b) The same quantity reported as a function of the noise characteristic time. The average amplitude ofthe shots is a~0.3.
321
Volume
128. number
PHYSICS
6.7
LETTERS
I 1 April 1988
A
0.2
a
C
A
x0 In t-
0
10
20
30
40
3
0
50
am+
IL20
10
0
40
30
50
FPT
FPT 0.7 y
1.0-
Dichotomic
E
D
I x0 In I-
_
A)
z,
noise
= 0.01
6)
f c
= 0.1
C)
7,
= 1.0
D)
Tc
= 10.
E)
2,
= loo.
OS, O
10
20
40
30
50
0
10
20
30
40
50
FPT
FPT
0.14
b
C s 2
15c 1
0 5o
FPT
loo
0
150 5o
-1
Gaussian
FPT
loo
noise
E
0
0
5o FPT
‘O”
L-
I I
A)
2,
= 0.01
B) C)
zc z,
w El
T’, 7,
= 0.1 = 1.0 = 10. = 100.
150
So FPT
‘0°
Fig. 5. (a) First passage times distributions, in the case of dichotomic noise. The parameter D is fixed to the value D= 0.1. (b ) First passage times distributions, in the case of gaussian noise. The experimental parameters are the same as in the case of dichotomic noise.
322
1I April 1988
PHYSICS LETTERS A
Volume 128,number6,7
a and -a with a > 0, as schematically shown in fig. lb. The sojourn time in the states 5(t) =a or t(t) = -a is governed by a distribution Q(t), which in this case can be taken in the exponential form cP(t)=yexp(-yt).
(13)
It can be easily demonstrated that the two-times correlation function Q(t- t’ ) in this case becomes
Q(t-t’)=Eexp(-It-f)/r,),
(14)
sults have then been compared with the corresponding results obtained with a gaussian noise statistics (case C). In fig. 5a are shown the FPT distributions obtained in the case of dichotomic noise statistics, as a function of the correlation time T,. In figs. 6 and 7 are reported the values of the mean first passage time ( T( 0 I 1) ) and of the standard deviation of the FPT distribution (6)) respectively. In fig. 6 are also shown the predictions of the theory of ref. [ 11, proving a remarkably good agreement between theory and experimental results.
where 2.3. Case C: gaussian noise
D=a2q
(15)
and T,=1/2y.
(16)
Dichotomic noise is theoretically easier to study than gaussian noise. In fact, its time behaviour is characterized only by the two eigenstates with eigenvalues a and -a, whilst a gaussian noise can take all the values between --co and +co. This property motivated some theoretical tractates [ 1,4,5,9,10 ] and the experimental work of Faetti et al. [ 71. Under the effect of the dichotomic noise, the representative variable x of the system can assume negative as well as positive values, so that in this case also the escape through the -X boundary is allowed. An exact analytic solution for the mean first passage time ( T(x,~x) ) has been obtained by the authors of ref. [ I] in the form (V-G
IX) >
= (X2-x;
I/20+
(X-x,)m.
(17)
This result has also been obtained, by using a different procedure, by HHnggi and Talkner [ 5 1. It should be noted that the only corrections to the white noise limit, lim (7(x,, ? -0
lx))
= 1X2-x:
l/20,
(18)
are of O( r’j2 ). Therefore, these terms are of the same general form found by Doering et al. [ 6 ] for gaussian coloured noise. The digital simulation experiment in the case of dichotomic noise has been performed by keeping fixed the value of the parameter D, while varying the correlation time Tc of the noise; the experimental re-
The theory of FPT is well known in the limit case of a white gaussian process 5(t) only: in this case. the two-times correlation function Q( t- 1’ ) is given by
Q(t-t’)=2Dd(t-t’),
(19)
and the correlation time of the noise is T,=O. This fact allows an equivalent description of the dynamical system of eq. (2) in terms of a unidimensional Fokker-Planck equation, given the correct boundary and initial conditions. More physical interest is in the study of gaussian noise with correlation time T, # 0 (coloured noise). The two-times correlation function Q(t- t' ) of a coloured gaussian noise is usually taken in the exponential form
Q(t-t’)=
Eexp(-It-f
I/r,).
(20)
A gaussian noise correlated in this way is of the Ornstein-Uhlenbeck form [ 111, with
D=
(t’(t))T,
.
(21)
A number of theoretical approaches have been introduced that lead to approximated solutions for the static and dynamical properties of systems described by specific Langevin equations under the action of gaussian coloured noise [ 3,12- 141 (for an exhaustive review of these arguments, see ref. [ I] ). In particular, if t(t) is exponentially correlated one can write the equation that governs the time behaviour of the noise in the form 4(t)=-
fi(f)+n(t),
(22) 323
Volume
128. number
PHYSICS
6,7
LETTERS
A
I I April 1988
YI q Dichotomic ,
0
’
, , , ,, ,,
, , , ,,,,,
Fig. 6. Mean first passage time as a function of the continuous line represents the theoretical precision predictions of the phenomenological eq. (24) in the first passage time in the case of white gaussian noise
, , , ,, ,,,
noise correlation time of ref. [I ] in the case case of gaussian noise. (z,= 0). The parameter
where q(t) is a gaussian white noise with correlation function (~(t)q(t’)>=206(t-1’).
, , , , ,,,,
, , , ,, ,,
Gaussian
, , rm
q in the case of dichotomic noise and gaussian noise. The of dichotomic noise. whilst the dotted one represents the The arrow denotes the theoretical predictions for the mean D is fixed to the value D= 0. I
Eq. (22), together with eq. (2), leads to a formally exact, bidimensional Fokker-Planck equation. However, difficulties arise when treating the first passage time properties of multidimentional systems [ 1 1, so
(23)
q Dichotomic l Gaussian
b _
0
Fig. 7. Standard deviation of the first passage times distribution in the case of dichotomic of the noise correlation time r,. The parameter D is fixed to the value D= 0.1.
324
noise and gaussian
noise reported
as a function
Volume
128, number
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PHYSICS
that most authors have developed approximated unidimensional techniques to obtain “reduced” Fokker-Planck equations [ 3,8,12- 14 I. The limitations of these methods have been recently demonstrated by the authors of ref. [ 7 1; as a matter of fact, the entire Fokker-Planck structure may break down when the contribution of the neglected terms becomes important. This effect strongly depends on noise statistics, because the neglected terms yield corrections to the reduced Fokker-Planck equation of O(Drf ) for gaussian noise and of O(Dr,) for a non-gaussian (dichotomic) noise. In principle, the corrections due to higher order terms beyond the Fokker-Plank structure can arise for any values of the parameters D and rc,. This fact highly motivates experimental studies of the static and dynamical properties of systems described by eq. (2); unfortunately, only few controlled experimental results are up to now available. In particular the FPT problem has been experimentally analyzed by Moss et al. [ 21 by the use of analog simulation and by Hanggi et al. [ 151 with a computer simulation experiment; these results have been recently updated by Mannella and Palleschi [ 16 1. The experimental results for the free process described by eq. ( 1 ), in the case of a gaussian coloured noise, are shown in figs. 5b, 6 and 7. The effect of changing noise statistics is evident: the mean first passage time calculated in the case of gaussian noise r(t) is always higher than the corresponding one in the case of dichotomic noise; these results are shown in fig. 6. This characteristic becomes more evident by increasing the noise correlation time rc.The same trend is shown for the standard deviation m for the value T,= 100 (strongly coloured noise), the standard deviation of the FPT distribution is about four times higher in the case of gaussian noise than in the case of dichotomic noise (see fig. 7). A least-squares lit of the experimental results shows that in the gausSian noise case the mean FPT dependence on T, can be well reproduced by the phenomenological relation ( T(x,, IX)) = [x2--x:, //2D+Am,
(24)
with AZ 1.4. Remarkably, the same dependence of MFPT on T has been predicted by the authors of ref. [ 61 who, at the lowest order in the correlation time, obtain
LETTERS
A
(T(x,)~))=(x2-x~(/2D+I1,Jr,ID,
1 I April 1988
(25)
where A, is the Milne extrapolation lenght given in terms of the Riemann zeta function by IIM= -c( l/2)= 1.46035... . Therefore, it can be concluded that the qualitative dependence of the mean FPT on rc is the same, in the case of gaussian or dichotomic noise statistics. However, the approximation of a gaussian noise with a dichotomic noise cannot furnish quantitative previsions if not in the region of almost white noise (r,=O). When, otherwise, one is interested in the dynamical properties of systems subjected to highly coloured noises, specific theories must be applied which take into account the precise characteristic of the noise statistics.
3. Conclusions The results of the digital simulation experiment reported in this paper clearly show the drastic effect of changing noise statistics in the case of strongly coloured noise, This effect, if evident in the simple syscould become even more tem under study, pronounced in more complicated systems. Therefore, an accurate theoretical study of the complex interplays between noise statistics and colour, and the dynamical properties of such systems would be highly useful, in the author’s opinion, for shedding light over the first passage times theory and all the related problems.
References [ 1] K. Lindenberg, B.J. West and J. Masoliver, in: Noise in nonlinear dynamical systems, eds. P.V.E. McClintock and F. Moss (Cambridge Univ. Press, Cambridge), to be published. [2] P. Hanggi, T.J. Mroczkowski, F. Moss and P.V.E. McClintock, Phys. Rev. A 32 (1985) 695. [ 31J.M. Sancho, F. Sagues and M. San Miguel, Phys. Rev. A 33 (1968) 3399. [41 J. Masoliver, K. Lindenberg and B.J. West, Phys. Rev. A 34 (1986) 1481. [S ] P. Hanggi and P. Talkner, Phys. Rev. Lett. 49 ( 1982) 423. [ 6 ] C.R. Doering, P.S. Hagan and C.D. Levermore, Phys. Rev. Lett. 59 (1987) 2129. [ 71 S. Faetti, L. Fronzoni, P. Grigolini, V. Palleschi and G. Tropiano, submitted to J. Stat. Phys.
32.5
Volume
I28, number
6,7
PHYSICS
[ 81 J.M. Sancho, M. San Miguel. S.L. Katz and J. Gunton. Phys. Rev. A 26 ( 1982) 1589. [9] F. Sagucsand M. San Miguel, Phys. Rev. A 32 (1985) 1843. [ lo] W. Horsthemke and R. Lefever. Noise induced transition. in: Springer series in synergetics, Vol. 15 (Springer. Berlin. 1984). [ 111 M.C. Wang and G.E. Uhlenbeck. Rev. Mod. Phys. (1945) 323.
37-6
LETTERS
I I April 1988
A
[ 121 P. Grigolini. to be published in Phys. Lett. A. [ 131 R.F. Fox, Phys. Rev. A 33 (1986) 467. [ 141 P. HPnggi and P. Riseborough. Phys. Rev. A 27 ( 1983) 3379. [ 151 P. Hinggi. F. Marchesoni and P. Grigolini. 2. Phys. B 56 (1984)
333.
[ 161 R. Mannella and V. Palleschi. submitted
to Phys. Lett. .A.
128, number
6,7
PHYSICS
LETTERS
A
I I Aprd 1988
FIRST PASSAGE TIMES DISTRIBUTION DEPENDENCE AND COLOUR IN A SIMPLE DYNAMICAL SYSTEM Vincenzo PALLESCHI Istliuro di Fisica .4towca c Molecolare
de1 CNR,
Received 9 November 1987; revised manuscript Communicated by A.R. Bishop
Cia d~l Glardino
received
ON NOISE
STATISTICS
No. 3. 56100 Plsa. Ital?
27 January
1988: accepted
for publication
5 February
1988
The effect of changing noise statistics on first passage times (FPT) distribution 1s studied by means of d&al simulation m a simple dynamical system. The first experimental determination of the FPT distribution is given in three different situations. involving “shot”, dichotomic and gaussian noise statistics. In the first two cases, the experimental results are compared with an exact theory recently developed by Lindenberg et al., whilst the results obtained with gaussian noise statistics are compared with the theoretical prevision of Doering et al. A remarkably good agreement between theory and experiment is found. Important effects on the FPT distribution are found by increasing the correlation time 5,. of the noise.
1. Introduction
x
There has recently been a great deal of interest in the study of systems driven by stochastic fluctuations with a finite correlation time To. In the last few years a number of theoretical techniques has been developed to understand the equilibrium and dynamical properties of such systems [ l-6 1. The main purpose of this paper is the study of the dynamical properties of a system defined by the stochastic differential equation i=((t)
.
(1)
The “free process” described by eq. ( 1) can be considered as the simplest limit case of the general Langevin equation i-=4x)
+&x)<(t)
9
(2)
with v(x) =O and g(x) = 1. The fluctuating term r(t) is a zero-centered stochastic quantity of prescribed statistics. The two-times correlation function of the noise t(t) is written as (<(t)T(t’)>=Q(t-t’).
(3)
The intensity D of the noise and its correlation time ~~ can be defined in terms of the two-times correlation function Q( t- t’ ), 318
(4) XI
q.=;
s
tQ(t)dt.
(5)
0
The time that a physical process described by the general Langevin equation ( 2 ) takes to reach a given value x starting from the initial state x0 is called the first passage time (FPT). In the particular case of the system described by eq. ( 1 ), we shall define the FPT as the time needed to cross either the level +Z or the level --x. This quantity in this case thus coincides with the escape time of the system out of the interval t--X, +Z]. The first passage time, as well as its probability distribution T( x0 (X), depends greatly on the statistics and on the correlation properties of the variable c(t) [ 2,3 1. The main reason for comparing noises with the same intensity D and correlation time TV, but governed by different statistics, lies in the fact that even in the simple process described by eq. ( 1) no explicit exact solution for the first passage time distribution is known, when the coloured noise obeys gaussian statistics. As clearly pointed out in a recent paper by Lindenberg et al. [ 11, the main the-
Volume
128, number
PHYSICS
6,7
oretical difficulties arise from the fact that the FPT problem for non-markovian processes cannot generally be formulated as a boundary value problem, as usual in a Markov process (a promising alternative approach to these arguments has been recently developed by Doering et al. [ 6 ] ) . The difficulties in dealing with gaussian coloured noise have motivated studies of models of non-gaussian coloured noises. In particular, exact theories have been developed for the mean first passage time by Lindenberg et al. [ 1,4], for a system described by eq. ( 1) and driven by “shot” or dichotomic noise. The same results have been independently obtained by Hanggi and Talkner [ 5 1, in the case of dichotomic noise. However, the effects due to a change of noise statistics can often be dramatic, as will be shown in the following section by using the results of a digital simulation experiment. These effects become more and more pronounced by increasing the noise correlation time 5,. A similar study has been performed in a previous paper by Faetti et al. [ 71, for what concerns the equilibrium static properties of a different system described by a Langevin equation of the general form of eq. (2). The authors in that case studied the effect on the equilibrium probability distribution p(x) of varying the correlation time 7c of the noise, a) Shot noise
+X
xll -F
LETTERS
11April 1988
A
in the case of gaussian tomic) noise statistics.
or non-gaussian
(dicho-
2. Experiment The numerical simulation of eq. ( 1) has been performed by using the algorithm proposed by Sancho et al. [ 8 1. The system is prepared at the time t = 0 in the state x=0, k=O. Because of the absence of deterministic forces, there is no “natural” length scale for the variable x. Therefore, we can calculate the FPT distribution r(x,,l_?) by arbitrarily fixing the lower and upper boundaries at the values x0 = 0 and X= 1, respectively. The first passage time probability distribution T( 0 ) 1) between the two states x0 = 0 and X= 1 has been experimentally obtained by histogramming the results of 5000 independent realizations. Some typical trajectories x( t) are shown in fig. . 1.
The first moment of the FPT distribution first passage time) is defined by the relation
(mean
This quantity, together with the second moment the distribution
b) Dichotomic
noise
c) Gaussian
of
noise
iEJFr:~:~ FPT
-
.Y
-Y _
\ FPT
-
0
Fig. 1. Some typical realization of the escape from the region [ -2, +X], under the effect of shot, dichotomic lower part of the figure are reported the corresponding forms of the noise t(t).
and gaussian
noise. In the
319
Volume 128,
number 6,7
PHYSICS
LETTERS
ourselves to the case of positive a,, assuming the variables a, and t, to be random uncorrelated variables with distribution functions
is obtained by numerical integration directly from the experimental distribution T(xo IX). The numerical simulation of eq. ( 1) has been performed in three different cases, characterized by the different statistics which the noise t(t) obeys.
n(4)=
a random
process c(t) defined
.
r(t) = c G(t-r,) I
kexp(-4/a),
(9)
Wt,)=yexp(-X).
(10)
The parameters a and y represent the average jump size and the inverse of the average time between two successive shots, respectively. The particular choice of the distribution function Y( t,) given in eq. ( 10) always yields a delta-correlated noise r(t), i.e. Q(t-t’)=s(t-t’). Therefore, in this case the FPT distribution depends on the characteristic time 1/;J of the noise only via a simple time scale factor. Nevertheless, the comparison between experimental results and theoretical prediction can be considered a valid test of the reliability of the simulation programme. It must be noticed that, as a consequence of the particular choice of amplitude probability dis-
2. I. Case A: “shot” noise Let us consider
11 April 1988
A
as (8)
The noise consists of random impulses of amplitude a, at the times t,, as shown in fig. 1a. This fluctuating process is a kind of “shot” noise; it has a physical correspondence in the noise that often affects some electronic circuits, producing random amplitude spikes casually distributed in time. Following the theoretical tractation of ref. [ 11, we shall restrict
C
2
-Ii”\. I*
I-
.
a00
0
0
”
FPT
loo
”
150
So FPT loo
Shot
A) az0.05 B) az0.1 C) az0.2
0
5o
loo FPT
Fig. 2. First passage times distribution at the value 1 /y= 3.
320
0 150
as a function
5o
100
.
1 150
5o
E
D
,
0
FPT
loo
Noise
D) a=03 E) az0.4
150
FPT
of the shot mean amplitude
a. The characteristic
time 1 /y of the noise is kept fixed
Volume
128, number
PHYSICS
6,7
tribution ii( only the escape through the +X boundary is, in this case, allowed. The stochastic trajectory analysis technique (STAT) developed by the authors of ref. [ 1 ] leads to the following relation, for the mean first passage time between the two limits x0 and X,
(11) The results of the digital simulation of the system described by eq. (1 ), subjected to the shot noise characterized by eqs. (8)-( lo), are shown in figs. 2-4. In fig. 2 is reported the experimental dependence of the FPT distribution T(0) 1) on the shot mean amplitude a for a given value of the characteristic time 1/y of the noise. This is the first expera)
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A
imental determination of the FPT distribution for a system described by a Langevin equation of the kind of eq. (2 ). In figs. 3a and 3b the mean first passage time ( T( 0 I 1) ) is reported as a function of the shot mean amplitude a and of the characteristic time of the noise. A comparison of these experimental results with the theoretical predictions of ref. [ 1 ] shows a close agreement in the whole range of parameters chosen for the experiment. In figs. 4a and 4b the standard deviation
~=J(~2(011))-(~T(OI~))2
(12)
is plotted as a function of the noise mean amplitude and characteristic time. 2.2. Case B: dichotomic noise In the case of a dichotomous stochastic process the fluctuating quantity C(t) alternately takes the values
8ow
a)
20 •I
0
B’O
0 3 0
a
’ ‘Y Fig. 3. (a) Mean first passage time as a function of the shot average amplitude a. The characteristic time ofthe noise is 1/y= 3. (b) The same quantity reported as a function of the noise characteristic time. The average amplitude ofthe shots is a~0.3. The continuous lines correspond to the predictions of the theory developed in ref. [ 11,
Fig. 4. (a) Standard deviation of the first passage times distribution as a function of shot average amplitude. The characteristic time of the noise is 1 /y= 3. (b) The same quantity reported as a function of the noise characteristic time. The average amplitude ofthe shots is a~0.3.
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A
0.2
a
C
A
x0 In t-
0
10
20
30
40
3
0
50
am+
IL20
10
0
40
30
50
FPT
FPT 0.7 y
1.0-
Dichotomic
E
D
I x0 In I-
_
A)
z,
noise
= 0.01
6)
f c
= 0.1
C)
7,
= 1.0
D)
Tc
= 10.
E)
2,
= loo.
OS, O
10
20
40
30
50
0
10
20
30
40
50
FPT
FPT
0.14
b
C s 2
15c 1
0 5o
FPT
loo
0
150 5o
-1
Gaussian
FPT
loo
noise
E
0
0
5o FPT
‘O”
L-
I I
A)
2,
= 0.01
B) C)
zc z,
w El
T’, 7,
= 0.1 = 1.0 = 10. = 100.
150
So FPT
‘0°
Fig. 5. (a) First passage times distributions, in the case of dichotomic noise. The parameter D is fixed to the value D= 0.1. (b ) First passage times distributions, in the case of gaussian noise. The experimental parameters are the same as in the case of dichotomic noise.
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a and -a with a > 0, as schematically shown in fig. lb. The sojourn time in the states 5(t) =a or t(t) = -a is governed by a distribution Q(t), which in this case can be taken in the exponential form cP(t)=yexp(-yt).
(13)
It can be easily demonstrated that the two-times correlation function Q(t- t’ ) in this case becomes
Q(t-t’)=Eexp(-It-f)/r,),
(14)
sults have then been compared with the corresponding results obtained with a gaussian noise statistics (case C). In fig. 5a are shown the FPT distributions obtained in the case of dichotomic noise statistics, as a function of the correlation time T,. In figs. 6 and 7 are reported the values of the mean first passage time ( T( 0 I 1) ) and of the standard deviation of the FPT distribution (6)) respectively. In fig. 6 are also shown the predictions of the theory of ref. [ 11, proving a remarkably good agreement between theory and experimental results.
where 2.3. Case C: gaussian noise
D=a2q
(15)
and T,=1/2y.
(16)
Dichotomic noise is theoretically easier to study than gaussian noise. In fact, its time behaviour is characterized only by the two eigenstates with eigenvalues a and -a, whilst a gaussian noise can take all the values between --co and +co. This property motivated some theoretical tractates [ 1,4,5,9,10 ] and the experimental work of Faetti et al. [ 71. Under the effect of the dichotomic noise, the representative variable x of the system can assume negative as well as positive values, so that in this case also the escape through the -X boundary is allowed. An exact analytic solution for the mean first passage time ( T(x,~x) ) has been obtained by the authors of ref. [ I] in the form (V-G
IX) >
= (X2-x;
I/20+
(X-x,)m.
(17)
This result has also been obtained, by using a different procedure, by HHnggi and Talkner [ 5 1. It should be noted that the only corrections to the white noise limit, lim (7(x,, ? -0
lx))
= 1X2-x:
l/20,
(18)
are of O( r’j2 ). Therefore, these terms are of the same general form found by Doering et al. [ 6 ] for gaussian coloured noise. The digital simulation experiment in the case of dichotomic noise has been performed by keeping fixed the value of the parameter D, while varying the correlation time Tc of the noise; the experimental re-
The theory of FPT is well known in the limit case of a white gaussian process 5(t) only: in this case. the two-times correlation function Q( t- 1’ ) is given by
Q(t-t’)=2Dd(t-t’),
(19)
and the correlation time of the noise is T,=O. This fact allows an equivalent description of the dynamical system of eq. (2) in terms of a unidimensional Fokker-Planck equation, given the correct boundary and initial conditions. More physical interest is in the study of gaussian noise with correlation time T, # 0 (coloured noise). The two-times correlation function Q(t- t' ) of a coloured gaussian noise is usually taken in the exponential form
Q(t-t’)=
Eexp(-It-f
I/r,).
(20)
A gaussian noise correlated in this way is of the Ornstein-Uhlenbeck form [ 111, with
D=
(t’(t))T,
.
(21)
A number of theoretical approaches have been introduced that lead to approximated solutions for the static and dynamical properties of systems described by specific Langevin equations under the action of gaussian coloured noise [ 3,12- 141 (for an exhaustive review of these arguments, see ref. [ I] ). In particular, if t(t) is exponentially correlated one can write the equation that governs the time behaviour of the noise in the form 4(t)=-
fi(f)+n(t),
(22) 323
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I I April 1988
YI q Dichotomic ,
0
’
, , , ,, ,,
, , , ,,,,,
Fig. 6. Mean first passage time as a function of the continuous line represents the theoretical precision predictions of the phenomenological eq. (24) in the first passage time in the case of white gaussian noise
, , , ,, ,,,
noise correlation time of ref. [I ] in the case case of gaussian noise. (z,= 0). The parameter
where q(t) is a gaussian white noise with correlation function (~(t)q(t’)>=206(t-1’).
, , , , ,,,,
, , , ,, ,,
Gaussian
, , rm
q in the case of dichotomic noise and gaussian noise. The of dichotomic noise. whilst the dotted one represents the The arrow denotes the theoretical predictions for the mean D is fixed to the value D= 0. I
Eq. (22), together with eq. (2), leads to a formally exact, bidimensional Fokker-Planck equation. However, difficulties arise when treating the first passage time properties of multidimentional systems [ 1 1, so
(23)
q Dichotomic l Gaussian
b _
0
Fig. 7. Standard deviation of the first passage times distribution in the case of dichotomic of the noise correlation time r,. The parameter D is fixed to the value D= 0.1.
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that most authors have developed approximated unidimensional techniques to obtain “reduced” Fokker-Planck equations [ 3,8,12- 14 I. The limitations of these methods have been recently demonstrated by the authors of ref. [ 7 1; as a matter of fact, the entire Fokker-Planck structure may break down when the contribution of the neglected terms becomes important. This effect strongly depends on noise statistics, because the neglected terms yield corrections to the reduced Fokker-Planck equation of O(Drf ) for gaussian noise and of O(Dr,) for a non-gaussian (dichotomic) noise. In principle, the corrections due to higher order terms beyond the Fokker-Plank structure can arise for any values of the parameters D and rc,. This fact highly motivates experimental studies of the static and dynamical properties of systems described by eq. (2); unfortunately, only few controlled experimental results are up to now available. In particular the FPT problem has been experimentally analyzed by Moss et al. [ 21 by the use of analog simulation and by Hanggi et al. [ 151 with a computer simulation experiment; these results have been recently updated by Mannella and Palleschi [ 16 1. The experimental results for the free process described by eq. ( 1 ), in the case of a gaussian coloured noise, are shown in figs. 5b, 6 and 7. The effect of changing noise statistics is evident: the mean first passage time calculated in the case of gaussian noise r(t) is always higher than the corresponding one in the case of dichotomic noise; these results are shown in fig. 6. This characteristic becomes more evident by increasing the noise correlation time rc.The same trend is shown for the standard deviation m for the value T,= 100 (strongly coloured noise), the standard deviation of the FPT distribution is about four times higher in the case of gaussian noise than in the case of dichotomic noise (see fig. 7). A least-squares lit of the experimental results shows that in the gausSian noise case the mean FPT dependence on T, can be well reproduced by the phenomenological relation ( T(x,, IX)) = [x2--x:, //2D+Am,
(24)
with AZ 1.4. Remarkably, the same dependence of MFPT on T has been predicted by the authors of ref. [ 61 who, at the lowest order in the correlation time, obtain
LETTERS
A
(T(x,)~))=(x2-x~(/2D+I1,Jr,ID,
1 I April 1988
(25)
where A, is the Milne extrapolation lenght given in terms of the Riemann zeta function by IIM= -c( l/2)= 1.46035... . Therefore, it can be concluded that the qualitative dependence of the mean FPT on rc is the same, in the case of gaussian or dichotomic noise statistics. However, the approximation of a gaussian noise with a dichotomic noise cannot furnish quantitative previsions if not in the region of almost white noise (r,=O). When, otherwise, one is interested in the dynamical properties of systems subjected to highly coloured noises, specific theories must be applied which take into account the precise characteristic of the noise statistics.
3. Conclusions The results of the digital simulation experiment reported in this paper clearly show the drastic effect of changing noise statistics in the case of strongly coloured noise, This effect, if evident in the simple syscould become even more tem under study, pronounced in more complicated systems. Therefore, an accurate theoretical study of the complex interplays between noise statistics and colour, and the dynamical properties of such systems would be highly useful, in the author’s opinion, for shedding light over the first passage times theory and all the related problems.
References [ 1] K. Lindenberg, B.J. West and J. Masoliver, in: Noise in nonlinear dynamical systems, eds. P.V.E. McClintock and F. Moss (Cambridge Univ. Press, Cambridge), to be published. [2] P. Hanggi, T.J. Mroczkowski, F. Moss and P.V.E. McClintock, Phys. Rev. A 32 (1985) 695. [ 31J.M. Sancho, F. Sagues and M. San Miguel, Phys. Rev. A 33 (1968) 3399. [41 J. Masoliver, K. Lindenberg and B.J. West, Phys. Rev. A 34 (1986) 1481. [S ] P. Hanggi and P. Talkner, Phys. Rev. Lett. 49 ( 1982) 423. [ 6 ] C.R. Doering, P.S. Hagan and C.D. Levermore, Phys. Rev. Lett. 59 (1987) 2129. [ 71 S. Faetti, L. Fronzoni, P. Grigolini, V. Palleschi and G. Tropiano, submitted to J. Stat. Phys.
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[ 81 J.M. Sancho, M. San Miguel. S.L. Katz and J. Gunton. Phys. Rev. A 26 ( 1982) 1589. [9] F. Sagucsand M. San Miguel, Phys. Rev. A 32 (1985) 1843. [ lo] W. Horsthemke and R. Lefever. Noise induced transition. in: Springer series in synergetics, Vol. 15 (Springer. Berlin. 1984). [ 111 M.C. Wang and G.E. Uhlenbeck. Rev. Mod. Phys. (1945) 323.
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[ 121 P. Grigolini. to be published in Phys. Lett. A. [ 131 R.F. Fox, Phys. Rev. A 33 (1986) 467. [ 141 P. HPnggi and P. Riseborough. Phys. Rev. A 27 ( 1983) 3379. [ 151 P. Hinggi. F. Marchesoni and P. Grigolini. 2. Phys. B 56 (1984)
333.
[ 161 R. Mannella and V. Palleschi. submitted
to Phys. Lett. .A.