Barrier Grossing Induced by Fractional Gaussian Noise
A problem of the rate of escape of a particle under the influence of the external fractional Gaussian noise is studied by using the method of numerical integration of an overdamped Langevin equation. Considering a truncated harmonic potential, the dependences of the mean escape time on the noise int...
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| Цитувати: | Barrier Grossing Induced by Fractional Gaussian Noise / O.Yu. Sliusarenko, V.Yu. Gonchar, A.V. Chechkin // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 579-585. — Бібліогр.: 51 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860121912698470400 |
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| author | Sliusarenko, O.Yu. Gonchar, V.Yu. Chechkin, A.V. |
| author_facet | Sliusarenko, O.Yu. Gonchar, V.Yu. Chechkin, A.V. |
| citation_txt | Barrier Grossing Induced by Fractional Gaussian Noise / O.Yu. Sliusarenko, V.Yu. Gonchar, A.V. Chechkin // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 579-585. — Бібліогр.: 51 назв. — англ. |
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| container_title | Український фізичний журнал |
| description | A problem of the rate of escape of a particle under the influence of the external fractional Gaussian noise is studied by using the method of numerical integration of an overdamped Langevin equation. Considering a truncated harmonic potential, the dependences of the mean escape time on the noise intensity and Hurst index are evaluated, together with the probability density functions for the escape times. It is found that, like the corresponding classical problem with white Gaussian noise, they both obey an exponential law.
За допомогою чисельного iнтегрування передемпфованого рiвняння Ланжевена дослiджено задачу про швидкiсть вильоту частинки iз потенцiальної ями пiд дiєю дробового гаусового шуму. На прикладi обрiзаного гармонiчного потенцiалу отримано залежностi середнього часу вильоту вiд iнтенсивностi шуму та показника Херста, а також обчислено функцiї розподiлу часiв вильоту. Зроблено висновок, що, як i у випадку класичної задачi з бiлим гаусовим шумом, цi величини є експоненцiальними функцiями вiдповiдних параметрiв.
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BARRIER CROSSING INDUCED BY FRACTIONAL GAUSSIAN NOISE
BARRIER CROSSING INDUCED BY FRACTIONAL
GAUSSIAN NOISE
O.YU. SLIUSARENKO,1 V.YU. GONCHAR,1 A.V. CHECHKIN1, 2
1National Science Center “Kharkiv Institute of Physics and Technology”,
Akhiezer Institute for Theoretical Physics
(1, Akademichna Str., Kharkiv 61108, Ukraine)
2School of Chemistry, Tel Aviv University
(Ramat Aviv 69978, Tel Aviv, Israel)
PACS 05.40.Fb, 02.50.Ey,
82.20.-w
c©2010
A problem of the rate of escape of a particle under the influ-
ence of the external fractional Gaussian noise is studied by using
the method of numerical integration of an overdamped Langevin
equation. Considering a truncated harmonic potential, the depen-
dences of the mean escape time on the noise intensity and Hurst
index are evaluated, together with the probability density func-
tions for the escape times. It is found that, like the corresponding
classical problem with white Gaussian noise, they both obey an
exponential law.
1. Introduction
The problem of a Brownian particle’s escape rate arises
in a lot of natural processes in physics, chemistry, and
biology, such as diffusion in solids, homogeneous nucle-
ation, electrical transport theory, chemical kinetics, un-
folding of proteins etc. (see, e.g., [1, 2] and references
therein). The first attempt to describe the process was
made by S. Arrhenius [3], who introduced the rate coef-
ficient k and noticed the dependence
k = ν exp(−βEb), (1)
where ν and β are constants, and Eb is the activation
energy. Since is was realized that the escapes could hap-
pen due to thermal noise, the further development of
the problem awaited the creation of a consistent fluctu-
ation theory, attaining the second birth after the works
of Smoluchowski [4]. First, the problem was studied in
[5], the seminal papers on the rate theory were written
by H. Kramers [6] (now the problem of the escape rate
is also known as the Kramers problem) and S. Chan-
drasekhar [7], nowadays being included in almost every
textbook on statistical physics in that “classical” form.
Later on, it was re-considered using different approaches
and in more details (see e.g. [2, 8]).
However, the theory of Brownian motion cannot ex-
plain anomalous diffusion phenomena which are widely
observed in various physical (Sinai diffusion [9], turbu-
lent Richardson flow [10,11], motion of charge carriers in
amorphous semiconductors [12, 13]), biological (motion
in biological cells [14, 15]), biochemical (the spreading
of tracer molecules in subsurface hydrology [16]), chem-
ical, and geophysical systems [17]. In such systems, the
mean squared displacement of a particle does not obey
a regular diffusion law〈
x2(t)
〉
= 2DHt
2H , (2)
where DH is a generalized diffusion coefficient, H is
Hurst exponent varying between 0 and 1. The case where
H > 1/2 (the mean squared displacement grows faster
than t1) is called superdiffusion; when H < 1/2, we have
a subdiffusion phenomenon.
Despite the “symptoms” of systems may be the same,
the anomalous diffusion has several mechanisms. The
most discussed ones nowadays are the continuous time
random walks and the fractional Brownian motion mod-
els. The former implies either subdiffusion (while walk-
ing, the particle experiences long periods of rest, so that
the waiting times have an infinite characteristic time) or
superdiffusion (e.g., Lévy flights, when the mean squared
displacement diverges, but the waiting times are finite).
The Kramers problem for Lévy flights was considered in
[17–20].
The second model (fractional Brownian motion) was
suggested by Kolmogorov in 1940 [21] and later reconsid-
ered by Mandelbrot and van Ness [22]. They defined the
fractional Brownian motion as a self-similar stochastic
process, whose formal derivative ξH(t) called the frac-
tional Gaussian noise is a stationary random process
with long memory effects. Namely, its autocorrelation
function in the discrete time approximation has the form
〈ξH(0)ξH(j)〉=DH
(
|j+1|2H−2|j|2H +|j−1|2H ,
)
. (3)
where j is an integer. At large j corresponding to a
long-time asymptotics, the autocorrelation function de-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 579
O.YU. SLIUSARENKO, V.YU. GONCHAR, A.V. CHECHKIN
cays as 〈ξH(0)ξH(t)〉 ' 2DHH(2H − 1)t2H−2, H 6= 1/2,
thus showing a long power-law memory. The “ordi-
nary” Brownian limit corresponds to H = 1/2, and〈
ξ1/2(t1)ξ1/2(t2)
〉
= 2Dδ(t1 − t2).
Despite its wide use, the fractional Brownian motion is
not completely understood, since still there are no con-
sistent analytical methods. However, the development
of modern computational devices allows one to investi-
gate the stochastic systems with such diffusion mecha-
nisms, by using various numerical simulation techniques.
Thus, the fractional Brownian motion is used to model
a variety of processes including the monomer diffusion
in a polymer chain [23], single file diffusion [24], diffu-
sion of biopolymers in the crowded environment inside
biological cells [25], long-term storage capacity in reser-
voirs [26], climate fluctuations [27], econophysics [28],
and teletraffic [29].
Due to such substantial quantity of applications, the
further development of both analytical and simulation
approaches is promising.
In this paper, we consider a generalization of the bar-
rier crossing problem by using a numerical method of
integrating the overdamped Langevin equation with a
fractional Gaussian random source ξH(t):
dx
dt
= −dU
dx
+D1/2ξH (t) , (4)
where x(t) is particle’s coordinate, U(x) is the potential,
ξH(t) is the fractional Gaussian noise with intensity D,
and H stands for the Hurst index.
We also stress that, in our Langevin description, the
fluctuation–dissipation theorem does not hold, as it will
be seen below. Therefore, our model is different from
that analyzed in [30] and [31]. On the other hand, our
approach is similar to that of paper [32]; however, the
autocorrelation function of the long-correlated Gaussian
noise used there is different from that for the fractional
Gaussian noise.
2. Simulation Details
For a simulation, we will need a reliable fast generator of
random fractional Gaussian numbers ξH(t) (see Eq. (4)).
Since the generators provide usually good results either
for H < 1/2 (antipersistent case) or for H > 1/2 (per-
sistent case), we choose two separate ones for each of
cases.
The fastest, precise enough (see the tests below), and
free of edge effects fractional Gaussian noise generator
for the antipersistent case is described in [33]. In brief,
the idea is as follows.
First, we define a function
Rx(n) =
{
2−1
[
1− (n/N)2H
]
, for 0 ≤ n ≤ N,
Rx(2N − n), forN < n < 2N,
(5)
where H is the Hurst parameter, 0 < H < 1/2; n is the
step number, and N is the random sample length.
Second, we perform the Fourier transformation of
Eq. (5): Sx(k) = F {Rx(n)} .
Next, we define
X(k) =
0, for k = 0,
exp(iθk)ξ(k)
√
Sx(k), for 0 < k < N,
ξ(k)
√
Sx(k), for k = N,
X∗(2N − k), forN < k < 2N,
(6)
where ∗ stands for the complex conjugation, θk are uni-
form random numbers from [0, 2π), ξ(k) are Gaussian
random variables with the zero mean and the variance
equal to 2, and all random variables are independent of
one another.
Finally, y(n) = x(n) − x(0), where x(n) = F−1X(k)
is the inverse Fourier transformation of Eq. (6), repre-
sents a free fractional Brownian trajectory which is to
be differentiated with respect to the time in order to get
fractional Gaussian random numbers. Since the vari-
ance
〈
ξ2
〉
depends on N, it should be normalized so that〈
ξ2
〉
= 2.
For the persistent case, we use a generator exploit-
ing the spectral properties of a fractional Gaussian noise
[34].
– First, take a white Gaussian noise ξ(t), t is an integer.
– Take the Fourier transformation of it: S(k) = F{ξ(t)}.
– Multiply it by 1/ωH−1/2, 1/2 < H < 1.
– Make the inverse Fourier transformation, so that
ξH(t) = F−1{S(k)/ωH−1/2} is supposed to approximate
a fractional Gaussian noise with index H.
– Normalize it.
A set of tests of the generators was performed to ver-
ify the correctness of the program and to determine the
validity limits of the algorithm itself at various values of
parameters.
The first and the most natural is the verification of
the autocovariance function of the noise (Eq. (3)) with
D = 1 and C(j) ≡ 〈ξH(0)ξH(j)〉:
The second test is the calculation of the mean squared
displacement of a free fractional Brownian particle,
whose results are omitted in the present paper as trivial.
More spectacular is the test of the mean squared dis-
placement of a particle in an infinite harmonic potential
well. We start from the overdamped Langevin equation
580 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
BARRIER CROSSING INDUCED BY FRACTIONAL GAUSSIAN NOISE
Fig. 1. Autocovariance function of a fractional Gaussian noise,
persistent and antipersistent cases, a log-log scale. Squares, circles,
triangles, and rhombi with error bars are the simulation data for
H=0.8, 0.6, 0.45, and 0.1, respectively. Solid lines provide the
linear fitting with Eq. (3)
(4) with the potential U(x) = ax2/2. In dimensionless
variables, we get
xn+1 − xn = −xnδt+D1/2δtHξH(n). (7)
The results of simulations and their comparison with
analytical asymptotes are shown in Fig. 2.
Finally, we verify whether the particle’s mean escape
time from a semiaxis matches the analytical scaling sug-
gested in [35]:
p(t) ∝ t−2+H . (8)
Setting U(x) ≡ 0 in Eq. (4), we come to the following
discrete-time dimensionless Langevin equation:
xn+1 − xn = δtHξH(n). (9)
Now, the simulation procedure for the mean escape
time is as follows (see the sketch in the inset in Fig. 3):
– Place a “particle” into the starting point (x = 0).
– Begin the iterations of Eq. (9).
– Stop the iterations when the “particle” reaches the ab-
sorbing boundary at x0 = 1.
– Remember the time of this escape event.
– Re-execute these steps for 100,000 times and average
the escape times.
The results demonstrate a good coincidence with
Eq. (8) (see Fig. 3). Here, the time step δt= 0.01.
Thus, after ascertaining the work of the algorithm and
the generators properly, we pass directly to the main
Fig. 2. Mean squared displacement of a particle inside a har-
monic potential well. Main graph: antipersistent case, H = 0.25
and D=0.1, 0.25, 0.5, and 1.0 (upward). Inset: persistent case,
D=1.0 and H=0.8, 0.7, and 0.6 (upward). Points are the simula-
tion data; dashed lines are the asymptotes limt→0
〈
x2(t)
〉
∝ t2H
and limt→∞
〈
x2(t)
〉
= DΓ(1 + 2H)
Fig. 3. Verification of the analytical scaling of the free semiaxis
mean escape time for a fractional Brownian motion suggested in
[35] (solid lines). Circles and triangles represent the simulation
results forH = 0.25 andH = 0.75, respectively. The inset explains
the simulation algorithm
problem. Due to the presence of two different random
fractional Gaussian noise generators and different typical
time scales for the persistent and antipersistent cases, it
is natural to subdivide the further description into two
parts.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 581
O.YU. SLIUSARENKO, V.YU. GONCHAR, A.V. CHECHKIN
Fig. 4. Mean escape time for a fractional Gaussian noise in the
antipersistent case as a function of the noise intensity, a log plot
(main panel) and a linear plot (inset). The points show the sim-
ulation data (the inaccuracies are of points’ size magnitude), the
solid lines demonstrate the fitting with Eq. (10)
3. Antipersistent Case
When dealing with a simulation of the escape problem,
two moments are of major importance: the proper se-
lection of a time step (δt) and a set capacity (N). The
latter is the generator’s parameter that determines the
number of random variables in a sample, within which
they are correlated with one another. The only way to
evaluate them correctly is to simulate some sample paths
with arbitrary δt (small enough compared to time scales)
and N (large enough). Varying them, we should achieve
a satisfactory relationship between the time needed for
the simulations and the accuracy. Certainly, they may
be different for each pair of the noise intensity and the
Hurst parameter.
In the following simulations, we take the time step
δt=0.001, and the set capacity varies from N = 213 to
N = 220. Again, starting from Eq. (7), we perform the
common procedure of evaluating the mean escape time:
– Place a “particle” into the starting point (x = 0).
– Begin the iterations of Eq. (7).
– Stop the iterations, when the “particle” reaches the
edge of the potential x0 =
√
2.
– Remember the time of this escape event.
– Re-execute these steps for 100,000 times and average
the escape times.
The results are shown in Figs. 4 and 5 below.
Fig. 5. Mean escape time for a fractional Gaussian noise in the
antipersistent case as a function of the Hurst index, a log plot. The
points show the simulation data (the inaccuracies are of points’ size
magnitude), the solid lines demonstrate the fitting with Eq. (10)
As clearly seen from Fig. 4, the data points of the
mean escape time dependence on the noise intensity may
be nicely fitted with an exponential function, so we in-
troduce the coefficients aA(H) and bA(H) (the indices A
here indicate the antipersistent case):
Tesc = exp(aA + bA/D),
or
lnTesc = aA(H) + bA(H)
1
D
. (10)
The quantity aA(H) may be fitted well with a linear
dependence
aA(H) = a′A + a′′AH, (11)
while bA(H) is better fitted with a function
bA(H) = b′A + b′′AH + b′′′AH
2, (12)
where a′A = −3.019, a′′A = 7.296, b′A = 0.705, b′′A = 1.489
and b′′′A = −2.281.
The escape times probability density function is sim-
ulated almost in the same way as the mean escape time.
But, instead of averaging the escape times at the last
step, we handle them with a routine that constructs the
probability density function. Again, like the classical
Kramers problem with a white Gaussian noise source,
the escape times probability density function obeys an
exponential law (see Fig. 6):
p(t) =
1
Tesc
exp(−t/Tesc). (13)
582 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
BARRIER CROSSING INDUCED BY FRACTIONAL GAUSSIAN NOISE
Fig. 6. Escape times probability density function as a function
of the walking time, the antipersistent case. The points show
the simulation data, the solid lines demonstrate the fitting with
Eq. (13)
Fig. 7. Mean escape time for a fractional Gaussian noise in the
persistent case as a function of the noise intensity, a log plot. The
points show the simulation data (the inaccuracies are of points’ size
magnitude), the solid lines demonstrate the fitting with Eq. (14)
4. Persistent Case
Here, the procedures are completely the same, with only
slight differences in fittings.
Figure 7 shows the mean escape time dependence on
the noise intensity, which is again exponential:
lnTesc = aP (H) + bP (H)
1
D
. (14)
Now, both aP (H) and bP (H) are, with a good preci-
sion, linear functions of H (the subscript “P ” indicates
Fig. 8. Mean escape time for a fractional Gaussian noise in the
persistent case as a function of the Hurst index, a log plot. The
points show the simulation data (the inaccuracies are of points’ size
magnitude), the solid lines demonstrate the fitting with Eq. (14)
Fig. 9. Escape times probability density function as a function of
the walking time, the persistent case. The points show the simu-
lation data, the solid lines demonstrate the fitting with Eq. (13)
the relation to the persistent case):
aP (H) = a′P + a′′PH,
bP (H) = b′P + b′′PH,
where a′P = −1.680, a′′P = 4.869, b′P = 1.051 and b′′P =
−0.399.
The mean escape time dependence on the Hurst pa-
rameter is shown in Fig. 8.
As expected, the escape times probability density
function in the persistent case is also exponential of the
form of Eq (13) (see Fig. 9).
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 583
O.YU. SLIUSARENKO, V.YU. GONCHAR, A.V. CHECHKIN
Fig. 10. Relative escape rate k (H) /k (H = 1/2) as a function of
µ = |H − 1/2| in the persistent case (squares) and the antipersis-
tent case (circles). The solid parabolic and straight lines stand for
fitting the results with Eqs. (10) and (14), respectively
5. Conclusions
We have considered the Kramers problem for the frac-
tional Brownian motion. Using the method of numerical
integration of the overdamped Langevin equation with a
fractional Gaussian random source, we have shown that,
similarly to the classical result presented, e.g., in [7],
the dependence of the mean escape time from a trun-
cated harmonic potential on the noise intensity is expo-
nential, both for the persistent and antipersistent cases.
This contradicts the conclusion made in [32], where the
stretched exponential behavior of the escape time prob-
ability density function was reported. The escape time
probability density function also behaves qualitatively
in the same manner as that of the classical Brownian
particle.
An important phenomenon is revealed when consid-
ering the diffusion rate. Let us introduce the escape
rate k = 1/Tesc and make a comparison of the escape
rate with the classical Kramers one. Figure 10 indicates
that, in the persistent case, the escape rate is smaller
than the classical one, and we have a subdiffusion phe-
nomenon. On the contrary, when dealing with the free
fractional Brownian motion, the mean squared displace-
ment is
〈
x2(t)
〉
= 2D|t|2H (the larger the value of H, the
faster the particle) and thus, the persistent noise gives
birth to the superdiffusivity. The same picture arises in
the antipersistent case, but vice versa: for the free frac-
tional Brownian motion, we have the subdiffusion, while
the walks inside a harmonic potential are superdiffusive.
However, such an event is in accordance with Molchan’s
analytics [35] discussed in Section 2.
At the end, we would like to mention some very
promising applications of the numerical results obtained
here. Namely, the model studied in this paper is relevant
for a number of real physical systems described with con-
tinuum elastic models such as flexible and semiflexible
polymers [36–38], membranes [37, 39–41], growing in-
terfaces [42–46], fluctuating surfaces [47], and diffusion-
noise systems [48]. Indeed, it was shown very recently
that the Generalized Elastic Model suited for the de-
scription of the systems listed above yields the Langevin
description with a fractional Gaussian noise [49–51].
The authors acknowledge the discussions with
J. Klafter, R. Metzler, and I. Sokolov. AVC acknowl-
edges the financial support from the MC IIF Programme,
grant “LeFrac”.
1. Noise in Nonlinear Dynamical Systems, edited by
F. Moss and McClintock (Cambridge University Press,
Cambridge, 1989).
2. P. Hänggi, P. Talkner, and M. Bokrovec, Rev. Mod. Phys.
62, 251 (1990).
3. S. Arrhenius, Z. Phys. Chem. 4, 226 (1889).
4. M.V. Smoluchowski, Ann. Phys. 21, 756 (1906).
5. L.A. Pontryagin, A.A. Andronov, and A.A. Vitt, Zh.
Eksp. Teor. Fiz. 3, 165 (1933).
6. H.A. Kramers, Physica A 7, 284 (1940).
7. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
8. A.N. Malakhov, Chaos 7, 3 (1997).
9. Y. Sinai, Theor. Prob. Appl. 27, 256 (1982).
10. L.F. Richardson, Proc. Roy. Soc. London A 110, 709
(1926).
11. G. Boffetta and I.M. Sokolov, Phys. Rev. Lett. 88, 094501
(2002).
12. H. Scher and E.W. Montroll, Phys. Rev. B 12, 2455
(1975).
13. G. Pfister and H. Scher, Adv. Phys. 27, 747 (1978);
Q. Gu, E.A. Schiff, S. Grebner, and R. Schwartz, Phys.
Rev. Lett. 76, 3196 (1996).
14. A. Caspi, R. Granek, and M. Elbaum, Phys. Rev. Lett.
85, 5655 (2000).
15. A. Caspi, R. Granek, and M. Elbaum, Phys. Rev. E 66,
011916 (2002).
16. H. Scher, G. Margolin, R. Metzler, J. Klafter, and
B. Berkowitz, Geophys. Res. Lett. 29, 1061 (2002);
B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, Rev.
Geophys. 44, RG2003 (2006).
17. P.D. Ditlevsen, Phys. Rev. E 60, 172 (1999).
584 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
BARRIER CROSSING INDUCED BY FRACTIONAL GAUSSIAN NOISE
18. A.V. Chechkin, J. Klafter, and I.M Sokolov, Europhys.
Lett. 63, 3 (2003).
19. P. Imkeller and I.J. Pavlyukevich, Phys. A 39, L237
(2006); P. Imkeller and I. Pavlyukevich, Stoch. Proc.
Appl. 116, 611 (2006).
20. A.V. Chechkin, O.Yu. Sliusarenko, R. Metzler, and
J. Klafter, Phys. Rev. E 75, 041101 (2007).
21. A.N. Kolmogorov, Dokl. Acad. Sci. USSR 26 115 (1940).
22. B.B. Mandelbrot and J.W. van Ness, SIAM Rev. 1 422
(1968).
23. D. Panja, E-print arXiv:0912.2331.
24. L. Lizana and T. Ambjörnsson, Phys. Rev. Lett. 100,
200601 (2008); Phys. Rev. E 80, 051103 (2009).
25. G. Guigas and M. Weiss, Biophys. J. 94, 90 (2008);
J. Szymanski and M. Weiss, Phys. Rev. Lett. 103, 038102
(2009); V. Tejedor et al., Biophys J. (at press).
26. H.E. Hurst, Trans. Amer. Soc. Civil Eng. 116, 400
(1951).
27. T.N. Palmer, G.J. Shutts, R. Hagedorn, F.J. Doblas-
Reyes, T. Jung, and M. Leutbecher, Ann. Rev. Earth
Planet. Sci. 33, 163 (2005).
28. I. Simonsen, Physica A 322, 597 (2003); N.E. Frangos,
S.D. Vrontos, and A.N. Yannacopoulos, Appl. Stochast.
Models in Business and Industry 23, 403 (2007).
29. T. Mikosch, S. Rednick, H. Rootzén, and A. Stegemann,
Ann. Appl. Prob. 12, 23 (2002).
30. I. Goychuk and P. Hänggi, Phys. Rev. Lett. 99, 200601
(2007).
31. I. Goychuk, arXiv:0905.0826v3 (2009).
32. A.H. Romero, J.M. Sancho, and K. Lindenberg, Fluct.
and Noise Lett. 2, 2 (2002).
33. B.S. Lowen, Meth. Comput. Applied Probab. 1:4, 445
(1999).
34. G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian
Random Processes (Chapman & Hall, New York, 1994).
35. G.M. Molchan, Commun. Math. Phys. 205 97 (1999).
36. M. Doi and S.F. Edwards, The Theory of Polymer Dy-
namics (Clarendon Press, Oxford, 1986).
37. R. Granek, J. Phys. II France 7, 1761 (1997).
38. E. Farge and A.C. Maggs, Macromol. 26, 5041 (1993);
A. Caspi et al., Phys. Rev. Lett. 80, 1106 (1998); F.
Amblard et al., Phys. Rev. Lett. 77, 4470 (1996).
39. E. Freyssingeas, D. Roux, and F. Nallet, J. Phys. II
France 7, 913 (1997); E. Helfer et al., Phys. Rev. Lett.
85, 457 (2000).
40. R. Granek and J. Klafter, Europhys. Lett. 56, 15 (2001).
41. A.G. Zilman and R. Granek, Chem. Phys. 284, 195
(2002).
42. S.N. Majumdar and A. Bray, Phys. Rev. Lett. 86, 3700
(2001).
43. J. Krug et al., Phys. Rev. E 56, 2702 (1997).
44. S.F. Edwards and D.R. Wilkingson, Proc. R. Soc. London
A 381, 17 (1982).
45. H. Gao and J.R. Rice, J. Appl. Mech. 65, 828 (1989).
46. J.F. Joanny and P.G. de Gennes, J. Chem. Phys. 81, 552
(1984).
47. Z. Toroczkai and E.D. Williams, Phys. Today 52, No. 12,
24 (1998).
48. N.G. van Kampen, Stochastic Processes in Chemistry and
Physics (North-Holland, Amsterdam, 1981).
49. A. Taloni and M.A. Lomholt, Phys. Rev E 78, 051116
(2008).
50. L. Lizana et al., ArXiv preprint arXiv:0909.0881 (2009).
51. A. Taloni, A.Chechkin, and J. Klafter, Phys. Rev. Lett.
(2010) (accepted).
Received 08.10.09
ПРОХОДЖЕННЯ ЧЕРЕЗ БАР’ЄР, ЗУМОВЛЕНЕ
ДРОБОВИМ ГАУСОВИМ ШУМОМ
О.Ю. Слюсаренко, В.Ю. Гончар, О.В. Чечкiн
Р е з ю м е
За допомогою чисельного iнтегрування передемпфованого рiв-
няння Ланжевена дослiджено задачу про швидкiсть вильоту
частинки iз потенцiальної ями пiд дiєю дробового гаусового
шуму. На прикладi обрiзаного гармонiчного потенцiалу отри-
мано залежностi середнього часу вильоту вiд iнтенсивностi шу-
му та показника Херста, а також обчислено функцiї розподiлу
часiв вильоту. Зроблено висновок, що, як i у випадку класичної
задачi з бiлим гаусовим шумом, цi величини є експоненцiаль-
ними функцiями вiдповiдних параметрiв.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 585
|
| id | nasplib_isofts_kiev_ua-123456789-56201 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T17:39:28Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Sliusarenko, O.Yu. Gonchar, V.Yu. Chechkin, A.V. 2014-02-13T20:34:52Z 2014-02-13T20:34:52Z 2010 Barrier Grossing Induced by Fractional Gaussian Noise / O.Yu. Sliusarenko, V.Yu. Gonchar, A.V. Chechkin // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 579-585. — Бібліогр.: 51 назв. — англ. 2071-0194 PACS 05.40.Fb, 02.50.Ey, 82.20.-w https://nasplib.isofts.kiev.ua/handle/123456789/56201 A problem of the rate of escape of a particle under the influence of the external fractional Gaussian noise is studied by using the method of numerical integration of an overdamped Langevin equation. Considering a truncated harmonic potential, the dependences of the mean escape time on the noise intensity and Hurst index are evaluated, together with the probability density functions for the escape times. It is found that, like the corresponding classical problem with white Gaussian noise, they both obey an exponential law. За допомогою чисельного iнтегрування передемпфованого рiвняння Ланжевена дослiджено задачу про швидкiсть вильоту частинки iз потенцiальної ями пiд дiєю дробового гаусового шуму. На прикладi обрiзаного гармонiчного потенцiалу отримано залежностi середнього часу вильоту вiд iнтенсивностi шуму та показника Херста, а також обчислено функцiї розподiлу часiв вильоту. Зроблено висновок, що, як i у випадку класичної задачi з бiлим гаусовим шумом, цi величини є експоненцiальними функцiями вiдповiдних параметрiв. The authors acknowledge the discussions with J. Klafter, R. Metzler, and I. Sokolov. AVC acknowledges the financial support from the MC IIF Programme, grant “LeFrac”. en Відділення фізики і астрономії НАН України Український фізичний журнал Загальні питання теоретичної фізики Barrier Grossing Induced by Fractional Gaussian Noise Проходження через бар’єр, зумовлене дробовим гаусовим шумом Article published earlier |
| spellingShingle | Barrier Grossing Induced by Fractional Gaussian Noise Sliusarenko, O.Yu. Gonchar, V.Yu. Chechkin, A.V. Загальні питання теоретичної фізики |
| title | Barrier Grossing Induced by Fractional Gaussian Noise |
| title_alt | Проходження через бар’єр, зумовлене дробовим гаусовим шумом |
| title_full | Barrier Grossing Induced by Fractional Gaussian Noise |
| title_fullStr | Barrier Grossing Induced by Fractional Gaussian Noise |
| title_full_unstemmed | Barrier Grossing Induced by Fractional Gaussian Noise |
| title_short | Barrier Grossing Induced by Fractional Gaussian Noise |
| title_sort | barrier grossing induced by fractional gaussian noise |
| topic | Загальні питання теоретичної фізики |
| topic_facet | Загальні питання теоретичної фізики |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/56201 |
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