- Email: [email protected]
On certain time characteristics of dynamical systems driven by noise
__ B
6 October
2%
1997
r
PHYSICS
_-
ELSEVIER
Physics Letters A 234 (1997)
LETTERS
A
329-335
On certain time characteristics of dynamical systems driven by noise A.L. Pankratov I Institute
@forPhysics
of Microstructures
of the Russian Acudemy of Scienc~es. GSP 105. Nihp
Received
10 July 1997; accepted for publication Communicated by CR. Doering
Novgorod
603600.
Ru.ssim Fedewriorl
18 July 1997
Abstract The moments of the transition time of non-linear dynamical systems driven by noise are introduced. It is demonstrated that the well-known recurrence formula for the moments of the first passage time (FP) of the absorbing boundary by a Brownian particle can also be used to obtain the exact values of the moments of the transition time in non-linear dynamical systems with noise, described by a symmetric potential profile. @ 1997 Elsevier Science B.V. Keywords:
Brownian
motion; Kramers’
problem; Moments
of first passage time
1. Introduction The investigation of temporal scales of transition in various polystable systems driven by noise is a subject of great theoretical and practical importance, e.g. phase transitions, stochastic resonance, Josephson electronics, etc. The pioneering work of the problem of determining time scales of transition processes was carried out by Kramers [ 11 who considered transition processes as a process of Brownian diffusion and used the FokkerPlanck equation to obtain several approximate expressions for the desired time characteristics. The main approach of Kramers’ method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact time characprocesses
’ E-mail: [email protected]. 037%9601/97/$17.00 P/I SO375-9601
@ 1997 Elsevier (97)00599-9
Science B.V. All rights reserved
teristics it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the common problem to find the time scales of diffusion transition processes. In recent years this problem has been reexamined by some authors (see e.g. Refs. [ 2-4 1) but almost all analytical results obtained are approximate and valid only in the limit of a large barrier height. Considering the one-dimensional Brownian diffusion in the overdamped limit, we note that there are known many different time characteristics, defined in different ways, e.g. the decay time of a metastable state or the relaxation time to a steady state. The usual method of eigenfunction analysis, when the time scale (the relaxation time) is supposed to be equal to the inverse minimal non-zero eigenvalue, is not applicable for the case of a large noise intensity because then higher eigenvalues should be also taken into account. Only the moments of the FPT may be obtained exactly for arbitrary potentials with absorbing bound-
330
A.L. Pankratov/Physics
aries and arbitrary noise intensities. However, almost any concrete physical situation (some examples are listed above) is described by smooth potentials that do not have absorbing boundaries and the moments of the FPT may not give correct values of the time scales in those cases. Our approach based on the Laplace transform method gives an opportunity to demonstrate the conformity (duality) between time characteristics [ 51, which allows one to obtain the exact relaxation time of a system described by an arbitrary symmetric potential profile. According to this duality (we call it the principle of conformity), the relaxation time of the “symmetric system” may be written in terms of quadratures simply using the dimensionless potential profile. We have introduced a new time characteristic, the moments of the transition time, extended the principle of conformity and proved that any moments of time of the transition processes in arbitrary symmetric potential profiles are equal to the corresponding moments of the FPT and may be written in terms of quadrature& On the basis of the obtained results the complete analysis may be done of the stability to fluctuations of bistable memory cells (e.g. based on the parametric quantron [6]), described by symmetric potential profiles.
2. Main equations and set up of the problem 2.1. The transitian probab~ii~ Consider a Brownian diffusion in a potential profile @p(x), which is an adequate model of various transition processes in non-line~dynami~al systems driven by noise. Let the coordinate x(t) of the Brownian particle described by the probability density W(x, t) at an initial instant of time have a fixed value x( 0) = x0 within the interval (c, d), i.e. the initial probability density is the delta function, W(x,O)
=6(x-x0),
x0 E (C,d).
(1)
In this case ( 1) the one-dimensional probability density W( x, t) is the transition probability density from the point no to the point x: W(x, t) = W(x, t; x0,0). It is known that the probability density W( x, t) of the
Letter.? A 234 (1997) 329-335
Brownian particle in the overd~ped Fokker-Planck equation (FPE),
limit satisfies the
d’W(x, t> =-- dG(x, t) at 13X
+
a2w(x, t) ,&
1
(2)
with the initial condition ( 1) and with the boundary conditions W(IIZ:OO, f) = 0. Here B = h/kT, C(x, t) is the probability current, h is the viscosity, T is the temperature, k is the Boltzmann constant and p(x) = @(x)/kT is the dimensionless potential profile with $9(Z&o) = 03. It is necessary to find the probability Q,d( t, x0) = Q(t,xa) of the transition of the Brownian particle from the point x0 to outside of the considered interval (c,d) during the time t > 0: Q(t,xo) = j-:, W(x, t)dx + Jdfc””W(x, t)dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary ]7,8]. We suppose that c and d here are ~bitr~ily chosen points of an arbitrary potential profile @J(X) and the boundary conditions at these points may be arbitrary: W(c, t) & 0, W(d, t) & 0. When considering the absorbing boundary one should suppose zero boundary conditions: W(c, t) = W(d, t) = 0. The main distinction of the transition probability from the probability to pass the absorbing boundary is the possibility for a Brownian particle to come back in the considered interval (c, d) after crossing boundary points. This possibility may lead to a situation when in spite of the Brownian particle having already crossed the points c or d, at the time t --+ 00, this Brownian partide may be located within the interval (c, d). Thus, the set of transition events may be not complete, i.e. at the time t --+ 03 the probability Q(t,ro) may tend to a constant, smaller than unity: lim ,_.,oo Q( t, x0) < 1, as it is in the case when there is a steady-state distribution for the probability density lim f--Ku W(x,t) = W,,(x) f 0. 2.2. Moments of the transition time Consider the probability Q( t, x0) of a Brownian particle located at the point x0 within the interval (c, d) to be at the time f > 0 outside of the consid-
A.L. Punkratov/Physics
ered interval. Then we can decompose this probability to a set of moments. On the other hand, if we know all moments, we can in some cases construct a probability as the set of moments. Thus, analogically to the moments of the FTP [ 7-91 we can introduce the moments of the transition time 8,( c, x0, d) taking into account that the set of transition events may be not complete, i.e. lim,,, Q( t, xc) < 1,
(3) Here we can formally denote the derivative of the probability divided by the normalization factor as w,( t, x0) and thus introduce the probability density of the transition time wC,d( t,xa) = w,(t,~e) in the following way. W,(r,Xo)
=
aQ(tt x01 at
1
Q(~.xo)
- Q(O,xo)’
(4)
It is easy to check that the normalization condition is satisfied for such a definition, sooo w,( t, xa)dt = 1. The condition of non-negativity of the probability density w,( t, x0) 2 0 is, actually, the condition of monotonicity of the probability Q( t, x0). In the case when c and d are absorbing boundaries the probability density of the transition time coincides with the probability density of the FPT WT( t, xc), w7-(t,xo) =
dQ(t,xo) at
.
We denoted here w,( t, x0) and wr( t, x0) by different indexes r and T to note again that there are two different functions and w,( t, x0) = WT(t, x0) in the case of absorbing boundaries only. The moments of the FPT are T,,(c.xo,d)
=.i’t
= (t”)
x
0
,,aQ(wo)dr r?t
=s 00
t”Wr(t,xO)dt.
0
Integrating (3) by parts one can get the expression for the mean transition time (MTT) 61 (c, x0, d) = (t).
Letters A 234 (I 997) 329-335
331
This definition completely coincides with the characteristic time of the probability evolution introduced in Ref. [ lo] from geometrical considerations, when the characteristic scale of the evolution time was defined as the length of the rectangle with equal square, and the same definition was later used in Refs. [5,11,12]. Analogically to the MTT (6) the mean square 82 (c, ~0, d) = (t’) of the transition time may be also defined as
%(c,xo,d)
=
2s00c (s,"[Q(
00,
Q(~,xo)
x01 - Q(7, xo) ldr) dt - Q(O,xo) (7)
Note that the previously known time characteristics, such as the moments of the FPT, the decay time of a metastable state or the relaxation time to a steady state follow from the moments of the transition time if a concrete potential is assumed: a potential with an absorbing boundary, a potential describing a metastable state or a potential, within which a non-zero steadystate distributionmay exist, respectively. Besides, such a general representation of the moments 6,, (c, ~0, d) (3) gives an opportunity to apply the new approach proposed by Malakhov [ 12,131 to obtain the mean transition time and to extend easily it to obtain any moments of the transition time in arbitrary potentials, so 6,, (c, x0, d) may be expressed in terms of quadratures as it is known for the moments of the FPT. The principle of conformity, presented earlier in Ref. [ 51 for the relaxation time to steady state in an arbitrary symmetric potential, may be now also extended for any moments of the transition time.
3. The principle of conformity There is a recurrence formula [ 9] for the moments of the FPT of the absorbing boundary located at the point x = d > x0 for the Brownian particle (40(-o) = -too),
A.L. Pankratov/Physics
332
Letters A 234 (1997) 329-335
ary, but also the moments of the transition time of the system with noise, described by an arbitrary potential profile symmetric with respect to the point d. Superpose the point of symmetry of the dimensionless potential profile on the origin of the coordinate z = x - d such that 9(-z) = p(z) and put zo = x0 - d < 0. For the Laplace transformed probability density, Y( z, s) = so” W( z, t) ees’dt from Eq. (2) may be written as follows, d’Y(z, s) dz 2 = -BS( z - zo).
b)
(9)
Note that the probability Laplace transform is
current
in terms
of the
M
c)
G(z,s)
=
J
G( z, t) e-“‘dt
0
ddz)y(z Fig.
1. Potential profiles, illustrating the principle of conformity.
T,,t-oo,xo,d)
=nBj
eP(X)]
IO
7’n_1 (-co,
u, d) e-cD(“)du dx,
-cc (8)
which presents the nth moment of the FPT directly from the function of the potential profile q(x) and the (n - 1)th moment. Here To( -00,x0, d) = 1 and ri (-co,xu, d) is the mean FPT. We will consider the case when the boundary c = --co is natural (i.e. p( -co) = t-co) and the boundary d is an arbitrarily chosen point, which is, in addition, the point of symmetry of the considered potential profile p(x) . Let us formulate the principle of conformity: The moments of the transition time of a dynamical system with noise, described by an arbitrary potential p(x) such that p( +oo) = co, symmetric relative to some point x = d, with an initial delta-shaped distribution, located at the point x0 < d (Fig. la), coincides with the corresponding moments of the FTP for the same potential, having an absorbing boundary at the point of symmetry of the original potential profile (Fig. 1b) . Let us prove that formula (8) gives not only the values of the moments of FPT of the absorbing bound-
s>
+
dYtz,s)
’
dz
(10)
Suppose that we know two linearly independent solutions U(z) = U(z,s) and V(z) = V(z,s) of the homogeneous equation corresponding to (9) (i.e. when the right-hand side of Eq. (9) is equal to zero), such that U(z) + 0 at z -+ +co and V(z) 4 0 at z --) -cc. Because of symmetry of the function p( z ) these independent solutions may be also chosen as symmetrical, such that U(-z) = V(z), U(0) = V(O),
In this case the general solution represented as follows, Y(z,s)=Yl(z) =&(z)
of Eq. (9) may be
+y-(z),
z < LO,
+-Y+(z),
zo < z < 0,
= Y?(z),
z 3 0,
where F(z)
= C,V(z>,
h(z)
= C2U(z)*
L(ltzo)Vtz),
Y_(Z)
=
Y+(z)
=Lvtzo)Ll(z). Wzol
w[zo]
(11)
A.L. Pankratov/Physics
Here dV(z) = U(z) -- dz
Wlzl
dU(z) dz
V(z)
is the Wronskian, Ct and C2 are arbitrary constants which may be found from the continuity condition of the probability density and the probability current at the origin, Yl(0) + y+(o) &
= -0,s)
= SO,s).
(12)
Calculating from ( 12) the values of arbitrary and putting them into ( 11) , one can get the value for the probability current Laplace e( z, S) ( IO) at the point of symmetry z =
-V(zo)
1
G’“‘s’=W[zo]
[
constants following transform 0,
1
dV(z 1 dz &
(13)
In accordance with Refs. [ 5,10,1 I] the mean transition time in an arbitrary potential profile is
Q(mzo)
-
if~i-~x~~o~o)=~~,~Q(,,,o)
=
zo) - Q(Ov
[Q(m,
lim
dhzo)
./
zo> 1 - &O,s> ’
(14)
W(z.,t)dz,
0
s
M
M
$(s, 3))
=
Q( f,ZO) e+‘dt
=
Y( z, s)dz. s
0
0
The same formula ( 14) may be derived, naturally, taking the Laplace transform from the definition (6). Using this approach one can also get the formula for the mean square of the transition time (t’) taking the Laplace transform from the definition (7)) &-mO.zO,O)
= !L”,2
= (t2)
os
s$(s,zo) - Q(O, zo) Q(cQ, zo) - Q
m = Q
$1
Q(oo, zo> - s$(s,zo) s2tQ(~,zo) - Q(O,zo,l
Actually, we can prove the principle of conformity for all moments of the transition time ( 14), (15) and
(16)
- Q
In our particular case Q(0, zo) = 0, Q(oo, zn) = l/2, because the steady-state probability density W( z, co) will spread symmetrically from both sides of the point of symmetry z = 0. The time scale 61 (-co, zo,O) ( 14) has here the sense of the relaxation time to the steady state. Thus, combining ( 16) and ( 13) we get the following formula for the Laplace transformed probability density of the transition time,
dV(z)
2V(zo)
7
w,zo,
[
m =
WAS,zo) =
w,(s,zo) =
where
Q(t.zo)
w,(s,zo),
-Q(O,zo)]
sTQ(m, zo) - Q
.ShO
so on, but it is more simple to prove it for the formally introduced probability density of the transition time w7 ( t , zo) . Taking the Laplace transform from expression (4) and noting that s$( S, za) - Q(0, zo) = &(O, s) [ 111, one can get the following formula for
= yz(O), =G(z
333
Letters A 234 (1997) 329-335
.
(17)
1 -4
Before finding the Laplace transformed probability density wr( S, za) of the FPT for the potential depicted in Fig. lb, let us determine the Laplace transformed probability density w,( S, zo) of the transition time for the system whose potential is depicted in Fig. lc. This potential is transformed from the original profile (Fig. la) by a vertical shift of the right part of the profile arbitrary in value and a sign step /3. As in this case the derivative dp( z )/dz in Eq. (9) is the same for all points besides z = 0, we can again use the linearly independent solutions U( z ) and V(: ), and the potential jump equals p at the point z = 0 may be taken into account by the new joint condition at z = 0. The probability current at this point is continuous as before, but the probability density W(z. t) has now the step, so the second condition of ( 12 1 is the same, but instead of the first one we should write 6 (0) I_ v+ (0) = fi(0) eeP. It gives new values of the arbitrary constants Ct and C2 and a new value of the probability current at the point z = 0. Now the Laplace transform of the probability current is
&O,s) =
2V(zo) WIz01(1
+e-P)
(18)
334
AL.
Punkratr,v/Plly.~ics
One can find that for the potential depicted in Fig. Ic Q(o0, zo) has also been changed and equals now Q(m zo> = I/( 1 + e-p), while Q(0, zc), certainly, as before, is Q(0, 20) = 0. It is easy to check that the substitution of the new values of C?(O, s), Q( 03, ze) into formula ( 16) gives the same formula ( 17) for w,( s, ~0). Putting now p = 00, i.e. locating the absorbing boundary at the point z = 0 (X = d), we get the same formula ( 17), already not for the probability density of the transition time but for the probability density of the FPT wr(s, za). It is known that, if the Laplace transformations of two functions coincide, then their origins coincide also. So, if we substitute the coinciding probability densities w,( t, za) and ~r(t,zo) into formula (3) (see formulae (4), (5)) for the cases of the symmetric potential profile and the profile with the absorbing boundary at the point of symmetry, we should get equal values for the moments of the transition time and the FPT. Thus, we have proved the principle of conformity for probability densities and the moments of the transition time of a symmetrical potential profile and the FPT of the absorbing boundary located at the point of symmetry.
4. Conclusions In this paper the moments of the transition time have been introduced and the principle of confo~ity has been proved. It is obvious that the moments of FPT with respect to the point 0 are the same for the profiles depicted in Fig. 1. For the moments of the transition time this coincidence is not so easy to understand and the fact that the principle of conformity for the potential depicted in Fig. lc leads to an unusual conclusion: neither the quantity, nor the sign of the potential step p influences the moments of the transition time. For the mean transition time this fact may be explained in the following way: - if the transition process is going from up to down, then the probability current is large, but it is necessary to fill the lower minimum by the larger part of the probability to reach the steady state; - if the transition process is going from down to up, then the probability current is small, and it is necessary to fill the upper minimum by the smaller part of the
Metiers A 234 (1997) 329-335
probability to reach the steady state. The difference in the quantities of currents is completety compensated by the final probabilities on reaching the steady state. Howw does the relaxation time obtained by the method of eigenfunction analysis correspond to the introduced mean transition time? In Ref. [S] we have considered the “quartic potential” and demonstrated that in the case of small noise intensity the mean transition time absolutely coincides with the relaxation time obtained in Ref. [3] by eigenfunction analysis. Thus, the approach proposed gives an opportunity to obtain time scales of diffusion processes in an easier way than the method of eigenfunction analysis and besides the higher moments of the transition time may be also obtained by the approach presented. Note finally that the principle of conformity, proved for a delta-shaped initial distribution of the probability density ( 1), may be also extended to arbitrary initial distributions located within the considered interval W(.X,O) = W;,,(X), X E (C,d).
Acknowledgement The author wishes to thank Professor A.N. Malakhov, Professor A.I. Saichev and Dr. A.A. Dubkov for helpful discussions, constructive comments and support. This work has been supported by the Russian Foundation for Fund~ental Research (Project N 96-02- 16772-a and in part Project N 9702- 16928-a).
References [ 11 H. Kramers. Physica 7
( 1940)
284.
C. Blomberg, Physica (Utrecht)
1:
R.S. Larson and M.D. (1978)
A 86 ( 1977) 49.
Kostin, J. Chem.
Phys. 69 N
II
4821.
14
R.S. Larson, J. Chem. Phys. 81 (1984)
15
A.N.
Malakhov
and A.L.
Electr. 38 (1995)
167.
16
K.K. Likharev, Int. J. Theor. Phys. 21
17
L.A.
Pontryagin, A.A.
Tear. Fiz. 3 (1933) reproduced,
31
I.
Vitt, 2%. Eksp.
165 [translated by J.B. Barbour and
in: Noise 1989)
( 1982)
Andronov and A.A. in Non-linear
Moss and P.V.E. McClintock Cambridge,
1731.
Pankratov, Radiophys. Quantum
Vol.
Dynamics.
(Cambridge
eds. F.
University Press,
I, p. 3291.
[ 8 I P Hanggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. 62 (1990)
2.51.
A.L. Pankratov/Physics 191 V.I.
Tikhonov
(Soviet IO/ A.N. Eiectr.
Radio,
and
M.A.
Moscow,
Malakhov 36 (1993)
Mironov, 1978)
and N.V.
[in
Agoudov,
Markovian
processes,
Letters A 234 (1997)
[ I I 1 A.N.
Russian]. Radiophys.
329-335
335
( 1996)
Malakhov
and A.L.
Pankratov,
Physica A 229
Malakhov
and A.L.
Pankratov.
Physica C 269 ( 1996)
Malakhov,
CHAOS
7 ( 1997)
109. Quantum
[ 12 / A.N.
97.
46.
113/
A.N.
N 3. in press.
6 October
2%
1997
r
PHYSICS
_-
ELSEVIER
Physics Letters A 234 (1997)
LETTERS
A
329-335
On certain time characteristics of dynamical systems driven by noise A.L. Pankratov I Institute
@forPhysics
of Microstructures
of the Russian Acudemy of Scienc~es. GSP 105. Nihp
Received
10 July 1997; accepted for publication Communicated by CR. Doering
Novgorod
603600.
Ru.ssim Fedewriorl
18 July 1997
Abstract The moments of the transition time of non-linear dynamical systems driven by noise are introduced. It is demonstrated that the well-known recurrence formula for the moments of the first passage time (FP) of the absorbing boundary by a Brownian particle can also be used to obtain the exact values of the moments of the transition time in non-linear dynamical systems with noise, described by a symmetric potential profile. @ 1997 Elsevier Science B.V. Keywords:
Brownian
motion; Kramers’
problem; Moments
of first passage time
1. Introduction The investigation of temporal scales of transition in various polystable systems driven by noise is a subject of great theoretical and practical importance, e.g. phase transitions, stochastic resonance, Josephson electronics, etc. The pioneering work of the problem of determining time scales of transition processes was carried out by Kramers [ 11 who considered transition processes as a process of Brownian diffusion and used the FokkerPlanck equation to obtain several approximate expressions for the desired time characteristics. The main approach of Kramers’ method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact time characprocesses
’ E-mail: [email protected]. 037%9601/97/$17.00 P/I SO375-9601
@ 1997 Elsevier (97)00599-9
Science B.V. All rights reserved
teristics it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the common problem to find the time scales of diffusion transition processes. In recent years this problem has been reexamined by some authors (see e.g. Refs. [ 2-4 1) but almost all analytical results obtained are approximate and valid only in the limit of a large barrier height. Considering the one-dimensional Brownian diffusion in the overdamped limit, we note that there are known many different time characteristics, defined in different ways, e.g. the decay time of a metastable state or the relaxation time to a steady state. The usual method of eigenfunction analysis, when the time scale (the relaxation time) is supposed to be equal to the inverse minimal non-zero eigenvalue, is not applicable for the case of a large noise intensity because then higher eigenvalues should be also taken into account. Only the moments of the FPT may be obtained exactly for arbitrary potentials with absorbing bound-
330
A.L. Pankratov/Physics
aries and arbitrary noise intensities. However, almost any concrete physical situation (some examples are listed above) is described by smooth potentials that do not have absorbing boundaries and the moments of the FPT may not give correct values of the time scales in those cases. Our approach based on the Laplace transform method gives an opportunity to demonstrate the conformity (duality) between time characteristics [ 51, which allows one to obtain the exact relaxation time of a system described by an arbitrary symmetric potential profile. According to this duality (we call it the principle of conformity), the relaxation time of the “symmetric system” may be written in terms of quadratures simply using the dimensionless potential profile. We have introduced a new time characteristic, the moments of the transition time, extended the principle of conformity and proved that any moments of time of the transition processes in arbitrary symmetric potential profiles are equal to the corresponding moments of the FPT and may be written in terms of quadrature& On the basis of the obtained results the complete analysis may be done of the stability to fluctuations of bistable memory cells (e.g. based on the parametric quantron [6]), described by symmetric potential profiles.
2. Main equations and set up of the problem 2.1. The transitian probab~ii~ Consider a Brownian diffusion in a potential profile @p(x), which is an adequate model of various transition processes in non-line~dynami~al systems driven by noise. Let the coordinate x(t) of the Brownian particle described by the probability density W(x, t) at an initial instant of time have a fixed value x( 0) = x0 within the interval (c, d), i.e. the initial probability density is the delta function, W(x,O)
=6(x-x0),
x0 E (C,d).
(1)
In this case ( 1) the one-dimensional probability density W( x, t) is the transition probability density from the point no to the point x: W(x, t) = W(x, t; x0,0). It is known that the probability density W( x, t) of the
Letter.? A 234 (1997) 329-335
Brownian particle in the overd~ped Fokker-Planck equation (FPE),
limit satisfies the
d’W(x, t> =-- dG(x, t) at 13X
+
a2w(x, t) ,&
1
(2)
with the initial condition ( 1) and with the boundary conditions W(IIZ:OO, f) = 0. Here B = h/kT, C(x, t) is the probability current, h is the viscosity, T is the temperature, k is the Boltzmann constant and p(x) = @(x)/kT is the dimensionless potential profile with $9(Z&o) = 03. It is necessary to find the probability Q,d( t, x0) = Q(t,xa) of the transition of the Brownian particle from the point x0 to outside of the considered interval (c,d) during the time t > 0: Q(t,xo) = j-:, W(x, t)dx + Jdfc””W(x, t)dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary ]7,8]. We suppose that c and d here are ~bitr~ily chosen points of an arbitrary potential profile @J(X) and the boundary conditions at these points may be arbitrary: W(c, t) & 0, W(d, t) & 0. When considering the absorbing boundary one should suppose zero boundary conditions: W(c, t) = W(d, t) = 0. The main distinction of the transition probability from the probability to pass the absorbing boundary is the possibility for a Brownian particle to come back in the considered interval (c, d) after crossing boundary points. This possibility may lead to a situation when in spite of the Brownian particle having already crossed the points c or d, at the time t --+ 00, this Brownian partide may be located within the interval (c, d). Thus, the set of transition events may be not complete, i.e. at the time t --+ 03 the probability Q(t,ro) may tend to a constant, smaller than unity: lim ,_.,oo Q( t, x0) < 1, as it is in the case when there is a steady-state distribution for the probability density lim f--Ku W(x,t) = W,,(x) f 0. 2.2. Moments of the transition time Consider the probability Q( t, x0) of a Brownian particle located at the point x0 within the interval (c, d) to be at the time f > 0 outside of the consid-
A.L. Punkratov/Physics
ered interval. Then we can decompose this probability to a set of moments. On the other hand, if we know all moments, we can in some cases construct a probability as the set of moments. Thus, analogically to the moments of the FTP [ 7-91 we can introduce the moments of the transition time 8,( c, x0, d) taking into account that the set of transition events may be not complete, i.e. lim,,, Q( t, xc) < 1,
(3) Here we can formally denote the derivative of the probability divided by the normalization factor as w,( t, x0) and thus introduce the probability density of the transition time wC,d( t,xa) = w,(t,~e) in the following way. W,(r,Xo)
=
aQ(tt x01 at
1
Q(~.xo)
- Q(O,xo)’
(4)
It is easy to check that the normalization condition is satisfied for such a definition, sooo w,( t, xa)dt = 1. The condition of non-negativity of the probability density w,( t, x0) 2 0 is, actually, the condition of monotonicity of the probability Q( t, x0). In the case when c and d are absorbing boundaries the probability density of the transition time coincides with the probability density of the FPT WT( t, xc), w7-(t,xo) =
dQ(t,xo) at
.
We denoted here w,( t, x0) and wr( t, x0) by different indexes r and T to note again that there are two different functions and w,( t, x0) = WT(t, x0) in the case of absorbing boundaries only. The moments of the FPT are T,,(c.xo,d)
=.i’t
= (t”)
x
0
,,aQ(wo)dr r?t
=s 00
t”Wr(t,xO)dt.
0
Integrating (3) by parts one can get the expression for the mean transition time (MTT) 61 (c, x0, d) = (t).
Letters A 234 (I 997) 329-335
331
This definition completely coincides with the characteristic time of the probability evolution introduced in Ref. [ lo] from geometrical considerations, when the characteristic scale of the evolution time was defined as the length of the rectangle with equal square, and the same definition was later used in Refs. [5,11,12]. Analogically to the MTT (6) the mean square 82 (c, ~0, d) = (t’) of the transition time may be also defined as
%(c,xo,d)
=
2s00c (s,"[Q(
00,
Q(~,xo)
x01 - Q(7, xo) ldr) dt - Q(O,xo) (7)
Note that the previously known time characteristics, such as the moments of the FPT, the decay time of a metastable state or the relaxation time to a steady state follow from the moments of the transition time if a concrete potential is assumed: a potential with an absorbing boundary, a potential describing a metastable state or a potential, within which a non-zero steadystate distributionmay exist, respectively. Besides, such a general representation of the moments 6,, (c, ~0, d) (3) gives an opportunity to apply the new approach proposed by Malakhov [ 12,131 to obtain the mean transition time and to extend easily it to obtain any moments of the transition time in arbitrary potentials, so 6,, (c, x0, d) may be expressed in terms of quadratures as it is known for the moments of the FPT. The principle of conformity, presented earlier in Ref. [ 51 for the relaxation time to steady state in an arbitrary symmetric potential, may be now also extended for any moments of the transition time.
3. The principle of conformity There is a recurrence formula [ 9] for the moments of the FPT of the absorbing boundary located at the point x = d > x0 for the Brownian particle (40(-o) = -too),
A.L. Pankratov/Physics
332
Letters A 234 (1997) 329-335
ary, but also the moments of the transition time of the system with noise, described by an arbitrary potential profile symmetric with respect to the point d. Superpose the point of symmetry of the dimensionless potential profile on the origin of the coordinate z = x - d such that 9(-z) = p(z) and put zo = x0 - d < 0. For the Laplace transformed probability density, Y( z, s) = so” W( z, t) ees’dt from Eq. (2) may be written as follows, d’Y(z, s) dz 2 = -BS( z - zo).
b)
(9)
Note that the probability Laplace transform is
current
in terms
of the
M
c)
G(z,s)
=
J
G( z, t) e-“‘dt
0
ddz)y(z Fig.
1. Potential profiles, illustrating the principle of conformity.
T,,t-oo,xo,d)
=nBj
eP(X)]
IO
7’n_1 (-co,
u, d) e-cD(“)du dx,
-cc (8)
which presents the nth moment of the FPT directly from the function of the potential profile q(x) and the (n - 1)th moment. Here To( -00,x0, d) = 1 and ri (-co,xu, d) is the mean FPT. We will consider the case when the boundary c = --co is natural (i.e. p( -co) = t-co) and the boundary d is an arbitrarily chosen point, which is, in addition, the point of symmetry of the considered potential profile p(x) . Let us formulate the principle of conformity: The moments of the transition time of a dynamical system with noise, described by an arbitrary potential p(x) such that p( +oo) = co, symmetric relative to some point x = d, with an initial delta-shaped distribution, located at the point x0 < d (Fig. la), coincides with the corresponding moments of the FTP for the same potential, having an absorbing boundary at the point of symmetry of the original potential profile (Fig. 1b) . Let us prove that formula (8) gives not only the values of the moments of FPT of the absorbing bound-
s>
+
dYtz,s)
’
dz
(10)
Suppose that we know two linearly independent solutions U(z) = U(z,s) and V(z) = V(z,s) of the homogeneous equation corresponding to (9) (i.e. when the right-hand side of Eq. (9) is equal to zero), such that U(z) + 0 at z -+ +co and V(z) 4 0 at z --) -cc. Because of symmetry of the function p( z ) these independent solutions may be also chosen as symmetrical, such that U(-z) = V(z), U(0) = V(O),
In this case the general solution represented as follows, Y(z,s)=Yl(z) =&(z)
of Eq. (9) may be
+y-(z),
z < LO,
+-Y+(z),
zo < z < 0,
= Y?(z),
z 3 0,
where F(z)
= C,V(z>,
h(z)
= C2U(z)*
L(ltzo)Vtz),
Y_(Z)
=
Y+(z)
=Lvtzo)Ll(z). Wzol
w[zo]
(11)
A.L. Pankratov/Physics
Here dV(z) = U(z) -- dz
Wlzl
dU(z) dz
V(z)
is the Wronskian, Ct and C2 are arbitrary constants which may be found from the continuity condition of the probability density and the probability current at the origin, Yl(0) + y+(o) &
= -0,s)
= SO,s).
(12)
Calculating from ( 12) the values of arbitrary and putting them into ( 11) , one can get the value for the probability current Laplace e( z, S) ( IO) at the point of symmetry z =
-V(zo)
1
G’“‘s’=W[zo]
[
constants following transform 0,
1
dV(z 1 dz &
(13)
In accordance with Refs. [ 5,10,1 I] the mean transition time in an arbitrary potential profile is
Q(mzo)
-
if~i-~x~~o~o)=~~,~Q(,,,o)
=
zo) - Q(Ov
[Q(m,
lim
dhzo)
./
zo> 1 - &O,s> ’
(14)
W(z.,t)dz,
0
s
M
M
$(s, 3))
=
Q( f,ZO) e+‘dt
=
Y( z, s)dz. s
0
0
The same formula ( 14) may be derived, naturally, taking the Laplace transform from the definition (6). Using this approach one can also get the formula for the mean square of the transition time (t’) taking the Laplace transform from the definition (7)) &-mO.zO,O)
= !L”,2
= (t2)
os
s$(s,zo) - Q(O, zo) Q(cQ, zo) - Q
m = Q
$1
Q(oo, zo> - s$(s,zo) s2tQ(~,zo) - Q(O,zo,l
Actually, we can prove the principle of conformity for all moments of the transition time ( 14), (15) and
(16)
- Q
In our particular case Q(0, zo) = 0, Q(oo, zn) = l/2, because the steady-state probability density W( z, co) will spread symmetrically from both sides of the point of symmetry z = 0. The time scale 61 (-co, zo,O) ( 14) has here the sense of the relaxation time to the steady state. Thus, combining ( 16) and ( 13) we get the following formula for the Laplace transformed probability density of the transition time,
dV(z)
2V(zo)
7
w,zo,
[
m =
WAS,zo) =
w,(s,zo) =
where
Q(t.zo)
w,(s,zo),
-Q(O,zo)]
sTQ(m, zo) - Q
.ShO
so on, but it is more simple to prove it for the formally introduced probability density of the transition time w7 ( t , zo) . Taking the Laplace transform from expression (4) and noting that s$( S, za) - Q(0, zo) = &(O, s) [ 111, one can get the following formula for
= yz(O), =G(z
333
Letters A 234 (1997) 329-335
.
(17)
1 -4
Before finding the Laplace transformed probability density wr( S, za) of the FPT for the potential depicted in Fig. lb, let us determine the Laplace transformed probability density w,( S, zo) of the transition time for the system whose potential is depicted in Fig. lc. This potential is transformed from the original profile (Fig. la) by a vertical shift of the right part of the profile arbitrary in value and a sign step /3. As in this case the derivative dp( z )/dz in Eq. (9) is the same for all points besides z = 0, we can again use the linearly independent solutions U( z ) and V(: ), and the potential jump equals p at the point z = 0 may be taken into account by the new joint condition at z = 0. The probability current at this point is continuous as before, but the probability density W(z. t) has now the step, so the second condition of ( 12 1 is the same, but instead of the first one we should write 6 (0) I_ v+ (0) = fi(0) eeP. It gives new values of the arbitrary constants Ct and C2 and a new value of the probability current at the point z = 0. Now the Laplace transform of the probability current is
&O,s) =
2V(zo) WIz01(1
+e-P)
(18)
334
AL.
Punkratr,v/Plly.~ics
One can find that for the potential depicted in Fig. Ic Q(o0, zo) has also been changed and equals now Q(m zo> = I/( 1 + e-p), while Q(0, zc), certainly, as before, is Q(0, 20) = 0. It is easy to check that the substitution of the new values of C?(O, s), Q( 03, ze) into formula ( 16) gives the same formula ( 17) for w,( s, ~0). Putting now p = 00, i.e. locating the absorbing boundary at the point z = 0 (X = d), we get the same formula ( 17), already not for the probability density of the transition time but for the probability density of the FPT wr(s, za). It is known that, if the Laplace transformations of two functions coincide, then their origins coincide also. So, if we substitute the coinciding probability densities w,( t, za) and ~r(t,zo) into formula (3) (see formulae (4), (5)) for the cases of the symmetric potential profile and the profile with the absorbing boundary at the point of symmetry, we should get equal values for the moments of the transition time and the FPT. Thus, we have proved the principle of conformity for probability densities and the moments of the transition time of a symmetrical potential profile and the FPT of the absorbing boundary located at the point of symmetry.
4. Conclusions In this paper the moments of the transition time have been introduced and the principle of confo~ity has been proved. It is obvious that the moments of FPT with respect to the point 0 are the same for the profiles depicted in Fig. 1. For the moments of the transition time this coincidence is not so easy to understand and the fact that the principle of conformity for the potential depicted in Fig. lc leads to an unusual conclusion: neither the quantity, nor the sign of the potential step p influences the moments of the transition time. For the mean transition time this fact may be explained in the following way: - if the transition process is going from up to down, then the probability current is large, but it is necessary to fill the lower minimum by the larger part of the probability to reach the steady state; - if the transition process is going from down to up, then the probability current is small, and it is necessary to fill the upper minimum by the smaller part of the
Metiers A 234 (1997) 329-335
probability to reach the steady state. The difference in the quantities of currents is completety compensated by the final probabilities on reaching the steady state. Howw does the relaxation time obtained by the method of eigenfunction analysis correspond to the introduced mean transition time? In Ref. [S] we have considered the “quartic potential” and demonstrated that in the case of small noise intensity the mean transition time absolutely coincides with the relaxation time obtained in Ref. [3] by eigenfunction analysis. Thus, the approach proposed gives an opportunity to obtain time scales of diffusion processes in an easier way than the method of eigenfunction analysis and besides the higher moments of the transition time may be also obtained by the approach presented. Note finally that the principle of conformity, proved for a delta-shaped initial distribution of the probability density ( 1), may be also extended to arbitrary initial distributions located within the considered interval W(.X,O) = W;,,(X), X E (C,d).
Acknowledgement The author wishes to thank Professor A.N. Malakhov, Professor A.I. Saichev and Dr. A.A. Dubkov for helpful discussions, constructive comments and support. This work has been supported by the Russian Foundation for Fund~ental Research (Project N 96-02- 16772-a and in part Project N 9702- 16928-a).
References [ 11 H. Kramers. Physica 7
( 1940)
284.
C. Blomberg, Physica (Utrecht)
1:
R.S. Larson and M.D. (1978)
A 86 ( 1977) 49.
Kostin, J. Chem.
Phys. 69 N
II
4821.
14
R.S. Larson, J. Chem. Phys. 81 (1984)
15
A.N.
Malakhov
and A.L.
Electr. 38 (1995)
167.
16
K.K. Likharev, Int. J. Theor. Phys. 21
17
L.A.
Pontryagin, A.A.
Tear. Fiz. 3 (1933) reproduced,
31
I.
Vitt, 2%. Eksp.
165 [translated by J.B. Barbour and
in: Noise 1989)
( 1982)
Andronov and A.A. in Non-linear
Moss and P.V.E. McClintock Cambridge,
1731.
Pankratov, Radiophys. Quantum
Vol.
Dynamics.
(Cambridge
eds. F.
University Press,
I, p. 3291.
[ 8 I P Hanggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. 62 (1990)
2.51.
A.L. Pankratov/Physics 191 V.I.
Tikhonov
(Soviet IO/ A.N. Eiectr.
Radio,
and
M.A.
Moscow,
Malakhov 36 (1993)
Mironov, 1978)
and N.V.
[in
Agoudov,
Markovian
processes,
Letters A 234 (1997)
[ I I 1 A.N.
Russian]. Radiophys.
329-335
335
( 1996)
Malakhov
and A.L.
Pankratov,
Physica A 229
Malakhov
and A.L.
Pankratov.
Physica C 269 ( 1996)
Malakhov,
CHAOS
7 ( 1997)
109. Quantum
[ 12 / A.N.
97.
46.
113/
A.N.
N 3. in press.