Colored Noise in Dynamical Systems: Some Exact Solutions

Peter Jung

School of Physics, Georgia Institute of Technology, Atlanta GA 30332, USA

Abstract . I consider one dimensional nonlinear systems driven by colored Gaus- sian noise. A framework is presented which allows for the construction of models that can be exactly solved in terms of single variable Fokker-Planck equations. I further discuss an exactly solvable linear model with multiplicative colored noise, where the noise color mediates a stochastic-resonance fike behavior.

1 In t roduct ion

The general subject of colored noise driven dynamical flows is rooted in the study of the motion of small particles suspended in a fluid and moving under the influence of random forces which result from collisions with molecules of the fluid induced by thermal fluctuations, or in short the phenomenon of Brownian motion (Einstein 1956). In the earliest studies of Brownian motion, the damping of the motion of the suspended particles was very large compared to that of the fluid molecules, so that inertial effects could be neglected. Moreover, the thermal fluctuations occur on a time scale which is very much shorter than that of the Brownian particle. It is then a good approximation to assume that the random forces are uncorrelated as perceived by the particle on its own, much slower time scale. This assumption considerably simplifies the problem, because it allows one to treat the stochastic dynamical motions as a Markov process for which many methods and approximation schemes are available. The fluctuations which can be treated under this assumption have often been termed "white noise". Thus, white noise fluctuations ~(t), are those for which the autocorrelation function is given by,

< ~(t)~(s) >= 2D~(t - s), (1)

with the noise intensity D. A large body of literature on white noise appli- cations exists, and appropriate starting points are the now classic reviews of Chandrasekhar, Uhlenbeck and Ornstein, and others in the collection of Wax (1954), the texts by Stratonovich (1963), van Kampen (1992), Risken (1984) and Horsthemke and Lefever (1984) or the reports by Hnggi and Thomas (1982), and Fox (1978).

In the physical world this idealization, however, is sometimes not justified. For example, the photon statistics of a dye-laser (Zhou et al. 1986) and the motional narrowing of magnetic resonance lineshapes (Kubo 1962) are con- siderably affected by the finite correlation time of fluctuations. It has become

24 Peter Jung

customary to study colored noise driven systems 1 in terms of a Langevin equation

= h(x) + g(x)~(t), (2)

where x is the observable of interest, and h(x) and g(x) are continuous dif- ferentiable functions. The no~se ~(t) is considered stationary and Gaussian distributed with zero mean and correlation function

< ~(t)~(s) >= ~(t - s). (3)

The spectral density, given by the Fourier-transform of the correlation func- tion is frequency dependent, hence the term colored noise.

In many studies (for a recent review, see (Hnggi and Jung 1995)), an exponential correlation function ~(s) = (D/To)exp(--N/rc ) has been used, with r~ describing the time-scale of the fluctuations and D/T its variance. The power spectrum is given by a Lorentzian with corner-frequency 1/T~. For this Ornstein- Uhlenbeck noise, the white-noise limit is recovered by decreasing the correlation-time r~ and leaving D constant. While Ornstein-Uhlenbeck noise is generated by applying white noise to a first order linear filter with one real-valued pole of the Greens function, a whole class of other Gaussian noise sources can be constructed by applying white noise to linear filters of all orders with complex-valued poles of the Greens functions. Another example is the so called harmonic noise (Schimansky-Geier and Zfilicke 1990) where the Greens function of the filter has two complex conjugate poles.

In section 2 we derive exact equations of motions of the Fokker-Planck type for the probability density P(x, t) corresponding to the Langevin equation(2) for a specified class of functions g(x) and h(x). In section 3, we discuss some of these exactly solvable models explicitly. In section 4, we consider a linear model, driven by multiplicative colored noise and periodic forcing, which is exactly solvable and shows noise-color induced stochastic resonance.

2 Equation of motion for the probability density

Starting from (1), the stochastic Liouville equation for the density p(x, t) = 5(x - x[r(t)]), where x[r(t)] is the solution of (1) with r(t) being a realization of the noise source ~(t), can be obtained as

h~ (x, t) = (x + ~r(t)) pr (x, 0, (4)

I One has to distinguish systems, where the randomness of the observable of inter- est results from the interaction with a large number of other intrinsic degrees of freedom (internal noise) and systems which are driven by noise, e.g. by a noisy parameter (external noise).

Colored Noise in Dynamical Systems 25

with the operators

.A = -~h(x) 0 (5)

In the interaction-picture, i.e.

Pr (x, t) --= exp(-.At)p~ (x, t), (6)

the equation of motion simplifies to

[9~(x, t) = r(t)w(t)p~(x, t) Lw (t) = exp( -At)u exp(t), (7)

with the formal solution

[/0 ] pr(x,t) = Texp r(t')Ew(t')dt' p~(x,O), (8) where 7- is the backward chronological operator. In the next step we have to average over realizations of the noise r(t). Since p~(x, 0) in general cannot be factorized out of the average, it is clear that a closed equation of motion for the average P(x,t) - (p~(x,t))~ cannot be derived in general from Eq.(8). For the specific initial condition P(x, O) = 5(x - xo), however, the average factorizes. For a general initial condition, we can still factorize the average for times larger than the correlation time of the noise. Taking the average over all realizations r(t) and factorizing yields the result

[L ] P(x,t) = (p~(x,t)}~ = T(exp r(t').w(t')dt' ),.P(x,0). (9) The expression in the bracket can be evaluated exactly, if the time-evolution operator commutes at arbitrary times t and t ~, i.e.

[/:~(t), w( t ' ) ] - w(t)E~(t') -~(t ' )~(t) - O. (10)

Using the condition (10), Eq.(3), and the Gaussian properties of the noise, Eq.(9) can be written as

P(x , t )=exp{ l ffot fotw(tl)w(t~)(tl-t2)dtldt2} P(x,O). (11)

Taking the time derivative, we obtain the equation of motion in the interaction- picture

/o ig(x,t) = w(t) dt'~(t')O(t - t ' )P(x,t ) . (12)

26 Peter Jung

In the original representation p(x,t) = exp(.At)P(x, t), one obtains the equation of motion

~0 t ~(x,t) = .Ap(x,t) + 13 dt' exp(At')/3exp(-.At')(t ')p(x,t). (13)

Some comments on this Fokker-Planck equation are in order:

1. It is exact only for initial conditions of the type P(x, O) = 5(x - xo) 2. The second term depends via the lower integration limit exp l i c i t ly on

the initial time to = 0, a hint that we are dealing with a non-Markovian process. For times t large against the correlation time of the noise, this dependence will disappear.

3. In the white noise limit, i.e. (t) = 2Da(t), the standard Fokker-Planck equation is recovered.

To construct exactly solvable models with colored noise, we have to find functions h(x) and g(x) such, that the condition (10) is fulfilled. Here, the Baker-Haussdorf relation

exp(At)/3 exp( -At ) =/3 + [A,/3]t + I[A, [.4, B]]t 2 + ... (14)

will be particularly useful.

3 Exact ly solvable models

In this section we discuss how one can use the results of the preceding sec- tion to construct exactly solvable models. An exactly solvable model in my notation is a model where the time evolution of the variable x is described by a single variable Fokker-Planck equation. Although, one cannot find in gen- eral explicit time-dependent solutions for such an equation, the stationary probability density and moments can be computed.

3.1 Trans format ions to l inear sys tems

The most simple way to fulfill (10) is to require that the operators A and/3 are commuting, i.e. [A,/3] = 0. Then, the equation of motion (13) becomes

~(x,t) = Ap(x,t) +/32 dt'(t')p(x,t), (15)

which is identical to the usual Fokker-Planck equation, but with a time de- pendent diffusion coefficient. In order that the operators .4 and/3 commute, the functions h(x) and g(z) have to fulfill the ordinary differential equation

Colored Noise in Dynamical Systems 27

h(x)g ' (x ) - g(x)h'(x) = 0. The solution h(x) = cg(x) with the arbitrary constant c suggests the simple coordinate transform

f X

y = c 1/h(x')dx' - ct, (16)

to the linear Langevin equation ~) = ~(t), which describes a free part icle driven by colored Gaussian noise. The requirement of commuting operators .4 and t3 has thus lead us to a class of exactly solvable models which can be transformed to a linear model. The Hongler-models (see e.g. (Horsthemke and Lefever 1984)) are a more general class of such systems.

3.2 Exactly solvable nonlinear systems

More interesting exactly solvable models which cannot be transformed to a linear process can be found by requiring

[..4, 13] = aB, (17)

where ol is an arbitrary constant. Using the Baker-Hausdorff relation (14), the evolution operator in the interaction picture can be brought to the simpler form

Zw (t) = exp(--`4t)B exp(`4t) = B exp(-cd) . (18)

which fulfills- as required- the commutation-relation [w (t), :w (t~)] = 0 for all times t and tq The equation of motion for the probability density becomes

t) = .4p(x, t) + 132 at' t), (19)

which is again of the Fokker-Planck type with a time dependent diffusion coefficient. The condition (17) requires that the functions g(x) and h(x) obey the ordinary differential equation h(x)g'(x) -g (x )h ' (x ) = -~g(x) . For the specific choice g(x) = 1, the solution of the differential equation for h(x) reads h(x) = -c~x + t5 with an arbitrary constant /3. The only with our scheme exactly solvable model with additive colored noise is the Ornstein-Uhlenbeck process.

We can, however, construct a variety of exactly solvable multiplicative models. For a given function h(x), the correct function g(x) which leads to a solvable model reads

g(x) = gh(x ) exp ((~V(x)), (20)

where V(x) is the integral of (1/h(x)) and c~ and K are arbitrary constants. For the special choice h(x) = -'),x, we find g(x) = K3,x ~/~+1. This special exactly solvable model has been identified by Hgnggi (H/inggi 1981) by using functional calculus.

28 Peter Jung

4 Co lored noise med ia ted s tochast i c resonance: an exact so lu t ion

In this section, we present a periodically driven system with multiplicative colored noise, for which we can calculate the moments of the distribution function exactly (Fulinsky 1995, Berdichevsky and Gitterman 1996). The system is given by

= - (a + ~(t))x + A sin(12t) (21)

where we assume ~(t) to be Gaussian, colored noise with zero mean. Although the stochastic differential equation is linear, i.e. linear superposition applies for two solutions, the response to the colored noise and the periodic forcing is not decomposable as in the case of a periodically driven, additive stochastic differential equation (Jung and H/inggi 1990). This has the consequence, that although equation (21) is linear, the amplitude of the systems response to the periodic forcing actually depends on the intensity of the noise. Fulinsky (1995) and Berdichevsky and Gitterman (1996) have shown that the exact solution of (21) with colored dichotomous noise shows stochastic resonance, i.e. the amplitude of the response to the periodic forcing as a function of the noise intensity goes through a maximum (for a review on stochastic resonance, see (Jung 1993)).

The solution of (21) is written as

x(t) = x(O)G(t) + A sin(Y2t')G(t - t')dt', (22)

with the Greens-function

G(t) = exp ( -a t - fot~(t')dt') . (23)

For Gaussian noise, ((t) the averages of exponentials can be carried through yielding

(G(t)) = exp ( -a t + lcr2(t)) , (24)

with the variance of F(t) = f t ~(t')dt', given by

cr2(t) = dtl dt2(tl - t2). (25)

The time dependent mean value (x(t)} is then written as

/0 t

Colored Noise in Dynamical Systems 29

Supplementing the response function (G(t)) by a Heaviside step function O(t), i.e. R(t) _~ O(t)(G(t)), the response is described by thecomplex-valued susceptibility R(w), given by the one-sided Fourier-transform of R(t).

To become more specific, we are choosing now exponentially correlated noise, i.e. (t) = (D/T~)exp(-It l/r~ ). Inserting this correlation function into (25) yields for the variance c~2(t)

cr2(t )=2Dt+2Dr~ exp - -1 , (27)

and the Greens function

(G(t)) = exp [ - (a - D)t + DT(exp(--t/rc) -- 1)]. (28)

For D > a, the Greens function increases exponentially since the noise in- duced drift Dx becomes larger than the binding force -ax . We restrict our- selves to D < a, where the system approaches a steady state for large times. The susceptibility can be written as the Fourier integral

/ / [ ( ( ' ) ) ] [/(w) = exp (D-a - iw) t+Drc exp -~-~ -1 dt. (29) As mentioned in the beginning of this section, although the system is linear, the response of the system to periodic forcing actually depends on the noise intensity D.

4.1 The wh i te noise l imit

In the limit of white noise, i.e. Tc --+ 0, this above described mixing of exter- nal noise and periodic forcing is readily understood as the consequence of the renormalization of the drift due to the noise induced term Dx (Horsthemke and Lefever 1984). The response function and the susceptibility assume the simple forms (G(t)) = exp ( - (a - D)t) and/~o(w) = 1/(a - D + iw), respec- tively. For the amplitude 5 of the response to the periodic forcing, given by the absolute value of the susceptibility at the driving frequency tg, we find in turn

1 52(D' ~2) = (a - D) 2 + Y22" (30)

The response amplitude increases monotonically for increasing noise strength D until it reaches at the critical value Dc = a its maximum 1/w - the response amplitude of a free particle.

30 Peter Jung

0.50

0.48

0.46

0.44

0.42

0.40 0.0

I i I .4..+..V,..t.--F I .+~ "~' 'F~

- ..+' , ,~ . ;~. ;~.~.~ .-F ~.~'"

_ . i i i , , ,,, ,,, B _

_ - . . . . . - - - . 15] -E ] -

ave = 0.00 + . . . . - - . . [ ]~ arc = 0.05 ~ . . . . . . -ave = 0 . i0 "" "--. " arc = 0.15 [] . . . . a'rc = 1.00 . . . . . . .

I I I I

0.2 0.4 0.6 0.8 1.0

D/a

Fig. 1. The response amplitude ~ is shown as a function of the noise intensity D/a at the driving frequency .Q/a -- 2 and various values of the correlation time of the noise arc. The curves for arc ---- 0,0.05,0.1 and arc ---- 0.15 have been plotted according to the approximation (31) while the curve for a~'c = 1 has been obtained from a numerical integration of Eq.(29).

4.2 Co lored noise and s tochast i c resonance

The integral in (29) can not be evaluated analyt ica l ly in general. For smal l correlat ion t imes of the noise re, however, the exponent ia l in (29) can be expanded and we obta in in leading order

R((2) =/~/0((2)(1 - Dye) (31)

where f/0(w) is the suscept ibi l i ty in the white noise l imit , given above. The response ampl i tude 32 (D, (2) -- (1 - Drc) / ( (a - 0 ) 2 + (22) shows a max imum as funct ion of the noise strength at D = a - re(2 2 in leading order in rc. Measur ing the frequency in units of 1/re, i.e. J~ = (27c and the re laxat ion t ime vr - (a - D) -1 in units of re, i.e. ~ = v~/7"c, the resonance condit ion can be wr i t ten in the appeal ing form

7"r = 1 /~2- (32)

In F ig . l , we have p lotted the response ampl i tude a~ as a funct ion of the noise strength D at (2/a = 2. For smal l noise color ave, e.g. arc = 0.1, the response ampl i tude shows a max imum at D/a = 1 - aT-c((2/a) 2 -- 0.6. In the white noise l imit the max imum shifts towards the crit ical value Dc ----a. For

Colored Noise in Dynamical Systems 31

increasing correlation times arc, the maximum shifts to smaller values of the noise strength D/a to eventually disappear.

5 Acknowledgments

I want to thank my friends from the Humboldt University Lutz Schimansky- Geier and Thorsten Pgschel for inviting me to write this article in occasion of Werner Ebeling's 60th birthday and for their infinite patience in receiving my late manuscript. This work has been supported by the Deutsche Forschungs- gemeinschaft within the Heisenberg program.

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