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Physica A 215 (1995)495 510
Multiplicative random walks Claude Aslangul t Groupe de Physique des Solides. Laboratoire associb au CNRS (UA n 17) et aux Universitbs Paris ~71 et Paris" V1, Tour 23, Place Jussieu, 75251 Paris Cedex 05, France
Received 15 December 1994
Abstract Simple examples of multiplicative random walks are considered in which the random variable X is multiplied by a given scaling factor at each step of the process. Several cases are analyzed, either pure or disordered, showing how the disorder can affect the variation in time of various expectation values. It is seen that unstable (inflating) exponentially diverging cases are only slightly "renormalized" by disorder, even strong. On the contrary, for the deflating regimes, increasing disorder turns the asymptotic regime from exponential into algebraic decay in time.
1. Introduction Ordinary random walk is ubiquitous in science and can be used in a variety of fields, ranging from quarks physics [1] to biology [2]. On a physical level, such a framework is sound every time that the problem under consideration contains two classes of dynamical variables, namely slow and fast variables. An universally known example is that of a massive particle subjected to rapid collisions from the small particles of a surrounding fluid and thus undergoes Brownian motion; here, the random variable X denotes the coordinate of the heavy particle. Use of the Chapm a n - K o l m o g o r o v equation with standard assumptions allows to go from the Langevin equation (in the overdamped limit) to a master equation for the distribution function of the random variable X. Other examples are chemical reactions or, more generally, evolution in time of populations (birth-death problems) [3]. Usually, such a framework implies an a d d i t i v e stochastic process defined by stating that, at each step in time, the "coordinate" X gets an increment 5X with a given probability. For instance, in the simplest case (1 D lattice with spacing a), one writes
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the following:
X(t + At) =
X(t) + a X(t) - a X(t)
(probability p), (probability q),
(1.1)
(probability 1 - p - q).
W h e n p = q, this is a symmetric (unbiased) walk; in the opposite extreme case p ~ 0, q = 0 (resp. p = 0, q :/: 0), this represents a process in which X can only increase (resp. decrease), one celebrated example being shot noise as considered by Schottky [3]; in terms of p o p u l a t i o n evolution, this corresponds to a pure birth (resp. death) process [2]; when viewed as a walk on a lattice, one speaks of a directed walk [3, 4]. O n e can also conceive a process in which, at each step in time, the r a n d o m variable scales with a definite factor )~. Generally speaking, such a "walk" can occur in a self-similar structure in which, as c o m p a r e d to an ordinary one, translational s y m m e t r y is replaced by s y m m e t r y under scaling. O n e thus writes for the r a n d o m variable X (not necessarily a coordinate in ordinary space):
X(t + At) =
2X(t) 2-1X(t)
(probability p), (probability q),
X(t)
(probability 1 - p - q).
(1.2)
This defines a multiplicative stochastic process. Clearly, changing )~ into )~- 1 a m o u n t s to exchanging p and q; thus, with (1.2), it is enough to consider for instance the 2 > 1 case (the case 2 = 1 is clearly of no interest). O n e can also define the extreme directed version of (1.2), namely:
X(t + At) = ~,~X(t) (X(t)
(probability p), (probability 1 - p).
(1.3)
N o w , the two cases 2 > 1 and 2 < 1 have to be explicitly analyzed. When 2 > 1, the process m a y be said to be inflating; in the opposite case (0 < 2 < 1), it m a y be called deflating (or contracting). The situation where 2 is negative displays interesting specific features but will not be considered in this paper. Equivalently, a process such as (1.3) can be defined by introducing a two-valued stationary r a n d o m variable ).t taking the value 2 with probability p and the value 1 with probability 1 - p; then:
X(t + At) = 2t X(t),
{~ 2t =
(probability p), (probability 1 - p).
(1.4)
Clearly, the variable X(t) is a p r o d u c t of N = t/At r a n d o m variables. As an example, let us consider the energy loss of a d a m p e d particle undergoing friction due to interaction with a bath. The most simple scheme yields an exponential decay (time constant T) of the mechanical energy E, E being reduced by a factor 1/e between t and t + ~; this is a deterministic d a m p i n g since, at each step, the energy is reduced by a given factor with probability one, resulting from the fact that the surrounding m e d i u m is assumed to be homogeneous. O n the other hand, one can
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consider a strongly inhomogeneous medium, resulting for instance from rapid quenching of a system far from equilibrium. Let us assume that the medium consists of well separated islands (bubbles) of high density, the location of each being at random. An incoming particle can either move rather freely (between the bubbles) or in a damped manner by interaction with the high-density islands. During the motion of the particle, the time development of its mechanical energy can be crudely pictured as resulting from a random walk in which, at each step in time, either the energy is constant or is reduced by a given factor. The resulting process is of the form (1.3) with ). < 1, it may be said to describe probabilistic friction. In addition, if the bubbles have some intrinsic random feature, the probability p itself becomes a random variable. A process such as (1.3) can also appear, even in a homogeneous medium, when each scattering center can produce either an elastic (probability p) or an inelastic diffusion (probability 1 - p ) , according to the energy and/or the impact parameter of the incoming particle. This is the case when the interaction between the projectile and the target consists of two parts, one short-ranged, the other long-ranged, defining two different regions often separated by an energy barrier; roughly speaking, an ~ particle with low energy and large impact parameter cannot excite internal degrees of freedom of a nucleus and is elastically scattered. The case in which p is a random quantity can be realized when each scattering center is shielded ("dressed") at random thus displaying a barrier of random thickness. Another example of such multiplicative processes is the evolution in time of the population of individuals being able to procreate at each generation with probability p, each of them having a fertility rate equal to 2. If for some reason (ages, health . . . . t each individual has a variable (random) probability to procreate, again, p is a random variable. Other examples can be found in cascade [5] (or fragmentation) processes arising whenever a globally conserved quantity (the energy for instance) is, at each step, redistributed in a fraction of the members of an ensemble. An example is the scenario for turbulence within the so-called /~-model: at each step, the energy is transferred downwards smaller and smaller length scales in such a way that a fraction 1/,:. < I of the eddies has its energy increased by a factor ~ whereas the energy of the other becomes zero. As a final example, connection can be made with the driving of a dynamical variable by an external random one in the vicinity (just above or just below) of a bifurcation [6]. The onset of intermittency can be analyzed by linearizing the chaotic map, leading to a recurrence of the form:
X(t + At) = 2,X(t).
(1.5)
Now, inflation means instability and conversely. It will be seen that, for the contracting case (), = 1 or 2 < 1), the disorder can change the exponential regime into a power-law decay; this means that the negative Lyapunov exponents of the pure case disappear as such and that the stability becomes less robust, a phenomenon which can be called stability weakening. Needless to say, such non-linear maps can arise as
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recursion relations for more formal variables, such as the Riccati variables in disordered systems [7]. In this paper, I will analyse such multiplicative processes, considering first the very simple case where the probabilities p and q have fixed (non-random) values (the so-called pure, or ordered, case). This will serve as a basis for the more interesting case in which p and q are themselves random variables (the disordered case). It could appear that the link between additive and multiplicative processes is trivial. Indeed, assuming that X can take on real positive values, the random variable Y -- In X is another stochastic process; when X is a multiplicative one according to Eq. (1.3), Y is an additive process in the log domain with an increment a = In 2, so that going from additive processes to multiplicative ones may seem to be a mere fact of transcription. This is not true since the expectation value of In X has no obvious relation with the logarithm of the expectation value of X, all the more when disorder is present. Thus, multiplicative processes are worthy to investigate for themselves. It will be shown that, according to the value of the scaling parameter 2 (smaller or greater than 1), the disorder can either only slightly modify the dynamical regime or strongly alter it. Indeed, it will be shown that the inflating case is rather insensitive to disorder, even strong (the dynamical laws remain basically unchanged and are merely "renormalized"); on the contrary, a strong disorder can qualitatively modify the asymptotic regime in the deflating case, turning from exponential to power-law decay in time. On a technical level, it will be shown that a commonly used expansion (with respect to the inverse moments) here nearly completely fails to produce the correct behaviours, in the sense that, even when all these inverse moments exist and are finite, the corresponding expansion has a very narrow range of validity with respect to the variation of the scaling parameter 2.
2. Pure system The stochastic process defined by Eq. (1.2) can be described by its repartition function, P(x, t), allowing to write the probability, that at time t, X has its value between x and x + dx, an initial state being given. P(x, t) satisfies the dynamical equation:
P(x, t + At) dx = (1 - p - q) P(x, t) dx + p P(x/2, t) d(x/2) + q P(2x, t) d (2x). (2.1) When 2 > 1, p is responsible for inflation, whereas q forces the random variable to decrease. Clearly, the process obeying Eq. (2.1) embodies competition between inflation and deflation. By going to continuous time with transition probabilities per unit time Wd = p/At and Wg = q/At, the density P(x, t) is found to satisfy the first-order differential equation: ~P ~t
1
-
(W d
+ Wg)P(x,t) + ~ WdP(X/~,t ) +/.
WgP(2x, t),
(2.2)
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which obviously entails that the total probability is constant in time. Since 2 is here assumed to be positive, it is readily seen that, the two semi-infinite lines x > 0 and x < 0 are decoupled one from the other. This means that, given an initial P(x, t = O) which is non-zero on both sides of the origin x = 0, its positive (x > 0) and negative (x < 0) parts develop in time independently (when 2 > 1, P(x, t) splits into two parts which move away one from the other). In addition, it is evident that if P(x, 0) = 6(x), then P(x, t) remains equal to 6(x) for all times; thus 6(x) may be said to be as a fixed point (in the space of solutions) of the evolution in time, separating the positive and negative parts of any P(x, t). Finally, since eq. (2.2) is linear, it is enough to consider an initial repartition of the form: P(x,0) = 6(x - Xo),
(2.3)
where Xo will be taken as positive in the following. As a simple exercise, let us consider the directed version of the model, for which one has: OP ~t
W P ( x , t) + 1 W P ( x / ) , t) ,~
(2.4)
together with the initial state defined by Eq. (2.3). It is nearly obvious that P(x, t) will consist of a-functions located at points with abscissa ),"Xo and this is proved in details in Appendix A. It results:
P(x,t)
e wt +~ l ( w t ) ,
6( x
~,
(2.5)
-- A XO).
n=O
With this, we can compute all the moments xk:
xk=
(2.6)
x k P(x, t) dx. 0
They have the expression: __
xk=xkoe
+oo
1
w, ~ ~ . ( W t ) , ) , k = x ~ e , a ~
l,w,
(2.7)
n=O
Note that this result can be quickly obtained by multiplying both sides of eq. (2.4) by x k and by performing integration over x; the above derivation was given for future reference only (see Section 3). In particular, one has: .~(t) = xoe I~ l~w,,
Ax2=x-5(t)_[y,(t)]Z=x~[e(~2-11W,
(2.8)
e2(Z l~w,]~x2e(~2 1)w,,
(2.9a)
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C, Aslangul / Physica A 215 (1995) 495-510
where the last approximate expression is valid for W t ~> 1 (and for any 2 > 0). The relative fluctuations are measured by: Ax Y
--
~
e (~-
(2.9b)
1)2 wt/2
As a product of r a n d o m variables, X has relative fluctuations which diverge at large times. It is seen that, in this crude version, all the moments of X either go to infinity ()~ > 1, inflating case) or converge toward zero (2 < 1, deflating case). For completeness, I give here the solution (see Appendix B) of the same problem in its discrete time version as defined by Eq. (1.4); one then has: xk(t) = X~[1 + (2 k -- 1)p] t'/aq,
(2.10)
where It~At] = integer part of t/At. Clearly, the envelope of (2.10) obtained by coarse-graining on a scale At reproduces (2.7), which is also recovered by taking the proper limit (p = W At, At --* 0). Let us now consider the more general case as described by Eq. (2.2). It turns out convenient to introduce another scaling parameter A and a bias parameter ~ such that: )~ = e A,
Wg = We ~,
Wo = W e +~
(2.11)
can be positive or negative. As noted above, the problem is symmetric under the substitution A ~ - A , e ~ - e , so that it is enough to consider the case A > 0 (i.e. 2 > 1) and e, positive or negative. For A > 0 and ~ > 0, the bias acts in the same direction as the inflating transition rate and enhances the expansive character of the motion. On the contrary, and more interestingly, if c < 0 (and still A > 0), the bias competes with the inflating scaling. The moments are easily obtained by multiplying by x k and by integrating both sides of eq. (2.2). I thus find: d x~ dt
- -
=
W ( e +~+nm + e (t~+na) _
_
2che)x k
(2.12)
so that: xk(t) = X~ exp[4 W t s h ( ½ k A ) sh (e + ½kA)].
(2.13)
1 The various moments have all the same kind of behaviour when e + 5A > 0 (remember that A is positive); when this inequality is satisfied - this may be used to define the strong bias case they are all diverging exponentially, as in the directed limit with 2 > 1 (see Eq. (2.7)). As contrasted, when the field acts against inflation (e < 0) and is strong enough, the lowest moments tend toward zero whereas the highest ones still diverge. Indeed, for a given e, an integer ko can be defined such that:
- (ko + 1 ) ~ A < e < l
--ko½A.
(2.14)
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With this definition, all the m o m e n t s x k > ko diverge, whereas all the m o m e n t s x k ~ k,, go to zero. It is interesting to note that, for a given scaling p a r a m e t e r 2, the external bias e can be adjusted to control ko. F o r instance, with c, < - ~1 A, the average value of X goes to zero whereas its mean square dispersion diverges, both following an exponential law. Needless to say, c, can even be chosen so that a given m o m e n t remains constant in time: by taking r, = - ~ 1A , the expectation value of X is a constant, whereas all the other m o m e n t s go to infinity at infinite times.
3. The disordered case
I now consider the m o r e interesting case where the transition probabilities themselves are r a n d o m functions of x. This is now a much involved p r o b l e m and only the directed version of the model will be analyzed in what follows. With this restriction, the following equation is to be solved: 5P 5t
-
1 W(x)P(x,t) +=W(x/,~)P(x/)~,t). ,~
(3.1)
W ( x ) is assumed to be a r a n d o m function with short-ranged correlations, so that, for a given 2 :~ 1, the two values W ( x ) and W ( x / 2 ) are taken as statistically uncorrelated. Using the same a r g u m e n t as the one yielding Eq. (2.5), P(x, t) turns out to consist of atomic c o m p o n e n t s located at the points 2"Xo; we thus write: +3C,
P(x,t) = ~
r t , ( t ) 6 ( x - 2"Xo),
(3.2)
n=O
so that the expectation value of x k (the kth m o m e n t ) is given by:
x~(t) = X~o X ,~"~Tr.(t).
I3.3)
n=O
By substituting the expansion (3.2) in Eq. (3.1), the 7t, are seen to satisfy the following system: ~o = - W ( x o ) g o ,
~c, = - W ( x , ) T z , + W ( x ,
1)~,-1
(n >~ 1).
(3.4)
By introducing the Laplace transforms ~,(z) of g, (t), one easily finds (W, --- W O? x o)): !
~O(Z)--Z-~-Wo
~.(z) =
1
fi
Z -]- W o m = 1
wm (n/> I). z~-W m
(3.51
Using (3.2), the Laplace transform of P(x, t), P(x, z), is found as:
P(x,z)- . 1 6(x . z + Wo
Xo) .+ ~ ~(x. ,=1
)~"Xo)
m=l z + Wm
,
I3.6)
C. Aslangul/Physica A 215 (1995) 495 510
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and, due to the assumption that the W's are statistically uncorrelated, the disorderaverage of P writes:
(P(x,z)>
=
~
~
a(x -
;~"Xo).
(3.7)
n=O
This allows to obtain the Laplace transform (xk)(z) of the disorder-averaged moment of X, (xk)(t), as a geometrical series which is summed to yield:
(3.8)
(Xk)(Z) = 2kz + (1 -- 2k)/(1/(Z + W ) ) " In this expression, the still-standing average is the Hilbert transform of p: Wo+Wc
f
Wo
--p(W)dW, z+W
(3.9)
where p(W) is the W repartition function, which is non-zero on the interval (possibly infinite) extending from Wo >/0 to I410 + We. For a non-disordered system, Eq. (3.8) gives back the Laplace transform of expression (2.7), as it must; note also that expression (3.8) is even valid for k = 0. More interestingly, this equation shows that the following functional relation holds true:
(Xk)(Z;;'~)=Xko I(Xl)(Z;Ak)
(3.10)
( k ~ 1)9
where the explicit dependence upon 2 has been temporarily displayed. This means that it is enough to analyze (Xl) in details to obtain all the other moments. At this point, it is better for clarity to successively consider various classes of disorder, going from weak disorder to strong disorder, the precise meaning of this being given in due time.
3.1. Weak disorder case I now consider the case where all the moments ( W ~) exist for any real ~, positive or negative. This defines the weak disorder case and implies that either Wo > 0 and Wc < + ~ or p has the proper behaviours near W = 0 and W = + oo. Whenever necessary, I will for definiteness specifically choose a uniform distribution between Wo > 0 and W o + We: p(W)=~
1
ifWo2. This entails that the average of
exp L1 + (1
~)(r~/w ( W
1
•
(3.16)
Clearly, such an expression, obtained through an apparently safe and widely accepted procedure, cannot have here an unrestricted range of validity; namely, provided that )~ is greater that 1 and that the dimensionless dispersion Ch/w is large enough, (.'~) becomes negative, which simply does not make sense. As a whole, (3.16) is at best an asymptotic expression valid in the limit of vanishingly small disorder. I n order to analyze this difficulty, explicit use of the distribution defined in Eq. (3.10) is now made. This yields the exact expression for (xl)(z): (z) :
Xo
2z + ( 1 - ;O Wc/ln
(3.17) l+z~
The inverse Laplace transform ~2)(t) is determined by all the singularities of (xl)(z): clearly, (x~) possesses a cut going from - ( W o + We) to - I4/o; this is always true,
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C. Aslangul/Physica A 215 (1995) 495 510
independent of the specific choice for p and directly comes out from the fact that ((z + W ) - I ) is clearly singular whenever - z belongs to the interval on which p is nonzero. In addition, it is easy to see that ( x l ) always has a just one simple pole zl in the first Riemann sheet, defined as the root of the equation: 2z~ + ( 1 -
2)Wc/ln (1 + += W0~ w . o2) 1
(3.18)
This real pole has the same sign as 2 - 1; it lies between - Wo and 0 for 2 < 1 and is thus positive for 2 > 1. In either case, this is the only relevant singularity in the large-time limit since contributions from the other singularities are exponentially small as compared to the former. It is not difficult to find approximate expressions for z~ as a function of 2; namely, for 2 ~ 1 and ~ Wc/Wo:
zl=-Wo+(Wo+Wc)exp~-~-~oo)+...
01, weak or strong disorder), relative fluctuations diverge at large times, as expected for a product of random variables, the mean square dispersion (Ax 2) being dominated by its first term (x2). This divergence is always exponential in the inflating case. In the deflating case, exponentially divergent fluctuations - already present in the pure case, see Eq. (2.9b) - are found only in the weak-disorder situation. For the intermediate or strong disorder, one has: (Ax)
- (~)
~ [f(0)]
1
1- 2 1 -- 22
(Wct)u
"
(3.36)
As a whole, in any case, the random variable X has no self-averaging moments.
4. S u m m a r y
and conclusions
In this paper, I aimed to study the influence of disorder on the asymptotic dynamics of simple multiplicative random random walks which, generally speaking, can arise when symmetry under scaling is substituted for translational symmetry. The pure case trivially displays exponential dynamics (either deflating or inflating) and the point was
C. Aslangul /Physica A 215 (1995) 495-510
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to analyze how disorder can modify this regime. For the directed version of the process, it was shown that the inflating case is rather robust and gives rise at most to (renormalized) exponential variations in time, even when disorder is strong; this means that the corresponding (positive) Lyapunov exponents still exist, even in the presence of strong disorder. On the contrary, the deflating case retains an exponential decay only in the presence of weak disorder. A qualitative change occurs when disorder is strong enough, in which case the decay follows an algebraic law, with an exponent simply related to the basic repartition function describing disorder; in such a case, the Lyapunov exponents disappear as such and the stability is weakened due to the transition from exponential to power-law decay. In passing, it was observed that a conventional expansion in terms of the inverse moments here nearly completely fails. Work on the more general (both-way) model is in progress and will be the subject of a future paper.
Appendix A I here give the details about the derivation of Eq. (2.5). Let the function Q defined as: P(x,t)=Q(y=lnx,
t)[d~]
xl Q(y,t)
e-;'Q(y,t).
IA1)
From this definition, it results that: P(x/)~,t) = e ~ , - t , ~ Q ( y
(A2j
_ ln)~,t)
and, from Eq. (2.4), the dynamical equation for Q is:
~Q ~t
-
W Q ( y , t ) + W Q ( y - In 2, t),
(A3t
which represents a directed (additive) walk on a lattice with spacing In £. The initial condition for P(see Eq. (2.3)) now reads: Q(y, O) = 6(y - lnxo).
(A4)
Eq. (A3t can be easily integrated by introducing the Fourier transform of Q: +x
(~(K,t)= ~ dyelt~'Q(y,t).
{A51
By substituting in Eq. (A3) and by taking the initial condition Q(K,0) = e iK~..... into account, one readily finds: Q_(K,t) = e x p [ - W t ( 1 - e iKl"~) + i K l n x o ] .
(A61
From this, Q(y, t) is obtained as: Q(y,t) = e
w, ~ ~-~-e dK J
iK(y-I ..... ~ e x p ( W t e
..... ).
(A7)
C. Aslangul/PhysicaA 215 (1995) 495-510
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By now expanding the right exponential in its power series, one has:
f ~dKe_iK(y_lnxo) einKln;~ Q(y,t) = e -w' +~ ~ (Wt)" n! n=O
+~ (Wt)"
=e -w' ~ --~.
~5
5(y-lnxo-nln2),
(A8)
n=O
which immediately leads to the Eq. (2.5) of the text.
Appendix B F o r the time-discretized version of the process as defined by Eq. (1.3) or, equivalently, Eq. (1.4), the distribution function of X, R(x, t) is:
R(x,t) - P r o b [ X ( t ) < x]
(B1)
and, from Eq. (2.4), obeys the following equation:
R(x,t + At) = pR(x/2, t) + (1 - p)R(x,t).
(B2)
This is readily solved by recurrence, starting with the initial condition deduced from Eq. (2.3), namely R(x,O) = O(x -Xo), 0 being the unit-step function; one eventually finds: N
R[x, N A t < t < (N + 1)At] = ~ C~p'(1 - p)N "O(x - )~'Xo).
(B3)
r=0
This allows to obtain the m o m e n t s of the r a n d o m variable X as: N
xk(t)= ~ C~pr(l__p)N ~ ( 2 " X ~ ) = X ko [1 + ()~k _ 1)p]IntU/A,),
(B4)
r=O
where Int(.) denotes the integer part; this is Eq. (2.10) of the main text.
References [1] J. Alam, S. Raha and B. Sihna, Phys. Rev. Lett. 73, (1994) 1895. [2] J. D. Murray, Mathematical Biology (Springer, 1990). [3] C. W. Gardiner, Handbook of Stochastic Methods (Springer, 1985). [4] C. Aslangul, M. Barth616my, N. Pottier and D. Saint-James, J. Stat. Phys. 65, (1991) 673. I-5] G. Paladin and A. Vulpiani, Phys. Rep. 156 (1987) 148. 1-6] J. F. Heagy, N. Platt and S. M. Hammel, Phys. Rev. E 49 (1994) 1140. 1-7] J. M. Luck, Syst6ms desordonn~s unidimensionnels (AI6a, Saclay, 1992). 1-8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, 1980).
