Ann. Henri Poincaré 4, Suppl. 1 (2003) S385 – S394
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S385-10
DOI 10.1007/s00023-003-0929-7
Annales Henri Poincaré
First-Passage Problems in Spatially Extended Systems
Alan J. Bray
Abstract. First-passage problems are related to such questions as the statistics of
first-crossing times (across a boundary, or set of boundaries) of a stochastic process
x(t), e.g. a random walker. In the ”spatially extended” generalization, the single
variable x(t) is replaced by a field f (x, t), with specified dynamics, and one considers
the first passage of f at a given point x. Such problems are, in general, non-trivial
even for rather simple processes such as diffusion. This class of problems will be
introduced through coarsening systems and reaction-diffusion problems. Analytic
results will be presented for the asymptotics of the one-dimensional two-species
annihilation process, A + B → 0, with different initial densities. The asymptotic
decay of the density of the minority species maps onto the survival probability, P (t),
of a diffusing particle moving in an infinite sea of diffusing traps. We show that
1/2 ), where ρ is the trap density, D the trap diffusion constant
P (t) ∼ exp(−λρ[Dt]
√
and λ = 4/ π, independent of the particle diffusion constant. The corresponding
results for general dimensions d ≤ 2 will also be presented.
1 Introduction
There is a long history of the study of first-passage problems in many different
areas of science [1, 2]. The simplest example is that of a random walker, starting
at position x, with an absorbing boundary at x = 0. If the walker’s diffusion
constant is D, √
the probability that it has not yet been absorbed at time t is [1]
P (t) = erf(|x|/ 4Dt), where erf(x) is the error function. The derivative of this
function, P1 (t) = −dP/dt = (x2 /4πDt3 )1/2 exp(−x2 /4Dt), gives the probability
distribution of the first-passage time, i.e. the time at which the particle first reaches
the origin. The persistence exponent, θ, is defined by the asymptotic large-time
behavior P (t) ∼ t−θ , giving θ = 1/2 for the random walker.
The determination of θ for more complex processes is in general very difficult.
The random walker discussed above is described (in continuous time and space)
by the Langevin equation dx/dt = η(t), where η(t) is Gaussian white noise with
mean zero and correlator η(t)η(t ) = 2Dδ(t − t ). The next simplest process
is the random acceleration problem, d2 x/dt2 = η(t). This process is known to
have θ = 1/4 [3], though the calculation is much more involved than the random
walk [4]. The higher derivative processes, dn x/dtn = η(t), have thus far proved
intractable for n > 2. It seems, however, that the exponent θ is not a simple
fraction: numerical computations give [5] θ(3) = 0.22022(3), θ(4) = 0.20958(3)
and θ(5) = 0.20413(3). The feature of all the processes with n > 1 that makes
the calculation of θ difficult is that they are non-Markovian [2]. Of course, these
processes can be written as Markovian processes in a larger number of variables,
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e.g. d2 x/dt2 = η(t) can be expressed as dx/dt = v, dv/dt = η(t), i.e. it is a Markov
process with two degrees of freedom. The “spatially extended” processes which we
now discuss involve infinitely many degrees of freedom. We discuss these under
two related headings: coarsening problems, and reaction-diffusion problems, with
special attention being paid to the two-species annihilation process A + B → 0.
2 Coarsening Processes
Consider a system which, under cooling, undergoes a phase transition from a disordered phase to an ordered phase. The transition can correspond to either an
order-disorder transition or to phase separation, according to whether the order
parameter is nonconserved or conserved. The essential point is that there is a
unique Gibbs state (and the dynamics is ergodic) in the high-temperature phase,
while in the low temperature phase there is more than one Gibbs state and ergodicity is broken.
If such a system is cooled rapidly through the phase transition, domains of
the two (or more) possible ordered phases form and grow (“coarsen”) with time
[6], the driving force for the coarsening being the elimination of interfaces (domain
walls) between the phases so as to reduce the overall (free) energy of the system.
One can then pose the following persistence (or first-passage) problem: what is the
fraction of space, at a given time t, which has always been in the same phase? This
corresponds to the fraction of space which has never been crossed by a domain wall.
In most cases, this decays as a power-law in time, P (t) ∼ t−θ , where θ is in general a
nontrivial number. Note that the order parameter, φ(x, t), of the phase transition
is a field, i.e. it corresponds to an infinite number of degrees of freedom. Even
though the dynamics of the entire field can be Markovian, e.g. the system might be
described (for a nonconserved order parameter) by the time-dependent GinzburgLandau equation ∂t φ = −Γ(δF/δφ), where F [φ] is a Ginzburg-Landau free-energy
functional, the effective dynamics of the field at a single point, φ(x, t), is nonMarkovian [2]. Measurements of θ have been performed for a number of coarsening
systems, e.g. Ising models [7], twisted nematic liquid crystals [8], gas diffusion [9],
soap froths [10], and breath figures [11] (formed by vapor condensation on a cold
plate). The last three examples show that coarsening phenomena encompass a
much larger class of systems than phase transitions.
In general, the exponent θ is a non-trivial number, even for simple diffusion
[12] (where one considers regions where the diffusion field has not crossed its
mean value). One notable exception is the one-dimensional (1D) Potts model, for
which exact results have been obtained [13]. This will be discussed in the following
section.
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3 Reaction-Diffusion Processes
Domain coarsening in the 1D Ising model at temperature T = 0 takes place
through the elimination of domain walls. In this case the walls are independent
random walkers which annihilate on contact, so the coarsening can be described
by the reaction-diffusion process A + A → 0. The “site” (or spin) persistence problem posed above then amounts to calculating the fraction of sites that have not
not been crossed by an A-particle or, in a continuum description, the fraction of
the line which has not been traversed by an A-particle. The generalization to the
q-state Potts model is immediate. In this model there are q possible phases. If
the domain walls are represented by A-particles, and the system coarsens from a
random initial state, two domain walls either annihilate (A + A → 0) on contact
or coalesce (A + A → A), with probabilities 1/(q − 1) and (q − 2)/(q − 1), these
being the probabilities that the Potts states on either side of the walls are the
same or different respectively. The persistence (fraction of space not crossed by
the exponent is given exactly by [13]
a walker) decays as P (t) ∼ t−θ(q) where
√
θ(q) = −1/8 + (2/π 2 ){cos−1 ([2 − q]/ 2q)}2 .
A related problem, which might be termed “walker persistence” (as opposed
to “site” persistence) is the following. What is the fraction of walkers (i.e. Aparticles) which have never met another walker? The connection with the previous
problem is as follows. Let us tag one particle and give it a different diffusion
constant (D ) from the other particles (D). Then we calculate the probability,
P (t), that the tagged particle has not met another particle up to time t. For D = 0
this problem reduces to the site persistence problem, while D = D corresponds
to the walker persistence problem. An early analytical attack on the general D
problem was made by Monthus [14], who found P (t) ∼ t−φ(q) and computed φ(q)
to first order in D /D, for all q, and to first order in (q − 1) for all D /D. Two
trivial cases are q = 2, D = D, and q = ∞, D /D arbitrary. In the former, all
surviving walkers are persistent (since all reactions lead to annihilation), so the
persistence is proportional to the particle density which is known [1] to decay as
t−1/2 , i.e. φ(2) = 1/2. In the latter, all reactions lead to coalescence, so only the
closest walkers to the left and right of the tagged particle play a role. The problem
then reduces to a three-particle problem that can be solved exactly [15] for any
D /D, with P (t) ∼ t−3/2 for D = D, i.e. φ(∞) = 3/2. For values of q in the
range 2 < q < ∞, however, only simulation results are available up to now [16].
One point worth noting is that φ(q) > θ(q) for all q, i.e. the tagged particle has a
greater survival probability if it stays still (D = 0) than if it moves like the other
particles (D = D). We will return to this theme below.
A grossly simplified version of this model, which turns out to be equivalent
to a much-studied problem in chemical kinetics, is as follows. Consider a set of
completely non-interacting random walkers, with equal diffusion constants D, initially randomly distributed with density ρ. What is the fraction, P (t), of walkers
that have never met another walker up to time t? Clearly this will decay faster
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than for the Potts model, because no walkers ever disappear. In fact it decays as
√
P (t) = exp(−λ1 t)
(1)
in 1D, as first shown rigorously by Bramson and Lebowitz [17] in another context.
For analytical study it is convenient to tag one particle, give it a different diffusion
constant, D , and treat the other particles as traps for the tagged particle. The
idea is then to calculate the survival probability, P (t), of the tagged particle in
the presence of the moving traps.
Now consider a seemingly very different problem, the two-species annihilation
process A + B → 0, where the A and B particles have diffusion constants DA and
DB , and are initially randomly distributed with densities ρA (0), ρB (0). What is
the asymptotic time dependence of the particle densities? This problem was first
posed almost 20 years ago in the seminal paper of Toussaint and Wilczek (TW)
[18], using the process as a model of monopole-antimonopole annihilation in the
early universe (though applications to chemical kinetics and condensed matter
physics are more numerous [19]). TW showed that if ρA (0) = ρB (0) the densities
decay asymptotically as t−d/4 in space dimensions d < 4. If the initial densities are
different, however, the density of the minority species (A, say) decays much more
rapidly. The connection with the trapping problem is as follows. At late times,
ρA (t) ρB (t) and the A particles can be regarded as independently diffusing in a
background of the majority B particles, which act as traps for the A particles by
virtue of the annihilation reaction. An equivalent problem, therefore, is to consider
a single A particle moving among B particles (which do not interact with each
other) and ask for the probability, P (t), that the A particle survives to time t. In
the context of the original A + B → 0 process (or A + B → B [20], which has the
same asymptotics [17]) the “particle” A and “traps” B are taken to have the same
diffusion constants, but for generality we will take them to be different, DA = D
and DB = D.
Until recently, no analytical result was available for the constant λ1 in Eq.
(1). Furthermore, attempts to determine λ1 by numerical simulations (or even to
confirm the predicted stretched-exponential decay with exponent 1/2) are severely
hampered by large, slowly decaying transients [21, 22, 23, 24]. In ref. [23], a sophisticated numerical approach enabled data to be obtained down to P (t) ∼ 10−35 ,
but still the asymptotic time dependence could not be unambiguously established.
In the following section we outline a derivation of an exact result for λ1 and present
the general result for dimensions in the range 1 ≤ d ≤ 2. The principal results are
presented in ref.[25], with full details in ref.[26].
4 Diffusing Particle with Mobile Traps
Consider a particle with diffusion constant D moving in a sea of mobile traps
with diffusion constant D. The particle starts at the origin, while the traps are
initially placed randomly in space with density ρ. We derive an exact result for the
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First-Passage Problems in Spatially Extended Systems
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constant λ1 in the 1D result (1), and present the generalization to space dimensions d ≤ 2. The results are obtained by deriving upper and lower bounds on the
survival probability of the particle and showing that they coincide asymptotically
for t → ∞. The upper bound is intuitive, but supported by numerical studies,
while the lower bound is rigorous.
1. Upper Bound. An intuitive upper bound, PU (t), on P (t), for any D , is provided
by the problem with D = 0, in which the particle stays at its initial position,
which we call x = 0. Although we have as yet been unable to make this bound
rigorous [27], it is intuitively clear that when, as here, the traps are (statistically)
symmetrically placed with respect to the particle, the particle will on average
survive longer if it stays still than if it diffuses. This assertion has been checked,
using the algorithm outlined in ref.[23], for all (lattice) walks up to time t = 28
[26]. For D = 0, P (t) is just the probability that none of the moving traps has
reached the origin up to time t. This problem is exactly soluble [28]. Since similar
techniques will be needed to derive the lower bound, we outline the solution here.
The N traps move independently according to the Langevin equations ẋi =
ηi (t) , i = 1, . . . , N , where ηi (t) is Gaussian white noise with mean zero and
correlator ηi (t)ηj (t ) = 2D δij δ(t − t ). The quantity P (t) is just the product
of the individual trap probabilities. For
√ a given trap starting at xi , the required
probability is [1] P1 (xi , t) = erf(|xi |/ 4Dt). So the upper bound is
N
√
PU (t) =
erf |xi |/ 4Dt
,
(2)
i=1
where . . . means an average over the initial positions of the traps. Since the xi
are also independent, the latter average also factors. Using N = ρL, where L is the
length of the system, and each xi is uniformly distributed in (−L/2, L/2), gives
PU (t) = 1 − 1/L
L/2
−L/2
√
ρL
dx erfc(|x|/ 4Dt)
→ exp[−4ρ(Dt/π)1/2 ] ,
(3)
where the final result follows on taking the limit L → ∞.
2. Lower Bound. Consider the same system as before but with a pair of absorbing
boundaries at x = ±l/2. We consider the subset of initial conditions in which
all the traps lie outside the interval (−l/2, l/2) (and the particle is at x = 0),
and trajectories in which neither the particle nor any of the traps has crossed a
boundary up to time t. We calculate the probability, PL (t), of such an occurrence
over the ensemble of all initial conditions and trajectories. These restricted initial
conditions and trajectories are a subset of all the possible initial conditions and
trajectories in which the particle never meets a trap. It follows that P (t) ≥ PL (t),
i.e. PL (t) is a lower bound.
The probability that there are no traps in the interval (−l/2, l/2) at t = 0 is
exp(−ρl). Given that there are no traps in this interval at t = 0, the probability
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Ann. Henri Poincaré
that no traps enter the interval up to time t is given by the same result as in the
derivation of the lower bound, namely exp[−4ρ(Dt/π)1/2 ]. Finally, the probability
that the particle, starting at x = 0, has not left the interval (−l/2, l/2) up to time
t is given, for times t l2 /D , by [1] (4/π) exp(−π 2 D t/l2 ). Assembling these
contributions gives
P (t) ≥ (4/π) exp[−4ρ(Dt/π)1/2 − (ρl + π 2 D t/l2 )] .
(4)
Since this inequality holds for all l, the best lower bound is obtained by
maximizing with respect to l. The optimum value is l = (2π 2 D t/ρ)1/3 , and the
best lower bound is
PL (t) =
4
exp[−4ρ(Dt/π)1/2 − 3(π 2 ρ2 D t/4)1/3 ] .
π
(5)
Since the second term in the exponent is negligible compared to the first as t → ∞,
the two bounds converge to yield the asymptotic form
P (t) ∼ exp[−4ρ(Dt/π)1/2 ] .
(6)
for P (t).√More precisely, we can take the logarithm of P (t) and divide out the
leading t dependence to get
π 2/3 (D /D)1/3
4
ln P (t)
4
√ ≤− 2
√
+
3
≤
,
(7)
π
π
2
(ρ Dt)1/2
(ρ2 Dt)1/6
√
giving limt→∞ −[ln P (t)]/(ρ2 Dt)1/2 = 4/ π. It is striking that the asymptotic
form (6) depends only on the density ρ and diffusion constant D of the traps, and
is independent of the diffusion constant D of the particle.
As an aside we note that, while the left-hand inequality in (7) holds for all
t, since Eq. (3) does, the right-hand inequality is strictly a large t result. This is
because the factor (4/π) exp(−π 2 D t/l2 ) in Eq. (4) comes from the lowest mode in
the Fourier decomposition of the survival probability of the particle in the interval
(−l/2, l/2). This mode dominates for D t l2 , which requires ρ2 D t 1. A lower
bound on P (t) valid for all t can easily be written down by including all Fourier
modes, but the large-t form (5) is sufficient for present purposes. A comparison
of the predicted bounds with the numerical data of ref.[23] show that the data
lie between the bounds [25, 26] (except at early times where lattice effects are
important [24]). The approach to asymptotic behavior is slow, but consistent with
the slow convergence of the bounds. A clearer understanding of the subdominant
terms is necessary before a detailed comparison with numerical or experimental
data can be made.
The model can be generalized to include n diffusing particles starting from
the same point [23]. The bounds can easily be generalized to this case [26]. The
probability that all n particles survive at time t satisfies the same inequalities (7)
except that the final term on the right-hand side acquires an additional factor
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First-Passage Problems in Spatially Extended Systems
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n1/3 . So the bounds converge for t → ∞ and the same asymptotic form, Eq. (6),
is obtained for all n.
This approach is also readily generalized to dimensions d > 1 [26]. For d <
2 we find P (t) ∼ exp(−λd td/2 ), where λd = (2ρ/πd)(4πD)d/2 sin(πd/2) while,
for d = 2, P (t) ∼ exp[−λ2 t/ ln t] with λ2 = 4πρD. The latter agrees with the
functional form obtained in [17], but with a precise value for the constant λ2 .
For d > 2, simple exponential decay is obtained, in agreement with [17], but the
bounds no longer converge and the decay constant cannot be determined [26].
The optimal lengthscale l of the spatial region used to construct the lower
bound behaves as l ∼ t(2−d)/(4−d) for 1 < d < 2, and as l ∼ ln t for d = 2. The
fact that this lengthscale is large for t → ∞ suggests a degree of universality in
the results, e.g. it should not matter for the asymptotics whether the problem is
defined on a continuum or a lattice. Similarly the asymptotic expression for the
upper bound is insensitive to small-scale details of the model: one obtains the same
asymptotic behavior on the lattice and the continuum. For d > 2 this is no longer
true: the optimal l tends to a constant determined by the range of interaction of
the particle and traps (a nonzero interaction range is necessary for all d ≥ 2, or
the interaction rate is zero), and the two bounds no longer converge. We therefore
expect the asymptotics to be non-universal for d > 2. The role of d = 2 as a kind of
critical dimension for this problem is directly related to the recurrence of random
walks in d < 2.
We conclude this section by quoting some results for the 1D trapping problem
with a finite number of traps. In this case the particle’s survival probability decays
asymptotically as a power-law, P (t) ∼ t−θ , where the exponent θ is a function of
the numbers NL and NR of traps respectively to the left and right of the particle.
The only analytical results are for N ≡ NL + NR ≤ 2 [15, 29], but one can obtain θ
perturbatively for general NL and NR as a power series in the expansion parameter
αN = D /(D + (N − 1)D ) [25, 30]. To second order the result is
θ=
N 1
1
+ (N −∆2 )αN + 2 [(N −2)(N −∆2 )(π −8(1−ln 2))+2N (N −1)]α2N (8)
2 π
π
where ∆ = NL −NR . One noteworthy feature
√ of this result is that for the symmetric
case ∆ = 0 (or, more generally, when ∆ ≤ N ), θ increases with D for small D .
This lends additional support to our assertion that setting D = 0 should give an
upper bound on the survival probability.
5 Conclusion
A number of examples of persistence, or first-passage, phenomena in spatially
extended systems have been presented. In coarsening phenomena, the persistence
usually decays as a power-law of the time, t, whereas in the trap model discussed
in the latter part of this work it decays as a stretched exponential for d < 2 (and a
simple exponential for d > 2 [17]). It is striking that in the trap model the leading
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Ann. Henri Poincaré
asymptotic decay of the particle’s survival probability is, for d ≤ 2, independent
of the diffusion constant, D . By contrast, in the coarsening dynamics of the 1D
Potts model, for example, it depends in a non-trivial way on the ratio, D /D, of
the particle and trap diffusion constants [14]. In particular, the exponent θ(q),
which characterizes the survival probability of a static particle (D = 0) is smaller
than the exponent φ(q) that describes the case D = D [16]. In an attempt to gain
some insight into why D is important in some models and not in others we can
employ the “toy model” of ref.[31].
In this model one considers a static particle (D = 0) and moving traps. The
density of traps can decrease with time, e.g. through annihilation or coalescence
events, but in the toy model one treats the traps as disappearing randomly in such
a way as to mimic the time-dependence of the density, ρ(t), of the original model.
In d = 1, for example, the survival probability of the particle is given by [31]
D t ds
ρ(s) .
(9)
Q(t) = exp −2
π 0 s1/2
For ρ(t) = ρ, a constant, this approach is, of course, exact and one recovers the
result (3). If ρ(t) ∼ t−a for large t, however, (9) predicts a stretched exponential
decay Q(t) ∼ exp(−const t1/2−a ) for 0 < a < 1/2 and a power-law decay for
a = 1/2, the latter result being in qualitative agreement (but with the wrong
powers!) with the known power-law decay of persistence in, for example, 1D Potts
models.
Consider, now, the stretched exponential regime and let the particle now
move, with diffusion constant D . Within the toy model one can exploit the same
type of bounding arguments as employed in section 4. Eq. (9), corresponding to
D = 0, gives the intuitive upper bound, while the derivation of a rigorous lower
bound follows the same pattern as Eq. (4), i.e. the result of the upper bound,
Eq. (9) is multiplied by the factor exp[−(ρl + π 2 D t/l2 )], which has to be maximized with respect to the “box size” l. As before, this second factor has the form
exp[−const (D t)1/3 ], depending explicitly on D . The convergence of the bounds
for large t requires that this latter factor be subdominant with respect to the
nominally dominant factor, Eq. (9). If ρ(t) ∼ t−a , with a < 1/2, for large t this
dominant factor decays, as we have seen, as Q(t) ∼ exp(−constt1/2−a ), so the
relevant condition for the convergence of the bounds is a < 1/6. Within the toy
model, therefore, the leading asymptotic form for the survival probability is definitely independent of D for a < 1/6, but not necessarily for a > 1/6. Although
a better understanding of the D -dependence is clearly desirable, the results from
this very simple toy model are rather suggestive.
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Ann. Henri Poincaré
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Alan J. Bray
Department of Physics and Astronomy
The University
Manchester M13 9PL
UK