Escape through an unstable limit cycle: Resonant activation
Bidhan Chandra Bag
*
and Chin-Kun Hu
†
Institute of Physics, Academica Sinica, Taipei 11529, Taiwan
Received 13 April 2006; published 26 June 2006
We consider a Brownian particle acted on by a linear conservative force, a nonlinear frictional force, and
multiplicative colored and additive white noises; the frictional force can be negative when the external energy
supply is large enough. We numerically calculate the mean first passage time MFPT for the particle to escape
from an unstable limit cycle and find resonant activation, i.e., the MFPT first decreases, followed by a rise after
passing through a minimum with increasing noise correlation time for a fixed noise variance. For fixed noise
strength of the multiplicative noise the MFPT increases linearly with . This is in sharp contrast to the case of
fluctuations of nonlinear potentials, in which the MFPT first increases nonlinearly before reaching a limiting
value. Our model could be useful for understanding some biological processes.
DOI: 10.1103/PhysRevE.73.061107 PACS numbers: 05.40.Jc, 05.45.-a, 05.20.-y, 05.70.Ln
The processes of noise-driven escape of particles over po-
tential barriers, i.e., the well-known Kramers’ problem 1,
are ubiquitous in a wide variety of physical, chemical, and
biological contexts 2. Kramers considered a model Brown-
ian particle trapped in a one-dimensional well representing
the reactant state, which is separated by a barrier of finite
height from a deeper well, signifying the product state. The
particle was supposed to be immersed in a medium such that
the medium exerts a frictional force on the particle but ther-
mally activates it so that the particle may gain enough energy
to cross the barrier. Over several decades the model and
many of its variants have served as standard paradigms in
various problems of physical and chemical kinetics to under-
stand the decay rate of metastable systems in the over-
damped and underdamped limits 3, the effect of anharmo-
nicities 4, the role of the relaxing bath 5, the signature of
non-Markovian effects 6, and quantum and semiclassical
corrections to the classical rate and related similar aspects
7. The vast body of literature has been the subject of sev-
eral reviews 2,8. In the majority of these studies the focus
lies on the competing attractors of the dynamical system,
which are separated by a separatrix containing a saddle point.
However, models where the separatrix is instead an unstable
limit cycle often arise in the context of chemical reactions
constrained to occur far from a equilibrium 9. The rate of
escape through the unstable limit cycle in the weak noise
limit has been studied 10 and noise-driven transitions in
models with an unstable limit cycle can be found in Ref.
11.
Most of the works mentioned above have considered the
static barrier. However, a surge of fresh interest in this topic
was triggered, not long ago, by Doering and Gadoua 12
who studied how the interwell mean first passage time
MFPT of a Brownian particle in a bistable potential de-
pends on the correlation time of the barrier fluctuations.
They observed that this dependence may be nonmonotonic
and called it resonant activation RA. This interesting phe-
nomenon 12 has been further investigated 13–19. In the
present paper we examine a related issue. We show that the
phenomenon of RA appears in the problem of escape through
an unstable limit cycle in the presence of multiplicative col-
ored and additive white noises. Although traditionally RA
appears due to fluctuations in a nonlinear potential, the
present analysis suggests that it may be observed even in the
presence of a linear potential as a result of strange behavior
of the limit cycle. The limit cycle is a conspicuous feature in
a variety of models in biology and chemistry that deal with
situations far from thermal equilibrium. On the other hand, in
Refs. 17,19 it was emphasized that “far from equilibrium”
is one of the necessary conditions for the RA phenomenon.
To start with, we consider a simple model defined by
v ˙ =- aq + bv
2
-1v + qt + t , 1
where q and v q ˙ represent the coordinate and the velocity
of the Brownian particle, and t and t are Gaussian
colored and white noises, respectively. In general, we ex-
press the thermal fluctuation of the system as additive noise
and the effect of the external environmental fluctuation on
the system as multiplicative noise. Thus t and t in Eq.
1 correspond to internal thermal noise and external noise,
respectively. In a complex system multiplicative noise is
very relevant and it makes the system far from equilibrium.
The two noises are characterized by the relations t =0,
t =0,
tt' =
D
0
e
-|t-t'|/
, 2
tt' =2Dt - t' . 3
Here D
0
and are the strength and correlation time of the
multiplicative noise; D is the strength of the additive white
noise. Equation 2 shows that in Eq. 1 is the Ornstein-
*On leave from Department of Chemistry, Visva-Bharati, Santini-
ketan, India.
†
Corresponding author. Electronic address:
huck@phys.sinica.edu.tw
PHYSICAL REVIEW E 73, 061107 2006
1539-3755/2006/736/0611074 ©2006 The American Physical Society 061107-1