Driven activation versus thermal activation
Driven activation versus thermal activation
Patrick Ilg∗ and Jean-Louis Barrat
Université de Lyon; Univ. Lyon I, Laboratoire de Physique de la Matière Condensée et des Nanostructures; CNRS,
UMR 5586, 43 Bvd. du 11 Nov. 1918, 69622 Villeurbanne Cedex, France
(Dated: November 2, 2018)
Activated dynamics in a glassy system undergoing steady shear deformation is studied by numerical simula-
tions. Our results show that the external driving force has a strong influence on the barrier crossing rate, even
though the reaction coordinate is only weakly coupled to the nonequilibrium system. This ”driven activation”
can be quantified by introducing in the Arrhenius expression an effective temperature, which is close to the one
determined from the fluctuation-dissipation relation. This conclusion is supported by analytical results for a
simplified model system.
PACS numbers: 64.70.Pf,05.40.-a,05.70.Ln
Activated rate theory is ubiquitous in the description and
understanding of dynamical processes in condensed matter,
physical chemistry or materials science. The basic problem,
known as the ”barrier crossing” or ”Kramers problem”, is that
of a single degree of freedom, coupled to a heat bath, and
moving in a double well potential. The ”barrier crossing rate”
is defined as the average time taken by the system to switch
from a potential well to the other, under the influence of ther-
mal noise. In general, the single degree of freedom, often
called ”reaction coordinate”, is coupled to a complex, fluc-
tuating environment. The ”thermal noise” is a schematic de-
scription of the interaction with this environment.
This approach has been applied to a wealth of different
problems. We can for example mention diffusion in solids, in
which case the reaction coordinate is an atomic position, and
the noise is associated with thermal vibrations. In isomeriza-
tion reactions, the reaction coordinate is an internal coordinate
of the molecule, coupled to a liquid solvent. In nucleation the-
ory, the internal coordinate describes a collective fluctuation
of an order parameter, and the ”barrier” is interpreted as a free
energy, rather than energy, barrier. Other examples involve
the Eyring theory of plasticity in solids, in which the activated
process is associated with a local strain change.
The analysis of the barrier crossing problem is often asso-
ciated with the names of Eyring, who proposed the so called
”transition state approximation” [1], and of Kramers, who
made the first complete analysis of the problem in the lim-
its of low and high friction [2]. Since then, many refinements
of the theory have been studied and are reviewed in reference
[3]. In all cases, it turns out that an essential factor in the re-
action rate, which to a large extent governs the variation with
temperature T , is the Arrhenius contribution:
r(T ) ∼ exp(−∆E/kBT ) (1)
where ∆E is the energy barrier to overcome. The exponential
variation of the Arrhenius factor (1) is, in fact, the hallmark of
activated processes.
∗Present address: ETH Zürich, Polymer Physics, HCI H541, CH-8093
Zürich, Switzerland
As discussed above, activated processes are often invoked
in the description of the dynamical response of condensed
matter systems. As such, they will typically take place un-
der nonequilibrium conditions. The deviation from equilib-
rium can be weak, e.g. during the flow of a Newtonian liquid,
in which case the applicability of equation (1) is straightfor-
ward. In other cases, however, the same equation is applied to
systems that are strongly out of equilibrium, in the sense that
their response to an external driving force is strongly nonlin-
ear, or that their phase space distribution is very different from
the equilibrium, Gibbs-Boltzmann distribution.
A prototypical example of such a strongly nonequilibrium
situation is the flow of a glassy system. Such a flow can be in-
duced only by stresses larger than the yield stress (see e.g. [4]
for the effect of strain and temperature in glassy solids). In
the absence of flow, the relaxation is very slow, and the sys-
tem is out of equilibrium and non-stationary [5, 6]. The flow
produces a nonequilibrium steady state [7, 8, 9], with a typi-
cal relaxation time that is fixed internally by the applied stress
or the strain rate. This situation has attracted a considerable
amount of theoretical and experimental interest, in two differ-
ent contexts. The first one is the rheology of ”soft glasses”
(emulsions, pastes, colloidal glasses, foams). The second one
is the plastic deformation of bulk metallic glasses. In both
cases, approaches have been proposed that introduce a ”noise
temperature” [10] or ”disorder temperature” [11, 12]. In [10],
this noise temperature replaces the actual temperature in equa-
tion (1). In such models, the effective temperature is intro-
duced in a somewhat empirical manner.
Another concept of effective temperature, rooted in sta-
tistical mechanics ideas, was introduced in [13, 14], based
on the ”fluctuation-dissipation ratio”. At equilibrium, the
fluctuation-dissipation theorem states that the ratio between
integrated response and correlation functions (FDR) is equal
to the temperature. Cugliandolo et al. [13] showed how this
concept could be extended to out-of-equilibrium system, by
defining the effective temperature from the FDR, which now
differs from the thermal bath temperature. It was proposed
that a thermometer probing a nonequilibrium system on long
time scales would actually be sensitive to this effective tem-
perature, and this result was checked numerically on simple
models [8, 9, 15]. Experimental evidence supporting this
definition of an effective temperature has been found e.g. in
http://arxiv.org/abs/0704.0738v1
[16, 17].
In this contribution, we explore the influence of an external
driving force on the rate of a simple activated process. Our
primary objective is here to check how the external drive, and
the ”noise” it generates, can influence the dynamics of an in-
ternal degree of freedom, which is not directly coupled to the
driving force. A very standard way of quantifying the results
is to use the Arrhenius representation, which provides an op-
erational way of introducing an ”activation temperature”, that
can be compared to other calculations of effective tempera-
tures in nonequilibrium systems.
Our approach involves the simulation of the classical Kob-
Andersen ”binary Lennard-Jones” model undergoing shear
flow, similar to the one used in ref. [8]. In order to probe ac-
tivated dynamics, one appealing possibility would be to iden-
tify and study the activated events that actually give rise to
the flow at low temperature, in the spirit of [10]. This ap-
proach, however, is difficult and could yield ambiguous re-
sults, as the flow is self consistently coupled to these events.
We therefore make use of the flexibility of numerical model-
ing to devise a very simple ”activated degree of freedom” that
has only a weak coupling to the existing flow in our system.
This is achieved by replacing each particle of the minority
species rBj by a peanut shaped ”dumbbell” with coordinates
j ± (uj/2)e
z, with fixed orientation along ez , the direc-
tion perpendicular to the shear plane. Each center of force
in the dumbbell carries half of the particle interaction, and
the separation between the two centers of force u is small
enough that the perturbation of the surrounding fluid can be
neglected. The important feature of the model is the fact that
the two centers of force are related through an internal ”reac-
tion coordinate” u, which evolves in a bistable intramolecular
potential V (u) = (V0/u
2 − u20)
2, where u0 = 0.1 (in
Lennard-Jones units) is the equilibrium dumbbell separation
(see fig. 1b). Each dumbbell is therefore a simple ”two-state”
system which can undergo, under the influence of the interac-
tions with the surrounding fluid, an ”isomerization reaction”.
This reaction corresponds to exchanging the positions of the
two centers of force (see fig. 1).
This ”isomerization” will be the focus of our study. Its rate
can be studied as a function of the imposed barrier height, of
the external temperature T and on the driving force, which is
here quantified by the shear rate γ̇. We have chosen to work
under conditions for T and γ̇ that have been well character-
ized previously [8] (T = 0.3 and γ̇ = 10−3, in Lennard-Jones
units) and to concentrate on the influence of barrier height
∆E = V0. At this temperature, the system would not undergo
structural relaxation on the time scales that can be achieved
using computer simulation. Under the influence of the exter-
nal drive, a relaxation on a time scale τα ≃ 100 is observed.
This time scale is very well separated from microscopic, vi-
brational time scale, so that our system is a practical realiza-
tion of the theoretical concepts described in ref. [13].
Determination of reaction rates is a notoriously difficult
challenge for numerical simulations, as the activated events
typically take place on much larger time scales than the short
time vibrations of the intramolecular bonds. A number of so-
phisticated methods [18] have been developed to bypass this
intrinsic difficulty, either from biased simulations, or by mak-
ing use directly of the rate formula 1. Unfortunately, such
methods always assume that the system is close to thermal
equilibrium, and are therefore inapplicable in our case. The
forward flux method recently proposed in [19] is applicable
to nonequilibrium systems. However, only a single reaction
coordinate per system can be treated with this method, which
is impractical for the present situation. As a result, we have
to use ”brute force” simulations to obtain reaction rates from
the study of individual trajectories, which seriously limits the
range of barrier heights that can be considered.
The Sllod equations of motion appropriate for a fluid under-
going simple shear were integrated with a leapfrog algorithm
using a time step of ∆t = 5×10−4 in reduced Lennard-Jones
units [8]. For the dumbbell particles, the leapfrog algorithm
is applied to the center-of-mass and relative positions and mo-
menta. Lees-Edwards periodic boundary conditions [8] are
employed in order to minimize effects due to the finite system
size. Constant temperature conditions are ensured by rescal-
ing the velocity components in the neutral direction of all par-
ticles at each time step.
The reaction rate r is determined from the number correla-
tion function C(t) by
C(t) =
〈δn(t)δn(0)〉
〈δn(0)2〉
≈ exp [−rt/〈n〉] (2)
where δn(t) = n(t) − 〈n〉 and n(t) equals one if u(t) > uB
and zero else [20]. The systems studied contain N = 2048
particles, 410 of which are dumbbells. Eq. (2) is evaluated
from an ensemble average over 20− 40 independent systems.
The results are found to be independent of the exact location
of the dividing surface uB in the vicinity of the barrier max-
imum uB = 0. The fast initial decay of C(t) is well de-
scribed by transition state theory. Escape rates are extracted
from fits to Eq. (2) for intermediate times 5 ≤ t ≤ 10.
We verified that very similar results are found within a broad
range 1 ≤ t ≤ 30, before the correlation function finally de-
cays to zero, in full agreement with theoretical expectation
[20]. For relatively low barrier heights V0/T . 3, C(t) de-
cays more rapidly, so that we extracted rates for shorter times,
0.5 ≤ t ≤ 1.
In the following, we present results for the reaction rate as a
function of V0 based on the study of the decay in the number-
number correlation function [20]. We adopt common practice
by giving all temperature and energy values in terms of the
depth of the Lennard-Jones potential ǫ.
Figure 2 illustrates the difficulty of the approach, by show-
ing the trajectories of selected dumbbells for different values
of the barrier at T = 0.3. For V0/kT = 1, barrier crossings
are so common that describing them trough classical rate the-
ory is problematic. For V0/kT = 10, the crossings become
very unlikely, so that the determination of the rate becomes
difficult. This leaves us with typically two decades in terms of
variation of the reaction rate.
The corresponding reaction rates, determined from the cor-
relation function of the dumbbell internal coordinate, are
shown in figure 3. At T = 0.8, γ̇ = 10−3, the rates obey
the equilibrium Arrhenius law (1), showing that under these
conditions the drive is only a weak perturbation to the system.
We now concentrate on the rates obtained at T = 0.3 and
γ̇ = 10−3. The reaction rates are clearly influenced by the
external driving imposed to the system. To show this, we use
the rates obtained at a rather high temperature, req(T = 0.8),
to extrapolate to T = 0.3. The equilibrium extrapolation rext
is achieved using the Arrhenius formula, i.e. rext(T = 0.3) =
req(T = 0.8)×exp(+V0/0.8−V0/0.3). Clearly, the extrapo-
lated rates are significantly lower than those actually observed
under shear, except at low barrier heights (high rates) were the
two estimates almost coincide. The difference between the ex-
trapolated rates and the measured ones is an indicator of the
inadequacy of the standard Arrhenius formula, using the ther-
mal bath temperature, in the driven system.
In spite of the limited range of accessible rates, it is clear
from figure 3 that the rates in a glassy system under shear do
not obey Arrhenius behavior of the form exp(−V0/T ) over
the whole range of barrier heights under study. While this
law is relatively well obeyed at low barriers and large cross-
ing rates, it would significantly underestimate the rate for high
barriers. Instead, at high barrier rates, the crossing rate is con-
siderably increased. If an attempt is made to fit the results
to an ”effective Arrhenius factor”, a value of Teff ≃ 0.6 is
obtained.
Under the same conditions, a completely different de-
termination of the effective temperature [8], based on the
fluctuation-dissipation approach mentioned above, yields
T ∗ ≃ 0.65. This is in good agreement with the present fit
to an Arrhenius law. The determination of Teff based on re-
action rates is of limited accuracy, such that we cannot ex-
clude that Teff and T
∗ actually differ slightly or that Teff is
slightly dependent on barrier height. A more precise determi-
nation of Teff would require larger barrier heights, which is
computationally quite demanding. Note, that V0 = 3 corre-
sponds for T = 0.3 already to the rather high barrier height of
V0/kT = 10.
In figure 3, we also display the results obtained for the rates
at a slightly higher value of the shear, γ̇ = 10−2. The sepa-
ration of time scales between relaxation time and microscopic
times is less marked than for the low shear rates (τα ≃ 10
in this case). It appears that the increase in shear induces a
change in the prefactor for the rates, rather than in the bar-
rier height dependence. This is consistent with the relatively
weak influence of shear rate on effective temperature reported
earlier [8].
It is interesting to discuss the time scale at which the
crossover between the two Arrhenius laws, characterized ei-
ther by the bath temperature or an effective temperature, takes
place. A natural guess would be to associate this crossover
with a value of the rate that corresponds to the inverse of the
α relaxation time. The general idea is, that fluctuations taking
place on longer time scales will be associated with a higher
temperature [8, 13]. In figure 3 we see that this guess over-
estimates the crossover rate by a factor of 5 in the case of
γ̇ = 10−3. It is not clear at this point, whether this differ-
ence is significant or reflects merely some arbitrariness in the
definition of relaxation times.
The simulation results presented above suggest that the
activated dynamics is governed by an elevated temperature
Teff ≃ 0.6 > T . This temperature is consistent with the ef-
fective temperature T ∗ = 0.65 found in extensive simulation
studies on the fluctuation-dissipation relation in this system
[8]. In order to investigate the relation between Teff and T
and to rationalize our simulation results, we study to following
toy model proposed in [21].
Consider a particle of mass m at position x moving in an
external potential V (x) under the influence of two thermal
baths. One bath, associated with the fast degrees of freedom,
is kept at temperature Tfast and exerts an instantaneous fric-
tion force of strength Γ0. The second bath, which mimics the
slow degrees of freedom is held at temperature Tslow and is
described by the retarded friction coefficient (memory kernel)
Γ(t). The equations of motion read ẋ = v,
mv̇ = −V ′(x)−
dsΓ(t−s)v(s)−Γ0v(t)+ξ(t)+η(t) (3)
The fast bath is modeled as Gaussian white noise with
〈η(t)〉 = 0, 〈η(t)η(s)〉 = 2Tfastδ(t − s), whereas the ran-
dom force due to the slow bath is described by 〈ξ(t)〉 = 0,
〈ξ(t)ξ(s)〉 = 2TslowΓ(t − s). We use an exponentially de-
caying memory kernel Γ(t) = α−1e−t/(αγ) for which the
non-Markovian dynamics (3) can equivalently be rewritten as
Markovian dynamics in an extended set of variables [22].
Exact solutions of the model (3) for harmonic potentials
V are presented in [21]. For barrier crossing problems with
double-well potentials V , no analytical solutions to (3) are
known. We therefore extend the widely used transition state
approximation to the present model after adiabatic elimina-
tion of the fast degrees of freedom. The resulting expres-
sion for the rate rTST is rather lengthy and will be presented
elsewhere together with the (straightforward) procedure. For
the double-well potential V (x) considered above, the depen-
dence of the rate rTST on the barrier height V0 is again domi-
nated by the Arrhenius factor, however with an effective tem-
perature Teff,TST = Tfastw/[w + 4(Tfast − Tslow)], where
w = Tslow + αV
′′Tfast and V
′′ = 8V0/u
0. Thus, if the
slow and fast bath are both kept at the same temperature,
Tslow = Tfast = T , one recovers the usual Kramers re-
sult with Teff,TST = T . If, however, Tslow > Tfast, the
escape rate is enhanced due to Teff,TST > Tfast. Due to
the interplay between fast and slow dynamics in the barrier
crossing, the effective temperature is in general intermedi-
ate between the temperature of the slow and the fast bath.
These predictions are in agreement with the simulation re-
sults presented above. Furthermore, estimating the coefficient
α ≈ 0.01 from the inverse high frequency shear modulus for
the Lennard-Jones system [22], the predicted effective tem-
perature is Teff,TST ≈ 0.45. In view of the simplicity of the
model and the uncertainty in α, the order of magnitude agree-
ment with the observed Teff is reasonable.
In conclusion, we have shown that activated processes out
of equilibrium are influenced by an external driving, even if
the corresponding degree of freedom is weakly coupled to
the drive. Qualitatively, this increase is the essential result
from our simulations. From a more quantitative point of view,
the analysis of the Arrhenius plot allows one to define oper-
ationally an effective activation temperature. The link of this
activation temperature to other definitions of effective tem-
perature, and the time scale for the crossover from ”thermal
activation” to ”driven activation” will have to be explored fur-
ther. However, the results are consistent with a general picture
involving a degree of freedom coupled to two different heat
baths, one associated with short time vibrations and one asso-
ciated with shear induced fluctuations, taking place on longer
time scales and described by a higher temperature [21].
This ”driven activation” (as opposed to ”thermal” activa-
tion) could have interesting consequences for characterizing
the effective temperature of nonequilibrium systems, by pro-
viding a ”thermometer” based on activated processes. It can
also be of importance within the theory of plasticity of amor-
phous materials, by providing a self-consistent description of
the ”noise” that induces local plastic events, within a classical
statistical mechanics description involving a noise tempera-
ture [11, 23]. Inserting an effective temperature in Eyring’s
rate theory of plasticity, T. Haxton and A. J. Liu were recently
able to account for the flow curves of simple glassy systems
at low temperatures [24].
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dumbbell separation  u 
FIG. 1: (a) Schematic representation of a dumbbell particle in a sys-
tem undergoing shear flow with fixed orientation perpendicular to the
shear plane. Also shown is an isomerization reaction. The magnitude
of the separation between the two centers of force is considerably
exaggerated in this schematic representation. (b) Intramolecular po-
tential (characteristic of the nonlinear ”spring” shown in panel (a) )
between the two centers of force that define the dumbbell.
FIG. 2: (Color online) Trajectories of the internal dumbbell coor-
dinate for different barrier heights. The thermal bath temperature
is T = 0.3 in reduced Lennard-Jones units, and the shear rate is
γ̇ = 10
1 1.5 2 2.5 3 3.5
barrier height  V
τα, γ=10−2
τα, γ=10−3
1 / 0.8
1 / 0.3
1 / 0.6
FIG. 3: (Color online) Reaction rate as a function of barrier height,
for fixed temperature and shear rate. Full squares: results for T =
0.8 (red) at equilibrium. T = 0.3 (black and blue), and different
shear rates (full diamonds and circles correspond to γ̇ = 10−2 and
γ̇ = 10
−3, respectively). Open circles represent rext, an extrapola-
tion of the high temperature results to T = 0.3 as explained in the
text.
