Dissipating the Langevin equation in the presence of an external stochastic potential

0.34 MB·English

by  Jeremy M. Moix

Dissipating the Langevin equation in the presence of an external stochastic potential
Jeremy M. Moix and Rigoberto Hernandez
∗
Center for Computational and Molecular Science and Technology,
5 School of Chemistry and Biochemistry,
0 Georgia Institute of Technology,
0
Atlanta, GA 30332-0400
2
(Dated: February 2, 2008)

n Abstract
a
J In the Langevin formalism, the delicate balance maintained between the fluctuationsin the system and their corresponding
dissipation may be upset by the presence of a secondary, space-dependent stochastic force, particularly in the low friction
3
1 regime. Inpriorwork,thelatterwasdissipated self-consistently throughanadditional uniform (mean-field)friction [Shepherd
and Hernandez, J. Chem. Phys., 115, 2430-2438 (2001).] An alternative approach to ensure that equipartition is satisfied

] relies on the use of a space-dependent friction while ignoring nonlocal correlations. The approach is evaluated with respect to
h its ability to maintain constant temperature for two simple one-dimensional, stochastic potentials of mean force wherein the
c friction can be evaluated explicitly when there is no memory in the barriers. The use of a space-dependent friction is capable
e
of providingqualitatively similar results tothoseobtained previously,butin extremecases, deviationsfrom equipartition may
m
be observed dueto theneglect of thememory effects present in thestochastic potentials.
-
t
a
t I. INTRODUCTION dependent stochastic potential. The behavior of a
s
. Brownian particle diffusing across various subsets of
t
a this class of potentials has been the subject of intense
In the theory of diffusion processes over fixed barri-
m research.15,16,17,18,19,20,21,22,23 This activity has largely
ers, numerous studies have shown that the dissipative
been motivated by the discovery of resonant activation
-
term in the Langevin equation is rarely constant along
d in which the rate of transport over a stochastic barrier
n the reaction coordinate.1,2,3,4,5,6,7,8 A general rate the- exhibitsamaximumasafunctionofthecorrelationtime
o ory when the friction is both space- and time-dependent in the fluctuations of the barrier height.15 However, un-
c has beendevelopedto accountfor this phenomenonover
til recently, simulations of these systems have not been
[ theentirefrictionregime.9,10,11Onemightna¨ivelyexpect performed in the low friction regime, where deviations

1 that a space-dependent component must be included in
from equipartition may occur, due to an inability to ad-
v the friction kernel to capture the essential dynamics of
equately describe the friction in the presence of an ad-
7 a given system. However, this is not always the case.
ditional stochastic force.24,25 In previous work, the dis-
2 Several groups have shown that the average dynamical
3 sipation of this excess energy was achieved through a
properties may still be adequately described by a gener-
1 self-consistent approachin which the friction constant is
alizedLangevinequationwithspace-independentfriction
0 renormalizediteratively until equipartition is satisfied.24
5 even when the reaction coordinate has a strong spatial This renormalizationis approximate because it does not
0 dependence.2,4,6,12 AnanalysisbyHaynesandVothcon-
explicitly accountforthe correlationsbetweenthe exter-
/ cluded that the key factor is not whether the friction
t nal stochastic forces across space and time, but rather
a is space-dependent, since it generally will be, but rather
uses a single mean friction to dissipate theses forces at
m howthefrictionvariesalongthereactioncoordinate.13In
times longer than their correlation times. A possible
- particular,they suggestthat the symmetry of the space-
improvement to the self-consistent approach can be ob-
d
dependentfrictionwithrespecttothebarriercanbeused
n tained by allowing the friction to be space-dependent
as a metric for evaluating the role of the friction in the
o while explicitly ignoring the memory in the stochastic
dynamics. Similar product and reactant states will give
c potential, In the special case that the stochastic poten-
: risetosimilar(symmetric)frictioncomponentsaboutthe
v tial has no memory, then this treatment is exact. How-
transition state. Perhaps surprisingly, an antisymmetric
i ever,thisapproximationisoftennotjustifiedwhenmod-
X friction does not have a significant impact on the dy-
eling real systems and therefore, the model potentials
r namics, while a symmetric friction can result in large employed are chosen to have an exponentially decaying
a deviations from the predictions of standardrate theories
memory of their past states. In the most extreme cases,
for processes with space-independent friction.3,4,6,8,13,14
these correlations can result in deviations from equipar-
Thus, the Langevin model with a uniform effective fric-
tition during the course of the simulation, although the
tion can often approximate the dynamics of projected
space-dependentfrictiondissipatessuchfluctuationscor-
variables even if the formal projection would have re-
rectlyinmostsituations. Thegeneralconclusionappears
quired a space-dependent model.
to be that the more detailed space-dependent approach
The central question explored in this work is whether is in qualitative agreement with the self-consistent ap-
a single uniform effective friction suffices even when proach and hence, as in the fixed barrier case, Langevin
the Langevin system is subjected to an external space-
1
systems with stochastic forces may be dissipated by a and is specified by
single (though renormalized) uniform friction.
The conclusions ofthis workaresupportedby a study 1k (x x0 )2 for x0 <x x
2 0 − m m ≤ −m
of two different classes of one-dimensional problems in U(x,t)= V + 1k (x x )2 for x <x x+ ,
 m‡ 2 m‡ − ‡m −m ≤ m
which the particle diffuses across a periodic array of 1k (x x0 )2 for x+ <x x0
 2 0 − m+1 m ≤ m+1
coherent or incoherent barriers. These two cases can (4)
be specified by sinusoidal or merged-harmonic-oscillator where themth well and adjacent barrier are centered
potentials, respectively. For such simple forms of the at x0 = λ/2 + mλ and x = mλ, respectively.
m − ‡m
stochastic potential, analytic expressions for the friction The connection points are chosen to ensure continu-
as a function of the spatial coordinate can readily be ity in the potential and its first derivative such that
obtained and are presented in Sec. II. The resulting x = k λ/(2k 2k )+mλ. As opposed to the si-
±m ± 0 0− m‡
Langevin dynamics across these potentials dissipated ei- nusoidalpotential, the width of the MHO barriersvaries
ther uniformly or through the space-dependent friction stochastically in time according to the relation k =
m‡
are illustrated in Sec. III. (k +η(m,t)),which,inturn,definesthebarrierheight
0
−
V = k k λ2/(8k 8k ). The remainingparameters
m‡ − 0 m‡ 0− m‡
in the potentials are chosensuchthat the lattice spacing
II. LANGEVIN MODEL WITH STOCHASTIC is 4 and the thermal energy of the particle is 1/6 of the
POTENTIALS average value of the barrier heights.
The stochastic term, η(t), is defined as an Ornstein-
An equation of motion describing the diffusion of a Uhlenbeck process governed by the following differential
particle influenced by a stochastic potential of mean equation,
force can be adequately described by a phenomenolog-
ical Langevin equation of the form, η(t) 2σ2
η˙(t)= + ζ(t), (5)
− τc s τc
v˙ = γ(t)v+ξ(t)+F(x;t), (1)
−
with the probability distribution,
where F(x;t) U(x;t) is an external stochastic
x
≡ −∇
force, and γ(t) is the friction required to dissipate both
1 η(t)2
thethermalforcesandthoseduetotheexternalstochas- P(η(t))= exp , (6)
tic potential. The thermal bath is described by ξ(t), √2πσ2 (cid:18)− 2σ2 (cid:19)
whichisaGaussianwhitenoisesourcewithtimecorrela-
and time correlation,
tion givenby the fluctuation-dissipationrelation(FDR),
t t
hξ(t)ξ(t′)i=2kBTγthδ(t−t′). (2) hη(t)η(t′)i=σ2exp −| −τ ′| . (7)
(cid:18) c (cid:19)
In the limit that F(x;t) = F(x;0) for all t, these equa-
tions reduce to the Langevin equation with γ(t) = γth. The variance of the distribution is given by σ2, τc is the
Otherwise,the questionremainsastowhatisthe appro- correlation time, and ζ(t) is an additional white noise
priate form of γ(t). Two approaches for addressing this source. The distribution of barriers heights for the si-
questionarepresentedinSectionsIIBandIIC,afterfirst nusoidal potential is given directly by the distribution
describing the explicit forms ofthe stochastic potentials. of η(t), but due to the nature of the expression for the
barrier heights of the MHOs, the resulting distribution
for this potential takes on a more complex form that is
sharper and slightly skewed compared with Eq. 6. As a
A. Stochastic Potential Representation
result,amuchsmallerrangeoffluctuationsisallowedfor
theMHOthanthesinusoidalpotentialtoensurethatthe
The space-dependent friction (SDF) that arises from
distributiondoesnotbecomesignificantlynon-Gaussian.
the fluctuations in F(x;t) can readily be evaluated an-
More details on the exact behavior of the MHO barrier
alytically for two different classes of one-dimensional
heights are provided in Ref. 24.
stochastic potentials. The first of these is a sinusoidal
potential taking the general form,
1 πx B. Uniform Dissipation
U(x;t)= E + η(t) sin +1 , (3)
b
2 2
(cid:18) (cid:19)(cid:16) (cid:16) (cid:17) (cid:17) Inpreviouswork,24 aself-consistentprocedurewasde-
inwhichthebarriersfluctuatecoherentlywitheachother. veloped to ensure that the evolution of the system us-
The second is constructed using a series of merged har- ing Eq. 1 remains in thermal equilibrium. This was
monicoscillators(MHOs)inwhicheachbarrierisallowed accomplished through an iterative procedure in which
to fluctuate independently (incoherently) ofoneanother, the friction, given by the sum of the two contributions
2
from the thermal bath and the stochastic potential, i.e. uncorrelated, i.e. δF δF = 0, then Eq. (9), reduces
th U
h i
γ γ +γ , is renormalized according to the relation, to
th F
≡
v2(t) 2kBTγc(x;t)= ξ(t)ξ(t′) + δFU(x;t)2 , (10)
γ(n+1) =γ(n) h in . (8) h i h i
(cid:18) kbT (cid:19) The thermal fluctuations are ohmic as given in Eq. 2,
and the relationship for the fluctuations in the force is
The friction for the next iteration is determined from
δF (x;t) F (x;t) F (x;t) , where the average is
U U U η
the value of the friction at the current step scaled by ≡ −h i
taken with respect to the auxiliary stochastic variable,
the magnitude of the deviation from equipartition seen
η. The average value of the force can be determined
in the dynamics until convergenceis reached to within a
according to the usual integrals,
desiredaccuracy. Themaincriticismtothisapproachlies
in the approximationmade in developing Eq. 8 in which ∞ dηP(η) xU(x;t)
the stochasticpotentialis treatedasalocalnoisesource, FU(x;t) = − −∞ ∇ , (11)
h i R ∞ dηP(η)
γF, obeying a fluctuation-dissipation relation equivalent −∞
to Eq. 2. However, the stochastic potentials have mem- where the fluctuations in thRe force are governed by the
ory and are therefore nonlocal in nature leading to non- stochastic Ornstein-Uhlenbeck process, η, whose proba-
vanishing cumulants at third and higher orders. These bility distribution is given by Eq. 6.
effects are included, but only in an average manner, to The remaining steps of the derivation rely upon the
second order in this approach. specific form of the potential. As an illustration, the
SDF is evaluated explicitly below for the simpler sinu-
soidal (coherent) stochastic potential. (The results for
C. Space-Dependent Dissipation the incoherent MHO potential can be found in the Ap-
pendix.) The derivation begins by direct evaluation of
Eq. 10 for the specific class of potentials. As remarked
An alternative approach to dissipating the external
above, the first term reproduces the FDR, Eq. 2, for the
stochastic force relies on replacing the space- and time-
thermal forces. Ignoring the correlation in the forces at
dependentfriction,γ(x,t),byaspace-dependentfriction,
different times, the second reduces to:
γ(x(t)), satisfying a local FDR. Given that the size of
the fluctuations in F(x;t) depend on x at a given t, a π2 πx
Brownianparticle moving quickly acrossthe surface will δFU(x;t)2 = cos2 ;t
h i 4 2 ×
edxeppeernidenocnetahesepraiersticolfe’fsorvceelsocwithy.osHeorweleavteirvewmheangnthiteudthees ∞dη (cid:16)E +(cid:17)1η 2P(η)
b
Bonrloywtnhiaenlopcaarltiflculectmuaotvieosnssloowfltyh,ethsteopchaartsitcilcepwoitllenstaimalpilne Z−π∞ (cid:18)πx 2 (cid:19) −
cos ;t
the vicinity of its local position x. In this regime, the 2 2 ×
particle arrives at a local quasi-equilibrium which must h ∞ (cid:16) (cid:17) 1 2
dη E + η P(η) . (12)
necessarily satisfy the FDR locally. This suggests that b
2
the dissipation should notbe uniform, but rather should Z−∞ (cid:18) (cid:19) (cid:21)
depend on position, and therefore indirectly on time. It The Gaussian integrals are readily evaluated to yield:
should be noted that while the mean-field approach de-
σ2π2 πx
scribedinthe previoussubsectioniscapableofincluding δF (x;t)2 = cos2( ;t). (13)
U
the average of the correlations between the fluctuations, h i 16 2
the approximation made here does not account for any Upon substitution into Eq. 9, the explicit form of the
of the memory effects. However, in the limit that there SDF is
is no memory in the external stochastic potential, the
σ2π2 πx
following results are exact. γ (x;t)=γ δ(t t)+ cos2( ;t). (14)
c th ′
− 32k T 2
The question now arises of how to explicitly describe B
thefrictionconstantinthepresenceofanadditionalfluc- This is the simplest possible form for this result, and is
tuating force resulting from the potentials of mean force due to the separability of the potential into a sum of
given in Eqns. 3 and 4. The friction constant must dis- deterministic and linear stochastic terms. In fact, it is
sipate the excess energy that arises from the fluctuating easilyshownthatforanyseparablepotentialoftheform,
forces through a local space-dependent FDR,
U(x;t)=U¯(x)+η(t)W(x), (15)
2k Tγ (x;t)= δF (x;t)2 , (9)
B c h c i whereU¯(x)is the deterministic componentofthe poten-
tialofmeanforce,thenthe additionalfrictiondue tothe
wherethecumulativeforceissimplythesumofthether-
stochastic potential is given by
mal Gaussian noise and the stochastic force arising from
the external potential, F = F + F . Assuming the
c th U δF (x;t)2 =( W(x))2 dη(η2 η)P(η), (16)
respective fluctuations in the bath and the potential are h U i ∇x −
Z
3
provided the distribution is normalized. The MHO does
4
not satisfy the condition of Eq. 15 and hence its friction
4
correction can not be obtained by Eq. 16. The form of 2
tthaiensfrmictoiroentecromrrse,cbtiuotntfhoerrtehqeuiMsitHeOapcpornoxseimquaetniotnly(tchoant- x;t) 0 x;t)
the forces are uncorrelated at different times) enters the U(2 F(
-2
derivation in a conceptually equivalent way.
-4
0.2 0.2
D. Mean First-Passage Times
>
The dynamics ofthe systemwerecharacterizedby the 2
)
t
mean first passage time (MFPT) of a particle to es- x;0.1 0.1
(
cape its initial minima and establish a quasi-equilibrium F
δ
within another well. With periodic, stochastic poten- <
tials, this may be accomplished by defining a region of
the phase space of the particle bounded by an energetic
0 0
constraint.26 The MFPT is simply the average of a suf- -2 1 0 1 2
x
ficient number of first passage processes into this region,
with the corresponding rate given by the inverse of the
MFPT. While the incorporation of a space-dependent
FIG. 1: Representative fluctuations over one period of the
friction in the algorithm for the numerical integration
MHOpotentialandforce(toppanel),andtheresultingspace-
of the equations of motion would seemingly result in a
dependentfriction(bottompanel). Thenumericalcomponent
dramatic increase in computational expense, the actual
in the bottom panel is displayed as the solid black line, with
effort is comparable to the previous mean-field approach theanalyticresult,givenintheAppendix,asthedottedwhite
because the preliminary convergence procedure for the line. The temperature is 2/3, the variance is 0.22, and the
friction constant is now unnecessary. thermal friction is 0.08.
III. RESULTS AND DISCUSSION illustrated below, this effect does not have a dramatic
effect on the resulting dynamics.
The analytic and numerical space-dependent compo- Values of the friction corrections calculated from the
nents of the friction over one period of the MHO and iterative and space-dependent approaches for the MHO
sinusoidal potentials can be seen in the bottom panel of and sinusoidal potentials are displayed in Table I with
Figs. 1 and 2, respectively, with the numerical results the values of the thermal friction listed in the left-most
averagedover 500 representative trajectories. column.
The top panels display the fluctuations in the poten- The variance and correlation time for both potentials
tial and the resulting forces that give rise to the space- is 0.22 and 1, respectively. The resulting temperatures,
dependent friction. The analytic forms of the SDF, dis- (k T v2 ), are also listed for the space-dependent
B
≡ h i
played as the dotted white line, agree with the corre- approach. The friction correction in the self-consistent
sponding numerical results, and exact agreement is ob- method ensures equipartition by definition, and there-
tained upon further averaging. The fluctuations in the fore, is not listed. The magnitude of the SDF for all val-
forces reach a maximum at approximately the midpoint uesofτ followaccordingly;howeverthisistheonlyvalue
c
betweentheminimaandmaxima,wheredeviationsfrom withrespecttothegivenvarianceforwhichanydeviation
the average force take on the largest values. The fluctu- from equipartition is observed. As can be seen, both the
ations in the potential are largest at the barriers, while self-consistent and space-dependent components of the
theforcesarezeroattheselocations. Thisleadstoavan- total frictionfor eachpotential provide negligible contri-
ishing contribution to the total friction from the space- butions for this variancesince the magnitude ofthe fluc-
dependentcomponentatthesepoints. Inthewellregion, tuationsinthebarrierheightarerelativelysmall. There-
the behaviorofthe SDFforthe sinusoidalandMHO po- fore the total friction is a sum of a large thermal com-
tentials is inherently different. The SDF for the MHO is ponent, and a space-dependent contribution. The slight
zero outside of the barrier region since the wells do not differences in the magnitudes ofthe SDF for the two po-
fluctuatebyconstruction. However,thesinusoidalpoten- tentials can be attributed to the piecewise nature of the
tial fluctuates continuously throughout leading to a fric- MHO potential. The particles spend mostof the simula-
tioncorrectionalongtheentirereactioncoordinate. Con- tiontimeinthewellswhichdonotfluctuate. Acontribu-
sequently,themagnitudeofthefrictioncorrectioninsim- tion to the total friction from the space-dependent term
ulations employing the sinusoidal potential are slightly is included only when the energetically-limited particle
larger than that in those employing the MHO. But, as accumulatesenoughenergy to explore the upper portion
4
MHO Sin
γth hγFi0 hγFisdf hv2isdf hγFi0 hγFisdf hv2isdf
0.08 0.00 0.01 0.67 0.00 0.03 0.69
0.2 0.00 0.01 0.67 0.00 0.03 0.68
0.4 0.00 0.01 0.67 0.01 0.03 0.67
TABLE I: The average of the friction corrections, γF, calculated by theiterative self-consistent (0) and space-dependent (sdf)
approaches for the MHO and sinusoidal potentials. The resulting temperatures are also included for the space-dependent
friction. In all cases the temperature is 2/3 (in units of a standard temperature, kbT0), the variance, σ2 = 0.22, and the
correlation time, τc=1.
in a correction that is roughly constant for all values of
4
the correlation time, while the iterative approach does
4
2 exhibit some variation with τc. This is the expected re-
x;t) 0 x;t) stuualttisoinnsceinthtehesppaoctee-ndteipaelnadreenltocfrailctainodntahsesuremfoerse,thigenflourecs-
U(2 F( any correlation in the barrier heights. The iterative ap-
-2
proach,however,is capableofincorporatingthe memory
ofthepotentialintothefrictioncorrection,butonlyinan
-4
averagemanner. Asaconsequence,significantdeviations
0.4 0.4
fromequipartitionmaybeobservedwhensimulationsare
performed with a space-dependent friction that ignores
0.3 0.3
> the correlation effects, as illustrated by this extreme ex-
2
t) ample.
;
x0.2 0.2
F( Figs. 3 and 4 display the MFPTs obtained for
δ
< the MHO potential with the results from the space-
0.1 0.1
dependent and self-consistent approaches in the top and
bottom panels, respectively.
0 0
-2 -1 0 1 2 TheresultsinFig.3havebeencalculatedusingavari-
x
ance of σ2 = 0.05, while those in Fig. 4 use σ2 = 0.22.
Thecorrespondingresultsforthesinusoidalpotentialus-
ing a variance of 0.22 can be seen in Fig. 5. The values
FIG. 2: Representative fluctuations over one period of the
on the broken axis represent the numerically calculated
sinusoidal potential and force (top panel), and the resulting
space-dependent friction (bottom panel). The numerical re- MFPTs in the limits of correlationtime, τc. In the zero-
sult is displayed as the solid black line, with the analytic re- correlation time limit, the fluctuations in the potential
sult, given by Eq. 14, shown as the dotted white line. The are so rapid that the particle effectively experiences the
parameters used are thesame as in Fig. 1. average, stationary potential, from which the dynamics
werecalculated. Inthe limitofinfinite correlations,fluc-
tuations in the potential are nonexistent, and therefore
of the MHO potential. the particle experiences a single realization of the po-
Tofurtherexploretheaccuracyofthespace-dependent tential with constant barrier heights determined by the
approach,the sinusoidalpotential has been studied with initial value sampled from the distribution. The MFPTs
a ten-fold increase in the variance from 0.22 to 2.2. The displayedin Fig. 4 obtained with a largervariance alters
values of the friction correction from these simulations the magnitude of the resonant activation, but influences
are listed in Table II. the results for the two approaches equally. The results
The displayed correlation times, τ , are those that ex- fromthe simulations with a space-dependentfrictionare
c
hibit the largest resonant activation. Consequently, if systematicallyshiftedtolowerMFPTsasseeninallthree
memoryeffectsinthebarrierheightsareimportantinde- figures. This trend is most readily explained through by
termining the friction constant, it should be manifested the trends in Table. I. In the low friction regime, an
here. Although not shown for brevity, outside this re- increase in the friction increases the corresponding rate
gionof the correlationtime, the magnitude of the devia- of transport. The average space-dependent contribution
tions from equipartition decrease rapidly, but the size of is always larger than its respective mean field counter-
the space-dependent components remains roughly con- part,andis expectedto havethe largesteffectonthe re-
stant. Similarly, the corresponding corrections arising in sultswiththesmallestthermalfriction. Thefluctuations
theself-consistentmethodalsoapproachzero. Ascanbe present along the entire reaction coordinate of the sinu-
seenfromTableII,thespace-dependentapproachresults soidal potential do not appear to have a dramatic effect
5
τc=10−1 τc =100 τc =101
γth hγFi0 hγFisdf hv2isdf hγFi0 hγFisdf hv2isdf hγFi0 hγFisdf hv2isdf
0.08 0.04 0.28 0.72 0.05 0.29 0.74 0.01 0.28 0.68
0.2 0.04 0.28 0.71 0.05 0.29 0.72 0.01 0.28 0.68
0.4 0.04 0.28 0.70 0.06 0.29 0.71 0.01 0.28 0.67
TABLE II: The average of the friction corrections, γF, calculated by the iterative self-consistent (0) and space-dependent
approaches (sdf) for the sinusoidal potential. The resulting temperatures are also included for the space-dependent friction
method. The temperature is 2/3 in all cases and the variance, σ2 =2.2.
2000 2000
1500 1500
1000 1000
500 500
T T
P P
F 20000 F 20000
M M
1500 1500
1000 1000
500 500
0 0
0 10-3 10-1 101τ 103 105 8 0 10-3 10-1 101τ 103 105 8
FIG. 3: The mean first-passage times (MFPT) for a particle FIG. 4: The mean first-passage times (MFPT) for a particle
diffusing across the MHO stochastic potential are displayed diffusing across the MHO stochastic potential are displayed
for two possible scenarios of the dissipative mechanism. The for two possible scenarios of the dissipative mechanism. The
top panel uses space-dependentfriction, and thebottom dis- parameters are the same as in Fig. 3, except the variance is
plays the uniform friction determined by the self-consistent 0.22.
method. Thevarianceforbothis0.05,andthethreelinescor-
respond to values of the thermal friction of 0.08 (solid curve
with x symbols), 0.2 (dashed curve with triangles), and 0.4
(dot-dashedcurvewithsquares). Thesymbolson thebroken
tage is gained by using the self-consistent method be-
axisrepresentthenumericallycalculatedMFPTsatthelimits
cause it ensures the system is kept at constant temper-
of the correlation time.
ature for all values of the correlation time throughout
thesimulation,while thespace-dependentapproachmay
lead to deviations in extreme cases. The most signifi-
on the dynamics. The results in Fig. 5 for the sinusoidal cant difference between the two methods can be seen at
potential follow the same trend as those in Figs. 3 and 4 intermediatecorrelationtimes, inwhichthe resonantac-
fortheMHOpotentialindicatingthattheSDFapproach tivation observed from the iterative approach is slightly
iscapableofadequatelydescribingthefluctuationsinthe more pronounced. This can particularly be seen in the
system. Aside fromthe shift, the generalbehaviorofthe MFPTswhenthefrictioncasetakesonthesmallestvalue
MFPT is adequately reproduced by both methods, par- of γ = 0.08. Since the resonant activation arises from
th
ticularly at larger values of the thermal friction when correlationsinthebarrierheights,itisnotsurprisingthat
thespace-dependentcomponentbecomeslesssignificant. simulations incorporating a friction capable of account-
At this level of description, each of the two approaches ing for this phenomenon can have a noticeable impact
for constructing the frictionare capable of capturing the on the dynamics, even if it does so only in an average
essentialdynamics of the system. However,some advan- manner.
6
2500 using an alternate approach in which a uniform correc-
tion is calculated self-consistently. Although the latter
2000 approachdoeseffectivelyincludethetimecorrelationbe-
tween the barrier fluctuations at long times, the former
1500
does not in any sense. This neglect may result in devia-
tionsfromequipartitioninsomeextremecases. However,
1000
both approaches are capable of capturing the essential
500 dynamics of the system and lead to the now-expected
T
resonantactivationphenomenon. Consequently,the cen-
P
F 25000 tral result of this paper is that the Langevin dynamics
M of a particle under external stochastic potentials can be
2000 properlydissipatedbyasingleuniformrenormalizedfric-
tion without loss of qualitative (and often quantitative)
1500
accuracy.
The role of the memory time in an external stochas-
1000
tic potential acting on a particle described by a general-
500 ized Langevin equation of motion is still an open ques-
tion. In this limit, there would presumably be an inter-
0 play between the memory time of the thermal friction
0 10-3 10-1 101τ 103 105 8 and that of the stochastic potential. When the latter is
smallcomparedtotheformer,thequasi-equilibriumcon-
ditioncentraltothisworkwouldnolongerbesatisfiedby
the particle,andhence it is expectedthata non-uniform
FIG. 5: The mean first-passage times (MFPT) for a parti-
(and time-dependent) friction correction would then be
clediffusingacrossthesinusoidalstochasticpotentialaredis-
needed.
playedfortwopossiblescenariosofthedissipativemechanism.
Other than for the change from the MHO to the sinusoidal
potential, theparameters are thesame as in Fig. 4.
V. ACKNOWLEDGMENTS
IV. CONCLUSIONS
RHgratefullyacknowledgesAbrahamNitzanforanin-
The space-dependent friction arising from the pres- sightful question whose answer became this paper. This
ence ofasecondary(external)stochasticpotentialinthe work has been partially supported by a National Sci-
Langevin equation has been explicitly derived for two enceFoundationGrant,No.NSF02-123320. TheCenter
simpleclassesofthestochasticpotentials. Thenumerical for Computational Science and Technology is supported
results are in excellent agreement with analytic expres- througha SharedUniversity Research(SUR) grantfrom
sions describing the space-dependent friction. The re- IBMandGeorgiaTech. Additionally,RHistheGoizueta
sulting dynamics have been compared to those obtained Foundation Junior Professor.
VI. APPENDIX
ThepiecewisenatureoftheMHOpotentialresultsinapiecewiseformfortheassociatedSDF.Althoughincoherent,
everybarriergivesrisetothesameaverages,andhencethe procedureneedstobe carriedoutonlyoverasmallregion
defined by the closed interval, [x0 ,x ]. The limits of integration over this region can be determined from the
m ‡m
expression for the connection points
k λ
0
x−m =−2k0 2km‡ +mλ, (17)
−
where k = (k +η(t)). This can equivalently be expressed as
m‡ − 0
k λ
0
η(t)= 2k . (18)
0
−2(x−m mλ) −
−
At the top of the barrier, when x =x , η(t)= . In the intermediate region for arbitraryx,
−m ‡m ∞
k λ
0
η(t) = 2k
0
−2(x mλ) −
−
η . (19)
∗
≡
7
Otherwise, at the minimum when x =x0 , η(t)= k .
−m m − 0
Although it is apparent from the expression for the barrier height that the corresponding distribution is non-
Gaussian, the resulting forces are Gaussian with the probability given by Eq. 6. The average force for a given x is
simplytheweightedaverageoftheforceswhenxisintherespectiveregions,(x0 ,x )and(x ,x ),whichcorrespond
m −m −m ‡m
to η regions of ( k ,η ) and (η , ). The resulting integral for the average value of F(x;t) is now:
0 ∗ ∗
− ∞
η∗
FU(x;t) = −k0dηF(x)P(η)+ η∞∗dηF(x)P(η) . (20)
h i R ∞k0dηPR(η)
−
R
Here, one must be carefulin determining which portion of the force to use in the above equation. For example, when
η <η , the majority of the force is due to the barrier portion of the potential, not the well component. The average
∗
can thus be expressed as
η∗
F (x;t) = ∞dηk (x x0 )P (η)+ dη(k +η)(x x )P (η), (21)
h U i −Zη∗ 0 − m ′ Z−k0 0 − ‡m ′
where P (η) is defined through the normalization condition, i.e.,
′
η∗
∞
dηP (η)+ dηP (η) 1, (22)
′ ′
Z−k0 Zη∗ ≡
which leads to the probability distribution
exp η2
2 −2σ2
P (η)= , (23)
′ √2πσ21+er(cid:16)f k0(cid:17)
√2σ2
(cid:16) (cid:17)
where erf(x) is the standard error function. Use of the normalization condition reduces the average force to
η∗
F (x;t) =k (x x ) [k (x x0 )+k (x x )] ∞dηP (η)+(x x ) dηηP (η). (24)
h U i 0 − ‡m − 0 − m 0 − ‡m Zη∗ ′ − ‡m Z−k0 ′
The remaining integrals are readily computed; the explicit form of the average force is
1 erf η∗
F (x;t) = k (x x ) k (2x x0 x ) − √2σ2
h U i 0 − ‡m − 0 − m− ‡m 1+erf(cid:16) k0 (cid:17)
(cid:0) (cid:1) √2σ2
exp k02 exp (η(cid:16)∗)2 (cid:17)
2σ2 −2σ2 − − 2σ2
+(x−x‡m)r π  (cid:16) 1+(cid:17)erf k0(cid:16) (cid:17) . (25)
√2σ2
 (cid:16) (cid:17) 
Thesecondquantitytobecomputedistheaverageofthesquareoftheforce,andthederivationfollowsthat(above)
of the averageforce. The limits of integration are the same and the Gaussian integrals can be calculated in the same
manner. Again using the normalization requirement, the first integral is eliminated such that
F (x;t)2 = k2(x x )2+[k2(x x0 )2 k2(x x )2] ∞dηP (η)
h U i 0 − ‡m 0 − m − 0 − ‡m Zη∗ ′
η∗ η∗
+2k (x x )2 dηηP (η)+(x x )2 dηη2P (η). (26)
0 − ‡m ′ − ‡m ′
Z−k0 Z−k0
The first two integrals are the same as before, and the third can be obtained with little effort. The resulting mean
squared force is
1 erf η∗
F (x;t)2 = k2(x x )2+ k2(x x0 )2 k2(x x )2 − √2σ2
h U i 0 − ‡m 0 − m − 0 − ‡m 1+erf(cid:16) k0 (cid:17)
(cid:0) (cid:1) √2σ2
 (cid:16) (cid:17)
8
+ √4k20πσσ22(x−x‡m)2exp(cid:16)−12kσ+022(cid:17)er−f exkp0(cid:16)−(2ησ∗)22(cid:17)
√2σ2
erf η∗ (cid:16)+erf (cid:16)k0 (cid:17)(cid:17) 
+σ2(x x )2 √2σ2 √2σ2
− ‡m  (cid:16)1+e(cid:17)rf k0(cid:16) (cid:17)
√2σ2
2σ2(x x)2 η∗exp (cid:16)−(2ησ∗)22(cid:17) +k0exp −2kσ022 . (27)
−r π − ‡m  (cid:16) 1+e(cid:17)rf 2kσ02 (cid:16) (cid:17)
 (cid:0) (cid:1) 
The SDF for the MHO potential is then obtained by appropriate substitutions into Eq. 9.
∗
Author to whom correspondence should be addressed; (1995).
Electronic address: [email protected]. 14 J.E.Straub,M.Borkovec,andB.J.Berne,J.Phys.Chem.
1 B. Carmeli and A. Nitzan, Chem. Phys. Lett. 102, 517 91, 4995 (1987).
(1983). 15 C.R.DoeringandJ.C.Gadoua,Phys.Rev.Lett.69,2318
2 J.E.Straub,M.Borkovec,andB.J.Berne,J.Chem.Phys. (1992).
89, 4833 (1988). 16 L. Gammaitoni, P. H¨anggi, P. Jung, and F. Marchesoni,
3 S.-B. Zhu and G. W. Robinson, J. Phys. Chem. 93, 164 Rev.Mod. Phys.70, 223 (1998).
(1988). 17 E. Pollak, J. Bader, B. J. Berne, and P. Talkner, Phys.
4 R.Krishnan,S.Singh,andG.W.Robinson,Phys.Rev.A Rev.Lett. 70, 3299 (1993).
45, 5408 (1992). 18 P.Talkner,E.Hershkovitz,E.Pollak,andP.H¨anggi,Surf.
5 E. Pollak and A. M. Berezhkovskii, J. Chem. Phys. 99, Sci. 437, 198 (1999).
1344 (1993). 19 R.Astumian and F. Moss, Chaos 8, 533 (1998).
6 J. B. Straus, J. M. Gomez–Llorente, and G. A. Voth, J. 20 C. V. den Broeck, Phys.Rev.E 47, 4579 (1993).
Chem. Phys. 98, 4082 (1993). 21 P. Reimann,Phys. Rev.E 52, 1579 (1995).
7 E. Neria and M. Karplus, J. Chem. Phys. 105, 10812 22 R. Bartussek, A. J. R. Madureira, and P. H¨anggi, Phys.
(1996). Rev.E 52, R2149 (1995).
8 D.AntoniouandS.D.Schwartz,J.Chem.Phys.110,7359 23 J. M. Porr`a, J. Masoliver, and K. Lindenberg, Phys. Rev.
(1999). E 50, 1985 (1994).
9 G. R. Haynes,G. A. Voth,and E. Pollak, J. Chem. Phys. 24 T.ShepherdandR.Hernandez,J.Chem.Phys.115,2430
101, 7811 (1994). (2001).
10 G. R. Haynes, G. A. Voth, and E. Pollak, Chem. Phys. 25 T. Shepherd and R. Hernandez, J. Phys. Chem. B 106,
Lett. 207, 309 (1993). 8176 (2002).
11 G. A. Voth,J. Chem. Phys. 97, 5908 (1992). 26 T.ShepherdandR.Hernandez,J.Chem.Phys.117,9227
12 B. J. Gertner, J. P. Bergsma, K. R. Wilson, S. Lee, and (2002).
J. T. Hynes,J. Chem. Phys. 86, 1377 (1987).
13 G.R.HaynesandG.A.Voth,J.Chem.Phys.103,10176
9