September 2011
EPL, 95 (2011) 68006
doi: 10.1209/0295-5075/95/68006

www.epljournal.org

Frictional stick-slip oscillation as a first-passage problem
Y. Visell(a)UPMC Université Paris 06, Institut des Systèmes Intelligents et de Robotique - Paris, France, EU
received 6 June 2011; accepted in final form 8 August 2011
published online 8 September 201183.10.-y – Fundamentals and theoretical
05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion
PACS 83.10.Mj – Molecular dynamics, Brownian dynamics
PACS
PACS

Abstract – We study complex frictional stick-slip oscillations by formulating a novel first-passage
time problem whose solutions are used to compute distributions of stick-slip oscillation periods,
displacements, and slip durations for a generic family of stochastic friction models. Approximate
solutions are developed using a level-crossing expansion due to Rice and Stratonovich. Sample
results for a minimal Langevin sliding friction model agree with simulations over a range ofsystem parameters, and reproduce qualitative features of prior experimental studies of stick-slip
motion. The analysis also reveals a complex oscillatory regime near the transition from stick-slip
to continuous sliding, in which additional transient modes are excited through boundary-noiseinteractions.
c EPLA, 2011
Copyright !

Stick-slip oscillations, an important source of vibration
and acoustic emissions in natural and man-made mechanical systems, are observed in many frictional settings:
during macroscopic sliding between unlubricated solid
surfaces, such as vehicle brakes [1]; in the bowing of a violin
string [2]; in nanoscale friction on an atomic substrate [3];or in the shear response of a granular material [4,5].
Minimal sliding friction models (fig. 1), are able tocapture many salient phenomena, but dry frictional sliding at macroscopic scales also involves localized contact at
multiple asperities, which are brought into rapid collision
during sliding, contributing a source of noise. In granular friction, fluctuating forces can be highly pronounced[4,6–8], and in both solid and granular sliding, multiple
dynamical regimes are observed (fig. 2), including irregular and periodic stick-slip oscillation, and continuous
sliding [4,6,9,10], with slip distances, force fluctuations,and periods found to be broadly distributed. Althoughphenomenological models have been proposed for granular [4,5], solid friction [1,9,11–13], and related seismological processes [14,15], no theory is able to fully explain thecomplex oscillatory behavior that is observed.
In a stochastic model of sliding friction, random force
components due to friction noise lend the velocity v(t) ofa sliding body a random character, which can genericallybe described by a stochastic dynamical process. Transitions to static friction occur when this velocity decreases(a) E-mail: yon@zero-th.org

F

A.

Tsl Tst
vb

B.

C.
.
x
m

vb

x(t)

t

.
x(t)

t

Fig. 1: (A) A sliding body exhibits frictional stick-slip oscillations, modeled as (B). A block, spring, damper, and surface,which (C) oscillate with period τ = Tsl + Tst .

Fig. 2: Illustration of complex stick-slip oscillatory behavior,from high-speed free sliding to low-speed periodic motion.

sufficiently, due to elastic restorative forces or frictional
damping (fig. 1B), that the contacting surfaces become
stuck in a local energetic minimum. This condition associates the stick-slip motion with a first-passage problem
for the random variable v(t). If the body is externallydriven at constant velocity vb (fig. 1), a maximum of thestatic friction force will, after some time, be reached, sliding motion will recommence, and the process will repeat.

68006-p1

Y. Visell
The present work develops such a first-passage problem
for a generic family of stochastic sliding friction models,and solves it, obtaining analytical expressions for the
distributions of periods, slip durations and displacements,
which statistically characterize the motion as a function of
the dynamical parameters and noise characteristics of themodel. Lima et al. previously studied the distribution of
excursions of a block sliding down an incline before coming
to rest [11], modeling experiments of Brito and Gomes [16]via a stochastic friction law that could be expressed as abiased diffusion. The block sliding distribution was effectively reproduced by solutions of a first-passage problem,
obtained via a Fokker-Planck equation [11]. Such an analysis is also suitable for modeling first-passage processes in
overdamped settings, where the relaxation time tR of the
dynamics is shorter than the typical first-passage time [17].
In such cases, similar to Kramers’ problem for the noiseinduced escape of a particle from a metastable potential, a constant escape rate is soon achieved, and thefirst-passage time density decays exponentially for times
exceeding tR , behavior that is also consistent with experimentally observed waiting time densities in mechanicalstick-slip motion at high damping ratios [9].
However, in the quasiperiodic frictional regimes studied
here, the presence of underdamped oscillation modes
render the dynamics essentially non-Markovian [18], withboth position and velocity values required to determine
the evolution at future times. Further, transitions to static
friction result in a reset to sharp initial conditions oneach cycle, ensuring the system remains far away fromequilibrium, thus making it impossible to solve the FokkerPlanck equation using known techniques. Instead, thepresent work develops solutions based on an expansionin level crossing statistics, that originates with work of
Rice and Stratonovich [19,20] that has recently beenused to model distributions of firing intervals in resonant
neurons [17,21]. There are similarities between membranepotential oscillations of a spiking neuron and stick-slip
frictional motion, with the distribution of stick-slip periods
paralleling the neuronal interspike interval density. This
paper draws on the aforementioned methods, and extends
them so as to address the unique velocity-type boundary
relevant to stick-slip oscillation. Additionally, it extracts
multivariate first-passage densities required to compute
distributions of slip displacements and oscillation periods.
Finally, it applies this theoretical treatment to a minimal
physically plausible stochastic friction model.
First-passage problem and solutions. – In a minimal deterministic model of stick-slip motion (fig. 3A), theoscillation period, a function of the system parameters, isequal to the sum of time spent during stick and slip phases,
T = Tst +Tsl . In the coordinate frame moving with thedriving velocity vb (fig. 1B), the static friction grows until
the threshold is reached, at relative displacement x0 . Sliding follows until the two surfaces lock into a local energetic
minimum, matching velocities, v(t) = vb , at t = Tsl . The

Fig. 3: (A) Periodic phase-space orbit produced by a simple
deterministic friction law. (B) Dynamics in a stochastic friction
model with a distribution p(T ) of periods. The state follows atransition probability density P (x, v, t|x0 , v0 , t0 ). The ellipsoids
represent level sets (isolines) of this density at successive times.

cycle then repeats. Additional factors (e.g., contact aging,
granular dynamics, lubrication) can yield more complex or
multiperiodic oscillation [10,22]. Ignoring such factors, themotion is that of a non-smooth dynamical system having
two dynamical phases: stick and slip [23].
For the remainder, to account for random forces duringsliding, consider a frictional process in which sliding
is governed by a continuous time stochastic dynamicalsystem:ẋ = f (x(t); η(t)).

(1)

Here, η is a noise source, f (·) describes the (possibly
nonlinear) sliding dynamics, and the vector x(t) = (x, v =ẋ, . . .) accounts for x, v, and any other required statevariables. The presence of damped oscillations is assumed.
At the onset of sliding (fig. 3B), trajectories of the system
are localized near (x, v) = (x0 , vb ), and spread out overtime, as represented via a transition probability density
P (x, t|x0 , t0 ≡ 0). A transition to static friction is assumed
to occur when the velocity v(t) first reaches vb . During
static friction, the state evolves deterministically, withthe surfaces in stationary contact, until the static friction
force limit is reached. The dynamics of transitions between
static and kinetic friction are ignored here. Variations inthe static friction threshold (due to asperity size, contact
aging, infinitesimal creep [24], etc.) are neglected, but
could, to a first approximation, be incorporated in thepresent analysis by allowing the static friction threshold tobe randomly distributed. Thus, we study the distributionp(T |x0 , 0) of periods T , assuming a reset to sharp initialconditions x0 upon transition to sliding.
Let F (x, t) be the joint probability density for a transition to static friction (characterized by v(t) = vb ) to occur
at displacement x and time t = Tsl . Tsl is thus a randomvariable, while sticking time Tst = (x0 − xsl )/vb is a deterministic function of the (random) displacement xsl at stick
onset. The period T = Tst + Tsl is distributed as

68006-p2

p(T |x0 , 0) =! T
0dτ F(x̂, τ ), x̂ ≡ x0 − vb (T − τ ).

(2)

Frictional stick-slip oscillation as a first-passage problem
The slip duration Tsl and length d = x0 − x(Tsl ) are via F̂n(k) (t) = Fn(k) (t) Θ(t − tn ), where Θ(·) is the Heaviside#t(k)distributed according to the marginal densities
step function and tn is chosen so that 0 n Fn (t)dt = 1.
! ∞F(t) can also be expressed via a cumulant expansion
dx F (x, Tsl ),(3) [18,20], by writing F (t) = Ṡ(t) exp(−S(t)), wherep(Tsl ) = F(Tsl ) =
−∞
! T
!
∞
"
(−1)p T! ∞gp (t1 ,. . . ,tp )dt1 · · ·dtp ;···S(T ) = −p!
dt F (x, t).
(4)p(d) = F (x = x0 − d) =
0
0
p=1
0

Computing the probability densities F(x, t), F(x), andF(t), which characterize the random stick-slip transitionsin the system, requires the solution of a first-passage
problem in the velocity v(t) across the value vb .
The most common approach to solving for the transition
state density P (x, t|x0 , t0 ) proceeds via a Fokker-Planckpartial differential equation. For example, for frictional
dynamics modeled on a generalized Langevin equation,it has the form [25]∂P U # (x) ∂P
γ ∂
D ∂2P
∂P
= −v
+
+
vP + 2
,
∂t
∂x
m ∂v
m ∂v
m ∂v 2(5)where U (x) is the elastic potential energy and D is the
diffusion coefficient. The required densities are related tosolutions of this equation in the presence of an absorbing boundary condition at v = vb , i.e.P (x, vb , t) = 0 with∂P (x, vb , t)/∂v > 0 [25]. However, analytic solution methods are known only for limiting cases, in which the motion
is overdamped [26], the relaxation time is shorter thanthe mean escape time, or the dynamics can otherwise
be reduced to a Brownian motion in a single variable(Smoluchowsi approximation [18]), and are inapplicablefor non-Markovian processes with sustained oscillations,which is the case of present interest.
Instead, solutions may be developed based on a levelcrossing series expansion for the random variable v(t), as
first formulated by Rice and Stratonovich [17,19,20]. It has
the form!
! t∞"(−1)p tdt1 np+1 (t, tp , . . . , t1 ). (6)dtp · · ·F(t) =
p!
0
0
p=0

(7)
Ṡ(t) = dS/dt can be interpreted as a time-dependent
escape rate. The densities gk may be expressed throughnk of equal and lower order by comparing terms in thesummations. The first two are given by g1 (t) = n1 (t) andg2 (t1 , t2 ) = n2 (t1 , t2 ) − n1 (t1 )n1 (t2 ). If the crossing times
tk occur almost independently, the gk decrease rapidly inmagnitude [18]. This facilitates approximations for F (T )in which S(T ) is replaced by partial sums of order k. The(1)
first-order “decoupling” estimate, Fg (T ), replaces the nk
at each order by products of one-crossing distributions, n1 ,#T(1)which yields Fg (T ) = n1 (T ) exp(− 0 n1 (t)dt). Higher(k)order approximations Fg (T ) can be expressed in termsof the upcrossing densities nk [17,20], and are normalized
for all positive times if a positive escape rate is ensured.
Similar approximations can be obtained for the joint
density F (x, T ), based on a generalization of eq. (6):
! ∞
!
∞
"
(−1)p T
dt1
dx1 × · · ·F(x, T ) =
p!
0
−∞
p=0
! ∞
! T
dtp
dxp np+1 (x1 , t1 ; x2 , t2 ; . . .; x, T );×0−∞nk (xk , tk ; . . . ; x1 , t1 ) is the probability of v(t) performing
upcrossings at points (xi , ti ), i = 1, 2, . . . , k. Truncation(k)(k)
and cumulant approximations, Fn (x, T ) and Fg (x, T ),can be derived just as for the univariate densities. A decoupling approximation can likewise be obtained, yielding#T(1)
Fg (x, T ) = n1 (x, T ) exp(− 0 n1 (t)dt).

Example: Langevin model. – While the forgoingtreatment is applicable to a broad class of stochastic
friction models, as a concrete example, consider a miniThe first term, n1 (t), is the density of velocity upcross- mal friction model with sliding dynamics based on theings at time t, or passages of v(t) across the level v = vb Langevin equationfrom below. nk (tk , tk−1 , . . . , t1 ) are the joint densities forẋ < vb : a = v̇ = −ω02 x − ζv + vb ξ(t), ẋ = v, (8)v(t) to perform upcrossings at k successive time instants.
n1 (t) overestimates F(t) by including all trajectories that(9)ẋ = vb : a = v̇ = 0, ẋ = v(t) = vb .
crossed at time t and at previous times. The second
term corrects for trajectories that performed an upcross- Equation (8) describes sliding motion (which holds foring prior to t, and subsequent terms provide further ẋ < vb ) in a coordinate frame moving at the driving veloccorrections. Thus, the partial sums of order k, here ity vb . In this model, noise generated by the collision of(k)denoted Fn (t), can be regarded as successive approx- thousands of µm-scale asperities per second is captured
imations to F(t) [21]. The k-th estimate is valid for by the random force mvb ξ(t), which vanishes as vb tends
times at which fewer than k + 1 upcrossings are likely to to zero. This is assumed to constitute Gaussian white
occur —i.e., for T < k/(2πω), where ω is the oscillation noise, although the analysis can be extended to correfrequency. Over long time intervals it is not normalized, lated noise [17]. This minimal Langevin dynamics has been
due to contributions of trajectories that perform multi- used to model experimental observations of sliding fricple upcrossings, but a normalized density can be defined tion on solid and granular surfaces [5,9]. Here, we assume
68006-p3

Y. VisellFig. 4: (Colour on-line) First upcrossing density n1 (t). The
exact analytic result (red line) is compared with the Monte
Carlo estimate (×). Left: ω0 = 1, D = 0.4, vb = 1, ζ = 0.1. Right:
ω0 = 1, D = 0.04, vb = 1.3, ζ = 0.11.

Fig. 5: (Colour on-line) Comparison of analytic approximations for slip time density F(T ) with numerical estimates(1)(1)(marked ×). Left: Fn (T ). Right: Fg (T ). Parameters: ω0 = 1,
D = 0.4, vb = 1.0, ζ = 0.1.

the driving velocity vb to be constant, so define a velocityscaled force term to be η(t) ≡ vb ξ(t), with $η(τ )η(τ + t)% =2Dδ(t). This noise is not thermal, but due to inelastic collisions of surface asperities, and standard thermodynamic
concepts such as temperature or fluctuation-dissipation
relations are not straightforward to apply. The line v = vb
is an absorbing boundary condition; first passage to vb
yields a transition to static friction (eq. (9)). The latter
persists until the threshold x = x0 is reached, when sliding
resumes from a localized state (x0 , v0 = vb ). Although thisexample involves a simple piecewise-linear, non-smooth
stochastic dynamical system, it nonetheless qualitatively
captures a rich variety of oscillatory phenomena.
Absent the boundary at v = vb , the sliding dynamics islinear with Gaussian noise, so the transition state density
on the whole phase plane (i.e., without boundary), here
denoted P (x, v, t|x0 , v0 , t0 ) ≡ P10 , is a bivariate Gaussian
process; its solution is well known [27,28]. The componentsµx (t) and µv (t) of the mean are trajectories of thedamped harmonic oscillator. The variances Σxx (t), Σxv (t)and Σvv (t) grow over time, as qualitatively illustrated in
fig. 3B. The free-sliding transition density P10 may be used
to compute upcrossing and first-passage densities in thepresence of the transition boundary v = vb as follows.
For a velocity upcrossing to occur at time t, it must
be that vb − adt < v(t) < vb . The probability for this tooccur is! vb
dvP (x,v,a,t|x0 ,v0 ,t0 ) = P (x,vb ,a,t|x0 ,v0 ,t0 )|a|dt

The density of upcrossings across v = vb is thus
! ∞dxP10 (x, vb , t)Ia (x, vb ).
n1 (t) =(13)−∞

As shown in the appendix, this integral over x, which
involves factors of Gaussian or Error Function type in x
and vb , yields an expression for n1 (t) with similar terms
in vb alone, and a complicated time evolution, due to its
dependence on components of the mean µ(t) and varianceΣ(t) of P10 . From it, we directly compute the approxi(1)(1)
mations Fn (T ) and Fg (T ) to F(T ). The joint densi(1)(1)
ties Fn (x, T ) and Fg (x, T ) depend on n1 (x, T ) in (11).
They yield the oscillation period and slip length densities via eqs. (3) and (4). Higher-order approximations,based on the upcrossing densities nk , involve multiple integrals with similar factors to those arising above. Although
too complex to be performed analytically, they may beevaluated via repeated numerical integration.
Numerical estimates for stick-slip densities in theLangevin model were obtained from Monte Carlo simulations. Each estimate was based on 105 to 106 instances of(8) integrated in time, using the Euler-Maruyama method,from the configuration (x0 , v0 ) at slip onset (t0 ≡ 0) until
the time of first passage. The integration time step was
h = 0.002, with other parameters as noted in captions offigs. 4–6. The results broadly agree with the analyticalpredictions over a wide range of system parameters. They
evidence stick-slip motion with finite-width distributionsof periods, slip distances, and slip times.
vb −adt
Density of upcrossings: Figure 4 compares analytic= P10 P (a|x, vb ) |a|dt. (10)results for n1 (T ), which are exact, with corresponding
Here, P (a|x, vb ) ≡ P (a|x, v = vb ) is, due to (8), a normal numerical estimates, demonstrating nearly perfect agreedensity with mean µa = −ω02 x − ζvb and variance σ 2 = 2D. ment. The location of the maximum of the upcrossing
Since a > 0 is required for an upcrossing, n1 (x, vb , t) is density lies close to the value of the deterministic period
obtained by integrating eq. (10) over positive values of a, τ = 2π/ω, with subsequent peaks lying at multiples of this$! ∞%value.
da aP (a|x,vb ) ≡ P10 Ia (x, vb ).
n1 (x, vb , t) = P10
Slip duration: The density F (T ) of slip durations is0
shown
in fig. 5 (left panel). The truncation approximation(11)(1)
Fn (T ) = n1 (T ) to F (T ) provides good agreement near
The integral Ia (x, vb ) in parentheses yields
the first maximum of F (T ), but overestimates later peaks,
&
$
$
%
%
µ2a
µa
σ2
µa
exp − 2 − erfc √. (12) as it also counts all upcrossings after the first. As T → ∞,Ia (x, v) =2π
2σ2n1 (T ) overestimates F (T ) by the asymptotic upcrossing
2σ68006-p4

Frictional stick-slip oscillation as a first-passage problem

Fig. 7: Two phase-space trajectories of the Langevin system,near the critical speed for stick-slip motion, integrated from(x0 , v0 ) until first passage. Dashed line: fundamental mode(mean period τ ). Solid line: A double orbit (τ ! ≈ 2τ ). D = 3,vb = 2, ω0 = 1, ζ = 0.02.
Fig. 6: (Colour on-line) Numerical and analytical oscillation densities. Quasiperiodic regime: (A) period density p(T )(1)computed from the normalized truncation estimate F̂n .
Parameters: ω0 = 2, D = 3, vb = 1.25, ζ = 0.1. (B) Slip size
density p(d). Other details as in (A). Multiperiodic regime:(1)(C) period density p(T ), in decoupling approx. Fg , ω0 = 1,
ζ = 0.4, D = 0.1, vb = 1.0. (D) period p(T ) and slip distance(1)p(d) densities from Fg , ω0 = 1, ζ = 0.11, D = 0.04, vb = 1.0.

rate, known as the Rice frequency [19], and here given by
n0 = (ω0 /2π)exp(−vb ζ/2D). The decoupling approxima(1)tion, Fg (T ), displays no such error (fig. 5, right panel),but underestimates the first and overestimates later peaks.
Quasiperiodic motion: Figure 6 (top row) presents
results for the densities of oscillation periods p(T ) andslip distances p(d) (eq. (4)) obtained using the first-ordertruncation approximation in comparison with numericalresults. In this regime, the two densities are each broadly
distributed about a single maximum, evidencing quasiperiodic motion. In this regime, there is good qualitative
agreement with densities obtained from experimental
observations of solid-on-solid stick-slip motion (cf.,Johansen et al. [9], figs. 18, 20), also consistent withseveral studies that have found Gaussian-shaped distributions of force fluctuations in solid and granular stick-slip
motion [7,9,29].
Multiperiod oscillation: As the platform velocity vb
approaches the fundamental critical speed, which is thehighest speed at which stick-slip motion is expected tooccur deterministically [1], some trajectories are observed
to “miss” a slip-to-stick transition on one oscillation cycle,but, due to noise perturbations, to attain it on a subsequent pass (fig. 7). Trajectories that pass N cycles ofthe motion before crossing contribute to peaks of thefirst-passage density with mean periods of about τ = N T .
These transient oscillations occur near subharmonics ofthe fundamental period, but are randomly spaced intime, so do not correspond to any stationary oscillation
mode. Similar near-threshold subharmonic oscillations
are seen in real frictional systems [2], are evidenced by
anticorrelations in return maps found in experiments on

granular sliding [4], and are predicted by deterministic
friction models that account for multiple, coupled oscillation modes [1,22,30,31].
Multimodal period and slip densities: The period
density, p(T ), was computed using the decoupling(1)
approximation Fg (x, T ) (figs. 6C, D). At low damping,a decaying sequence of maxima is observed, revealing
quasiperiodic oscillations with transient slip periods
spaced at integer multiples of the free-sliding period.
As noted above, they do not correspond to any stationary subharmonic oscillation mode, as they are randomly
distributed in time. The corresponding density p(d) of slip
displacements p(d) is approximately Gaussian-shaped,
despite the multimodal nature of p(T ). Simulations yielda small probability of sticking immediately after slip onset(d ≈ 0), consistent with experiments (e.g., [9], fig. 18).
However, effects due to static friction variations andpre-sliding would affect such transitions [2,3], and are notaccounted for here.
Summary. – A novel formulation of frictional stickslip motion as a first-passage time problem was presented,
and used to investigate statistical distributions of stickslip oscillatory motion. Approximations were obtainedfor distributions of periods, slip times, and slip lengths
for a generic class of stochastic friction models, via alevel-crossing statistics expansion. Applied to a Langevinmodel of sliding friction, analytic solutions yielded good
agreement with simulations and prior experiments. While
this model ignored the dynamics of static to kinetic friction transitions (which can depend on asperity strengths,
contact aging, etc.), such effects could, for example, be
accommodated by allowing for random variations in thestatic threshold, or by applying this analysis to a friction
model (e.g., the Burridge-Knopoff model [15]) that
accounts for multiple interfacial degrees of freedom. The
presence of noise-stimulated transient oscillation modes
suggests an intriguing connection with the literature on
stochastic resonance and excitable systems [32]. While thecase example examined here involved a piecewise-linear

68006-p5

Y. Visell
non-smooth dynamics, qualitative arguments summarized above suggest similar near-threshold behavior
may be expected in disordered frictional systems with
non-linearities or additional dynamical states.
Appendix: Langevin upcrossing
density. – The#∞
upcrossing density is n1 (t) = −∞ n1 (x, vb , t) dx, wheren1 (x, v, t) = P10 (x, v, t) Ia (x, v) ≡ P10 (Ia1 + Ia2 ), (A.1)
&
$
$
%
%
−µ2a
µa
µa
σ2
√
exp
erfc
,
,
I
=
−
Ia1 =
a2
2π
2σ 2
2
2σ
(A.2)
P10 is a Gaussian process in x = (x, v), so the integral inn1 (t) involves factors of Gaussian or Error Function typein x and vb . One can write√
P10 (x, v, t) = N (x; µ, Σ), Ia1 (x, v) = σ 2π N (x; 0, S),
where N is the bivariate normal density and$ 4 2 %
ω0 ω0 ζS=σ 2 ≡ C −1 .
ω02 ζ ζ 2

(A.3)

The integrand P10 Ia1 is a product of bivariate Gaussianfunctions, and can be written P10 Ia1 = Z̃N (x; µ̃, Σ̃),√Z̃ = |S|σ 2π N (µ; 0, S Σ̃Σ−1 ),(A.4)Σ̃ = (Σ−1 + S)−1 ,µ̃ = Σ̃Σ−1 µ.

The integral over x then gives! ∞dx P10 Ia,1 = Z̃N (v; µ̃v , Σ̃vv ).

(A.5)

(A.6)

−∞

The integral of P10 Ia2 yields
! ∞
(
' √
2 2
dx P10 Ia,2 = Z # n π erfc (nq) +(q −2 − 1)e−n q
−∞

with

√ $ $
%
%
Cxv 2
2
v ζ−
ω0 + µx ω02 ,
n=
σ
Cxx
%− 12
$
1
4ω04
2 −1 − 2
m = 2ω0 σ Cxx ,,q = 1+ 2
σ Cxx
1

Z # = σ(Cxx Cvv |4πΣ|)− 2 N (v; µv , Cvv ).

(A.7)
(A.8)
(A.9)

With these definitions, n1 (t) can be written asn1 (t) = Z̃N (v; µ̃v , Σ̃vv )√2 2+Z # (n π erfc (nq) + (q −1 − q)e−n q ). (A.10)∗∗∗

This work was supported by a grant from the MDEIEof Quebec for the EU FP7 project NIW (No. 222107).
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