Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables

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by  Juan P. Garrahan

Simple bounds on fluctuations and uncertainty relations for first-passage times of
counting observables
Juan P. Garrahan
School of Physics and Astronomy and
Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,
University of Nottingham, Nottingham NG7 2RD, UK
(Dated: January 4, 2017)
Recent large deviation results have provided general lower bounds for the fluctuations of time-
integrated currents in the steady state of stochastic systems. A corollary are so-called thermo-
dynamic uncertainty relations connecting precision of estimation to average dissipation. Here we
7
1 considerthisproblembutforcountingobservables,i.e.,trajectoryobservableswhich,incontrastto
0 currents,arenon-negativeandnon-decreasingintime(andpossiblysymmetricundertimereversal).
2 In the steady state, their fluctuations to all orders are bound from below by a Conway-Maxwell-
Poissondistributiondependentonlyontheaveragesoftheobservableandofthedynamicalactivity.
n
Weshowhowtoobtainthecorrespondingboundsforfirst-passagetimes(timeswhenacertainvalue
a
ofthecountingvariableisfirstreached)andtheiruncertaintyrelations. Justlikeentropyproduction
J
does for currents, dynamical activity controls the bounds on fluctuations of counting observables.
2

]
h I. INTRODUCTION dynamical activity [5, 11, 24]. Here show that from the
c boundstotheratefunctionsofcountingobservables, via
e In this note we try to connect three recent develop- trajectory ensemble equivalence, we can derive the cor-
m
ments in the theory of stochastic systems. The first are responding bounds for arbitrary fluctuations of FPTs.
- generalboundsonthefluctuationsoftime-integratedcur- After introducing the basics of dynamical large de-
t
a rents[1–4]. Obtainedbymeans“Level2.5”[5–7]dynami- viations, in Sec. III we show that the rate functions
st callargedeviationmethods[8–12],theseresultsstipulate of counting observables are bounded from above by a
t. general lower bounds for fluctuations at any order of all Conway-Maxwell-Poisson distribution (a generalisation
a empirical currents in the stationary state of a stochastic ofthePoissondistributionthatallowsfornon-Poissonian
m process[1–4]. Acorolaryarethermodynamic uncertainty number fluctuations [25]). The corresponding bound
- relations[13–15]connectingtheestimationerroroftime- for the cumulant generating function was first found in
d
integrated currents to overall dissipation. Ref. [2] (called “exponential bound”); here we rederive
n
itstraightforwardlyviaLevel2.5largedeviations, cf.[1].
o The second development are fluctuation relations for
In Sec. IV we consider the large deviations of FPTs and
c first-passage times (FPTs) [16–18], similar to those of
[ more standard observables such as work or entropy pro- establish the correspondence between the FTP and ob-

1 duction. From these an uncertainty relation connecting servable generating functions. This allows, in Sec. V, to
derive a general bound on FPT rate functions from the
v dissipationtothetimeneededtodeterminethedirection
large deviations of the observable distributions. From
9 oftimecanbederived[16]. Theseresultsindicatearela-
3 tion between the fluctuations of observables in dynamics these bounds FPT uncertainty relations follow. An im-
5 over a fixed time, with fluctuations in stopping times. portantobservationisthattheboundsonfluctuationsof
0 acountingobservableanditsFPTsarecontrolledbythe
The third development is trajectory ensemble equiva-
0 average dynamical activity, in analogy to the role played
lence [19–22] between ensembles of long trajectories sub-
.
1 jecttodifferentconstraints. Forexample, forlongtimes, by the entropy production in the case of currents [1, 13].
0 We hope these results will add to the growing body of
the ensemble of trajectories conditioned on a fixed value
7 work applying large deviation ideas and methods to the
of a time-integrated quantity is equivalent to that con-
1
studyofnon-equilibriumdynamicsinclassicalandquan-
ditioned only on its average [19, 20] (cf. microcanoni-
:
v cal/canonical equivalence of equilibrium ensembles [23]). tum stochastic systems, see e.g., [26–61].
i
X Similarly, the ensemble of trajectories of fixed total time
and fluctuating number of jumps is equivalent to that of
r II. STOCHASTIC DYNAMICS AND LARGE
a fixed number of jumps but fluctuating time [21, 22] (cf.
DEVIATIONS OF COUNTING OBSERVABLES
fixed volume and fixed pressure static ensembles [23]).
The works in Refs. [1–4] and [13–18] focus on trajec-
We consider systems evolving as continuous time
tory observables asymmetric under time reversal, such
Markov chains [62], with master equation,
as empirical currents, which can be positive or nega-
tive and can increase and decrease with time. Here we (cid:88) (cid:88)
∂ P (x)= W P (y)− R P (x), (1)
consider instead trajectory observables which are always t t yx t x t
x,y(cid:54)=x x
non-negative and strictly non-decreasing with time. We
call these counting observables. An example is the to- where P (x) is the probability being in configuration x
t
tal number of configuration changes in a trajectory, or at time t, W the transition rate from x to y, and R =
xy x
2
(cid:80)
W the escape rate from x. In operator form the ForobservablessuchasEq.(7)themomentgenerating
y(cid:54)=x xy
master equation reads, function Eq. (5) can be written as
∂ |P (cid:105)=L|P (cid:105), (2) Z (s)=(cid:104)−|etLs|x (cid:105), (9)
t t t t 0
with probability vector |P (cid:105) = (cid:80) P (x)|x(cid:105), where {|x(cid:105)} where L is the tilted operator [8–12],
t x t s
is an orthonormal configuration basis. The master oper-
(cid:88)
ator is, L =W −R= e−αxysW |y(cid:105)(cid:104)x|−R, (10)
s s xy
x,y(cid:54)=x
(cid:88) (cid:88)
L=W −R= W |y(cid:105)(cid:104)x|− R |x(cid:105)(cid:104)x|, (3)
xy x (cid:80)
and(cid:104)−|= (cid:104)x|. Thefunctionθ(s)isthengivenbythe
x,y(cid:54)=x x x
largest eigenvalue of L .
s
where W and R indicate the off-diagonal and diagonal
parts of L, respectively. This dynamics is realised by
stochastic trajectories, such as ω = (x → x → ... → III. LEVEL 2.5 AND FLUCTUATION BOUNDS
0 t1
x ). This trajectory has K jumps, with the jump be-
tK
tween configurations x and x occurring at time t , The computation of large deviation functions as the
ti−1 ti i
with 0 ≤ t ≤ ···t ≤ t, and no jump between t and ones in Eqs. (4) and (5) for arbitrary observables and
1 K K
t. We denote by π (ω) the probability of ω within the dynamicsisdifficultingeneral. Thereishoweveronecase
t
ensemble of trajectories of total time t. where the rate function can be written down explicitly
Properties of the dynamics are encoded in trajec- [5–7].
tory observables, i.e., functions of the whole trajectory, If we denote by M (ω) the total residence time in
x
A(ω), which are extensive in time. Examples include configuration x throughout trajectory ω, then t−1M (ω)
x
time-integrated currents or dynamical activities. Time- is the empirical measure. Similarly, from the num-
exentisivity implies that at long times their probabilities ber of jumps Q (ω) we can define the empirical flux,
xy
and moment generating functions have large deviation t−1Q (ω). Since M(ω) and Q(ω) are extensive observ-
xy
forms [8–12], ables, their probability obeys a large deviation principle
at long times [8–12],
(cid:88)
P (A)= δ[A−A(ω)]π (ω)≈e−tϕ(A/t), (4)
t ω t Pt(q,m)=(cid:88)πt(ω)(cid:89)δ(cid:2)mx−t−1Mx(ω)(cid:3) (11)
Z (s)=(cid:88)e−sA(ω)π (ω)≈etθ(s), (5) ω x
t ω t ×(cid:89)δ(cid:2)qxy−t−1Qxy(ω)(cid:3)≈e−tI(q,m).
xy
where the rate function ϕ(a) and the scaled cumulant
generatingfunctionθ(s)arerelatedbyaLegendretrans- TheratefunctionI(q,m)hasanexplicitforminthesta-
form [8–12], tionary state dynamics of Eq. (2), known as “level 2.5”
of large deviations [5–7],
ϕ(a)=−min[θ(s)+sa]. (6)
s (cid:20) (cid:18) (cid:19) (cid:21)
I(q,m)=(cid:88)q ln qxy −1 +(cid:88)m R , (12)
In what follows we focus on trajectory observables de- xy m W x x
x xy
xy x
fined in terms of the jumps in a trajectory,
wheremandqmustobeytheprobabilityconservingcon-
(cid:88)
A(ω)= αxyQxy(ω), (7) ditions,
xy
(cid:88) (cid:88) (cid:88)
m =1, q = q . (13)
whereQ (ω)isthenumberofjumpsfromxtoy intra- x xy yx
xy x y y
jectory ω. We will assume all α ≥0. This means that
xy
A(ω) is non-negative and non-decreasing with time. We Thisratefunctionisminimised(itsminimumvaluebeing
call A(ω) a counting observable as it counts the number zero) when m and q take the stationary average values
ofcertainkindsofjumpsinthetrajectory. Furthermore,
when α = α these observables are symmetric un- m =ρ , q =ρ W , (14)
xy yx x x xy x xy
dertime-reversal, incontrasttotime-integrated currents
whichareantisymmetric(andthereforeneithernecessar- where ρx is the stationary distribution, L|ρ(cid:105) = 0. The
ilypositivenornon-decreasingwithtime). Animportant rate function for a trajectory observable such as Eq. (7)
example of a counting observable is the total number of can then be obtained by contraction [8–12],
jumps or dynamical activity [5, 11, 24],
ϕ(a)= min I(q,m), (15)
q,m : a=α·q
(cid:88)
K(ω)= Q (ω). (8)
xy (cid:80)
where α·q = α q and a=A/t.
xy xy xy xy
3
An upper bound for ϕ(a) can be obtained following
(a) (b)
the procedure of Ref. [1]. From Eq. (15), any pair of 1
| �
empirical measure m and flux q that satisfies Eq. (13)
(cid:80) � �
and has a = α q will give an upper bound to )
xy xy xy t )
/ s
ϕ(a). A convenient and simple choice is, A (
( 0 �
�
a | �
m∗ =ρ , q∗ = ρ W , (16)
x x xy (cid:104)a(cid:105) x xy
(cid:80)
where (cid:104)a(cid:105) = α ρ W . We then get, with I (a) =
I(q∗,m∗), xy xy x xy ∗ A/t s
(a) (b)
(cid:104)k(cid:105)(cid:20) (cid:18) a (cid:19) (cid:21) FIG. 1. Bounds on observable fluctuations for a 2-level sys-
ϕ(a)≤I∗(a)= (cid:104)a(cid:105) ln (cid:104)a(cid:105) −(a−(cid:104)a(cid:105)) , (17) tem. TransitionratesareW10 =γ andW01 =κ. Weconsider
asA)observableAthetotalnumber)of1→0jumps. Inthesta-
wnahmerieca(cid:104)lka(cid:105)c=tiv(cid:80)ityxy(pρxerWuxnyit=t(cid:80)imxe)ρ.xRTxheisrtahtee fauvnecratigoendoyn- tpieo��/(rnuarnyitsttiamtee(cid:104)ias(cid:105)(cid:104)k=(cid:105)(cid:104)=A2(cid:105)/(cid:104)at(cid:105)=. Pγκan/e(γlg�((+a)κs)h.oTwhsethaveerraatgeefaucntcitviiotny
ϕ(A/t)(full/black)forγ =5andκ=1.25. Theratefunction
the right side of Eq. (17) is that of a Conway-Maxwell-
is bounded from above everywhere by a CMP rate function,
Poisson (CMP) distribution [25], a generalisation of the
Eq. (17) (dashed/red). We also show for comparison a Pois-
Poisson distribution for a counting variable with non-
son rate function with mean (cid:104)a(cid:105) (dotted/blue). Panel (b)
Poissonian number fluctuations. showsthecorres�p/oAndingscaledcumulantgenerat�ingfunction
FromtheLegendretransformEq.(6),theupperbound θ(s) = 1(cid:104)(cid:112)(γ−κ)2+4γκe−s−(γ+κ)(cid:105) (full/black). It is
Eq. (17) also implies a lower bound for the scaled cumu- 2
monotonic in s since A ≥ 0, and is bounded from below,
lant generating function θ(s), Eq. (18), by θ (s) = 2γκ(e−s/2−1) (dashed/red). For the
∗ γ+κ
(cid:20) (cid:18) (cid:19) (cid:21) case κ=γ the bounds become exact in this simple model.
(cid:104)a(cid:105)
θ(s)≥θ (s)=(cid:104)k(cid:105) exp −s −1 . (18)
∗ (cid:104)k(cid:105)
The expression on the right is the scaled cumulant gen-
betweenprecisionanddissipation, theuncertaintyinthe
erating function of a CMP distribution. This last result
estimation of a counting observable is bounded generi-
wasfirstderivedinRef.[2]inaslightlydifferentmanner.
cally by the overall average activity in the process.
Figure 1 illustrates the bounds Eqs. (17) and (18) for
theelementaryexampleofatwo-levelsystem. Theexact
rate function ϕ(a) and the upper bound I (a) have the
∗
sameminimumat(cid:104)a(cid:105),butthefluctuationsofaarelarger IV. LARGE DEVIATIONS OF FIRST-PASSAGE
TIME DISTRIBUTIONS
than those given by I (a) for all a. The exact cumulant
∗
generating function θ(s) and its lower bound θ (s) have
∗
the same slope at s=0, but θ (s) has derivatives which We consider now the statistics of first-passage times
∗
are smaller in magnitude to all orders that those of θ(s), (FPT) (also called stopping times), the times at which
again indicating that the CMP approximation provides a certain trajectory observable first reaches a threshold
lower bounds for the size of fluctuations of a. value. This implies a change of focus from ensembles of
As occurs with the analogous bounds on time- trajectoriesoftotalfixedtimetoensemblesoftrajectories
integrated currents [1–4], an immediate consequence of of fluctuating overall time [21, 63, 64]. Recently, distri-
the bounds on the rate function or cumulant generat- butionsofFPTassociatedwithentropyproductionhave
ingfunctionarethethermodynamicuncertaintyrelations been shown to obey fluctuation relations [16–18] remi-
[13–15]. From Eq. (17) or Eq. (18) we get a lower bound niscent of those of current-like observables. This sug-
for the variance of the observable in terms of its average gests a duality between observable and FPT statistics,
and the average activity (cf. [2]) which in turn is connected to the equivalence between
fixedtimeandfluctuatingtimetrajectoryensembles, see
θ(cid:48)(cid:48)(0) θ(cid:48)(cid:48)(0) (cid:104)a(cid:105)2
var(a)= ≥ ∗ = . (19) e.g. [21, 22].
t t (cid:104)k(cid:105)t We focus on stopping times for counting observables
as defined in Eq. (7). For simplicity we assume that the
This in turn provides an upper bound for precision of
coefficients α are either 0 or 1, so that A(ω) counts
estimation of the observable A in terms of the signal-to- xy
a subset of all possible jumps in a trajectory and takes
noise ratio (i.e. inverse of the error),
integer values. (These assumptions can be relaxed at
(cid:104)A(cid:105) (cid:112) theexpenseofslightlymoreinvolvedexpressionswithout
SNR(A)= ≤ (cid:104)K(cid:105), (20)
(cid:112) changing the essence of the results.)
var(A)
Lets consider the structure of trajectories associated
where (cid:104)K(cid:105) = t(cid:104)k(cid:105). Just like in the case of integrated with FPT events for a fixed value A of the observable
currents [13–15], where there is an unavoidable tradeoff A(ω). Such a trajectory will have A jumps for which
4
α = 1, occurring at times 0 ≤ t ≤ ···t ≤ t = τ For the case where the counting observable is the dy-
xy 1 A−1 A
with τ being the FPT through A(ω) = A. In between namical activity, Eq. (8), the analysis above is that of
these jumps the evolution will be one where only jumps “x-ensemble” of Ref. [21], i.e., the ensemble of trajecto-
with α = 0 occur. The weight of this trajectory is ries of fixed total number of jumps but fluctuating time.
xy
related to the amplitude of a matrix product state [65], For the general problem of the FPTs for arbitrary
countingobservables,Eqs.(23-24)coincidewiththeFPT
(cid:104)y|W˜ e(tA−tA−1)L∞ ···W˜ e(t2−t1)L∞W˜ et1L∞|x(cid:105). (21) distributionsfirstobtainedinRef.[17]inadifferentway.
ThederivationinRef.[17]proceedsinthestandardman-
This expression is the weight of all trajectories starting
ner used for example in the proof of FPT distributions
in x and ending in y, after A jumps that contribute
for diffusion processes [62]. It relates the probability of
to the observable, occurring at the specified times t
i having accumulated A up to time t, to the probability of
(i = 1,...,A), and with an arbitrary number of the
reachingAattimeτ ≤tforthefirsttimefollowedbyno
other jumps. Here L is the tilted operator Eq. (10)
∞ increment in A from τ to t,
at s → ∞, so that all transitions associated to A(ω) are
suppressed. The factors e∆tL∞ encode dynamics which (cid:88)(cid:90) t
P (A|x)= dτP (0|y)F (τ|A), (29)
donotcontributetoincreasingtheobservableandwhich t t−τ xy
occur between the times t . The operator y 0
i
W˜ =L−L , (22) where Pt(A|x) is Eq. (4) with the initial condition made
∞
explicit, and F (τ|A) refers to the FPT distribution for
xy
includes all the transitions that increase A(ω) by one time τ and final configuration y. If we transform from A
unit, and Eq. (21) has A insertions of W˜. Integrating to s, cf. Eqs. (4), (5) and (9), we can rewrite Eq. (29) as
Eq. (21) over intermediate times and summing over the matrix elements of
final configuration formally yields the FPT distribution, (cid:90) t
(cid:90) etLs = dτe(t−τ)L∞Fˆs(τ), (30)
F (τ|A)= (cid:104)−|W˜ e(τ−tA−1)L∞ ···W˜ et1L∞|x(cid:105). 0
x
0≤t1···≤τ where (cid:104)y|Fˆ (τ)|x(cid:105)=(cid:80) e−sAF (τ|A). After a Laplace
s A xy
This expression simplifies via a Laplace transform, transform and rearraging we get,
Fˆ (µ|A)=(cid:90) ∞dte−µτF (τ|A)=(cid:104)−|FA|x(cid:105), (23) Fˆsµ =(µ−L∞)(µ−Ls)−1. (31)
x x µ
0
ThislastexpressionisthesameasthatinRef.[17]aftera
where the transfer operator reads discreteLaplacetransformfromAtos. Wecaninvertthe
A → s transformation as follows. The l.h.s. of Eq. (31)
F =W˜ (µ−L )−1. (24)
µ ∞ is,
When A is large, A (cid:29) 1, the Laplace transformed FTP ∞
distribution has a large deviation form, Fˆ = (cid:88)e−sAFˆ (A), (32)
sµ µ
Fˆ (µ|A)≈eAg(µ), (25) A=0
x
while the r.h.s. can be rewritten as,
where eg(µ) is the largest eigenvalue of F . Note the
µ
similarities between Eqs. (23-25) and Eqs. (5-10). (µ−L )(µ−L )−1
∞ s
FroTmheEeqisg.e(n1v0a)l,u(e2s2)ofanFdµ(2a4n)dwLescaanrewrdiitree,ctly related. =(cid:104)1−e−sW˜ (µ−L∞)−1(cid:105)−1
∞
e−sFµ =(Ls−µ)(µ−L∞)−1+1. (26) = (cid:88)e−sA(cid:104)W˜ (µ−L∞)−1(cid:105)A. (33)
A=0
Consider now a row vector (cid:104)l| which is a left eigenvector
both of Fµ and Ls, with eigenvalue eg(µ) and θ(s), re- Equating Eqs. (32) and (33) term by term we get that
spectively. Multiplying Eq. (26) by (cid:104)l| and rearranging
we get Fˆ (A)=FA, (34)
µ µ
(cid:16) (cid:17)
e−s+g(µ)−1 (cid:104)l|=[θ(s)−µ](cid:104)l|(µ−L∞)−1. (27) with Fµ given by Eq. (24), showing that our derivation
is equivalent to that of Ref. [17]. The advantage of ex-
We see that for (cid:104)l| to be a simultaneous eigenvector of pressing the FPT distribution in terms of its generating
F and L we need to have g(µ)=s and θ(s)=µ. That function Eq. (24) as we have done here is that it allows
µ s
is, g is the functional inverse of θ and vice-versa, for a direct extraction of its large deviation function, see
Eqs. (25) and (28), giving access to the full statistics of
θ(s)=g−1(s), g(µ)=θ−1(µ). (28) FPTs in the limit of large A.
(a) (b)
1
| �
) � �
t )
/ s
A (
�( � 0
| �
5
A/t s
an upper bound for the FPT rate function,
(a) (b)
φ(τ/A)≤ φ (τ/A)= (37)
∗
A) ) (cid:104)k(cid:105)(cid:20) (cid:18)τ(cid:104)a(cid:105)(cid:19) (cid:18)τ(cid:104)a(cid:105) (cid:19)(cid:21)
�/ �( − ln − −1 .
( g (cid:104)a(cid:105) A A
�
Figure 2 illustrates the upper bound of the FPT rate
function, Eq. (37), and the lower bound of the FPT cu-
mulantgeneratingfunction,Eq.(35),forthesame2-level
�/A � model of Fig. 1.
The bound function φ (τ/A) has its minimum at the
∗
FIG. 2. Bounds on first-passage time fluctuations for the 2- exact value of the average FPT,
level system of Fig. 1. The FPT τ is defined as the time
when a total A of up/down jumps 1 → 0 is reached. In A
(cid:104)τ(cid:105) = , (38)
the stationary state (cid:104)τ(cid:105) = A/(cid:104)a(cid:105) = A(γ +κ)/(γκ). Panel A (cid:104)a(cid:105)
(a) shows the rate function φ(τ/A) (full/black) for γ = 5
and κ = 1.25, and assuming the initial state is 0. The where(cid:104)·(cid:105) indicatesaverageintheFPTensembleoffixed
A
ratefunctionisboundedfromaboveeverywherebyφ (τ/A),
∗ A. That the average FPT is given by the inverse of
Eq.(37)(dashed/red). WealsoshowforcomparisontheFPT
the observable per unit time follows immediately from
rate function of a Poisson process with the same mean (dot-
Eq. (28). The second derivative of φ (τ/A) at its mini-
ted/blue). Panel (b) shows the FPT scaled cumulant gener- ∗
mumprovidesalowerboundforthevarianceoftheFPT.
atingfunctiong(µ)=ln(γκ)−ln[(γ+µ)(κ+µ)](full/black).
From Eq. (37), or alternatively Eq. (35), we get,
It is bounded from below, Eq. (35), by g (µ) (dashed/red).
∗
var(τ) 1
=g(cid:48)(cid:48)(0)≥g(cid:48)(cid:48)(0)= . (39)
A ∗ (cid:104)a(cid:105)(cid:104)k(cid:105)
V. BOUNDS ON FPT DISTRIBUTIONS
This in turn gives a bound on the precision with which
one can estimate the FPT,
Equations (23-28) establish a connection between the
statistics of a counting observable, at fixed overall time, SNR(τ)= (cid:104)τ(cid:105)A ≤(cid:112)(cid:104)K (cid:105). (40)
and the statistics of the FPT for a fixed value of said (cid:112)var(τ) A
observable. This connection is due to the equivalence
[21,22]betweentheensembleoftrajectoriesoffixedtime, where (cid:104)K (cid:105) = (cid:104)τ(cid:105) (cid:104)k(cid:105). As for case of the uncertainty
A A
but where the observable is allowed to fluctuate (in a for the observable, Eq. (20), the precision of estimation
mannercontrolledbythefieldsconjugatetotheobserv- oftheFPTislimitedbythetotalaverageactivity,inthis
able), with the ensemble of fixed observable but where case for trajectories of length t=(cid:104)τ(cid:105) .
A
the time extension of trajectories is allowed to fluctu-
ate (in a manner controlled by the field µ conjugate to
time). This equivalence holds in the limit of large ob- VI. CONCLUSIONS
servable/time,wheretherelationbetweenthecontrolling
fields is given by Eq. (28). We can now use the results We have discussed general bounds on fluctuations
of Sec. III on bounds on observable fluctuations to infer of counting observables, hopefully complementing the
the corresponding bounds on FPT fluctuations. more detailed recent results on current fluctuations [1–
The bound Eq. (18) on the cumulant generating func- 4]. While empirical currents are the natural trajec-
tion of A provides an lower bound to the FPT scaled tory observables to consider in driven problems [1–
cumulant generating function g(µ) through Eq. (28). In- 5, 26, 27, 29, 43, 47, 61], counting observables such as
verting θ in Eq. (18) we get the dynamical activity are central quantities for systems
∗
with complex equilibrium dynamics, such as glass form-
(cid:104)k(cid:105) (cid:18) µ (cid:19) ers [24, 28, 30, 32, 39]. (And even for driven systems it
g(µ)≥g∗(µ)=−(cid:104)a(cid:105)ln (cid:104)k(cid:105) +1 . (35) is revealing to study the dynamical phase behaviour in
terms of both empirical currents and activities, see e.g.
[29, 38, 47, 61].)
For large A the FPT distribution also has a large devia-
The bounds are a straightforward consequence of the
tion form,
Level 2.5 large deviation [5–7] description, Eq. (12),
which provides an explicit (and useful) minimisation
F(τ|A)≈e−Aφ(τ/A), (36) principleforratefunctions. Butasremarkedin[4],these
bounds may be more or less descriptive depending on
whereφ(τ/A)isobtainedfromg(µ)byaLegendretrans- whether they are tight or loose, which in turn depends
form similar to Eq. (6). From Eq. (35) we then obtain onhowgoodthevariationalansatzis. Asobservedin[2],
6
theansatzEq.(16)isakintoamean-fieldapproximation Eq. (29), is directly related to the tilted operator L ,
∞
that homogenises the connections between states. For leading to the ensemble correspondence, Eqs. (25-28).
any counting observable which is a subset of the over- For currents, however, a zero observable does not im-
all activity the rate function is bound by a CMP dis- ply the absence of jumps that contribute to the observ-
tribution with sub-Poissonian number fluctuations, see able (only that their contribution adds up to zero), and
Eqs.(17)and(18). FortheelementaryexampleofFig.1 thecorrespondencebreaksdown(oratleast wehavenot
the bound is tight, but more complex problems of inter- been able to relate the corresponding cumulant gener-
estoftendisplaylarge(thatis,super-Poissonian)number ating functions in that case). Just like in the case of
fluctuations [24, 28, 32, 35, 39, 42]. It would be inter- activities, the FPTs are bounded by the distribution of
esting to find alternative yet simple variational ansatzes timesofaCMPprocess,Eqs.(35)and(37),asillustrated
thatcancapturesuchstrongfluctuationbehaviour. Nev- in Fig. 2.
ertheless,therearestillimportantconsequencesthatfol- Asoccursforcurrents[13–15],theboundstoratefunc-
low even from these simple bounds. An immediate one tions give rise to thermodynamic uncertainty relations
is that the dynamical activity cannot be sub-Poissonian, constraining the precision of estimation of both observ-
which in turn implies an exponential in time complexity ables and FPTs, Eqs. (20) and (40). For empirical cur-
for the efficient sampling of trajectories conditioned on rents, whicharetime-asymmetric, precisionislimitedby
it, cf. [47]. theaverageentropyproduced[13–15]. Inturn,forcount-
ing observables and their FPTs, the corresponding limit
We have also shown how to obtain related general is set by the average dynamical activity, suggesting that
boundsonthedistributionsoffirst-passagetimes. Again this quantity might play as important a role in the dy-
this complements for counting observables, and gener- namics as the overall dissipation.
alises, recent results on FPTs for current-like quantities
[16–18]. We did this by exploiting the correspondence
between the large deviation functions of observables and ACKNOWLEDGMENTS
those of FPTs, Eqs. (25-28). Note that this correspon-
dence works for observables which are non-decreasing in This work was supported by EPSRC Grant
time. For these, the zero increment probability P (0|y), No. EP/M014266/1.
t
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