Inverse thermodynamic uncertainty relations general upper bounds on the fluctuations of trajectory observables George Bakewell-Smith1Federico Girotti1M ad alin Gut a1 2and Juan P. Garrahan3 2

2025-05-03 0 0 2.1MB 11 页 10玖币
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Inverse thermodynamic uncertainty relations: general upper bounds on the
fluctuations of trajectory observables
George Bakewell-Smith,1Federico Girotti,1M˘ad˘alin Gut¸˘a,1, 2 and Juan P. Garrahan3, 2
1School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
2Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,
University of Nottingham, Nottingham, NG7 2RD, UK
3School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK
Thermodynamic uncertainty relations (TURs) are general lower bounds on the size of fluctutations
of dynamical observables. They have important consequences, one being that the precision of
estimation of a current is limited by the amount of entropy production. Here we prove the existence
of general upper bounds on the size of fluctuations of any linear combination of fluxes (including
all time-integrated currents or dynamical activities) for continuous-time Markov chains. We obtain
these general relations by means of concentration bound techniques. These “inverse TURs” are
valid for all times and not only in the long time limit. We illustrate our analytical results with a
simple model, and discuss wider implications of these new relations.
Introduction. Thermodynamic uncertainty relations
(TURs) refer to a class of general lower bounds on the
size of fluctuations in the observables of trajectories of
stochastic systems. TURs were initially postulated as a
bound on the scaled variance of time-averaged currents
in the stationary state of continuous-time Markov chains
[1], and soon after proven (using “level 2.5” large devi-
ation methods [2–4]) to apply to the whole probability
distribution [5]. TURs were subsequently generalised to
various other dynamics and observables, including for fi-
nite times [6, 7], for discrete-time Markov dynamics [8],
for the fluctuations of first-passage times [9, 10] and for
open quantum systems [11–14], among many other ex-
tensions and alternative derivations (see for example [15–
24]). For a review see [25].
The most widely considered form of the TUR is for
the relative uncertainty (variance over mean squared) of
a time-integrated current being larger than (twice) the
inverse of the entropy production. This has immediate
dual consequences for inference and estimation [1, 25]:
increased precision in the estimation of the value of a
current from a stochastic trajectory requires increasing
the dissipation, or alternatively, the value of the entropy
production can be inferred from the fluctuations of one
or more specific currents which might be easier to ac-
cess. Similar uses of the TUR can be formulated using
the dynamical activity [26–28] for the estimation of time-
symmetric observables [9, 25].
Despite their success and generality, a limitation of
TURs is that they only provide lower bounds on the size
of fluctuations: except in the few cases where they are
tight, inference on the observable of interest is hindered
by the absence of a corresponding upper bound. Here
we correct this issue by introducing a class of general up-
per bounds for fluctuations of trajectory observables con-
sisting of linear combination of fluxes (number of jumps
between configurations [2]) of a continuous-time Markov
chain, which includes all currents and activities. For lack
of a better name, we call these “inverse thermodynamic
uncertainty relations”. The inverse TURs are valid for
all times, and like the large deviation formulation of the
TURs, they bound fluctuations at all levels. We prove
these general relations using concentration bound tech-
niques [29–34].
Notation and definitions. Let X:= (Xt)t≥0be a
continuous-time Markov chain taking values in the finite
state space Ewith generator W=Px6=ywxy|xihy| −
Pxwxx|xihx|, with x, y ∈E. If X0is distributed accord-
ing to some probability measure νon the state space,
we denote by Pνthe law of Xand we use Eνfor the
corresponding expected value. We assume that Xis ir-
reducible with unique invariant measure (i.e. stationary
state) π. We are interested in studying fluctuations of
observables of the trajectory Xof the form
A(t) = X
x6=y
axyNxy(t),
where axy are arbitrary real numbers with P|axy|>0,
and Nxy(t) are the elementary fluxes, that is, the num-
ber of jumps from xto yup to time tin X. For a
time-integrated current the coefficients are antisymmet-
ric, while for counting observables (such as the activity),
they are symmetric.
The fluctuations of A(t) in the long time satisfy the
following theorems [35]:
(i) Strong Law of Large Numbers (holds almost surely)
lim
t→+∞
A(t)
t=haiπ:= X
x6=y
πxwxyaxy.
(iii) Central Limit Theorem (small deviations; holds in
distribution)
lim
t→+∞
A(t)−thaiπ
√t=N(0, σ2
∞)
where σ2
∞= limt→+∞σ2
ν(t)/t and σ2
ν(t) is the variance of
A(t) if νis the initial distribution (notice, however, that
arXiv:2210.04983v1 [cond-mat.stat-mech] 10 Oct 2022
2
the limit does not depend on ν).
(iii) Large Deviation Principle
PνA(t)
t=haiπ+ ∆ae−tI(∆a)for every ∆a∈R
for some rate function I:R→[0,+∞] which in gen-
eral is hard to determine and admits an explicit analytic
expression only for particular models.
In order to state our main result below we need to
introduce the following quantities. In the stationary
state π, the average of Aper unit time is haiπ=
Px6=yπxwxyaxy, while its static approximate variance is
ha2iπ, with ha2iπ=Px6=yπxwxy a2
xy (it is the variance
of the random variable Px6=yaxy ˜
Nxy, where we approx-
imate Nxy(t)/t with independent Poisson random vari-
ables ˜
Nxy with intensity πxwxy). The maximum escape
rate is q= maxxwxx, and c= maxx6=y|axy |the max-
imum amplitude of the coefficients that define the ob-
servable. Since we do not assume that Wis reversible,
we denote by εthe spectral gap of the symmetrization
<(W)=(W+W†)/2, where the adjoint is taken with
respect to the inner product induced by the stationary
state π. Finally, the average dynamical activity per unit
of time at stationarity is hkiπ:= Px6=yπxwxy.
Main results. We now state our three main results:
(R1) The variance σ2
π(t) of any time-integrated current
or flux observable A(t) in the stationary state has the
general upper bound
σ2
π(t)≤tha2iπ1 + 2q
ε(1)
Note that this is valid for trajectories of any length t.
(R2) The distribution of A(t)/t after time tstarting from
an initial measure νobeys a concentration bound
PνA(t)
t≥ haiπ+ ∆a≤C(ν)e−t˜
I(∆a),(2)
where ∆a > 0 is the fluctuation of Aaway from the
stationary average, and C(ν) := (Pxν2
x/πx)1/2accounts
for the difference between νand the stationary π, with
C(π) = 1. The bounding rate function is given by
˜
I(∆a) = ∆a2
2hkiπc2+2qha2iπ
ε+5cq∆a
ε(3)
(R3) The rate function for A(t)/t is lower bounded by
Eq. (3) for every ∆a > 0, that is
˜
I(∆a)≤I(∆a).(4)
(R1)-(R3) are extended straightforwardly to ∆a < 0 by
considering the observable −A(t).
Inverse TURs and bounds on precision. The most
direct use of TURs is in bounding the precision for esti-
mating a current from its time-average over a trajectory
-6 -5 -4 -3 -2 -1
0
0.5
1
1.5
-6 -5 -4 -3 -2 -1
0
0.5
1
1.5
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A/t
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(A/t)
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Exact
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TUR
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TUR
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iTUR
FIG. 1. Upper bound on current fluctuations. The full
(black) curve shows the exact rate function I(A/t) for the
current defined by a12 = 0.9, a13 =−0.9, a14 =−0.9, a23 =
0.9, a24 =−0.9, a34 = 0.9. The rate function is upper
bounded by the TURs: the dotted (blue) curve is the stan-
dard TUR using the entropy production, while the dot-dashed
(pink) is the TUR with the dynamical activity. The dashed
(red) curve is the inverse TUR: it lowers bound the rate func-
tion, which corresponds to an upper bound to fluctuations of
the current at all orders. Inset: sketch of the 4-state model.
in a non-equilibrium stationary state (NESS) π. We de-
fine the relative error Aof observable Aas the ratio
between the variance of Aand its average squared mul-
tiplied by t
2
A=tσ2
π(t)
hAi2
π
=σ2
π(t)
thai2
π
.(5)
From the standard application of the TUR together with
the “inverse TUR” Eq. (1) we can bound the relative
error from below and above
2
Σπ≤2
A≤ha2iπ
hai2
π1 + 2q
ε(6)
where Σπ=Px6=yπxwxy log(πxwxy /πywyx) is the av-
erage entropy production rate in the NESS. The lower
bound is the known statement that a smaller error re-
quires larger dissipation. The upper bound states that
the error is controlled by the static uncertainty and by the
ratio between the largest and smallest relaxation rates in
the dynamics. Note that 2q/ε ≥1 (see [36, Lemma 1]).
As an illustration of Eqs. (4) and (6) we consider the
fluctuations of currents in the 4-state model of Ref. [5].
The network of elementary transitions is shown in the
inset of Fig. 1. The rates are as in Ref. [5], w12 = 3,
w13 = 10, w14 = 9, w21 = 10, w23 = 1, w24 = 2, w31 = 6,
w32 = 4, w34 = 1, w41 = 7, w42 = 9 and w43 = 5.
A current is defined by the values of the six coefficients
ax>y =−ax<y which we take in the range axy ∈[−1,1].
3
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Arank (⇥105)
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Arank (⇥105)
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Arank (⇥105)
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Arank (⇥105)
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t=102
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t=1
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t=102
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t=
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(a)
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(b)
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(c)
<latexit sha1_base64="ahFzC3ehXJ+7FKN2vFeCt7x9yzU=">AAACLXicbVDLSsNAFJ34rPHVqjs3g6VQNyXpQl0W3Lis2Be0oUwmk3bo5MHMTaGEfoJb/Qy/xoUgbv0NJ20WfXhg4HDuY849biy4Asv6MnZ29/YPDgtH5vHJ6dl5sXTRUVEiKWvTSESy5xLFBA9ZGzgI1oslI4ErWNedPGb17pRJxaOwBbOYOQEZhdznlICWXqre7bBYtmrWAnib2DkpoxzNYcm4GngRTQIWAhVEqb5txeCkRAKngs3NQaJYTOiEjFhf05AETDnpwuscV7TiYT+S+oWAF+rqREoCpWaBqzsDAmO1WcvE/2r9BPwHJ+VhnAAL6fIjPxEYIpwdjj0uGQUxw4RS7TwhoH3QMZGEgg5obVfLdtLMXrZo3YBk07lZWZWA6/PmpqlztDdT2yades2+q9nP9XKjkSdaQNfoBlWRje5RAz2hJmojikboFb2hd+PD+DS+jZ9l646Rz1yiNRi/f6dcp0U=</latexit>
(d)
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Exact
<latexit sha1_base64="GznQBFYliWoYDOm4H0i9qWnwkg8=">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</latexit>
iTUR
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2
A
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TUR
FIG. 2. Lower and upper bounds on the estimation error. (a) Relative error 2
Afor estimating a current Afrom a
trajectory of length t= 10−2in the NESS of the model of Fig. 1. We show results for 206different currents A∈T. The full
(black) curve is the exact error. The standard TUR, dotted (blue) line, and the activity TUR, dot-dashed (pink) line, provide
lower bounds to the error which are independent of A. The inverse TUR, dashed (red) curve, gives an upper bound to the error
which varies with A. (b-d) Same for times t= 1,102,∞, respectively. The data in all panels is ranked according to decreasing
values of the error at t=∞. For comparison, the Acorresponding to entropy production is shown according to the same
ranking: TUR bound (green circle), exact (white triangle), TUR (blue cross), activity TUR (yellow square).
To perform the analysis, we construct a mesh across the
space of current observables T= [−1,1]6. We discretise
with a grid spacing of 10−1. Each point on this six-
dimensional mesh corresponds to a different current, and
in this way we get a reasonable representation of T.
Figure 1 shows the bounds for the long-time limit rate
function I(A/t) for one such current A∈T. The full
(black) curve is the exact rate function. It is calculated
from the “tilted” generator Wu=Px6=yeuaxy wxy|xihy|−
Pxwxx|xihx|as follows [35]: (i) the moment generat-
ing function (MGF) of Ais Zπ,t(u) := Eπ[euA(t)] =
hπ|etWu|−i, where |−i =Px|xiis the “flat state”; (ii) at
long times Zπ,t(u)etΛ(u), where the scaled cumulant
generating function (SCGF) Λ(u) is the largest eigen-
value of Wu; (iii) the rate function is obtained via a Leg-
endre transform, I(a) = supu[ua −Λ(u)].
The dotted (blue) curve in Fig. 1 is the usual TUR
using the entropy production [5]. The dot-dashed (pink)
curve is the alternative TUR which instead of Σπuses
the average dynamical activity, hkiπ=Px6=yπxwxy [9,
37]. Both these curves are above the true rate function,
thus providing the usual lower bounds on the size of the
fluctuations of A. The dashed (red) curve is Eq. (4) and
lies below the rate function: this inverse TUR is an upper
bound on the size of fluctuations of Aat all orders.
Figure 2 shows the bounds (6) on the precision error
(5), for all currents in the grid that scans T, both at
finite and infinite t. The full (black) curves are the exact
error 2
A, where the first two moments of Aare obtained
from the first and second derivatives of the MGF Zπ,t(u)
evaluated at u= 0. The errors for all the currents are
plotted with rank ordered by their value at t=∞, so
that in Fig. 2(d) they are monotonically decreasing. The
dotted (blue) lines are the lower bounds from the TUR
at either finite [6] or infinite [1] times. The dot-dashed
(pink) lines are the activity TUR, where in the l.h.s. of
Eq. (6) Σπis replaced by 2hkiπ. As the TURs do not
depend on the details of the current that they bound,
these curves appear constant in Fig. 2.
Figure 2 also shows the inverse TUR from the r.h.s. of
Eq. (6) as full (red) curves. This gives an upper bound
to the error. As the inverse TUR contains information
about the specific current through its static average and
second moment, the bound tracks the change in shape
of the exact error. Furthermore, in many instances the
ratio of the relative value of the upper bound to the error
is smaller than that of the error to the lower bound (note
the log scale in the plots).
Derivation of results. We now give the main steps for
the proofs of results R1-R3. For details see [36].
Result R3 follows easily from R2: indeed from the def-
inition of large deviation principle and Eq. (2) is easy to
see that
inf
∆a0>∆aI(∆a0)≥liminf
t→+∞−1
tlog PνA(t)
t≥ haiπ+ ∆a
≥˜
I(∆a).(7)
The inequality holds for ∆a0≥∆adue to the continuity
of ˜
Iand inf∆a0≥∆aI(∆a0) = I(∆a) because it is non-
decreasing for ∆a≥0.
To obtain R1 and R2, the first step is upper bounding
the moment generating function of A(t): we show that
for every u≥0 the following holds true
Zν,t(u)≤C(ν)et˜
Λ(u),(8)
摘要：

Inversethermodynamicuncertaintyrelations:generalupperboundsontheuctuationsoftrajectoryobservablesGeorgeBakewell-Smith,1FedericoGirotti,1MadalinGuta,1,2andJuanP.Garrahan3,21SchoolofMathematicalSciences,UniversityofNottingham,Nottingham,NG72RD,UnitedKingdom2CentrefortheMathematicsandTheoreticalPhy...

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Inverse thermodynamic uncertainty relations general upper bounds on the fluctuations of trajectory observables George Bakewell-Smith1Federico Girotti1M ad alin Gut a1 2and Juan P. Garrahan3 2.pdf

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