††thanks: Corresponding author

# Thermodynamic uncertainty relation for first-passage times on Markov chains

Arnab Pal<EMAIL_ADDRESS>School of Chemistry, Raymond and Beverly
Sackler Faculty of Exact Sciences, & Center for the Physics and Chemistry of
Living Systems. Tel Aviv University, 6997801, Tel Aviv, Israel  Shlomi Reuveni
<EMAIL_ADDRESS>School of Chemistry, Raymond and Beverly Sackler
Faculty of Exact Sciences, & Center for the Physics and Chemistry of Living
Systems. Tel Aviv University, 6997801, Tel Aviv, Israel  Saar Rahav
<EMAIL_ADDRESS>Technion, Israel Institute of Technology, Haifa, Israel

###### Abstract

We derive a thermodynamic uncertainty relation (TUR) for first-passage times
(FPTs) on continuous time Markov chains. The TUR utilizes the entropy
production coming from bidirectional transitions, and the net flux coming from
unidirectional transitions, to provide a lower bound on FPT fluctuations. As
every bidirectional transition can also be seen as a pair of separate
unidirectional ones, our approach typically yields an ensemble of TURs. The
tightest bound on FPT fluctuations can then be obtained from this ensemble by
a simple and physically motivated optimization procedure. The results
presented herein are valid for arbitrary initial conditions, out-of-
equilibrium dynamics, and are therefore well suited to describe the inherently
irreversible first-passage event. They can thus be readily applied to a myriad
of first-passage problems that arise across a wide range of disciplines.

Figure 1: Schematic description of a Markov jump process on a network with an
absorbing state $A$. We develop a thermodynamic uncertainty relation that
bounds relative fluctuations in the first passage time to $A$.

Many interesting phenomena occurring at the molecular scale can be viewed as
first-passage processes redner2001guide . From chemical reactions
szabo1980first ; szabo1984localized ; Nitzan-book ; Geometry-controlled ;
Gating  to enzyme catalysis FP-EK ; Ninio ; FP-EK-2 ; Cao1 ; Enzyme1 ; Enzyme2
; Enzyme3  and molecular search processes Mol_Search1 ; Mol_Search2 ;
Mol_Search3 ; Mol_Search4 ; Mol_Search5  — one is interested in the exact
moment at which a distinguished event occurs: the first passage time (FPT)
redner2001guide ; Bray2013review ; metzler2014first ; klafter2011first .
Thermal agitations make FPTs random and much effort has been directed to
determining their governing statistics in theory and experiment CV-expt ; FPT-
Theo-EXP1 ; FPT-Theo-EXP2 ; FPT-Theo-EXP3 ; FPT-Theo-EXP4 ; FPT-Theo-EXP5 . Of
particular interest in that regard is the relative magnitude of FPT
fluctuations around their mean value. This central gauge of randomness has
found numerous uses and applications Gating ; Enzyme1 ; Enzyme2 ; Enzyme3 ;
Mol_Search5 ; FPT-Theo-EXP1 ; FPT-Theo-EXP2 ; FPT-Theo-EXP3 ; FPT-Theo-EXP4 ;
Fluctuations1 ; Fluctuations2 ; Fluctuations3 ; Fluctuations4 ; Fluctuations5
; Fluctuations6 ; Fluctuations7 ; Fluctuations8 ; Fluctuations9 ;
Fluctuations10 ; Fluctuations11 . Yet, owing to conceptual and technical
challenges, its relation with the underlying thermodynamics remains poorly
understood.

The first passage event is often described as a completely irreversible step,
and as a result such processes are inherently out-of-equilibrium. The theory
of stochastic thermodynamics was developed to describe the nonequilibrium
thermodynamics of small stochastic systems, such as molecular motors, enzymes,
and nano-machines Sekimoto-book ; ST-1 ; ST-2 ; ST-3 ; ST-4 ; ST-5 ;
Pezzato2017 . One of the celebrated results in the field is the thermodynamic
uncertainty relation (TUR) TUR-1 ; TUR-2 . First conjectured by Barato and
Seifert, the TUR states that the normalized fluctuations of currents are
bounded from below by the inverse entropy production. The fundamental nature
of the TUR has spurred an intensive research effort TUR-review ; Seifert2019 ;
Horowitz-2017 ; Pietzonka2017 ; Proesmans2017 ; Barato2018 ; Dechant2018 ;
Hasegawa2019 ; Dechant2020 ; nonSS5 ; Uni ; PRR-TUR ; TUR-3 ; TUR-4 .

There has only been a handful attempts to extend the TUR to FPTs. Garrahan
TUR-FP-2 , whose work was later on generalized by Gingrich and Horowitz TUR-
FP-1 , derived a TUR for a fluctuating current that crosses a pre-determined
threshold for the first time. More recently, Falasco and Esposito have derived
a dissipation based speed limit on the mean arrival time to a target MFPT-TUR
. Notably all the above-mentioned works focus on systems in steady state. In
contrast, in first-passage processes, such as the one depicted in Fig. 1, one
is primarily interested in the transient dynamics. Moreover, the transition to
the absorbing state is irreversible and therefore its entropy production is
ill defined. These conceptual issues make an extension of the TUR to FPT
processes both interesting and challenging.

In this letter, we derive a TUR for the first-passage time to a site on a
Markov chain. The result is derived from a more general set up that we have
recently introduced to study systems with unidirectional transitions PRR-TUR .
Like other TURs, the TUR derived herein bounds the relative fluctuations of
the first-passage time, but its validity is not limited by the steady-state
assumption and it can moreover be applied directly to systems with absorbing
states. As this manuscript was being prepared Hiura and Sasa presented a
kinetic TUR for first passage times TUR-FP-3  (see also KUR ; Dechant-Sasa
for other kinetic TURs for currents). At the end of this letter, we show that
the TUR derived in TUR-FP-3  follows from ours as a special case.

First passage on a continuous time Markov chain.— On a Markov network, the
probability to find the system in different states evolves according to the
master equation

$\frac{d\mathbb{P}}{dt}=\bm{\Gamma}\mathbb{P},$ (1)

where $\bm{\Gamma}$ is the transition rates matrix. Its off diagonal elements
satisfy, $\Gamma_{ij}=K_{ij}$, where $K_{ij}$ is the rate of transitions from
site $j$ to site $i$. The diagonal entries to $\bm{\Gamma}$ are given by
$\Gamma_{jj}=-\sum_{i\neq j}K_{ij}$, namely the escape rates out of state $j$.
For physical reasons one often assumes microreversibility, namely that if
$K_{ij}>0$ then the reversed transition is also possible and $K_{ji}>0$. This
principle is essential for the thermodynamical consistency of the model.
However, when studying the kinetics of first passage processes one can not
apply this principle to transitions that lead to the absorbing state. Thus the
challenge is to extend existing frameworks so as to take into account
unidirectional transitions PRR-TUR .

In what follows, we consider models with $N$ regular sites and one absorbing
site labeled $N+1$. We also assume that all the transitions between the
regular sites satisfy the principle of microreversibility. As a result, the
transition rate matrix has the block structure

$\displaystyle\bm{\Gamma}=\begin{bmatrix}\bar{\bm{\Gamma}}&\bm{0}\\\
\bm{K}_{\text{out}}&0\end{bmatrix},$ (2)

where $\bar{\bm{\Gamma}}$ accounts for the transitions between the regular
states. $\bm{K}_{\text{out}}=(K_{N+1,1},K_{N+1,2},\cdots,K_{N+1,N})$ is a row
vector containing the transition rates to the absorbing state. For this class
of models the distribution of first passage times is given by

$f_{T}(t)=\bm{K}_{\text{out}}e^{\bar{\bm{\Gamma}}t}\tilde{\mathbb{P}}(0),$ (3)

where $\tilde{\mathbb{P}}(0)$ is a vector containing the initial probabilities
to reside in the $N$ regular sites. Once $f_{T}(t)$ is known one can calculate
and study its moments. In the following we will focus on the mean and
variance, and derive a bound on $CV^{2}=\text{Var}(T)/\left\langle
T\right\rangle^{2}$, which is known as the coefficient of variation (CV) or
randomness parameter of the first-passage time.

TUR for first passage time.— A description of first passage that is in the
spirit of stochastic thermodynamics can be obtained by directly considering
stochastic realizations of the Markovian jump process discussed above. In each
such realization the system starts at some initial site $i_{0}$, then jumps to
a site $i_{1}$ at time $t_{1}$, etc. An ensemble of all the realizations of
duration $\mathcal{T}$ is obtained by assigning to each such realization its
probability density. Now, let us define the functional

$T\left[\omega\right]=\sum_{i\neq A}\tau_{i}\left[\omega\right],$ (4)

where $\tau_{i}\left[\omega\right]$ is the total time the realization $\omega$
spent in site $i$ throughout the observation time $\mathcal{T}$. One notices
that if the absorbing state is reached in a realization $\omega$, then
$T\left[\omega\right]$ is simply the first passage time. Alternatively
$T\left[\omega\right]=\mathcal{T}$. The latter occurs with the survival
probability $\mathcal{S}(\mathcal{T})$. In finite and connected networks
$\mathcal{S}(\mathcal{T})$ decays exponentially as
$\mathcal{T}\rightarrow\infty$. In this limit the statistics of
$T\left[\omega\right]$ is precisely that of the first passage time.

A TUR for the first passage time can be obtained with the help of a more
general inequality that we recently derived for Markovian jump processes in
systems with unidirectional transitions PRR-TUR . For the functional (4) it
takes the form

$\text{Var}_{\omega}\left[T(\omega)\right]\geq\frac{\left[\mathcal{T}\mathcal{S}(\mathcal{T})-\int_{0}^{\mathcal{T}}~{}dt~{}\mathcal{S}(t)\right]^{2}}{\int_{0}^{\mathcal{T}}dt~{}\left[\frac{1}{2}\sigma_{\text{rev}}(t)+J_{\text{uni}}(t)\right]},$
(5)

where survival probability can also be understood as the ensemble averaged
rate of change of the functional (4): $\mathcal{S}(t)=\frac{d}{dt}\left\langle
T[\omega]\right\rangle_{t}$. Notably, the bound in PRR-TUR  also holds for
functionals that count transitions between states, and for models with more
general transitions than the ones considered here.

To understand the denominator of Eq. (4) we first note that transitions in our
model can be divided into two sets. The set of unidirectional transitions,
$E_{1}$, and the set of bidirectional transitions $E_{2}$. Importantly, every
transition in the bidirectional set must have a reversed counterpart which is
also in $E_{2}$. So if $(i\rightarrow j)\in E_{2}$ then also $(j\rightarrow
i)\in E_{2}$. There is no such restriction on transitions in $E_{1}$. In
stochastic thermodynamics all the transitions are often considered to be
bidirectional due to microreversibility. One can then interpret

$\sigma_{\text{rev}}(t)=\sum_{(j\rightarrow i)\in
E_{2}}\frac{1}{2}\left[K_{ji}P_{i}(t)-K_{ij}P_{j}(t)\right]\ln\frac{K_{ji}P_{i}(t)}{K_{ij}P_{j}(t)},$
(6)

as the rate of entropy production from bidirectional transitions. In contrast,
the contribution of unidirectional transitions is

$J_{\text{uni}}(t)=\sum_{(j\rightarrow i)\in E_{1}}{K}_{ij}P_{j}(t)~{},$ (7)

which expresses their flux.

A crucial aspect of Eq. (5) is that it holds for arbitrary partitions of the
model’s transitions into groups $E_{1}$ and $E_{2}$. For the first passage
models we study here, the transitions to the absorbing state must belong to
the unidirectional set $E_{1}$ as they have no reverse counterpart. All other
pairs of transitions that satisfy microreversibility can be placed in either
$E_{1}$ or $E_{2}$. Thus, a reversible transition that would usually be placed
in $E_{2}$ can be treated as a pair of irreversible transitions and put in
$E_{1}$ instead. This freedom of choice gives rise to a set of bounds that
emanate from Eq. (5). We emphasize that all these bounds are equally valid,
and which one should be used is discussed at the end of this derivation.

To complete the derivation of a bound for the first passage time we consider
the $\mathcal{T}\rightarrow\infty$ limit of Eq. (5). Using
$\int_{0}^{\mathcal{T}}~{}dt~{}S(t)=\mathcal{T}S(\mathcal{T})+\int_{0}^{\mathcal{T}}~{}dt~{}t~{}f_{T}(t)$
and $\left\langle
T\right\rangle=\lim_{\mathcal{T}\rightarrow\infty}\int_{0}^{\mathcal{T}}~{}dt~{}t~{}f_{T}(t)$,
we recast Eq. (5) as

$CV^{2}\geq\frac{1}{\int_{0}^{\infty}dt~{}\left[\frac{1}{2}\sigma_{\text{rev}}(t)+J_{\text{uni}}(t)\right]}.$
(8)

Eq. (8) is the central result of this paper. It shows that the coefficient of
variation of the first passage time is bounded from below by an expression
that combines the entropy production from the transitions in $E_{2}$ and the
fluxes from the transitions in $E_{1}$. We now discuss different
representations of the bound that arise from the freedom to place transitions
in either of these groups.

Entropic bound.— This form of the bound is obtained when all the bidirectional
transitions are treated as such and put in group $E_{2}$. The remaining
transitions are thus those that lead to the absorbing site $A$. Given that the
system is initially distributed somewhere among the sites $i\neq A$, and that
$P_{A}(t\rightarrow\infty)=1$, one finds that

$\int_{0}^{\infty}dtJ_{\text{uni}}(t)=\int_{0}^{\infty}dP_{A}(t)=P_{A}(\infty)-P_{A}(0)=1.$
(9)

The entropy production from all other transitions is given by

$\displaystyle\Sigma_{\text{rev}}=\frac{1}{2}\sum_{(j\rightarrow i)\in
E_{2}}\int_{0}^{\infty}dt\left[K_{ij}P_{j}(t)-K_{ji}P_{i}(t)\right]\ln\left[\frac{K_{ij}P_{j}(t)}{K_{ji}P_{i}(t)}\right].$
(10)

The entropic version of the bound is then given by

$CV^{2}\geq\frac{1}{\frac{1}{2}\Sigma_{\text{rev}}+1}.$ (11)

Kinetic bound.— The kinetic bound is obtained when all the reversible
transitions, that were placed in $E_{2}$ in the entropic form of the bound,
are treated as pairs of irreversible transitions and put in $E_{1}$. As a
result, the reversible entropy production vanishes. In contrast,
$\Sigma_{\text{uni}}$ is the integrated flux from all transitions, namely,

$\Sigma_{\text{uni}}=\int_{0}^{\infty}dtJ_{\text{uni}}(t)=\sum_{(j\rightarrow
i)}\int_{0}^{\infty}dtK_{ij}P_{j}(t).$ (12)

In this case one obtains the bound

$CV^{2}\geq\frac{1}{\Sigma_{\text{uni}}}.$ (13)

Mixed bounds.— Clearly Eqs. (11) and (13) follow extreme cases in which one
considers the maximally possible number of transitions to be either
bidirectional or unidirectional. Yet, one may also utilize the freedom to
place some bidirectional transitions in $E_{2}$, while treating others as
pairs of unidirectional directional transitions to be placed in $E_{1}$. This
leads to a bound of the form

$CV^{2}\geq\frac{1}{\frac{1}{2}\Sigma_{\text{rev}}+\Sigma_{\text{uni}}}.$ (14)

Here $\Sigma_{\text{rev}}$ and $\Sigma_{\text{uni}}$ are similar to (10) and
(12), but with the summation restricted to the appropriate subset of
transitions.

Figure 2: (a) A model of a random walk on a finite 1d lattice. (b) The optimal
bound on the CV of the FPT transitions from entropic to kinetic as motion on
the lattice becomes heavily biased to the right (larger $\epsilon$, see main
text for details). The inset focuses on the crossover region. The crossover
from entropic to kinetic propagates from the edges of the system to its
center. (c) The optimal bound on the CV of the FPT transitions from entropic
to mixed, as the transition from the {1,2,3} block to the {4,5} block become
more unidirectional (larger $\epsilon$, see main text for details).

The optimal bound.— The validity of several bounds raises a natural question:
which bound should one use? We argue that the tightest bound is typically the
most useful one. A simple algorithm allows to obtain the best bound from the
family of bounds considered here. One has to scan over all pairs of sites
$i,j$. For every pair of transitions $i\rightarrow j$ and $j\rightarrow i$
that can belong either to $E_{1}$ or to $E_{2}$ one calculates the quantities
$\Sigma_{1}=\int_{0}^{\infty}dt\left[K_{ij}P_{j}(t)+K_{ji}P_{i}(t)\right]$ and
$\Sigma_{2}=\int_{0}^{\infty}dt\left[K_{ij}P_{j}(t)-K_{ji}P_{i}(t)\right]\ln\left[\frac{K_{ij}P_{j}(t)}{K_{ji}P_{i}(t)}\right]$.
If $\Sigma_{1}<\Sigma_{2}$ one includes a contribution of $\Sigma_{1}$ in
$\Sigma_{\text{uni}}$. Alternatively, one includes a contribution of
$\Sigma_{2}$ in $\Sigma_{\text{rev}}$.

We call this bound the optimal bound, where the term optimal refers to the
optimal partition of transitions into $E_{1}$ and $E_{2}$. Finding this
optimal partition is relatively easy since each transition can be treated
separately. Moreover, in many cases one can use physical intuition to figure
out what to do with a given transition. For example, if a transition is
expected to be almost detailed balanced, $K_{ij}P_{j}(t)\simeq
K_{ji}P_{i}(t)$, then it is highly likely that $\Sigma_{2}<\Sigma_{1}$ as
illustrated below.

An illustrative example.— A simple model of a one dimensional random walker is
used to elucidate our approach and to connect to related results. The model
consists of a linear chain of six sites, where the leftmost site is reflecting
and the rightmost is absorbing [Fig. 2(a)]. The walker starts at the leftmost
site, and jumps from site to site. All forward transitions in the model occur
with rate $k_{+}$, except the $3\rightarrow 4$ and $5\rightarrow 6$
transitions that occur with rates $k_{r}$ and $k_{f}$ respectively. All
backward transitions in the model occur with rate $k_{-}$, except the
$4\rightarrow 3$ transition that occurs with rate $k_{l}$. This
parameterization of rates is flexible enough to exhibit several types of
physically interesting behaviour.

In Fig. 2(b), we demonstrate that the optimal bound transitions from the
entropic form of Eq. (11) to the kinetic form of Eq. (13). To do so, we set
$k_{+}=k_{r}=k_{f}=1$, and $k_{-}=k_{l}=e^{-\epsilon}$; and observe that all
transitions are approximately unidirectional when $\epsilon$ is large. In this
limit entropy production is very high. It is thus no surprise that replacing
entropic with kinetic terms results in a tighter bound. On the other extreme,
for small values of $\epsilon$, there is almost no bias which results in a
quasi-equilibrium on the bulk of the lattice as probability gradually leaks to
the absorbing site. In this limit, entropy production is much lower and the
inclusion of entropic terms leads to tighter bound. (We use the term quasi-
equilibrium loosely, as the flux to the absorbing site is not small.) For
intermediate values of $\epsilon$, there is a crossover region where the
behaviour is more complex leading to a mixed optimal bound that includes both
entropic and kinetic terms (Fig. 2(b) inset).

To show that a mixed bound can be optimal in large regions of the phase space,
we make a different choice for the rates in Fig. 2(a): $k_{+}=k_{-}=1$,
$k_{r}=k_{f}=1/5$, and $k_{l}=e^{-\epsilon}/5$. The lattice is then
approximately divided into two blocks: that of sites {$1,2,3$} and that of
sites {$4,5$}, with less frequent transitions between the blocks. When
$\epsilon=0$, a lattice spanning quasi-equilibrium is formed and one expects
the optimal bound to be entropic. In contrast, when $\epsilon$ is large the
transition between lattice blocks is almost irreversible. In this case, a
quasi-equilibrium is formed in each block separately, and one expects that the
tightest bound will be achieved by taking entropy production terms for
transitions within blocks and flux terms for transitions between blocks. Fig.
2(c) shows that this is indeed the case. For all parameters in this panel the
kinetic bound performs poorly.

Summary and discussion.— We have derived a TUR for first passage times. Our
approach allows to consider models with completely irreversible transitions
which are commonly used to study first passage problems. Crucially, we derive
mixed bounds that combine kinetic and thermodynamic contributions from
different transitions on the Markov network at hand. While the manuscript was
being prepared, a different but related approach was used in TUR-FP-3  to
derive a TUR for first passage times. Their results also apply for models of
the type studied here. However, note that the TUR derived there is equivalent
to the fully kinetic version of our bound, and is thus a special case. The
results presented in Fig. 2, as well as the considerations below, suggest that
the ability to also consider mixed bounds is important.

Our results show an interesting connection to another well known bound

$CV^{2}\geq\frac{1}{N},$ (15)

where $N$ is again the number of kinetic states in the model (excluding the
absorbing state). The bound (15) was proven by Aldous and Shepp (AS) ASbound ,
and is often used in the field of statistical kinetics FPT-Theo-EXP2 ; FPT-
Theo-EXP3 ; FPT-Theo-EXP4 . Barato and Seifert Barato2015  have shown that
higher moments of first passage times also satisfy inequalities that depend on
the number of states of the model.

We now discuss the relation between the TUR and the AS bound, with the help of
the example depicted in Fig. 2. If all the transitions are irreversible,
meaning $k_{-}=k_{l}=0$, then one is forced to use the kinetic bound. In this
case $\Sigma_{\text{uni}}=N=5$ since the walker moves sequentially from the
first site to the second, and continues moving right in this manner until it
reaches the last site which is absorbing. We therefore recover the bound (15).

Using the kinetic bound when $k_{-}$ or $k_{l}$ are non-vanishing will give
$\Sigma_{\text{uni}}>N$, and thus a bound which is looser than the AS bound.
Yet, a mixed TUR bound can significantly improve on the AS result. Consider,
for example, a scenario with $k_{l}=0$ and $k_{r},k_{f}\ll k_{-},k_{+}$. These
dynamics consist of fast local equilibration in two blocks of states:
{$1,2,3$} and {$4,5$}, and rare transitions between the blocks. Here, a mixed
bound, which uses the entropy production from the blocks and the fluxes of the
unidirectional transitions, is tightest. In such a configuration, one expects
$\Sigma_{\text{uni}}=2$ and $\Sigma_{\text{rev}}<1$: The mixed TUR behaves as
if it is an AS bound of a coarse-grained model with two effective kinetic
states. This is consistent with the results depicted in Fig. 2(c).

Importantly, Markovian models are coarse-grained approximations of underlying
microscopic dynamics. When several states of a model are almost always in
local equilibrium with each other, thermodynamic consistency demands that we
can coarse-grain them into one effective state, to obtain an even simpler
model. The considerations above show how to get a TUR for coarse-grained
models by appropriately partitioning transitions into two groups: uni- and bi-
directional.

Finally, we note that several restrictions made on the model can be relaxed
without many difficulties. The approach works for models with several
physically distinct transitions between the same states. One simply needs to
view these as separate transitions. It is also possible to study models with
irreversible transitions inside the network of transitions (PRR-TUR ; IFT-R-0
; IFT-R ; Work ). They do not need to lead to an absorbing state. Finally, one
can also study models with several absorbing states. The derivation given in
Ref. PRR-TUR  generically shows how to handle such models.

Acknowledgements.— Arnab Pal gratefully acknowledges support from the Raymond
and Beverly Sackler Post-Doctoral Scholarship and the Ratner Center for Single
Molecule Science at Tel-Aviv University. Shlomi Reuveni acknowledges support
from the Azrieli Foundation, from the Raymond and Beverly Sackler Center for
Computational Molecular and Materials Science at Tel Aviv University, and from
the Israel Science Foundation (Grant No. 394/19).

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