# Thermodynamic uncertainty relation for underdamped dynamics driven by time-
dependent protocols

Chulan Kwon  Department of Physics, Myongji University, Yongin, Gyeonggi-Do,
17058, Korea<EMAIL_ADDRESS>Youngchae Kwon  Department of Physics, Myongji
University, Yongin, Gyeonggi-Do, 17058, Korea  Hyun Keun Lee  Department of
Physics, Sungkyunkwan University, Suwon, Gyeonggi-do 16419, Korea

###### Abstract

The thermodynamic uncertainty relation (TUR) for underdamped dynamics has
intriguing problems while its counterpart for overdamped dynamics has recently
been derived. Even for the case of steady states, a proper way to match
underdamped and overdamped TURs has not been found. We derive the TUR for
underdamped systems subject to general time-dependent protocols, that covers
steady states, by using the Cramér-Rao inequality. We show the resultant TUR
to give rise to the inequality of the product of the variance and entropy
production. We prove it to approach to the known overdamped result for large
viscosity limit. We present three examples to confirm our rigorous result.

###### pacs:

05.70.Ln, 02.50.-r, 05.40.-a

_Introduction_. Non-equilibrium fluctuation in time-accumulated observables
such as work, heat, and entropy production (EP) has been an important subject
since the discovery of the fluctuation theorem (FT) for deterministic systems
about two decades ago evans93 ; evans94 ; gallavotti . The FT with various
forms has been proved to hold for stochastic systems theoretically jarzynski1
; jarzynski2 ; crooks ; kurchan ; lebowitz ; seifert ; esposito  and
experimentally wang ; hummer ; liphard ; trepagnier ; collin ; garnier ;
hayashi ; dongyun . In recent years, another fundamental property on non-
equilibrium fluctuation, the thermodynamic uncertainty relation (TUR), has
been discovered barato1 ; gingrich1 . The TUR provides a universal trade-off
relation between the fluctuation of an arbitrary time-accumulated observable
and its dissipation. For a non-equilibrium steady state, the TUR for an
observable $\Phi$ accumulated over time $\tau$ states

$\frac{{\rm
Var}~{}\Phi}{\langle\Phi\rangle^{2}}\geq\frac{2k_{B}}{\tau\sigma}~{},$ (1)

where ${\rm Var}~{}\Phi$ is the variance of $\Phi$,
$\langle\Phi^{2}\rangle-\langle\Phi\rangle^{2}$, and $\sigma$ is the average
EP rate. The proof is known for the systems with even parity in time reversal.

There have been extensive studies on the TUR for various systems in different
dynamics. It has been studied for continuous-time Markov jumping processes for
long-time gingrich1  and finite-time horowitz ; gingrich2 ; pietzonka1 , also
for discrete-time jumping process proesmans , and for linear response systems
macieszazak . It has been studied for general overdamped Langevin systems
dechant ; hasegawa1  and heat engines pietzonka2 . For systems with time-
dependent protocols, it has also been extensively studied for periodically
driven systems in the absence of steady states barato2 ; barato3 ; koyuk1 .
There have been alternative studies to consider a different type of
inequality, the generalized TUR, in use of the FT hasegawa2 ; potts .

For general time-dependent protocols, the TUR has not been throughly
confirmed. For overdamped Langevin systems, the rigorous derivation of the TUR
has been found by Koyuk and Seifert koyuk2 , which states in terms of our
notation

$\frac{{\rm
Var}~{}\Phi}{\left[(\tau\partial_{\tau}-\omega\partial_{\omega})\langle\Phi\rangle\right]^{2}}\geq\frac{2k_{B}}{\langle\Delta
S_{\rm tot}\rangle}~{}.$ (2)

where $\omega$ is the speed of the protocol change in time $t$, that appears
as $\lambda(\omega t)$ for the protocol $\lambda$ and $\langle\Delta S_{\rm
tot}\rangle$ is the average total EP produced in the system and bath. It holds
for any systems prepared in arbitrary initial states and goes to the steady-
state TUR in Eq. (1) in the absence of time-dependent protocol. There have
been investigations for underdamped systems including momentum variables with
odd-parity in time reversal fischer ; chun ; vu ; lee , but the recent result
for steady states by Vu and Hasegawa vu  has left a controversy that it does
not approach to the known overdamped TUR in Eq. (2) and even gives rise to a
looser bound.

In this study, we investigate the TUR for underdamped dynamics subject to
general time-dependent protocols which goes to Eq. (2) for large viscosity.
First, we will derive the TUR rigorously and show it to go to the proper
overdamped limit. We will present a few examples to confirm our results.

_Derivation of the TUR_. We consider a particle of mass $m$ in underdamped
stochastic dynamics subject to a general time-dependent protocol. The
probability distribution function (PDF), $\rho(\mathbf{x},\mathbf{p},t)$, for
position $\mathbf{x}$ and momentum $\mathbf{p}$ satisfies the Kramers equation

$\partial_{t}\rho(\mathbf{x},\mathbf{p},t)=-\partial_{\mathbf{x}}\cdot\mathbf{j}_{\mathbf{x}}-\partial_{\mathbf{p}}\cdot\mathbf{j}_{\mathbf{p}}$
(3)

where $\partial_{t}$, $\partial_{\mathbf{x}}$, and $\partial_{\mathbf{p}}$
denote partial differentiations with respect to variables in subscript.
$\mathbf{j}_{\mathbf{x}}=(\mathbf{p}/m)\rho$ and
$\mathbf{j}_{\mathbf{p}}=[\mathbf{H}(\mathbf{x},\mathbf{p},t)-D\partial_{\mathbf{p}}]\rho$
for the drift term
$\mathbf{H}=-(\gamma/m)\mathbf{p}+\mathbf{f}(\mathbf{x},\omega t)$ are the
probability current densities in position and momentum spaces, respectively.
$\gamma$ and $D$ are the viscosity and diffusion coefficients, respectively,
satisfying the Einstein relation, $D=\gamma\beta^{-1}$ for the inverse
temperature $\beta$ of the medium (bath) maintained in equilibrium. For
simplicity of notation, we consider a single colloidal particle in isotropic
medium, which can be extended to a many-particle system in non-isotropic
medium with viscosity and diffusion coefficients in matrix forms. The time-
dependent protocol is included in the force $\mathbf{f}$ and $\omega$ is
introduced as a parameter to control the speed of protocol change in time
koyuk2 ; for example, it is the frequency of an oscillating driving force. The
force can be given by $\mathbf{f}=-\partial_{\mathbf{x}}V(\mathbf{x},\omega
t)$ for time-the dependent protocol in a potential $V$ or
$\mathbf{f}=-\partial_{\mathbf{x}}V(\mathbf{x})+\mathbf{f}_{\rm
nc}(\mathbf{x},\omega t)$ for the time-dependent protocol in a nonconservative
external force. In the absence of time-dependent protocol, the system reaches
a thermal equilibrium with the Boltzmann distribution, $\rho\propto e^{-\beta
E}$ for the energy $E=p^{2}/(2m)+V(\mathbf{x})$.

We consider a time-accumulated production $\Phi[\mathbf{q}(t)]$ which is a
functional of path $\mathbf{q}(t)=(\mathbf{x}(t),\mathbf{p}(t))$ for $0\leq
t\leq\tau$ in the phase space. We write the production rate $\dot{\Phi}$ per
time written in the form of either $\partial_{t}\chi(\mathbf{x},\omega t)$
(type I) or $\mathbf{g}(\mathbf{x},\omega t)\cdot\dot{\mathbf{x}}$ (type II).
For example, the work production rate is given by
$\partial_{t}V(\mathbf{x},\omega t)$ or $\mathbf{f}_{\rm nc}(\mathbf{x},\omega
t)\cdot\dot{\mathbf{x}}$. The average total production
$\langle\Phi(\mathbf{q}(t)\rangle$ over many trajectories can be computed in
principle by the path integral with the path probability

$P[\mathbf{q}(t)]\propto\rho_{\rm
in}(\mathbf{q}_{0})\prod_{t}\delta(\dot{\mathbf{x}}-\mathbf{p}/m)e^{-\int_{0}^{\tau}dt{\cal
L}}$ (4)

where the Lagrangian density is given by ${\cal
L}=(1/4)D^{-1}[\dot{\mathbf{p}}-\mathbf{H}]^{2}+\epsilon\partial_{\mathbf{p}}\cdot\mathbf{H}$
by using a discretization parametrized by $0\leq\epsilon\leq 1$ where
$\dot{\mathbf{p}}\simeq[\mathbf{p}(t)-\mathbf{p}(t-dt)]/dt$ and
$\mathbf{H}\simeq\epsilon\mathbf{H}(t)+(1-\epsilon)\mathbf{H}(t-dt)$ kwon_jkps
though the path integral is independent of discretizations; typically,
$\epsilon=0$ (prepoint, Ito), $1/2$ (midpoint, Stratonovich), $1$ (postpoint,
anti-Ito).

We follow the recent approach via the Cramér-Rao inequality to derive the TUR
hasegawa1 ; vu . It is crucial to choose the adjoint dynamics which differs
from the original one by the perturbation parameter $\theta$. Out of many
choices for the adjoint dynamics, we propose one to give rise to the TUR which
relates the variance of $\Phi$ and the total entropy production (EP), and goes
to that for overdamped dynamics for large $\gamma$ recently derived by Koyuk
and Seifert koyuk2 . The adjoint dynamics is governed by the chosen drift term

$\mathbf{H}_{\theta}=-\frac{\gamma}{m(1+\theta)}\mathbf{p}+\alpha\mathbf{f}(\mathbf{x},\omega
t)+\left(1-\frac{1}{1+\theta}\right)D\partial_{\mathbf{p}}\ln\rho_{\theta}$
(5)

where $\rho_{\theta}$ is the PDF of the adjoint dynamics. $\alpha$ is
introduced for a later use, which is set to be unity for the time being. Then,
Eqs. (3) and (4) are modified by $\mathbf{H}_{\theta}$. The Cramér-Rao
inequality is written as

$\frac{{\rm
Var}_{\theta}\Phi}{[\partial_{\theta}\langle\Phi\rangle_{\theta}]^{2}}\geq\frac{1}{{\cal
I}(\theta)}$ (6)

where ${\rm
Var}_{\theta}\Phi=\langle(\Phi-\langle\Phi\rangle)^{2}\rangle_{\theta}$ is the
variance of the production and ${\cal
I}(\theta)=-\langle\partial_{\theta}^{2}\ln P_{\theta}[\mathbf{q}]\rangle$ is
the Fischer information. It goes to a TUR in the limit $\theta\to 0$.

The Fischer information ${\cal I}(0)$ is found from Eq. (4). Using the
midpoint (Stratonovich) representation with $\epsilon=1/2$,

$\displaystyle{\cal I}(0)$ $\displaystyle=$
$\displaystyle\frac{D^{-1}}{2}\int_{0}^{\tau}dt\left\langle(\partial_{\theta}\mathbf{H}_{\theta})^{2}-(\dot{\mathbf{p}}-\mathbf{H}_{\theta})\cdot\partial_{\theta}^{2}\mathbf{H}_{\theta}\right.$
(7)
$\displaystyle\left.+D\partial_{\theta}^{2}\partial_{\mathbf{p}}\cdot\mathbf{H}_{\theta}\right\rangle\Big{|}_{\theta=0}$

where the second and third terms are shown to be cancelled by using the
property of midpoint discretization that
$\langle\partial_{\theta}^{2}\mathbf{H}_{\theta}\cdot(\dot{\mathbf{p}}-\mathbf{H})\rangle=\langle\partial_{\theta}^{2}\mathbf{H}\cdot(\mathbf{j}_{\mathbf{p}}/\rho-\mathbf{H})\rangle$
and the integration by parts $\langle\partial_{\theta}^{2}\mathbf{H}\cdot
D\partial_{\mathbf{p}}\rho/\rho\rangle=-\langle
D\partial_{\mathbf{p}}\cdot\partial_{\theta}^{2}\mathbf{H}_{\theta}\rangle$.
We find
$[\partial_{\theta}\mathbf{H}_{\theta}]_{\theta=0}=(\gamma/m)\mathbf{p}+D\partial_{\mathbf{p}}\ln\rho=-\mathbf{j}^{\rm
irr}_{\mathbf{p}}/\rho$ where $\mathbf{j}^{\rm irr}_{\mathbf{p}}$ is the
irreversible current density in the momentum space, known as the
irreversibility measure for non-equilibrium risken ; spinney ; yeo . Hence, we
have

${\cal
I}(0)=\frac{D^{-1}}{2}\int_{0}^{\tau}dt\left\langle\left(\mathbf{j}^{\rm
irr}/\rho\right)^{2}\right\rangle=\frac{1}{2k_{B}}\langle\Delta S_{\rm
tot}\rangle$ (8)

where $\langle\Delta S_{\rm tot}\rangle$ is the average total EP in the system
and bath. Note that ${\cal I}(0)$ from the different choice of
$\mathbf{H}_{\theta}$ by Vu and Hasegawa has extra terms besides the average
EP, which remains even for the overdamped limit.

The average total production is given by
$\langle\Phi\rangle_{\theta}=\int_{0}^{\tau}dt\int d\mathbf{x}\int
d\mathbf{p}\Gamma(\mathbf{x},\mathbf{p},\omega
t)\rho_{\theta}(\mathbf{x},\mathbf{p},t)$ where the rate function $\Gamma$
associated with $\dot{\Phi}$ is independent of $\theta$ and has either form

$\Gamma(\mathbf{x},\mathbf{p},\omega
t)=\left\\{\begin{array}[]{ll}\partial_{t}\chi(\mathbf{x},\omega
t)&(\textrm{type I})\\\ \mathbf{g}(\mathbf{x},\omega
t)\cdot\mathbf{p}/m&(\textrm{type II})\end{array}\right.$ (9)

With $\mathbf{H}_{\theta}$ in Eq. (5), $\rho_{\theta}$ satisfies the same form
of the Kramers equation with renamed viscosity coefficient as
$\bar{\gamma}=\gamma/(1+\theta)$. Then, we find

$\langle\Phi\rangle_{\theta}=\int_{0}^{\tau}\\!\\!\\!dt~{}\phi(t;\bar{\gamma})~{},$
(10)

where $\phi=\int d\mathbf{x}\int
d\mathbf{p}\Gamma(\mathbf{x},\mathbf{p},\omega
t)\rho(\mathbf{x},\mathbf{p},t;\bar{\gamma})$. Therefore, we get

$\partial_{\theta}\langle\Phi\rangle_{\theta}\Big{|}_{\theta=0}=-\gamma\partial_{\gamma}\langle\Phi\rangle~{},$
(11)

Equation (6) gives the TUR

$\frac{{\rm
Var}\Phi}{\left[-\gamma\partial_{\gamma}\langle\Phi\rangle\right]^{2}}\geq\frac{2k_{B}}{\langle\Delta
S_{\rm tot}\rangle}~{}.$ (12)

It is the rigorous result, while it leaves a practical problem that $\gamma$
might not be a proper control parameter in real experiments.

In order to express Eq. (11) in terms of experimentally controllable
parameters, we consider $H_{\theta}$ with $\alpha\neq 1$ in Eq. (5). We
introduce the change in variables and parameters such that
$\bar{t}=t/(1+\theta)$, $\bar{\omega}=(1+\theta)\omega$,
$\bar{\mathbf{p}}=(1+\theta)\mathbf{p}$, $\bar{\alpha}=(1+\theta)^{2}\alpha$,
and $\bar{\beta}=(1+\theta)^{-2}\beta$. In this change, $\omega t$ remains
having the same form as $\bar{\omega}\bar{t}$. The resultant Kramers equation
has an invariant form with these renamed variables and parameters. The initial
PDF of the original system is written as $\rho_{\rm
in}(\mathbf{x}_{0},\mathbf{p}_{0};\\{\lambda_{i}\\})$ with its own parameters,
$\\{\lambda_{i}\\}$. Then, the corresponding initial PDF of the adjoint system
is given by $\rho_{\theta}(\mathbf{x},\mathbf{p},0)=(1+\theta)^{d}\rho_{\rm
in}(\mathbf{x},\bar{\mathbf{p}};\\{\bar{\lambda}_{i}\\})$ where
$\bar{\lambda}_{i}$s are renamed parameters depending on the detailed form of
the initial PDF and $d$ is the dimensionality. As a result, we have

$\displaystyle\rho_{\theta}(\mathbf{x},\mathbf{p},t)$ $\displaystyle=$
$\displaystyle(1+\theta)^{d}\int d\mathbf{x}_{0}d\bar{\mathbf{p}}_{0}{\cal
T}(\mathbf{x},\bar{\mathbf{p}},\bar{t}|\mathbf{x}_{0},\bar{\mathbf{p}}_{0})$
(13) $\displaystyle\times\rho_{\rm
in}\left(\mathbf{x}_{0},\bar{\mathbf{p}}_{0};\\{\bar{\lambda}_{i}\\}\right)~{},$

where ${\cal T}$ is the transition probability which is the solution of the
renamed Kramers equation. It depends on renamed parameters, $\bar{\omega}$,
$\bar{\alpha}$, and $\bar{\beta}$, included in ${\cal T}$ and also on initial
parameters $\\{\bar{\lambda}_{i}\\}$. It is subtle to take into account the
dependence of the initial condition on $\theta$. There are special but
practical initial PDFs realized in experiments. One can usually prepare a
system in equilibrium or local equilibrium with an initial PDF

$\rho_{\rm in}\propto e^{\beta\alpha
V(\mathbf{x}-\mathbf{a},0)+\beta(\mathbf{p}-m\mathbf{v})^{2}/(2m)}=e^{\bar{\beta}\bar{\alpha}V+\bar{\beta}(\bar{\mathbf{p}}-m\bar{\mathbf{v}})^{2}/(2m)}~{}.$
(14)

Note that $\bar{\mathbf{v}}=(1+\theta)\mathbf{v}$ for the initial average
velocity $\mathbf{v}$ is the only additional renamed parameter to be
considered.

For those conditions, we find
$\rho_{\theta}=\rho(\mathbf{x},\bar{\mathbf{p}},\bar{t};\bar{\alpha},\bar{\beta},\bar{\omega},\bar{\mathbf{v}})(1+\theta)^{d}$.
The rate function reads $\Gamma(\mathbf{x},\mathbf{p},\omega
t)=\Gamma(\mathbf{x},\bar{\mathbf{p}},\bar{\omega}\bar{t})/(1+\theta)$ for
which we use the scaling such that
$\partial_{t}\chi=\partial_{\bar{t}}\chi/(1+\theta)$ (type I) and
$\mathbf{g}\cdot\mathbf{p}/m=(\mathbf{g}\cdot\bar{\mathbf{p}}/m)/(1+\theta)$
(type II). Then, we find

$\langle\Phi\rangle_{\theta}=\int_{0}^{\tau/(1+\theta)}\\!\\!\\!d\bar{t}~{}\bar{\phi}(\bar{t};\bar{\alpha},\bar{\beta},\bar{\omega},\bar{\mathbf{v}})$
(15)

where $\bar{\phi}=\int d\mathbf{x}\int
d\bar{\mathbf{p}}\Gamma(\mathbf{x},\bar{\mathbf{p}},\bar{\omega}\bar{t})\rho(\mathbf{x},\bar{\mathbf{p}},\bar{t};\bar{\alpha},\bar{\beta},\bar{\omega},\bar{\mathbf{v}})$.
Then, we find

$-\gamma\partial_{\gamma}\langle\Phi\rangle=\left(-\tau\partial_{\tau}+\omega\partial_{\omega}+2\alpha\partial_{\alpha}-2\beta\partial_{\beta}+\mathbf{v}\cdot\partial_{\mathbf{v}}\right)\langle\Phi\rangle~{},$
(16)

where $-\beta\partial_{\beta}$ can be replaced by $T\partial_{T}$ for the
temperature $T$ of the bath. The left-hand-side is useful for theoretical
studies, while the right-hand-side is measurable in experiments. Note that
this expression is valid for the practical initial condition given in Eq.
(14).

_Overdamped limit_. We can show that the found TUR for large $\gamma$ goes to
that recently derived by Koyuk and Seifert for the overdamped dynamics. The
rigorous proof is possible by using the PDF form for small $\gamma^{-1}$
obtained from an earlier work by one of us ao , given as

$\rho(\mathbf{x},\mathbf{p},t)\propto\exp\left[-(1+g)\frac{\beta}{2m}\left(\mathbf{p}-m\mathbf{u}\right)^{2}-{\cal
V}(\mathbf{x},t)\right]~{}.$ (17)

Here, ${\cal V}$ is the leading-order potential landscape,
$-\ln\rho(\mathbf{x},t)$, for the PDF in the position space given by $\int
d\mathbf{p}\rho(\mathbf{x},\mathbf{p},t)$.
$g=(m/\gamma)\partial_{\mathbf{x}}\cdot\mathbf{u}$ gives the lowest-order
correction to the overdamped limit, which we will neglect in the following
derivation.
$\mathbf{u}=\gamma^{-1}[\alpha\mathbf{f}+\beta^{-1}\partial_{\mathbf{x}}{\cal
V}]$ is the velocity field of the probability current in the position space,
$\mathbf{j}_{\mathbf{x}}=\mathbf{u}\rho(\mathbf{x},t)$. This equation
manifests our expectation that fast-varying velocity variable maintains a
local equilibrium distribution around an instantaneous average velocity
$\mathbf{u}$ along the trajectory of slowly varying position variable.
Averaging the Kramers equation (3) over momentum, we get the expected Fokker-
Planck equation

$\partial_{t}\rho(\mathbf{x},t)=-\gamma^{-1}\partial_{\mathbf{x}}\cdot\left[\alpha\mathbf{f}-\beta^{-1}\partial_{\mathbf{x}}\right]\rho(\mathbf{x},t)~{}.$
(18)

Note that $\mathbf{j}_{\mathbf{p}}=-(\beta^{-1}\partial_{\mathbf{x}}{\cal
V})\rho(\mathbf{x},\mathbf{p},t)$ using Eq. (17), so $\int
d\mathbf{p}~{}\partial_{\mathbf{p}}\cdot\mathbf{j}_{\mathbf{p}}=0$. Changing
variable as $s=t\alpha/\gamma$,
$\rho(\mathbf{x},t)=\rho(\mathbf{x},s;(\alpha\beta)^{-1},\omega\gamma\alpha^{-1})$.
Note that and $\langle\Gamma(\mathbf{x},\mathbf{p},\omega
t)\rangle_{\mathbf{p}}=\alpha\gamma^{-1}\partial_{s}\chi$ (type I) or
$\mathbf{g}\cdot\langle\mathbf{p}/m\rangle_{\mathbf{p}}=\alpha\gamma^{-1}\mathbf{g}\cdot[\mathbf{f}+(\alpha\beta)^{-1}\partial_{\mathbf{x}}{\cal
V}]$ (type II) where $\langle\cdot\rangle_{\mathbf{p}}$ denotes the average
over $\mathbf{p}$ withe the PDF in Eq. (17). Wring
$\langle\Gamma(\mathbf{x},\mathbf{p},\omega
t)\rangle_{\mathbf{p}}=\alpha\gamma^{-1}\bar{\Gamma}(\mathbf{x},(\omega\alpha\gamma^{-1})s)$,
the equation (15) for $\theta=0$ is given by

$\langle\Phi\rangle=\int_{0}^{\tau\alpha\gamma^{-1}}\\!\\!\\!ds~{}\phi^{\rm
OD}(s;(\alpha\beta)^{-1},\omega\alpha\gamma^{-1})~{},$ (19)

where $\phi^{\rm OD}=\int
d\mathbf{x}\bar{\Gamma}(\mathbf{x},(\omega\alpha\gamma^{-1})s)\rho(\mathbf{x},s;(\alpha\beta)^{-1},\omega\alpha\gamma^{-1})$.
Then, it is straightforward to find that Eq. (11) up to the leading order in
$\gamma^{-1}$ leads to

$-\gamma\partial_{\gamma}\langle\Phi\rangle=\left(\tau\partial_{\tau}-\omega\partial_{\omega}\right)\langle\Phi\rangle~{}.$
(20)

Note that the various terms in the right-hand-side of Eq. (16) with no
$\mathbf{v}$-derivative can be shown to collapse to the above simple form. It
is indeed equal to
$\partial_{\theta}\langle\Phi\rangle_{\theta}\Big{|}_{\theta=0}$ in the
overdamped TUR for time-dependent protocols derived by Koyuk and Seifert, as
seen in Eq. (2).

It is pedagogically interesting to see $\langle\Delta S_{\rm tot}\rangle$
approach to that for the overdamped limit. The irreversible current is found
as $\mathbf{j}^{\rm
irr}_{\mathbf{p}}=(-\gamma\mathbf{p}/m-D\partial_{\mathbf{p}})\rho=-\gamma\mathbf{u}\rho(\mathbf{x},\mathbf{p},t)$
in the leading order in $\gamma^{-1}$ by using Eq. (17). Then, the average
total EP rate is given in the leading order as

$D^{-1}\langle\left(\mathbf{j}^{\rm
irr}_{\mathbf{p}}\rho^{-1}\right)^{2}\rangle\simeq\gamma^{-1}\beta\langle\left(\gamma\mathbf{u}\right)^{2}\rangle=D_{\rm
od}^{-1}\langle\left(\mathbf{j}_{\mathbf{x}}\rho(\mathbf{x},t)^{-1}\right)^{2}\rangle~{},$
(21)

where $D_{\rm od}=(\beta\gamma)^{-1}$ is the diffusion coefficient for the
overdamped dynamics. It is equal to the average total EP for the overdamped
system.

Figure 1: $B(t)$ versus dimensionless time $\omega_{0}t$ for the pulled
harmonic oscillator with the angular frequency $\omega_{0}=\sqrt{k/m}$ above
the horizontal line indicating the minimum bound $2$. Graphs are drawn from
the analytic calculations for various values of $\gamma$ in unit of
$\sqrt{km}$ and pulling speed $u=2$ in unit of $\omega_{0}(\beta k)^{-1/2}$.
For small $t$, $B(t)$ is divergent as $t^{-3}$ due to the scaling: ${\rm
Var}W\sim t^{2}$, $EP\sim t$, and $DW\sim t^{3}$. Figure 2: $B(t)$ versus
$\omega_{0}t$ for the harmonic oscillator with the oscillating center above
the horizontal line of the minimum bound $2$. Graphs are drawn from the
analytical calculations for various values of $\gamma$ in unit of $\sqrt{km}$
and the frequency $\omega$ in unit of $\omega_{0}$. We use $A=1$, the root-
mean-square (rms) distance $a=0.1$ in unit of $(\beta k)^{-1/2}$, and the rms
velocity $v=0.1$ in unit of $(m\beta)^{-1/2}$, the equilibrium rms velocity.
We use an initial inverse temperature $\beta^{\prime}=0.5\beta$. The panel (a)
is for $\omega=1$ and (b) for $\omega=0.5,~{}2$. For small $t$, $B(t)\sim
t^{-3}$ because ${\rm Var}W\sim t^{2}$, $EP\sim t$, and $DW\sim t^{3}$, which
shows the same scaling behavior as in Fig. 1. There are divergent peaks at
specific times near $t=2$ for $\omega=1$, near $t=1$ for $\omega=2$, and near
$t=4$ for $\omega=0.5$ with an additional peak near $t=7$ for large
$\gamma=20$. It is due to the property that $DW=0$ at those times as it
oscillates and grows from zero, which is characteristic of oscillating
protocols and is not the case in Fig. 1.  Figure 3:  $B(t)$ versus
$\omega_{0}t$ for the harmonic potential with the time-varying stiffness.
Graphs are drawn from the computer simulations for (a) $\gamma=1,~{}\omega=1$
(b) $\gamma=2,~{}\omega=0.5$ (c) $\gamma=2,~{}\omega=1$ (d)
$\gamma=2,~{}\omega=2$ in the same units as in Fig. 2. We use $b=1/2$. Graphs
in each panel tend to approach to the black curve above the horizontal line of
the minimum bound $2$ as the number of samples denoted by a multiple of
$s=6.144\times 10^{7}$ increases. Due to large non-equilibrium fluctuations,
relatively the huge number of samples are needed. Divergence and infinite
peaks at small times shown in the previous figures are expected, but not
clearly observed for the present numbers of samples used in the simulations.

_Examples_. As the first example, we consider a particle in a harmonic trap
potential of which the center is perturbed in time by an external device. The
potential is given by $V(x,t)=k(x-a(t))^{2}/2$ for stiffness $k$. We consider
the two cases: (i) $a(t)=ut$ for a constant pulling velocity $u$ playing the
role of $\omega$ (ii) $a(t)=A\sin\omega t$. We investigate the TUR for work
production $W$ given by $\int_{0}^{\tau}dt~{}\partial_{t}V$ and the rate
function $\Gamma(x,\omega t)=-ku(x-ut)$ for (i) and $-kA\omega\cos\omega
t(x-A\sin\omega t)$ for (ii). An arbitrary initial state is given by the PDF
$\propto e^{-(\beta^{\prime}/2)(k(x-a)^{2}+(p-mv)^{2}/m)}$ for
$\beta^{\prime}\neq\beta$ where $a$ and $v$ are the average values of position
and velocity at $t=0$. The problem is simplified as we separate variables such
that $(x,p)=(z_{x},z_{p})+(X,P)$ for $X=\langle x\rangle$ and $P=\langle
p\rangle$. Then, $\mathbf{z}=(z_{x},z_{p})$ is evolved by the equilibration
process in the harmonic potential. Using $P=m\dot{X}$, $X$ satisfies
$m\ddot{X}+\gamma\dot{X}+\alpha kX=ka(t)$ so $\bar{\mathbf{q}}=\mathbf{(}X,P)$
can be found easily. Then, the PDF for $\mathbf{q}=(x,p)$ is written as

$P(\mathbf{q},t)=\frac{1}{Z(t)}\exp\left[-\frac{1}{2}\mathbf{[\mathbf{q}-\overline{\mathbf{q}}(t)}]^{\rm
t}\cdot\mathsf{A}(t)\cdot[\mathbf{\mathbf{q}-\overline{\mathbf{q}}(t)}]\right],$
(22)

where $Z(t)=\sqrt{(2\pi)^{2}/\det\mathsf{A}(t)}$. From our earlier study
kwon_2011 , the covariance matrix of the Gaussian distribution is given by
$\mathsf{A}(t)^{-1}=\mathsf{A}_{\rm
eq}^{-1}-e^{-\mathsf{F}t}\left[\mathsf{A}_{\rm
eq}^{-1}-\mathsf{A}(0)^{-1}\right]e^{-\mathsf{F}^{\rm t}t}$ , where
$\mathsf{A}_{\rm eq}=\beta\left(\begin{array}[]{cc}k&0\\\
0&1/m\end{array}\right)$ and $\mathsf{F}=\left(\begin{array}[]{cc}0&1/m\\\
k&\gamma/m\end{array}\right)$. The case (i) has been extensively studied for
the overdamped dynamics vanzon1 ; vanzon2 ; kwangmoo  and the underdamped
dynamics kwon_jkps2018 ; kwon_3heats . (ii) has recently been studied by our
group and the detailed calculation is to be presented elsewhere youngchae . We
check the inequality in Eq. (12) for the work production $W$ by plotting

$B(t)=\frac{({\rm Var}W)~{}EP}{DW^{2}}$ (23)

where $EP=k_{B}^{-1}\langle\Delta S_{\rm tot}\rangle$ and
$DW=-\gamma\partial_{\gamma}\langle W\rangle$. Figures 1 and 2 shows that all
$B(t)$’s for various values of parameters for $\gamma$ and $\omega$ ($u$) are
above the horizontal line of $B(t)=2$. We confirm that Eq. (16) holds for an
arbitrary initial velocity $\mathbf{v}=v$ in one dimension.

We consider another type of the time-dependent harmonic potential given by a
time-varying stiffness kwon_breathing ; jun1 ; jun2 , given by $V=k(t)x^{2}/2$
with $k(t)=k(1+b\sin\omega t)$. The work production rate is equal to
$\Gamma(x,\omega t)=\partial_{t}V(x,\omega t)=(kb\omega\cos\omega t)x^{2}/2$.
This problem is investigated by using the computer simulation. Figure 3 shows
that $B(t)\geq 2$ for various values of $\gamma$ and $\omega$.

_Summary_. We derive the TUR for the product of the variance and EP for the
time-accumulated production of an arbitrary observable in underdamped dynamics
subject to general time-dependent protocols by exploiting the Cramers-Rao
inequality. We show it to approach to the known TUR in overdamped dynamics for
large viscosity limit. Any scaling function $s(\theta)$ with $s(0)=0$ and non-
zero $s^{\prime}(0)$ for the place of $(1+\theta)^{-1}$ in Eq. (5) goes to the
unique TUR in the limit $\theta\to 0$, since ${\cal I}(\theta)$ and
$(\partial_{\theta}\langle\Phi\rangle_{\theta})^{2}$ will have the same factor
$s^{\prime}(0)^{2}$. Divergence for small times and infinite-peak for
oscillatory protocols present in the figures seem to be characteristic of the
present TUR for short times, which might imply that the inequality bound is
too loose. It is interesting to see whether there is an alternative TUR to
give a tighter bound for small times.

We thank Professor Hyunggyu Park and Dr. Jongmin Park at Korean Institute of
Advanced Studies for stimulating and helpful discussions. This work was
supported by Basic Science Research Program of the National Research
Foundation(NRF) funded by the Ministry of Education with Grant No.
2020R1A2C100976111 (CK) and 2018R1D1A1B07049254 (HKL).

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