# Inferring nonequilibrium thermodynamics from tilted equilibrium
using information-geometric Legendre transform

Naruo Ohga<EMAIL_ADDRESS>Department of Physics, Graduate
School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan  Sosuke Ito  Department of Physics, Graduate School of
Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033,
Japan  JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

###### Abstract

We propose a method to infer nonstationary thermodynamic quantities from
measurements in tilted equilibrium, i.e., the equilibrium distribution with
external fields applied. In nonstationary processes, thermodynamic quantities
are often hard to measure directly. Instead, our method infers nonstationary
quantities from the equilibrium free energy and the expectation values of
observables measured in tilted equilibrium. The method allows us to compute a
lower bound of the nonequilibrium free energy (cumulative entropy production).
We also discuss the inference of the nonequilibrium affinities and a new
constraint on relaxation processes. Our theory is based on a newly developed
Legendre transform over probability distributions and does not rely on large
deviation theory or any other asymptotics. Therefore, it applies to arbitrary
far-from-equilibrium regimes and systems with any small or large number of
degrees of freedom.

Nonstationary stochastic processes [1, 2] are ubiquitous in nonequilibrium
systems, which experience various nonstationary probability distributions
before they reach the stationary equilibrium distribution. Examples are noisy
biochemical signal transduction, in which a transient peak of a concentration
of a chemical substance conveys the information [3, 4, 5, 6], and various
nonmonotonic relaxations [7, 8]. Realizing a nonstationary distribution
requires a fundamental energetic cost, analyzed by the framework of stochastic
thermodynamics [9, 10, 11, 12, 13]. Stochastic thermodynamics associates
thermodynamic quantities with probability distributions rather than individual
states. Therefore, to obtain thermodynamic quantities in experiments and
numerical simulations, we need to measure the details of probability
distributions by accumulating many realizations of the same nonstationary
process, which is too demanding in practice for systems with many states.

To reduce this demand for nonstationary measurements, we require an inference
method of thermodynamic quantities from measurable partial information. Such
methods have been actively studied for nonequilibrium steady states,
collectively termed thermodynamic inference [14]. Previous studies have
utilized, for example, raw time series data [15, 16, 17], the local velocity
[18, 19], partial transitions [20, 21, 22, 23, 24], current observables [25,
26, 27, 28], the Fisher information [29], and the waiting time distributions
of transitions [30, 31, 32, 33] to infer thermodynamic quantities such as the
entropy production rate. However, these studies mostly concern steady states.
Here we consider a new approach to thermodynamic inference specialized to
nonstationary processes: inferring nonstationary quantities from a ‘tilted’
equilibrium, i.e., the equilibrium distribution under an external field that
alters the energies of the states, which has not been utilized in
thermodynamic inference.

To develop a mathematical connection between tilted equilibrium and
nonstationary quantities, we learn from similar tilting techniques in other
fields such as equilibrium statistical mechanics and the large deviation
theory [34]. In these fields, the properties of the tilted states are
translated into those of interest through a Legendre transform. Therefore, it
will be helpful to develop a Legendre transform at the level of probability
distributions without relying on asymptotics. Such ‘microscopic’ Legendre
transform has not appeared in thermodynamics but is found in other fields such
as information geometry [35, 36], which is a geometric theory of probability
distributions developed in statistics [37, 38, 39], and the foundation of
statistical mechanics [40].

In this Letter, we propose a method to infer the nonequilibrium free energy
(cumulative entropy production), which quantifies the thermodynamic cost of
realizing a nonstationary probability distribution. We consider that we can
only keep track of a few observables over nonstationary processes, which is
insufficient to calculate the nonequilibrium free energy. To fill this
shortage, we employ tilted equilibrium measurements, in which we apply several
kinds of external fields, equilibrate the system, and measure the equilibrium
free energy and the expectation values of observables conjugate to the fields.
From the tilted equilibrium measurements, we can compute the minimal
nonequilibrium free energy to realize a given set of expectation values of the
observables in a nonstationary process. Moreover, if we can apply enough kinds
of external fields, we can obtain exact values of the nonequilibrium free
energy and, moreover, additional two pieces of nonstationary information: the
nonequilibrium affinities and a new constraint on relaxation processes. We
demonstrate these results with a one-dimensional Brownian particle. The
mathematical background of these results is a new Legendre transform between
equilibrium and nonequilibrium free energies, which we formulate by modifying
the information-geometric Legendre transform.

#### Nonstationary system and tilted equilibrium

We consider a system in contact with a single heat bath at a constant
temperature $T$ that stochastically transitions around a continuous state
space $\mathcal{X}\subset\mathbb{R}^{N}$ or a discrete state space
$\mathcal{X}\;{\coloneqq}\;\\{1,\dots,N\\}$. Let
$p\;{\equiv}\;\\{p(x)\\}_{x\in\mathcal{X}}$ denote the probability density
function for a continuous case and the probability mass function for a
discrete case, satisfying $\int_{\mathcal{X}}dx\,p(x)\;{=}\;1$. Here and
hereafter, $\int_{\mathcal{X}}dx$ should always be replaced with
$\sum_{x\in\mathcal{X}}$ for a discrete case. The energies of the states are
given by $\\{\epsilon(x)\\}_{x\in\mathcal{X}}$, which determines the
equilibrium distribution
$p^{\mathrm{eq}}\;{\equiv}\;\\{p^{\mathrm{eq}}(x)\\}_{x\in\mathcal{X}}$ with
$p^{\mathrm{eq}}(x)\propto\exp[-\epsilon(x)/k_{\text{b}}T]$, where
$k_{\text{b}}$ is the Boltzmann constant. The system exhibits a nonstationary
relaxation process $p(t)\;{\equiv}\;\\{p(x;t)\\}_{x\in\mathcal{X}}$ toward
$p^{\mathrm{eq}}$, where $t$ is the time variable. We do not assume any
specific time evolution law unless otherwise noted.

We assume that we cannot measure the details of the nonstationary
distributions along the relaxation process, and we can only measure the
expectation values of $K$ observables $B_{1},\dots,B_{K}$ along the
relaxation. The number $K$ is an arbitrary finite number for a continuous
space $\mathcal{X}$ and a number smaller than $N$ for a discrete space
$\mathcal{X}$. An observable
$B_{\alpha}\;{\equiv}\;\\{B_{\alpha}(x)\\}_{x\in\mathcal{X}}$ is defined to
take the value $B_{\alpha}(x)$ when the system is in state $x$, and its
expectation value is $\langle
B_{\alpha}\rangle_{p}\;{\coloneqq}\;\int_{\mathcal{X}}dx\,B_{\alpha}(x)p(x)$
with the vector notation $\langle\bm{B}\rangle_{p}\;{\equiv}\;(\langle
B_{1}\rangle_{p},\cdots,\langle B_{K}\rangle_{p})$. Without loss of
generality, we assume that the $K$ observables are linearly independent and
vanish at the equilibrium $\langle
B_{\alpha}\rangle_{p^{\mathrm{eq}}}\;{=}\;0$. We write the measured
expectation values as
$\bm{\eta}(t)\;{\coloneqq}\;\langle\bm{B}\rangle_{p(t)}$. Measuring a few
observables is far from sufficient to calculate thermodynamic quantities that
are functionals of $p$.

To overcome this insufficiency and infer thermodynamic quantities, we employ
tilted equilibrium under external fields. Let
$v\;{\equiv}\;\\{v(x)\\}_{x\in\mathcal{X}}$ denote an external field modifying
the energy from $\epsilon(x)$ to $\epsilon(x)-v(x)$. We consider the external
fields conjugate to each observable $B_{\alpha}$ with a variable intensity
$\theta_{\alpha}$:

$v(x;\bm{\theta})=\sum_{\alpha=1}^{K}\theta_{\alpha}B_{\alpha}(x),$ (1)

which incurs the tilted equilibrium
$p^{\mathrm{TE}}(\bm{\theta})\;{\equiv}\;\\{p^{\mathrm{TE}}(x;\bm{\theta})\\}_{x\in\mathcal{X}}$
with
$p^{{\mathrm{TE}}}(x;\bm{\theta})\propto\exp\\{-[\epsilon(x)-v(x;\bm{\theta})]/k_{\text{b}}T\\}$,
and define the set of all these tilted distributions
$\mathcal{P}_{{\mathrm{TE}}}\;{\coloneqq}\;{\left\\{p^{{\mathrm{TE}}}(\bm{\theta})\,\middle|\,\bm{\theta}\right\\}}$.
We define the tilted equilibrium free energy
$\mathcal{F}_{\bm{B}}(\bm{\theta})$,

$\mathcal{F}_{\bm{B}}(\bm{\theta})=-k_{\text{b}}T\ln\left[\int_{\mathcal{X}}dx\exp\left(-\frac{\epsilon(x)-v(x;\bm{\theta})}{k_{\text{b}}T}\right)\right],$
(2)

its difference from the equilibrium
$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta})\;{\coloneqq}\;\mathcal{F}_{\bm{B}}(\bm{\theta})-\mathcal{F}_{\bm{B}}(0)$,
and the tilted expectation values
$\bm{\eta}(\bm{\theta})\;{\coloneqq}\;\langle\bm{B}\rangle_{p^{\mathrm{TE}}(\bm{\theta})}$.

For thermodynamic inference, we need to perform the following tilted
equilibrium measurements: realizing the tilted equilibrium distributions for
sufficiently many $\bm{\theta}$’s, measuring
$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta})$ and $\bm{\eta}(\bm{\theta})$, and
interpolating the measured values to obtain the functional dependencies
$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta})$ and $\bm{\eta}(\bm{\theta})$ and the
inverse function $\bm{\theta}(\bm{\eta})$. The inverse function gives the set
of external field intensities $\bm{\theta}$ required to realize a set of
expectation values $\bm{\eta}$. Such a set of intensities $\bm{\theta}$ is
unique as a consequence of our new Legendre duality, as will be discussed
later. The equilibrium free energy $\Delta\mathcal{F}_{\bm{B}}(\bm{\theta})$
is measured as the work of any quasi-static process from $p^{\mathrm{eq}}$ to
$p^{\mathrm{TE}}(\bm{\theta})$ [41, 12].

Figure 1: Schematic of the proposed inference method. (a) A system with the
energy $\epsilon(x)$ and a nonstationary distribution $p(t)$. We can only
measure the expectation values of a few observables
$\bm{\eta}(t)\;{=}\;\langle\bm{B}\rangle_{p(t)}$. (b) To infer thermodynamic
quantities at $p(t)$, we apply an external field with intensities
$\bm{\theta}\;{=}\;\bm{\theta}(\bm{\eta}(t))$ to the system to realize a
tilted equilibrium incurring the same set of expectation values and measure
the tilted equilibrium free energy
$\mathcal{F}{B}(\bm{\theta}(\bm{\eta}(t)))$.

#### Minimum nonequilibrium free energy

The first quantity to be inferred is the nonequilibrium free energy [42, 43,
12]:

$\mathcal{H}[p]\coloneqq k_{\text{b}}T\\!\int_{\mathcal{X}}dx\,p(x)\ln
p(x)+\int_{\mathcal{X}}dx\,\epsilon(x)p(x).$ (3)

It is $(-T)$ times the sum of the system entropy and the bath entropy up to an
additive constant. It quantifies the fundamental thermodynamic cost of being
in a nonstationary distribution $p$. A relaxation process from $p$ to
$p^{\mathrm{eq}}$ dissipates the energy
$\Delta\mathcal{H}[p]\;{\coloneqq}\;\mathcal{H}[p]-\mathcal{H}[p^{\mathrm{eq}}]$.

For a given set of expectation values $\bm{\eta}$, the minimum dissipation
incurred by realizing the expectation values $\bm{\eta}$ in a nonstationary
process is
$\min_{\left.q\;\middle|\,\langle\bm{B}\rangle_{q}=\bm{\eta}\right.}\Delta\mathcal{H}[q]$.
Our main result states that this minimum is evaluated from the tilted
equilibrium measurements:

$\min_{\left.q\;\middle|\,\langle\bm{B}\rangle_{q}=\bm{\eta}\right.}\Delta\mathcal{H}[q]=\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))+\sum_{\alpha=1}^{K}\theta_{\alpha}(\bm{\eta})\,\eta_{\alpha}.$
(4)

The right-hand side is computed from the inverse function
$\bm{\theta}(\bm{\eta})$ and the equilibrium free energy
$\Delta\mathcal{F}(\bm{\theta})$ measured at
$p^{\mathrm{TE}}(\bm{\theta}(\bm{\eta}))$, which is the unique tilted
equilibrium incurring the given expectation values $\bm{\eta}$ (Fig. 1).
Therefore, for a nonstationary process $\\{p(t)\\}$, by observing the
expectation values $\bm{\eta}(t)\;{=}\;\langle\bm{B}\rangle_{p(t)}$, we can
evaluate the following lower bound on the dissipation

$\Delta\mathcal{H}[p(t)]\geq\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}(t)))+\sum_{\alpha=1}^{K}\theta_{\alpha}(\bm{\eta}(t))\,\eta_{\alpha}(t),$
(5)

from the tilted equilibrium measurements.

The main result (4) is derived by combining two distinct results. First, the
minimum in the left-hand side of Eq. (4) is achieved at
$q\;{=}\;p^{{\mathrm{TE}}}(\bm{\theta}(\bm{\eta}))$ (See supplemental material
(SM) [44] for derivation). This is already a nontrivial result: a minimization
problem in nonequilibrium physics is automatically solved by the equilibration
under an external field. Second, the nonequilibrium free energy
$\Delta\mathcal{H}[p^{{\mathrm{TE}}}(\bm{\theta}(\bm{\eta}))]$ is given by the
right-hand side of Eq. (4). This is a consequence of our new Legendre duality,
as will be discussed later.

The inequality (5) is saturated if and only if the distribution satisfies
$p(t)\;{=}\;p^{{\mathrm{TE}}}(\bm{\theta}(\bm{\eta}(t)))$. This condition is
equivalent to $p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$, i.e., that $p(t)$ is
realizable as a tilted equilibrium with some $\bm{\theta}$. Though strong the
condition $p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$ may seem, there are several
sufficient conditions to ensure $p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$ (see
SM [44] for the detailed conditions and their proofs). For example, if the
states are lumped into groups, the transition rates within a group are
sufficiently larger than the rates between different groups, and we can track
any observables associated with the groups, then we can ensure
$p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$. As another example, if the system
admits a symmetry of permuting the states, the initial condition is compatible
with the symmetry, and we can track any observables obeying the symmetry, then
we can ensure $p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$. We will discuss yet
another example of such a condition in the example section.

If we have enough information on the system to ensure that the system meets
one of these sufficient conditions for
$p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$, we can use the right-hand side of
Eq. (5) to obtain the exact nonequilibrium free energy $\mathcal{H}[p(t)]$,
not only the lower bound. In this situation, we can also obtain two additional
pieces of information from tilted equilibrium measurements, as discussed
below. Another possible situation is that we can expect the system to
approximately satisfy one of the sufficient conditions. Then, we can expect
that $p(t)$ is close to $\mathcal{P}_{\mathrm{TE}}$, and therefore, we can use
the right-hand side of Eq. (5) as an approximation of $\mathcal{H}[p(t)]$. We
can also obtain the following additional two pieces of information
approximately.

#### Entropy production rate and affinity

If we can ensure $p(t)\;{\in}\;\mathcal{P}_{{\mathrm{TE}}}$, we can also
compute the entropy production rate and the nonequilibrium affinities
(thermodynamic forces) from the considered measurements, which provides finer
information on the cause of the dissipation. The entropy production rate
$\dot{\sigma}$ is defined by
$\dot{\sigma}(t)\;{\coloneqq}\;-T^{-1}\partial_{t}\mathcal{H}[p(t)]$ and
quantifies the instantaneous dissipation, where
$\partial_{t}\;{\equiv}\;\partial/\partial t$ is the time derivative. The
affinity between states $y$ and $x$ is defined by
$f(x,y)[p]\;{\coloneqq}\;[\epsilon(y)-\epsilon(x)]/T+k_{\text{b}}\ln[p(y)/p(x)]$
[12] and quantifies the dissipation due to the probability flux from $y$ to
$x$. To see this, let $J(x,y;t)$ be the probability flux from state $y$ to $x$
at time $t$, which incurs the time evolution
$\partial_{t}p(x;t)\;{=}\;\int_{\mathcal{X}}dy\,J(x,y;t)$ 111For a continuous
space with a time evolution
$\partial_{t}p(x;t)\;{=}\;-\nabla\cdot\bm{J}(x;t)$, where $\bm{J}(x;t)$ is an
$N$-dimensional probability flux, $J(x,y;t)$ is defined as
$J(x,y;t)\;{=}\;[\bm{J}(x;t)+\bm{J}(y;t)]\cdot\nabla\delta(y-x)$, where
$\nabla\delta$ is the spatial derivative of the $N$-dimensional delta
function.. Then, we can decompose the entropy production rate as
$\dot{\sigma}(t)\;{=}\;\frac{1}{2}\int_{\mathcal{X\times\mathcal{X}}}\\!dxdy\,J(x,y;t)f(x,y)[p(t)]$,
which shows that $f(x,y)$ quantifies the component of entropy production rate
due to the flux $J(x,y)$.

For a nonstationary distribution $p(t)\;{\in}\;\mathcal{P}_{\mathrm{TE}}$ with
the observed expectation values $\bm{\eta}(t)$, we can compute the affinity
$f(x,y)[p(t)]$ from the tilted equilibrium measurements (see SM [44] for
derivation):

$Tf(x,y)[p(t)]=\sum_{\alpha=1}^{K}\theta_{\alpha}(\bm{\eta}(t))\quantity[B_{\alpha}(y)-B_{\alpha}(x)].$
(6)

The right-hand side can be calculated from $\bm{\theta}(\bm{\eta})$ obtained
by tilted-equilibrium measurements, supposing we know the values of
observables $B_{\alpha}(x)$ at each state.

We can further compute the entropy production rate and its decomposition from
the considered measurements. From the Legendre duality discussed below, the
entropy production rate satisfies

$\dot{\sigma}(t)=\sum_{\alpha=1}^{K}\quantity(-\frac{{\theta_{\alpha}(\bm{\eta}(t))}}{T})\,\partial_{t}\eta_{\alpha}.$
(7)

This equation decomposes the entropy production rate into the dissipation due
to the change in the expectation value of each observable. From this equation,
we can regard $(-\theta_{\alpha}/T)$ as the thermodynamic force conjugate to
the probability flux incurring the change in the expectation values
$\partial_{t}\eta_{\alpha}$. On the right-hand side, we can obtain
$\partial_{t}\eta_{\alpha}$ from the nonstationary measurement of the
expectation values and $\bm{\theta}(\bm{\eta}(t))$ from the tilted equilibrium
measurements.

#### Monotonicity of the tilted equilibrium free energy

If we can ensure $\\{p(t)\\}\subset\mathcal{P}_{\mathrm{TE}}$ over a
trajectory, and if the time evolution is Markovian, we obtain yet another
piece of information from the considered measurements: for any such
trajectory, the observed trajectory of the observables $\\{\bm{\eta}(t)\\}$
satisfies (see SM [44] for derivation)

$\partial_{t}\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}(t)))=\partial_{t}\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}(t)))\geq
0,$ (8)

which imposes a new restriction over the possible trajectories on
$\mathcal{P}_{\mathrm{TE}}$. This restriction is complementary and not
equivalent to the second law of thermodynamics [10, 12], i.e., the
monotonicity of the nonequilibrium free energy
$\partial_{t}\mathcal{H}[p(t)]\leq 0$.

#### Example

We demonstrate our results with a one-dimensional Brownian particle in a
harmonic trapping potential plus a ragged potential
$\epsilon(x)\;{=}\;ax^{2}+\epsilon_{r}(x)$, where $a$ is the width of the
harmonic potential and $\epsilon_{r}(x)$ is an arbitrary ragged potential. It
is a model of an optically trapped particle linked to a biomolecule [46, 10],
for which $\epsilon_{r}(x)$ denotes the energy of the molecule pulled to a
displacement $x$. We need not know the potential $\epsilon_{r}(x)$. The time
evolution is governed by the Fokker–Planck equation with a uniform mobility
$\mu$: $\partial_{t}p(x;t)\;{=}\;\mu(\partial/\partial
x)[(\partial\epsilon/\partial x)p(x)+k_{\text{b}}T\partial p(x)/\partial x]$.

We assume that we can keep track of the mean and variance of the nonstationary
distributions. This assumption corresponds to choosing two observables
$B_{1}(x)\;{=}\;x-m_{0}$ and $B_{2}(x)\;{=}\;x^{2}-s_{0}^{2}$, where
$m_{0}\;{\coloneqq}\;\langle x\rangle_{p^{\mathrm{eq}}}$ and
$s_{0}^{2}\;{\coloneqq}\;\langle x^{2}\rangle_{p^{\mathrm{eq}}}$ are
equilibrium expectation values, subtracted to ensure $\langle
B_{\alpha}\rangle_{p^{\mathrm{eq}}}\;{=}\;0$. For the thermodynamic inference,
we need to modify the energy into

$\epsilon(x)-v(x;\bm{\theta})=(a-\theta_{2})x^{2}-\theta_{1}x+\theta_{1}m_{0}+\theta_{2}s_{0}^{2}+\epsilon_{r}(x),$
(9)

which is still a harmonic potential plus the ragged potential. Therefore, we
can realize the tilted equilibrium by modulating the center and the width of
the harmonic trapping potential. We need to measure the tilted equilibrium
free energy from work measurements [10] and the tilted expectation values of
$x$ and $x^{2}$.

We first demonstrate our results for the simple case of
$\epsilon_{r}(x)\;{=}\;0$. In this case, the subspace
$\mathcal{P}_{\mathrm{TE}}$ is the set of the Gaussian distributions, and if
the initial distribution is also a Gaussian distribution, the time evolution
is confined in $\mathcal{P}_{\mathrm{TE}}$. Thus, we can obtain exact
nonequilibrium quantities from Eqs. (5), (6), and (7) and a new constraint in
time evolution from Eq. (8). We demonstrate them in Fig. 2.

For a nonzero $\epsilon_{r}(x)$, the sole solid results are Eqs. (4) and (5),
which give the minimal cost of realizing a given mean and variance.
Nevertheless, if $\epsilon_{r}$ is small, we can expect that time trajectories
are close to $\mathcal{P}_{\mathrm{TE}}$ and that the right-hand side of Eq.
(5) gives a fairly accurate estimation of $\mathcal{H}[p(t)]$. In fact, this
is the case even if the ragged potential is as large as $2k_{\text{b}}T$ (see
Fig. S1 in SM).

Figure 2: A demonstration of our scheme with a one-dimensional Brownian
particle with $\epsilon(x)\;{=}\;ax^{2}$. We scale the energy by
$k_{\text{b}}T$, the length by $\sqrt{k_{\text{b}}T/a}$, and the time by
$1/a\mu$. See SM [44] for parameter values. (a) The trapping potential
$\epsilon(x)$ (gray) and probability distributions over a relaxation process
(blue; vertically shifted), which are both assumed to be unknown. (b) The
tilted free energy $\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))$
obtained from the tilted equilibrium measurements. It is a function of
$\bm{\eta}$, but we plotted them as the functions of mean $\eta_{1}$ and the
standard deviation (SD) $\sqrt{\eta_{2}+s_{0}^{2}-(\eta_{1})^{2}}$ for better
visibility. We also show the trajectory of the observables (blue)
corresponding to panel (a). (c) The actual value of the nonequilibrium free
energy $\mathcal{H}[p(t)]$ and the estimated value
$\Delta\mathcal{F}_{\bm{B}}(\bm{\eta}(t))+\bm{\theta}(\bm{\eta}(t))\cdot\bm{\eta}(t)$
[Eq. (5)]. The values exactly match for a harmonic system. (d) The actual
entropy production rate $T\dot{\sigma}$ and its exact estimation
$-\sum_{\alpha=1}^{2}\theta_{\alpha}(\bm{\eta}(t))\partial_{t}\eta_{\alpha}$,
together with the decomposed values
$-\theta_{\alpha}(\bm{\eta})\partial_{t}\eta_{\alpha}$ [Eq. (7)]. (e) The
tilted equilibrium free energy
$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}(t)))$, which monotonically
increases in time [Eq. (8)].

#### Mathematical background

Our main result (4) derives from a novel Legendre duality over probability
distributions. Let
$\mathcal{H}_{\bm{B}}(\bm{\eta})\;{\coloneqq}\;\mathcal{H}[p^{{\mathrm{TE}}}(\bm{\theta}(\bm{\eta}))]$
be the restriction of $\mathcal{H}[p]$ onto $\mathcal{P}_{\mathrm{TE}}$ as the
function of $\bm{\eta}$. Then, the two functions
$-\mathcal{F}_{\bm{B}}(\bm{\theta})$ and $\mathcal{H}_{\bm{B}}(\bm{\eta})$ are
strictly convex and connected through a Legendre transform (see SM [44] for
derivation):

$\displaystyle\eta_{\alpha}(\bm{\theta})=-\frac{\partial\mathcal{F}_{\bm{B}}}{\partial\theta_{\alpha}},\quad\theta_{\alpha}(\bm{\eta})=\frac{\partial\mathcal{H}_{\bm{B}}}{\partial\eta_{\alpha}},$
(10a)
$\displaystyle\mathcal{H}_{\bm{B}}(\bm{\eta})=\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))+\bm{\theta}(\bm{\eta})\cdot\bm{\eta},$
(10b)

where $\bm{\theta}(\bm{\eta})$ and $\bm{\eta}(\bm{\theta})$ have already been
defined. Mathematically, we can regard the external field intensities
$\bm{\theta}$ and the expectation values $\bm{\eta}$ as two coordinate systems
on $\mathcal{P}_{\mathrm{TE}}$. The distributions
$p\;{\in}\;\mathcal{P}_{\mathrm{TE}}$, the coordinates $\bm{\theta}$, and the
other coordinates $\bm{\eta}$ all admit one-to-one correspondences. Both the
coordinates take zero at the equilibrium
$p^{\mathrm{eq}}\;{\in}\;\mathcal{P}_{\mathrm{TE}}$. This fact and Eq. (10a)
give that the functions $-\mathcal{F}_{\bm{B}}(\bm{\theta})$ and
$\mathcal{H}_{\bm{B}}(\bm{\eta})$ take their minima at the equilibrium
$p^{\mathrm{eq}}$. From Eq. (10b), the values at the equilibrium satisfy
$\mathcal{H}_{\bm{B}}(0)\;{=}\;\mathcal{F}_{\bm{B}}(0)$.

We also discuss a characterization of the external field. Although an external
field $v(\bm{\theta})$ uniquely determines a tilted equilibrium
$p^{\mathrm{TE}}(\bm{\theta})$, the same tilted equilibrium is incurred by any
external fields $v(x)\;{=}\;\epsilon(x)+k_{\text{b}}T\ln
p^{\mathrm{TE}}(x;\bm{\theta})+\mathrm{const.}$ with any $x$-independent
constant. Among them, the external field $v(\bm{\theta})$ introduced in Eq.
(1) is the unique external field satisfying the gauge fixing condition

$\int_{\mathcal{X}}dx\,v(x;\bm{\theta})p^{\mathrm{eq}}(x)=0.$ (11)

This equation is easily checked from the assumption
$\langle\bm{B}\rangle_{p^{\mathrm{eq}}}\;{=}\;0$. Therefore, we can restate
Eq. (10) as the Legendre duality between the nonequilibrium free energy and
the tilted equilibrium free energy with the gauge fixing. We note that a
Legendre transform with a different gauge fixing has appeared in Ref. [40].

For derivations of the Legendre duality and the other results, refer to SM,
which includes Refs. [47, 48, 49, 50, 51, 52]. SM also discusses the
underlying information-geometric perspective, as well as the possible
generalization of the setup to incorporate the internal entropy of the states
[21], exchange of particles with a single particle reservoir, and multiple
baths.

#### Conclusion

In this Letter, we have proposed a method of utilizing tilted equilibrium
measurements to extract the minimal nonequilibrium free energy for a given set
of observables, i.e., the minimal dissipation incurred from the set of
observables in a nonstationary process. This method applies to any stochastic
system with any choice of observables. Furthermore, if we have enough
information to ensure $p\;{\in}\;\mathcal{P}_{\mathrm{TE}}$, we can extract
the exact nonequilibrium free energy, the affinity, a decomposition of the
entropy production rate, and a new constraint on time trajectories from the
considered measurements. Our method significantly reduces the demand for
nonstationary measurements: we need not measure the details of probability
distributions, including their tails, and only need the expectation values of
a few arbitrary observables.

Our results open up a new direction of thermodynamic inference: the inference
for nonstationary processes based on static measurements. Future directions
include utilizing the fluctuations in tilted equilibrium to extract more
information on nonstationary processes and utilizing tilted nonequilibrium
steady states to extend the method to systems coupled to multiple baths. From
the theoretical point of view, the newly introduced Legendre transform is a
new bridge between nonequilibrium thermodynamics and equilibrium statistical
mechanics, just as information geometry has provided new tools in statistical
physics [53, 54, 55, 56, 7, 57, 58, 59, 60, 61] and stochastic thermodynamics
[62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 29, 72, 73, 74, 75, 76, 77]. It would
help find nontrivial connections between equilibrium phenomena, such as phase
transitions, and nonequilibrium properties of a system.

###### Acknowledgements.

We thank Kohei Yoshimura, Shin-ichi Sasa, Zhiyue Lu, Koki Shiraishi, Ken
Hiura, and Ryuna Nagayama for fruitful discussions. We thank Artemy Kolchinsky
for valuable comments on the previous version of the manuscript. S. I. is
supported by JSPS KAKENHI Grants No. 19H05796 and No. 21H01560, JST Presto
Grant No. JPMJPR18M2, and UTEC-UTokyo FSI Research Grant Program.

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Supplemental Material for
Inferring nonequilibrium thermodynamics from tilted equilibrium
using information-geometric Legendre transform

Naruo Ohga and Sosuke Ito

In Sec. A, we provide supplemental discussions on the example. Sec. B is
devoted to the complete derivation of the results. In Sec. C, we discuss
generalizations of the setup. Sec. D discusses two situations where the
trajectories are confined in $\mathcal{P}_{\mathrm{TE}}$. In Sec. E, we
restate our Legendre duality from information-geometric perspectives.

##  Supplemental information on examples

###  Example with a nonzero ragged potential

Figure S1: A demonstration of our scheme with a one-dimensional Brownian
particle with $\epsilon(x)=ax^{2}+b\cos cx$. We scale the energy by
$k_{\text{b}}T$, the length by $\sqrt{k_{\text{b}}T/a}$, and the time by
$1/a\mu$. See Sec. 3 for parameter values. (a) The trapping potential
$\epsilon(x)$ (gray) and probability distributions over a relaxation process
(blue; vertically shifted), which are both assumed to be unknown. (b) The
tilted free energy $\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))$
obtained from the tilted equilibrium measurements. It is a function of
$\bm{\eta}$, but we plotted them as the functions of mean $\eta_{1}$ and the
standard deviation (SD) $\sqrt{\eta_{2}+s_{0}^{2}-(\eta_{1})^{2}}$ for better
visibility. We also show the trajectory of the observables (blue)
corresponding to panel (a). (c) The actual value of the nonequilibrium free
energy $\mathcal{H}[p(t)]$ and its lower estimation
$\Delta\mathcal{F}_{\bm{B}}(\bm{\eta}(t))+\bm{\theta}(\bm{\eta}(t))\cdot\bm{\eta}(t)$
[Eq. (5)]. (d) The actual entropy production rate $T\dot{\sigma}$ and its
estimation
$-\sum_{\alpha=1}^{2}\theta_{\alpha}(\bm{\eta}(t))\partial_{t}\eta_{\alpha}$,
together with the decomposed values
$-\theta_{\alpha}(\bm{\eta})\partial_{t}\eta_{\alpha}$ [Eq. (7)]. (e) The
tilted equilibrium free energy
$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}(t)))$, which should
approximately monotonically increase in time [Eq. (8)].

We show a numerical calculation for a system with energy
$\epsilon(x)=ax^{2}+\epsilon_{r}(x)$ with a nonzero ragged potential
$\epsilon_{r}=b\cos cx$ with $b=k_{\text{b}}T$ and $c=6\sqrt{k_{\text{b}}T/a}$
[Fig. S1]. As in the main text, the time evolution is given by the
Fokker–Planck equation with a uniform mobility $\mu$:
$\partial_{t}p(x;t)=\mu(\partial/\partial x)[(\partial\epsilon/\partial
x)p(x)+k_{\text{b}}T\partial p(x)/\partial x]$.

As discussed in the main text, even if we do not know the details of the
ragged potential, we can expect that a relaxation trajectory of this system
approximately satisfies $p(t)\in\mathcal{P}_{\mathrm{TE}}$, and therefore we
can abuse the inference methods developed for
$p(t)\in\mathcal{P}_{\mathrm{TE}}$ with a good approximation. First, the
right-hand side of Eq. (5)
$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))+\bm{\theta}(\bm{\eta})\cdot\bm{\eta}$
gives a good estimation of the nonequilibrium free energy
$\Delta\mathcal{H}[p]$ [Fig. S1(c)]. Second, the right-hand side of Eq. (7)
$-\bm{\theta}(\bm{\eta})\cdot\partial_{t}\bm{\eta}/T$ gives a good estimation
of the entropy production rate $\dot{\sigma}$ [Fig. S1(d)]. We also obtain a
decomposition of the estimated entropy production rate into the components due
to the changes in $\eta_{1}$ and in $\eta_{2}$. Third, the tilted equilibrium
free energy $\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))$ is expected
to show a monotonic increase approximately, and in fact, it does monotonically
increase in this example [Fig. S1(e)].

This example shows that, even if a system does not exactly satisfy
$p(t)\in\mathcal{P}_{\mathrm{TE}}$, we can apply all our results with a good
approximation. Of course, we can rigorously apply the lower bound of the
nonequilibrium free energy in Eq. (5), which holds for any system.

###  Analytical calculation for a system with a harmonic potential

We explicitly calculate the relevant quantities in the absence of the ragged
potential $\epsilon(x)=ax^{2}$ as a demonstration. As in the main text, the
probability distribution obeys the Fokker–Planck equation

$\frac{\partial}{\partial t}p(x;t)=2a\mu\frac{\partial}{\partial
x}[xp(x;t)]+\mu k_{\text{b}}T\frac{\partial^{2}}{\partial^{2}x}p(x;t).$ (S1)

The solution of this Fokker–Planck equation is known to be asymptotically
Gaussian. Indeed, the $n$th cumulant ($n=1,2,\dots$) decays exponentially in
time as $\exp(-2n\mu at)$ to the equilibrium value [2], and the equilibrium
values are zero except for $n=2$. Thus, only the first two cumulants remain
significant after the initial fast transition, which means the distribution is
close to Gaussian.

As in the main text, we assume that we can measure the two observables
$B_{1}(x)=x$ and $B_{2}(x)=x^{2}-k_{\text{b}}T/2a$, where we have used that
the variance is $s_{0}^{2}=k_{\text{b}}T/2a$. Therefore, the modified state
energy due to the external fields is

$\displaystyle\epsilon(x)-v(x;\bm{\theta})=ax^{2}-\theta_{1}x-\theta_{2}x^{2}+\theta_{2}\frac{k_{\text{b}}T}{2a}$
$\displaystyle=\quantity(a-\theta_{2})\quantity(x-\frac{\theta_{1}}{2\quantity(a-\theta_{2})})^{2}-\frac{(\theta_{1})^{2}}{4\quantity(a-\theta_{2})}+\theta_{2}\frac{k_{\text{b}}T}{2a}.$
(S2)

The modified state energy is harmonic, and thus the tilted equilibrium is a
Gaussian distribution with mean $\theta_{1}/2(a-\theta_{2})$ and variance
$k_{\text{b}}T/2(a-\theta_{2})$.

The subspace $\mathcal{P}_{\mathrm{TE}}$ corresponding to the potential (S2)
is the set of all the one-dimensional Gaussian distributions. By changing
$\theta_{1}$ and $\theta_{2}$, we can make a harmonic potential with any
center position and width, and therefore we can reproduce Gaussian
distribution with any mean and variance. Since any solution of Eq. (S1) is
asymptotically Gaussian, the time evolution is asymptotically confined in
$\mathcal{P}_{\mathrm{TE}}$ except for the initial fast transition.

In the tilted equilibrium measurements, we measure the tilted equilibrium free
energy and the expectation values of the observables. The tilted equilibrium
free energy is

$\Delta\mathcal{F}_{\bm{B}}(\bm{\theta})=\frac{1}{2}k_{\text{b}}T\ln\quantity(\frac{a-\theta_{2}}{a})-\frac{(\theta_{1})^{2}}{4\quantity(a-\theta_{2})}+\theta_{2}\frac{k_{\text{b}}T}{2a}.$
(S3)

The first term is from the Gaussian integral of Eq. (S2) and only depends on
the width of the potential. The expectation values at the tilted equilibrium
are

$\eta_{1}(\bm{\theta})=\frac{\theta_{1}}{2(a-\theta_{2})},\,\,\eta_{2}(\bm{\theta})=\frac{k_{\text{b}}T}{2(a-\theta_{2})}+\frac{(\theta_{1})^{2}}{4(a-\theta_{2})^{2}}-\frac{k_{\text{b}}T}{2a}.$
(S4)

These relations are inverted as

$\theta_{1}(\bm{\eta})=\frac{k_{\text{b}}T\eta_{1}}{\eta_{2}-(\eta_{1})^{2}+s_{0}^{2}},\quad\theta_{2}(\bm{\eta})=a-\frac{as_{0}^{2}}{[\eta_{2}-(\eta_{1})^{2}+s_{0}^{2}]},$
(S5)

where the term $\eta_{2}-(\eta_{1})^{2}+s_{0}^{2}$ is the variance of the
tilted equilibrium.

From these pieces of information, we can obtain the following nonequilibrium
properties. First, the minimal nonequilibrium free energy with a given
$\bm{\eta}$ [Eq. (4)], which is equal to the nonequilibrium free energy on
$\mathcal{P}_{\mathrm{TE}}$, is computed from the Legendre duality as

$\displaystyle\Delta\mathcal{F}_{\bm{B}}(\bm{\theta}(\bm{\eta}))+\theta_{1}(\bm{\eta})\eta_{1}+\theta_{2}(\bm{\eta})\eta_{2}$
$\displaystyle\quad=-\frac{1}{2}k_{\text{b}}T\ln\quantity(\frac{\eta_{2}-(\eta_{1})^{2}+s_{0}^{2}}{s_{0}^{2}})+a\eta_{2}.$
(S6)

One can directly check that Eq. (S6) equals the nonequilibrium free energy
$\Delta\mathcal{H}$ under $\epsilon(x)=ax^{2}$ of the Gaussian distribution
with $\langle x\rangle=\eta_{1}$ and $\langle
x^{2}\rangle=\eta_{2}+s_{0}^{2}$.

Second, we obtain a decomposition of the entropy production rate [Eq. (7)]. As
seen from Eq. (S5), the sign of $(-\theta_{1}/T)$ is opposite to the sign of
$\eta_{1}$. Therefore, changing $\eta_{1}$ toward zero always causes a
positive entropy production rate. The sign of $(-\theta_{2}/T)$ is determined
by comparing the equilibrium variance $s_{0}^{2}$ and the nonequilibrium
variance $\eta_{2}-(\eta_{1})^{2}+s_{0}^{2}$. When the nonequilibrium variance
is larger than the equilibrium one, $(-\theta_{2}/T)$ is negative. Therefore,
a decrease in $\langle x^{2}\rangle$ causes a positive entropy production
rate. When the nonequilibrium variance is smaller, $(-\theta_{2}/T)$ is
positive. Therefore, an increase in $\langle x^{2}\rangle$ causes a positive
entropy production rate.

Third, the tilted equilibrium free energy $\Delta\mathcal{F}_{\bm{B}}$ should
increase in time [Eq. (8)]. When the mean is far from zero, the tilted
equilibrium free energy is forced to have a low value due to the term
$(\theta_{1})^{2}/[4(a-\theta_{2})]$ in Eq. (S3), and therefore
$\Delta\mathcal{F}_{\bm{B}}$ takes a low value. As the distribution gets
closer to the equilibrium, $\Delta\mathcal{F}_{\bm{B}}$ gradually increases to
the equilibrium value.

###  Details of the numerical calculations

In the numerical calculations in Figs. 2 and S1, we have numerically solved
the Fokker–Planck equation,

$\frac{\partial}{\partial t}p(x;t)=\mu\frac{\partial}{\partial
x}\quantity[\frac{\partial\epsilon}{\partial x}p(x;t)]+\mu
k_{\text{b}}T\frac{\partial^{2}}{\partial^{2}x}p(x;t),$ (S7)

by discretizing the space and solving the resulting discrete-space continuous-
time Markov jump system. We have also calculated the nonequilibrium and
equilibrium free energies by replacing the integrals with the sums over the
discretized space. We have confirmed that the results do not depend on the
space mesh size and that the results agree with the analytical calculation
when $\epsilon_{r}=0$.

We scale the energy by $k_{\text{b}}T$, the length by
$\sqrt{k_{\text{b}}T/a}$, and the time by $1/a\mu$. We choose the initial
distribution to be the Gaussian distribution with mean
$3\sqrt{k_{\text{b}}T/a}$ and variance $0.15k_{\text{b}}T/a$ for both Figs. 2
and S1. In the absence of the ragged potential (Fig. 2), no further free
parameters remain. For the case of $\epsilon_{r}(x)=b\cos cx$ (Fig. S1), the
remaining free parameters are $b/k_{\text{b}}T$ and $c\sqrt{a/k_{\text{b}}T}$,
both determining the shape of the ragged potential. We choose
$b/k_{\text{b}}T=1$ and $c\sqrt{a/k_{\text{b}}T}=6$.

##  Derivation of the results

###  Notation

We introduce the Kullback–Leibler (KL) divergence [47] between arbitrary two
probability distributions $\\{p(x)\\}$ and $\\{p^{\prime}(x)\\}$

$D_{\text{kl}}[p\|p^{\prime}]\coloneqq\int_{\mathcal{X}}dx\,p(x)\ln\frac{p(x)}{p^{\prime}(x)},$
(S8)

which plays a key role in the derivation. We introduce the tilted equilibrium
free energy as a functional of an external field $v$:

$\mathcal{F}[v]\coloneqq-
k_{\text{b}}T\ln\quantity[\int_{\mathcal{X}}dx\exp\quantity(-\frac{\epsilon(x)-v(x)}{k_{\text{b}}T})].$
(S9)

The function $\mathcal{F}_{\bm{B}}$ introduced in the main text is given by
$\mathcal{F}_{\bm{B}}(\bm{\theta})=\mathcal{F}[v(\bm{\theta})]$. We also
introduce the value of the equilibrium free energy at $p^{\mathrm{eq}}$ by
$F^{\mathrm{eq}}\coloneqq\mathcal{F}[0]$, where $0$ denotes the external field
$v(x)=0$ for all $x$.

###  Basics of the nonequilibrium free energy

We briefly summarize the properties of the nonequilibrium free energy
$\mathcal{H}[p]$ [43, 42, 12]. First, the equilibrium value
$\mathcal{H}[p^{\mathrm{eq}}]$ equals the equilibrium free energy
$F^{\mathrm{eq}}$:

$\displaystyle\mathcal{H}[p^{\mathrm{eq}}]$
$\displaystyle=k_{\text{b}}T\int_{\mathcal{X}}dx\,p^{\mathrm{eq}}(x)\ln
p^{\mathrm{eq}}(x)+\int_{\mathcal{X}}dx\,\epsilon(x)p^{\mathrm{eq}}(x)$
$\displaystyle=-\int_{\mathcal{X}}dx\,\quantity[\epsilon(x)-F^{\mathrm{eq}}]p^{\mathrm{eq}}(x)+\int_{\mathcal{X}}dx\,\epsilon(x)p^{\mathrm{eq}}(x)$
$\displaystyle=F^{\mathrm{eq}},$ (S10)

where we have used
$p^{\mathrm{eq}}(x)=\exp\\{-[\epsilon(x)-F^{\mathrm{eq}}]/k_{\text{b}}T\\}$.
Next, we rewrite the KL divergence as

$\displaystyle
k_{\text{b}}TD_{\text{kl}}[p\|p^{\mathrm{eq}}]=k_{\text{b}}T\int_{\mathcal{X}}dx\,p(x)\ln\frac{p(x)}{p^{\mathrm{eq}}(x)}$
$\displaystyle\quad=k_{\text{b}}T\int_{\mathcal{X}}dx\,p(x)\ln
p(x)+\int_{\mathcal{X}}dx\,p(x)\quantity[\epsilon(x)-F^{\mathrm{eq}}]$
$\displaystyle\quad=\mathcal{H}[p]-F^{\mathrm{eq}}.$ (S11)

Therefore, we have an expression of the nonequilibrium free energy

$\mathcal{H}[p]=F^{\mathrm{eq}}+k_{\text{b}}T\,D_{\text{kl}}[p\|p^{\mathrm{eq}}],$
(S12)

which is a classical result [43]. Since KL divergence is non-negative and zero
if and only if the two arguments coincide, $\mathcal{H}[p]$ takes its unique
minimum at the equilibrium $p=p^{\mathrm{eq}}$.

###  Equilibrium free energy and the KL divergence

We start with rewriting the KL divergence with the tilted equilibrium free
energy. For two arbitrary external fields $v,v^{\prime}$ satisfying the gauge
fixing condition (11) and the corresponding tilted equilibrium distributions
$p,p^{\prime}$, we obtain

$\displaystyle
k_{\text{b}}TD_{\text{kl}}[p^{\prime}\|p]=k_{\text{b}}T\int_{\mathcal{X}}dx\,p^{\prime}(x)\ln\frac{p^{\prime}(x)}{p(x)}$
$\displaystyle=-\int_{\mathcal{X}}dx\,\quantity{\epsilon(x)-v^{\prime}(x)-\mathcal{F}[v^{\prime}]-\epsilon(x)+v(x)+\mathcal{F}[v]}p^{\prime}(x)$
$\displaystyle=\mathcal{F}[v^{\prime}]-\mathcal{F}[v]+\int_{\mathcal{X}}dx\,\quantity[{v^{\prime}}(x)-v(x)]p^{\prime}(x).$
(S13)

This expression of KL divergence is inspired by information geometry but is
novel in thermodynamics. It is crucial for the derivation below. Substituting
$v^{\prime}=0$ and $p^{\prime}=p^{\mathrm{eq}}$ and using the gauge fixing
condition (11) lead to

$\mathcal{F}[v]=F^{\mathrm{eq}}-k_{\text{b}}T\,D_{\text{kl}}[p^{\mathrm{eq}}\|p].$
(S14)

This expression of $\mathcal{F}[v]$ ensures that $\mathcal{F}[v]$ takes its
unique maximum at the equilibrium $v=0$. Note that a similar expression of the
KL divergence $D_{\text{kl}}[p^{\mathrm{eq}}\|p]$ is discussed in Ref. [48].

###  Legendre duality

We prove the Legendre duality between $\mathcal{H}_{\bm{B}}(\bm{\eta})$ and
$\mathcal{F}_{\bm{B}}(\bm{\theta})$ (Eqs. (10) in the main text). Let
$\bm{\theta}$ be an arbitrary set of external field intensities,
$v=v(\bm{\theta})$ be the corresponding external field,
$p=p^{\mathrm{TE}}(\bm{\theta})$ be the corresponding tilted equilibrium, and
$\bm{\eta}=\langle\bm{B}\rangle_{p}$ be the expectation values.

First, we prove the transform from $\mathcal{F}_{\bm{B}}(\bm{\theta})$ to
$\mathcal{H}_{\bm{B}}(\bm{\eta})$. The derivative relation
$\partial\mathcal{F}/\partial\theta_{\alpha}=-\eta_{\alpha}$ is proven as

$\displaystyle\frac{\partial\mathcal{F}_{\bm{B}}}{\partial\theta_{\alpha}}$
$\displaystyle=-k_{\text{b}}T\frac{\partial}{\partial\theta_{\alpha}}\ln\quantity[{\int_{\mathcal{X}}dx}\exp\quantity(-\frac{\epsilon(x)-\sum_{\alpha}\theta_{\alpha}B_{\alpha}(x)}{k_{\mathrm{B}}T})]$
$\displaystyle=-\frac{\int_{\mathcal{X}}dx\,B_{\alpha}(x)\exp\quantity(-\frac{\epsilon(x)-\sum_{\alpha}\theta_{\alpha}B_{\alpha}(x)}{k_{\mathrm{B}}T})}{{\int_{\mathcal{X}}dx}\exp\quantity(-\frac{\epsilon(x)-\sum_{\alpha}\theta_{\alpha}B_{\alpha}(x)}{k_{\mathrm{B}}T})}$
$\displaystyle=-\int_{\mathcal{X}}dx\,B_{\alpha}(x)p(x)=-\eta_{\alpha}.$ (S15)

This relation states that the derivative of the equilibrium free energy by an
external field yields the corresponding expectation value, which is a
classical result in statistical mechanics (e.g., Ref. [41]). The relation
between the two functions
$\mathcal{H}_{\bm{B}}(\bm{\eta})=\mathcal{F}_{\bm{B}}(\bm{\theta})+\sum_{\alpha}\theta_{\alpha}\eta_{\alpha}$
is proven as

$\displaystyle\mathcal{H}[p]$
$\displaystyle=F^{\mathrm{eq}}+k_{\text{b}}TD_{\text{kl}}[p\|p^{\mathrm{eq}}]$
$\displaystyle=\mathcal{F}[0]+\mathcal{F}[v]-\mathcal{F}[0]+\int_{\mathcal{X}}dx\,\quantity[v(x)-0]p(x)$
$\displaystyle=\mathcal{F}[v]+\sum_{\alpha}\int_{\mathcal{X}}dx\,\theta_{\alpha}B_{\alpha}(x)p(x)$
$\displaystyle=\mathcal{F}[v]+\sum_{\alpha}\theta_{\alpha}\eta_{\alpha},$
(S16)

where we have used Eqs, (S12), and (S13), as well as
$\mathcal{H}_{\bm{B}}(\bm{\eta})=\mathcal{H}[p]$ and
$\mathcal{F}_{\bm{B}}(\bm{\theta})=\mathcal{F}[v]$ for the considered
$\bm{\theta}$, $v$, $p$ and $\bm{\eta}$.

Next, we prove the inverse transform from $\mathcal{H}_{\bm{B}}(\bm{\eta})$ to
$\mathcal{F}_{\bm{B}}(\bm{\theta})$. To this end, it suffices to show that
$\mathcal{F}_{\bm{B}}(\bm{\theta})$ is a strictly concave function. Then, the
standard argument of Legendre duality for strictly convex or strictly concave
functions (e.g., Ref. [49]) leads to the inverse transform
$\partial\mathcal{H}/\partial\eta_{\alpha}=\theta_{\alpha}$ and
$\mathcal{F}_{\bm{B}}(\bm{\theta})=\mathcal{H}_{\bm{B}}(\bm{\eta})-\sum_{\alpha}\theta_{\alpha}\eta_{\alpha}$.
The standard argument also ensures that $\mathcal{H}_{\bm{B}}$ is strictly
convex. To prove the strict concavity of $\mathcal{F}_{\bm{B}}$, we fix an
arbitrary point $\bm{\theta}^{\prime}$ and let $v^{\prime}$, $p^{\prime}$, and
$\eta^{\prime}_{\alpha}$ be the corresponding external field, distribution,
and the expectation values. The tangent plane of
$\mathcal{F}_{\bm{B}}(\bm{\theta})$ at $\bm{\theta}^{\prime}$ is given by

$\mathcal{F}_{\bm{B}}(\bm{\theta}^{\prime})+\sum_{\alpha}(-\eta^{\prime}_{\alpha})(\theta_{\alpha}-\theta^{\prime}_{\alpha})$
(S17)

as a function of $\bm{\theta}$, where we have used Eq. (S15). This tangent
plane always lies above $\mathcal{F}_{\bm{B}}(\bm{\theta})$ because the
following difference is nonnegative:

$\displaystyle\quantity[\mathcal{F}_{\bm{B}}(\bm{\theta}^{\prime})+\sum_{\alpha}(-\eta^{\prime}_{\alpha})(\theta_{\alpha}-\theta^{\prime}_{\alpha})]-\mathcal{F}_{\bm{B}}(\bm{\theta})$
$\displaystyle\quad=\mathcal{F}[v^{\prime}]-\mathcal{F}[v]-\sum_{\alpha}\int_{\mathcal{X}}dx\,(\theta_{\alpha}-\theta^{\prime}_{\alpha})B_{\alpha}(x)p^{\prime}(x)$
$\displaystyle\quad=\mathcal{F}[v^{\prime}]-\mathcal{F}[v]+\int_{\mathcal{X}}dx\,\quantity[{v^{\prime}}(x)-v(x)]p^{\prime}(x)$
$\displaystyle\quad=k_{\text{b}}TD_{\text{kl}}[p^{\prime}\|p]\geq 0,$ (S18)

where we have used Eq. (S13). The equality holds if and only if
$p^{\prime}=p$, i.e., $\bm{\theta}^{\prime}=\bm{\theta}$. Thus, the function
$\mathcal{F}_{\bm{B}}(\bm{\theta})$ is strictly concave.

The strict concavity of $\mathcal{F}_{\bm{B}}$ also shows that the
correspondence between $\bm{\eta}$ and $\bm{\theta}$ in Eq. (10a) is one-to-
one. This fact ensures the existence of the inverse function
$\bm{\theta}(\bm{\eta})$ used in the main text.

###  Lower bound of the nonequilibrium free energy

The main result [Eq. (4)] is the combination of the Legendre duality and that
the minimization is achieved at the tilted equilibrium
$p^{\mathrm{TE}}(\bm{\theta}(\bm{\eta}))$. Here we prove the latter result.
Let $\tilde{p}$ be any distribution, which may be outside of
$\mathcal{P}_{\mathrm{TE}}$. Let
${\tilde{\bm{\eta}}}=\langle\bm{B}\rangle_{\tilde{p}}$ be the expectation
values, and $\tilde{v}$ be the unique external field that incurs $\tilde{p}$
as the tilted equilibrium and satisfies the gauge fixing condition. Although
$\tilde{p}$ may not be in $\mathcal{P}_{\mathrm{TE}}$, we can always find
$\tilde{v}$ by
$\tilde{v}(x)=\epsilon(x)+k_{\text{b}}T\ln\tilde{p}(x)+\mathrm{const.}$, where
the constant is determined by the gauge fixing condition. Next, we introduce
$p=p^{\mathrm{TE}}(\bm{\theta}({\tilde{\bm{\eta}}}))$ as the unique
distribution on $\mathcal{P}_{\mathrm{TE}}$ that incurs the same expectation
values ${\tilde{\bm{\eta}}}$. We also define
$\bm{\theta}=\bm{\theta}({\tilde{\bm{\eta}}})$ and
$v=v(\bm{\theta}({\tilde{\bm{\eta}}}))$ as the external field intensities and
the external field needed to realize $p$. Then, the three distributions
$\tilde{p},p,p^{\mathrm{eq}}$ satisfy the generalized Pythagorean theorem
[36]:

$D_{\text{kl}}[\tilde{p}\|p^{\mathrm{eq}}]=D_{\text{kl}}[\tilde{p}\|p]+D_{\text{kl}}[p\|p^{\mathrm{eq}}].$
(S19)

To prove this relation, we calculate the difference using Eq. (S13):

$\displaystyle
k_{\text{b}}T\quantity{D_{\text{kl}}[\tilde{p}\|p]+D_{\text{kl}}[p\|p^{\mathrm{eq}}]-D_{\text{kl}}[\tilde{p}\|p^{\mathrm{eq}}]}$
$\displaystyle\quad=\int_{\mathcal{X}}dx\,[\tilde{v}(x)-v(x)]\tilde{p}(x)+\int_{\mathcal{X}}dx\,[v(x)-0]p(x)$
$\displaystyle\quad\qquad-\int_{\mathcal{X}}dx\,[\tilde{v}(x)-0]\tilde{p}(x)$
$\displaystyle\quad=\sum_{\alpha}\int_{\mathcal{X}}dx\,\theta_{\alpha}B_{\alpha}(x)\quantity[p(x)-\tilde{p}(x)]=0$
(S20)

The last equality is due to the relation of expectation values
$\int_{\mathcal{X}}dx\,B_{\alpha}(x)p(x)=\int_{\mathcal{X}}dx\,B_{\alpha}(x)\tilde{p}(x)=\tilde{\eta}_{\alpha}$.
Using Eq. (S12), we can rewrite Eq. (S19) as

$\mathcal{H}[\tilde{p}]=D_{\text{kl}}[\tilde{p}\|p]+\mathcal{H}[p]\geq\mathcal{H}[p],$
(S21)

where we have used the nonnegativity of the KL divergence. Therefore, the
nonequilibrium free energy $\mathcal{H}[\tilde{p}]$ takes its minimum at
$\tilde{p}=p\in\mathcal{P}_{\mathrm{TE}}$ among the distributions incurring
the same set of expectation values. We remark that the inequality (S21)
combined with the Legendre transform gives an expression similar to the
Donsker–Varadahn variational representation [50], but the use of coordinates
$\bm{\theta}$ is specific to information geometry.

###  Affinity and entropy production rate

We derive the relation between affinity and the external field in Eq. (6). The
derivation is a direct calculation. For any $\bm{\theta}$, the corresponding
external field $v$, and the corresponding distribution $p$, we obtain

$\displaystyle
Tf(x,y)[p]={\epsilon(y)-\epsilon(x)}+k_{\text{b}}T\ln\frac{p(y)}{p(x)}$
$\displaystyle\quad={\epsilon(y)-\epsilon(x)}+k_{\text{b}}T\quantity[-\frac{\epsilon(y)-v(y)}{k_{\text{b}}T}+\frac{\epsilon(x)-v(x)}{k_{\text{b}}T}]$
$\displaystyle\quad=v(y)-v(x)$
$\displaystyle\quad=\sum_{\alpha}\theta_{\alpha}\quantity[B_{\alpha}(y)-B_{\alpha}(x)],$
(S22)

which proves Eq. (6).

###  Monotonicity

Finally, we prove the monotonicity of the tilted equilibrium free energy (8)
under Markovian time evolutions. For Markovian time evolutions, the KL
divergence has the contraction property [1, 11],

$\partial_{t}D_{\text{kl}}[p(t)\|p^{\prime}(t)]\leq 0,$ (S23)

for two arbitrary trajectories $\\{p(t)\\}$ and $\\{p^{\prime}(t)\\}$ obeying
the same Markovian time equation. By substituting $p^{\mathrm{eq}}$ for $p(t)$
and noting that $p^{\mathrm{eq}}$ is the fixed point of the time evolution, we
obtain $\partial_{t}D_{\text{kl}}[p^{\mathrm{eq}}\|p(t)]\leq 0$. Combining
this equation with the expression (S14) proves the desired monotonicity (8).

We note that substituting $p^{\mathrm{eq}}$ for $p^{\prime}(t)$ and using the
expression (S12) gives the monotonicity $\partial_{t}\mathcal{H}[p(t)]\leq 0$,
which is the second law of thermodynamics [12]. Therefore, the monotonicity of
$\mathcal{F}$ and of $\mathcal{H}$ have similar mathematical origins.

##  Generalizations of the setup

We discuss two generalizations of the setup: the inclusion of internal entropy
and a particle reservoir and a formal generalization for systems with multiple
baths.

###  Systems with internal entropy and a particle reservoir

We can generalize all our results for systems with internal entropy and in
contact with a single particle reservoir. Internal entropy
$s_{\mathrm{int}}(x)$ is a contribution to the system entropy independent of
the probability distribution. It enters the system entropy as
$\int_{\mathcal{X}}dx\,p(x)\quantity[s_{\mathrm{int}}(x)-k_{\text{b}}\ln
p(x)]$. Such contribution arises, for example, when the states are already
coarse-grained [21]. Moreover, we allow the system to exchange particles with
a single particle reservoir of chemical potential $\mu$ upon transitions. The
particle number of the system when in the state $x$ is $n(x)$.

These generalizations modify the definitions of the nonequilibrium and
equilibrium free energies as follows [21]:

$\displaystyle\mathcal{H}[p]$
$\displaystyle=k_{\text{b}}T\\!\int_{\mathcal{X}}dx\,p(x)\ln
p(x)+\int_{\mathcal{X}}dx\,\omega(x)p(x),$ (S24a)
$\displaystyle\mathcal{F}[v]$
$\displaystyle=-k_{\text{b}}T\ln\quantity[\int_{\mathcal{X}}dx\exp\quantity(-\frac{\omega(x)-v(x)}{k_{\text{b}}T})],$
(S24b)

where $\omega(x)\coloneqq\epsilon(x)-\mu n(x)-Ts_{\mathrm{int}}(x)$. This
definition of $\mathcal{H}[p]$ is justified because it equals $(-T)$ times the
system entropy
$\int_{\mathcal{X}}dx\,p(x)[-Ts_{\mathrm{int}}(x)+k_{\text{b}}T\ln p(x)]$ plus
$(-T)$ times the bath entropy
$\int_{\mathcal{X}}dx\,p(x)\quantity[\epsilon(x)-\mu n(x)]$ up to an additive
constant. The minimum of $\mathcal{H}[p]$ is achieved at the equilibrium
distribution of this system
$p^{\mathrm{eq}}(x)\propto\exp\quantity[-\omega(x)/k_{\text{b}}T]$. The
definition of $\mathcal{F}[v]$ is justified because it equals the minimum of
nonequilibrium free energy with energy $\epsilon(x)-v(x)$. The change in
$\mathcal{F}[v]$ upon changing $v$ is given by the quasi-static work, just as
the standard equilibrium free energy [21].

Since the modified definitions (S24) are formally obtained by replacing
$\epsilon(x)$ with $\omega(x)$, it is easy to check that all of our results
can be generalized with this replacement. The external field $v$, defined as
modifying the energy from $\epsilon(x)$ to $\epsilon(x)-v(x)$, can be equally
interpreted as modifying $\omega(x)$ to $\omega(x)-v(x)$.

###  Systems coupled with multiple baths

We can also formally generalize our results for systems in contact with
multiple heat baths or multiple particle reservoirs [12]. Consider a system
with a nonequilibrium steady distribution
$p^{\text{st}}\equiv\\{p(x)^{\text{st}}\\}_{x\in\mathcal{X}}$ with a nonzero
entropy production rate. The cumulative entropy production of this system is
no longer determined solely by the instantaneous probability distribution.

To generalize our results for such systems, we define $\phi(x)$ by
$p^{\text{st}}(x)=\exp[-\phi(x)]$ and replace $\epsilon(x)/k_{\text{b}}T$ with
$\phi(x)$ and $p^{\mathrm{eq}}(x)$ with $p^{\text{st}}(x)$. We introduce
${\tilde{\mathcal{H}}}$, ${\tilde{\mathcal{F}}}$, and $\tilde{v}$ that replace
$\mathcal{H}/k_{\text{b}}T$, $\mathcal{F}/k_{\text{b}}T$, and
$v/k_{\text{b}}T$, respectively, to eliminate $T$ from the theory because the
temperature $T$ is not defined if the system is in contact with multiple heat
baths. With these replacements, we can formally redefine $T^{-1}$ times the
free energies as

$\displaystyle{\tilde{\mathcal{H}}}[p]$
$\displaystyle=\int_{\mathcal{X}}dx\,p(x)\ln
p(x)+\int_{\mathcal{X}}dx\,\phi(x)p(x),$ (S25a)
$\displaystyle{\tilde{\mathcal{F}}}[\tilde{v}]$
$\displaystyle=-\ln\quantity[\int_{\mathcal{X}}dx\exp\quantity{-[\phi(x)-\tilde{v}(x)]}].$
(S25b)

The functional $-{\tilde{\mathcal{H}}}[p]$ is known as the Hatano–Sasa excess
entropy production [51] or the nonadiabatic entropy production [52]. We can
formally reproduce all of our results with these replacements. For example,
these functions satisfy the monotonicity
$\partial_{t}{\tilde{\mathcal{H}}}[p(t)]\leq 0$ and
$\partial_{t}{\tilde{\mathcal{F}}}[v(t)]\geq 0$ for a Markovian time
evolution. The former is the nonnegativity of the Hatano–Sasa excess entropy
production rate, and the latter gives a new constraint on time evolution.
However, this generalization remains formal and does not directly lead to a
thermodynamic inference because we cannot physically implement the
modification from $\phi(x)$ to $\phi(x)-\tilde{v}(x)$ for a designated
$\tilde{v}(x)$ generically.

##  Sufficient conditions for $p(t)\in\mathcal{P}_{\mathrm{TE}}$

Here we discuss two situations where we can ensure
$p(t)\in\mathcal{P}_{\mathrm{TE}}$. Similar discussions relating Markovian
time evolutions with information-geometric subspaces are found in Ref. [76].
As discussed in the main text, if one of these situations arises exactly, we
can ensure $p(t)\in\mathcal{P}_{\mathrm{TE}}$, and thus we can use our results
to obtain exact values of the nonequilibrium free energy, the entropy
production rate, and the affinity, as well as the monotonicity of
$\mathcal{F}$. If one of these situations arises approximately, then we can
expect that $p(t)$ is close to $\mathcal{P}_{\mathrm{TE}}$, and we can obtain
a good estimation of these quantities.

###  Time-scale separation

The first situation is the existence of a time-scale separation. Consider that
the states are grouped (coarse-grained) into ${K+1}$ disjoint sets of states
$G_{0},G_{1},\dots,G_{K}$. We assume the time-scale separation, i.e., that the
transitions between two states in a group are much faster than those between
two states in different groups [20, 21]. We further assume that we can keep
track of the expectation values of the observables
$\chi_{0},\chi_{1},\dots,\chi_{K}$, where $\chi_{\alpha}(x)$ is defined as
$\chi_{\alpha}(x)=1$ for $x\in G_{\alpha}$ and $\chi_{\alpha}(x)=0$ for
$x\notin G_{\alpha}$. The expectation value $\langle\chi_{\alpha}\rangle_{p}$
gives the total probability of the $\alpha$th group. To apply our formulation,
we choose
$B_{\alpha}(x)=\chi_{\alpha}(x)-\langle\chi_{\alpha}\rangle_{p^{\mathrm{eq}}}$
for $\alpha=1,\dots,K$ as the observables. Here, we have subtracted
$\langle\chi_{\alpha}\rangle_{p^{\mathrm{eq}}}$ to ensure $\langle
B_{\alpha}\rangle_{p^{\mathrm{eq}}}=0$. We have excluded $\alpha=0$ to make
the observables linearly independent because we have the linear dependence
$\sum_{\alpha=0}^{K}\quantity[\chi_{\alpha}(x)-\langle\chi_{\alpha}\rangle_{p^{\mathrm{eq}}}]=1-1=0$.

Under these assumptions, we show that the time trajectory is confined in
$\mathcal{P}_{\mathrm{TE}}$ except for the initial fast relaxation. To this
end, we introduce the coarse-grained probability
$Q_{\alpha}(t)\coloneqq\langle\chi_{\alpha}\rangle_{p(t)}$ and the conditional
probability $q_{\alpha}(x;t)\coloneqq p(x;t)/Q_{\alpha}(t)$. We also introduce
their equilibrium values as
$Q_{\alpha}^{\mathrm{eq}}\coloneqq\langle\chi_{\alpha}\rangle_{p^{\mathrm{eq}}}$
and $q_{\alpha}^{\mathrm{eq}}(x)\coloneqq
p^{\mathrm{eq}}(x)/Q^{\mathrm{eq}}_{\alpha}$. The time-scale separation
implies that the conditional probability relaxes very fast to the conditional
canonical distribution for $x\in G_{\alpha}$ [20, 21], and thus the
probability distribution has the form

$p(x;t)=q_{\alpha}^{\mathrm{eq}}(x)Q_{\alpha}(t)$ (S26)

along the trajectory, except for the initial fast relaxation. Using Eq. (S26)
and
$q^{\mathrm{eq}}_{\alpha}(x)=p^{\mathrm{eq}}(x)/Q_{\alpha}^{\mathrm{eq}}=\exp\\{-[\epsilon(x)-F^{\mathrm{eq}}]/k_{\text{b}}T\\}/Q_{\alpha}^{\mathrm{eq}}$,
we have

$k_{\text{b}}T\ln
p(x;t)=-\epsilon(x)+F^{\mathrm{eq}}+k_{\text{b}}T\sum_{\alpha=0}^{K}\quantity(\ln\frac{Q_{\alpha}(t)}{Q_{\alpha}^{\mathrm{eq}}})\chi_{\alpha}(x).$
(S27)

By inserting $\chi_{0}(x)=1-\sum_{\alpha=1}^{K}\chi_{\alpha}(x)$ and rewriting
$\chi_{\alpha}(x)$ in terms of $B_{\alpha}(x)$, we can rearrange Eq. (S27) in
the form of

$k_{\text{b}}T\ln
p(x;t)=-\epsilon(x)+\sum_{\alpha=1}^{K}\theta_{\alpha}B_{\alpha}(x)+\mathrm{const.},$
(S28)

using a set of $K$ numbers $(\theta_{\alpha})_{\alpha=1}^{K}$, where the
constant term is independent of $x$. Rearranging gives

$p(x;t)\propto\exp\quantity(-\frac{\epsilon(x)-\sum_{\alpha=1}^{N}\theta_{\alpha}B_{\alpha}(x)}{k_{\text{b}}T}).$
(S29)

Therefore, $p(x;t)$ is a canonical distribution under an external field of the
form (1) and thus in $\mathcal{P}_{\mathrm{TE}}$ along the trajectory.

###  Symmetry

Another situation is where the transition rates and the initial distribution
have a symmetry. Focusing on a discrete space $\mathcal{X}=\\{1,\dots,N\\}$,
we consider a symmetry expressed by a permutation group $\mathcal{G}$ on
$\mathcal{X}$, whose element $g\in\mathcal{G}$ is a bijection from
$\mathcal{X}$ to itself. We assume that the state energy and the transition
rates are invariant under the permutations. In other words, the energy
satisfies $\epsilon(x)=\epsilon(g(x))$ for all $g\in\mathcal{G}$, and the rate
from $x$ to $y$ equals the rate from $g(x)$ to $g(y)$ for all
$g\in\mathcal{G}$ and all $x,y\in\mathcal{X}$. We also assume the symmetry of
the initial distribution $p(x;t{=}0)=p(g(x);t{=}0)$. We assume we can keep
track of the expectation values of any observables that are invariant under
the permutation. Concretely, we assume that the observables
$B_{1}(x),\dots,B_{K}(x)$ and the constant function $1(x)\equiv 1$ form a
basis of the invariant subspace of $\mathbb{R}^{N}$ under the action of the
permutation matrices corresponding to the elements of $\mathcal{G}$. Note that
the constant function cannot be one of $B_{\alpha}$ because it cannot satisfy
$\langle B_{\alpha}\rangle_{p^{\mathrm{eq}}}=0$.

Under these conditions, the distribution obeys the symmetry for all $t>0$:

$p(x;t)=p(g(x);t).$ (S30)

The vector $\\{k_{\text{b}}T\ln p(x;t)+\epsilon(x)\\}{}_{x\in\mathcal{X}}$ has
the same symmetry and thus can be expanded by $B_{1},\dots,B_{K}$ and $1$ as

$k_{\text{b}}T\ln
p(x;t)+\epsilon(x)=\theta_{0}+\sum_{\alpha=1}^{K}\theta_{\alpha}B_{\alpha}(x)$
(S31)

with a set of numbers $(\theta_{\alpha})_{\alpha=0}^{K}$. Rearranging Eq.
(S31) gives

$p(x;t)\propto\exp\quantity(-\frac{\epsilon(x)-\sum_{\alpha=1}^{K}\theta_{\alpha}B_{\alpha}(x)}{k_{\text{b}}T}).$
(S32)

This is the form of the canonical distribution under the external field
$v=\sum_{\alpha}\theta_{\alpha}B_{\alpha}(x)$. Therefore, $p(t)$ for any $t>0$
is in the subspace $\mathcal{P}_{\mathrm{TE}}$.

##  Information-geometric interpretation

Our results are inspired by and deeply connected with the information-
geometric structures, although we have stated our results without referring to
information geometry. In this section, we retell our mathematical results in
the context of information geometry. We do not show any derivations in this
section because most equations are already derived in Sec. B.

Information geometry is a variation of differential geometry on the space of
probability distributions. Here we concentrate on a discrete space
$\mathcal{X}=\\{0,1,\dots,N\\}$ and discuss the information-geometric
structure over the space of all the nonzero probability distributions
$\mathcal{P}\coloneqq{\left\\{p\in\mathbb{R}^{N}_{>0}\,\middle|\,\sum_{x}p_{x}=1\right\\}}$.
Information geometry has a characteristic “dual” structure and can be regarded
as a generalization of the conventional Riemannian geometry.

###  Dual affine coordinates

The fundamental element of information geometry is the dual affine
coordinates, which induce a complete information-geometric structure on
$\mathcal{P}$. Dual affine coordinates are a pair of two coordinate systems
connected through a Legendre transform of two strictly convex functions. Any
such pairs of coordinates serve as dual affine coordinates, inducing various
geometric structures.

Among various dual affine coordinates on $\mathcal{P}$ used in information
geometry, here we discuss a special class of dual affine coordinates. Let
$\hat{a}\equiv\\{\hat{a}(x)\\}_{x=0}^{N}$ be an arbitrary set of constants
satisfying $\sum_{x}\hat{a}(x)=1$, $\hat{b}\equiv\\{\hat{b}(x)\\}_{x=0}^{N}$
be an arbitrary set of constants, $\hat{C}$ be an arbitrary constant,
${\hat{\bm{B}}}\equiv\\{\hat{B}_{\alpha}(x)\\}_{\alpha=0}^{N}{}_{x=0}^{N}$ be
an arbitrary set of $({N+1})^{2}$ constants that satisfies $\hat{B}_{0}(x)=1$
for all $x$ and is regular as an $({N+1})\times({N+1})$ square matrix, and
${\hat{\bm{A}}}\equiv\\{\hat{A}_{\alpha}(x)\\}_{x=0}^{N}{}_{\alpha=0}^{N}$ be
the inverse matrix of ${\hat{\bm{B}}}$ satisfying
$\sum_{x}\hat{A}_{\alpha}(x)\hat{B}_{\beta}(x)=\delta_{\alpha\beta}$ and
$\sum_{\alpha}\hat{A}_{\alpha}(x)\hat{B}_{\alpha}(y)=\delta_{xy}$. Here, we
use the hat mark ($\hat{\phantom{\cdot\cdot}}$) to avoid confusion with the
symbols introduced in thermodynamics. Using these constants, we introduce two
coordinate systems over $\mathcal{P}$,
${\hat{\bm{\eta}}}={\hat{\bm{\eta}}}[p]$ and
${\hat{\bm{\theta}}}={\hat{\bm{\theta}}}[p]$, and two convex functions
relating the two coordinates, ${\hat{\mathcal{H}}}(\bm{\eta})$ and
$-{\hat{\mathcal{F}}}(\bm{\theta})$, as follows:

$\displaystyle\hat{\eta}_{\alpha}[p]$
$\displaystyle=\sum_{x}\hat{B}_{\alpha}(x)\quantity[p(x)-\hat{a}(x)],$ (S33a)
$\displaystyle{\hat{\mathcal{H}}}({\hat{\bm{\eta}}}[p])$
$\displaystyle=\sum_{x}\quantity[\ln p(x)+\hat{b}(x)]p(x)+\hat{C},$ (S33b)
$\displaystyle\hat{\theta}_{\alpha}[p]$ $\displaystyle=\sum_{x}\quantity[\ln
p(x)+\hat{b}(x)]\hat{A}_{\alpha}(x),$ (S33c)
$\displaystyle{\hat{\mathcal{F}}}({\hat{\bm{\theta}}}[p])$
$\displaystyle=\sum_{x}\quantity[\ln p(x)+\hat{b}(x)]\hat{a}(x)+\hat{C},$
(S33d)

where $\alpha$ runs over $\\{1,\dots,N\\}$. We can invert the coordinates
(S33a) and (S33c) as

$\displaystyle
p(x)=\hat{a}(x)+\sum_{\alpha=1}^{N}\hat{A}_{\alpha}(x)\hat{\eta}_{\alpha}[p]$
(S34a) $\displaystyle
p(x)\propto\exp\quantity(-\hat{b}(x)+\sum_{\alpha=1}^{N}\hat{\theta}_{\alpha}[p]\hat{B}_{\alpha}(x)).$
(S34b)

In this calculation, we have used

$\sum_{\alpha=1}^{N}\hat{A}_{\alpha}(x)\hat{B}_{\alpha}(y)=\delta_{xy}-\hat{A}_{0}(x)\hat{B}_{0}(y)=\delta_{xy}-\hat{A}_{0}(x),$
(S35)

which follows from that $\\{\hat{A}_{\alpha}(x)\\}_{x=0}^{N}{}_{\alpha=0}^{N}$
and $\\{\hat{B}_{\alpha}(x)\\}_{x=0}^{N}{}_{\alpha=0}^{N}$ are inverse to each
other and $\hat{B}_{0}(y)=1$. The dual affine coordinates (S33) satisfy the
Legendre transform

$\displaystyle\frac{\partial{\hat{\mathcal{H}}}}{\partial\hat{\eta}_{\alpha}}=\hat{\theta}_{\alpha}({\hat{\bm{\eta}}}),\quad\frac{\partial{\hat{\mathcal{F}}}}{\partial\hat{\theta}_{\alpha}}=-\hat{\eta}_{\alpha}({\hat{\bm{\theta}}}),\hskip
10.00002pt$ (S36a)
$\displaystyle{\hat{\mathcal{H}}}({\hat{\bm{\eta}}}[p])-{\hat{\mathcal{F}}}({\hat{\bm{\theta}}}[p])=\sum_{\alpha=1}^{N}\hat{\theta}_{\alpha}[p]\hat{\eta}_{\alpha}[p],$
(S36b)

Equation (S36b) and the first equation of (S36a) can be easily checked by a
direct calculation using Eqs. (S33)–(S34). The second equation of (S36a) is
obtained as the inverse transform. Therefore, the set of coordinates and
functions in Eq. (S33) serves as dual affine coordinates.

The simplest set of dual affine coordinates belonging to the class (S33) is

$\displaystyle\hat{\eta}_{\alpha}[p]=p(\alpha),$
$\displaystyle{\hat{\mathcal{H}}}({\hat{\bm{\eta}}}[p])=\sum_{x}p(x)\ln
p(x),\quad$ (S37)
$\displaystyle\hat{\theta}_{\alpha}[p]=\ln\frac{p(\alpha)}{p(0)},$
$\displaystyle\mathcal{F}({\hat{\bm{\theta}}}[p])=\ln p(0).$

This set is obtained by setting ${\hat{\bm{a}}}=0$, ${\hat{\bm{b}}}=0$,
$\hat{C}=0$, $\hat{B}_{\alpha}(x)=\delta_{\alpha x}+\delta_{\alpha
0}-\delta_{\alpha x}\delta_{\alpha 0}$, and
$\hat{A}_{\alpha}(x)=\delta_{\alpha x}-\delta_{0x}+\delta_{\alpha
x}\delta_{0x}$. This choice of dual affine coordinates (S37) is standard in
information geometry [35, 36].

Going back to our theory, we can regard the external field intensities
$\bm{\theta}$, the external fields $\bm{\eta}$, the nonequilibrium free energy
$\mathcal{H}$, and the tilted equilibrium free energy $\mathcal{F}$ as
belonging to the class (S33) for the extreme case of $K=N$. The condition
$K=N$ means that we can keep track of the maximal number of observables and is
equivalent to $\mathcal{P}_{\mathrm{TE}}=\mathcal{P}$. Then, by choosing
$\hat{a}=p^{\mathrm{eq}}$, $\hat{b}=\epsilon/k_{\text{b}}T$, $\hat{C}=0$, and
${\hat{\bm{B}}}=\bm{B}$, we reproduce

${\hat{\bm{\eta}}}=\bm{\eta},\quad{\hat{\mathcal{H}}}=\frac{\mathcal{H}_{\bm{B}}}{k_{\text{b}}T},\quad{\hat{\bm{\theta}}}=\frac{\bm{\theta}}{k_{\text{b}}T},\quad{\hat{\mathcal{F}}}=\frac{\mathcal{F}_{\bm{B}}}{k_{\text{b}}T}.$
(S38)

With this choice of the constants, Eq. (S33a) becomes the relation
$\bm{\eta}=\langle\bm{B}\rangle_{p}$, Eq. (S33b) becomes the definition of the
nonequilibrium free energy, and Eq. (S34b) becomes the tilted equilibrium
under the external field intensities $\bm{\theta}$.

Dual affine coordinates induce all the information-geometric structures and
quantities. We discuss below that any dual affine coordinates belonging to the
class (S33) yield the same structure regardless of the choice of
${\hat{\bm{a}}}$, ${\hat{\bm{b}}}$, $\hat{C}$, and ${\hat{\bm{B}}}$. Thus, the
structure induced from the dual affine coordinates in Eq. (S38) is identical
to the structure induced from those in Eq. (S37), the standard one in
information geometry.

###  Information geometric quantities

We introduce two information-geometric quantities induced from the dual affine
coordinates (S33). One is the Bregman divergence, a two-point function between
two arbitrary points in the space, and the other is the information-geometric
metric with the associated line element (infinitesimal distance).

The general definition of the Bregman divergence for any two distributions
$p,q\in\mathcal{P}$ is

$\displaystyle D_{\mathrm{Breg.}}[p\|q]$
$\displaystyle\quad\\!\coloneqq{\hat{\mathcal{H}}}[p]-{\hat{\mathcal{F}}}[q]-{\hat{\bm{\theta}}}[q]\cdot{\hat{\bm{\eta}}}[p]$
(S39a)
$\displaystyle\quad={\hat{\mathcal{H}}}[p]-{\hat{\mathcal{H}}}[q]-{\hat{\bm{\theta}}}[q]\cdot\quantity({\hat{\bm{\eta}}}[p]-{\hat{\bm{\eta}}}[q])$
(S39b)
$\displaystyle\quad=-{\hat{\mathcal{F}}}[q]+{\hat{\mathcal{F}}}[p]-{\hat{\bm{\eta}}}[p]\cdot\quantity({\hat{\bm{\theta}}}[q]-{\hat{\bm{\theta}}}[p]),$
(S39c)

where the second and the third equality is due to the Legendre duality (S36).
It is explicitly calculated as

$D_{\mathrm{Breg.}}[p\|q]=\sum_{x}\,p(x)\ln\frac{p(x)}{q(x)},$ (S40)

which is independent of the choice of ${\hat{\bm{a}}}$, ${\hat{\bm{b}}}$,
$\hat{C}$, and ${\hat{\bm{B}}}$. The result (S40) coincides with the KL
divergence. Equations (S39c) and (S40) reproduces Eq. (S13), which is central
to deriving our results.

The information-geometric line element between infinitesimally close two
distributions $p$ and $p+dp$ is defined as, up to the second order of $dp$,

$\displaystyle dl^{2}$
$\displaystyle\\!\coloneqq\sum_{\alpha}d\hat{\theta}_{\alpha}d\hat{\eta}_{\alpha}=2D[p\|p\\!+\\!dp]=2D[p\\!+\\!dp\|p]$
$\displaystyle=\sum_{\alpha,\beta}\frac{\partial^{2}{\hat{\mathcal{H}}}}{\partial\hat{\eta}_{\alpha}\partial\hat{\eta}_{\beta}}d\hat{\eta}_{\alpha}d\hat{\eta}_{\beta}=-\sum_{\alpha,\beta}\frac{\partial^{2}{\hat{\mathcal{F}}}}{\partial\hat{\theta}_{\alpha}\partial\hat{\theta}_{\beta}}d\hat{\theta}_{\alpha}d\hat{\theta}_{\beta},$
(S41)

where $d{\hat{\bm{\eta}}}$ and $d{\hat{\bm{\theta}}}$ represent the same
displacement expressed with each coordinate system, and the equalities are
from the Legendre duality and the expressions of the Bregman divergence. By an
explicit calculation, the line element equals $k_{\text{b}}T$ times the Fisher
information:

$dl^{2}=k_{\text{b}}T\sum_{x}\frac{[dp(x)]^{2}}{p(x)}$ (S42)

and is independent of ${\hat{\bm{a}}}$, ${\hat{\bm{b}}}$, $\hat{C}$, and
${\hat{\bm{B}}}$.

In short, our set of the coordinates and the functions
$(\bm{\eta},\mathcal{H}_{\bm{B}}/k_{\text{b}}T,\bm{\theta}/k_{\text{b}}T,\mathcal{F}_{\bm{B}}/k_{\text{b}}T)$
serves as dual affine coordinates that induce an information-geometric
structure with the KL divergence and the Fisher information. The structure
agrees with the one induced from the standard dual affine coordinates in
information geometry [Eq. (S37)]. This agreement is because the two sets of
dual affine coordinates belong to the same class of dual affine coordinates in
Eq. (S33).

From the information-geometric viewpoint, the discussion in this section gives
a stochastic-thermodynamic interpretation of the information-geometric
coordinates and functions. The coordinate ${\hat{\bm{\eta}}}$ is the
expectation values of the observables, ${\hat{\bm{\theta}}}$ is the
thermodynamic force, the function ${\hat{\mathcal{H}}}$ is the nonequilibrium
free energy, and ${\hat{\mathcal{F}}}$ is the tilted equilibrium free energy.
This set of interpretations gives a new link between stochastic thermodynamics
and information geometry, which extends the previously found interpretations
[66] to the global structure of information geometry.