example of a colloidal Brownian particle trapped in a one-dimensional
potential $U(x,t)$ and immersed in a fluid at equilibrium with temperature
$T$. In the overdamped regime, the distribution function $\rho(x,t)$ obeys the
FP equation (11lbf) and the typical SST problem is the connection between two
equilibrium states, those corresponding to the initial potential
$U_{\scriptsize\textnormal{i}}(x)$ and the target one
$U_{\scriptsize\textnormal{f}}(x)$. An advantage of the equilibrium situation
is our perfect knowledge of the initial and target distributions,
$\rho_{\scriptsize\textnormal{i}}(x)\propto\exp[-\beta
U_{\scriptsize\textnormal{i}}(x)]$ and
$\rho_{\scriptsize\textnormal{f}}(x)\propto\exp[-\beta
U_{\scriptsize\textnormal{f}}(x)]$.

An especially simple case is that of harmonic confinement, in which the
potential $U(x,t)$ is harmonic for all times. As described in Sec. 3.1, the
PDF is Gaussian for all times therein, and thus completely determined by its
average and variance. Moreover, the evolution equations of the average and
variance are analytically solvable in closed form. Therefore, although the
intermediate states between the initial and target PDFs are indeed non-
equilibrium ones, SST connections can be exactly worked out, both non-optimal
[17] and optimal in some sense [90, 93, 94, 92]—as already analyzed in Sec.
4.1.1.

In principle, SST methods are transposable to situations in which the initial
and final states are NESSs, instead of equilibrium states. Still, some
problems emerge because there is not a general form, playing the role of the
canonical distribution, for the PDF corresponding to any NESS. As a
consequence, the initial and target states are not perfectly known in general;
this constitutes a first limitation for the catalogue of NESS that can be
considered as candidates to be SST-connected.

The SST connection between two NESS of the Brownian gyrator is the subject of
study in Ref. [161]. The Brownian gyrator is an overdamped particle moving in
a two-dimensional potential

$U(\mathbf{x},t)=\frac{1}{2}\kappa_{x}(t)x^{2}+\frac{1}{2}\kappa_{y}(t)y^{2}+u(t)xy,\quad\mathbf{x}\equiv(x,y),$
(11lbtbzcdct)

where $\kappa_{x}$ and $\kappa_{y}$ are both positive and
$\kappa_{x}\kappa_{y}-u^{2}>0$, in order to have a confining potential. The
gyrator is coupled to two heat baths with temperatures $T_{x}$ and $T_{y}$ in
the $x$ and $y$ directions, respectively. The two-dimensional position
$\mathbf{x}$ is a Markov process and the FP equation for its PDF
$\rho(\mathbf{x},t)$ reads

$\displaystyle\gamma\partial_{t}\rho=\partial_{x}\left(\rho\,\partial_{x}U\right)+\partial_{y}\left(\rho\,\partial_{y}U\right)+k_{B}\left(T_{x}\partial_{x}^{2}\rho+T_{y}\partial_{y}^{2}\rho\right).$
(11lbtbzcdcu)

If $T_{x}=T_{y}$, and $(\kappa_{x},\kappa_{y},u)$ are time-independent
parameters, the stationary solution of this FP equation is the canonical PDF
at temperature $T=T_{x}=T_{y}$ corresponding to the static potential
$U(\mathbf{x})$. If $T_{x}\neq T_{y}$, the stationary solution of this FP
equation can be exactly computed and is Gaussian, although it does not have
the canonical shape [161] and induces a current which is rotational.

The approach of Ref. [161] is quite similar to that of Ref. [17] for the
engineered swift equilibration of a Brownian particle moving in a one-
dimensional harmonic potential. A non-optimal SST connection of two NESS of
the Brownian gyrator, corresponding to different values of the triplet
$(\kappa_{x},\kappa_{y},u)$, can be built in a simple way, because the PDF is
Gaussian, not only in the initial and target states, but for all times.
Therefore, one can write

$\rho(\mathbf{x},t)=\frac{e^{-\frac{1}{2}\nu_{1}(t)x^{2}-\frac{1}{2}\nu_{2}(t)y^{2}-\nu_{3}(t)xy}}{2\pi\left[\nu_{1}(t)\nu_{2}(t)-\nu_{3}^{2}(t)\right]^{-1/2}}.$
(11lbtbzcdcv)

The functions $\boldsymbol{\nu}(t)\equiv(\nu_{1}(t),\nu_{2}(t),\nu_{3}(t))$
obey a closed system of first-order ODEs, in which
$\boldsymbol{\lambda}\equiv(\kappa_{x},\kappa_{y},u)$ play the role of control
functions and appear linearly. Thus, the controls $\boldsymbol{\lambda}$ can
be explicitly written as
$\mathbf{A}(\boldsymbol{\nu})+\mathbf{B}(\boldsymbol{\nu})\dot{\boldsymbol{\nu}}$,
where $\mathbf{A}$ and $\mathbf{B}$ are certain functions of
$\boldsymbol{\nu}$—the exact expression for which is not relevant here and can
be found in Ref. [161]. In this way, the “equilibrium swift equilibration”
technique introduced in Ref. [17] is generalized to the connection of two
NESS, taking advantage of the simplicity of the mathematical problem for the
Brownian gyrator. Although the problem is two-dimensional and the initial and
target states are NESS, the main characteristic features that facilitates the
SST connection of a harmonically trapped particle still hold: perfect
knowledge of the initial and target distributions, Gaussianity of the PDF for
all times, and simple enough evolution equations for the relevant variables
(making it possible to obtain exact analytical solutions thereof).
Interestingly, the Brownian gyrator has been used to build a thermodynamic
cycle, which is studied through the lens of thermodynamic geometry [162]. This
represents an alternative proposal to those described in Sec. 6.1 for the
construction of a mesoscopic heat engine.

Another physical situation in which the SST connection between two NESS has
been considered is the uniformly heated granular fluid [62]. This case is more
involved than the Brownian gyrator, due to the intrinsically dissipative
character of the dynamics. The evolution equation for the PDF
$P(\mathbf{v},t)$ has two terms: a diffusive, FP, term stemming from the
stochastic forcing and an inelastic Boltzmann collision term, which makes the
evolution equation non-linear in the PDF.

In the long-time limit, the granular fluid reaches a homogeneous NESS due to
the balance—in average—of the energy loss in collisions and the energy input
by the applied stochastic forcing. Even for this homogeneous NESS, the
velocity PDF is non-Gaussian and thus is not completely characterized by the
variance—the granular (kinetic) temperature. In addition, nor is the velocity
PDF perfectly known: it is necessary to resort to approximate schemes, keeping
track of non-Gaussianities that are essential. The simplest way of doing so is
through the so-called first Sonine approximation, in which the granular fluid
is assumed to be described by its granular (kinetic) temperature $T$ and the
excess kurtosis $a_{2}$ (the fourth cumulant of the velocity),

$P(\mathbf{v};t)=\frac{e^{-w^{2}}}{\left[\pi
v_{T}^{2}(t)\right]^{d/2}}\left[1+a_{2}(t)S_{2}\left(w^{2}\right)\right],\quad\mathbf{w}\equiv\frac{\mathbf{v}}{v_{T}(t)}.$
(11lbtbzcdcw)

The thermal velocity $v_{T}$ is defined by $v_{T}^{2}\equiv 2T/m$ in terms of
the granular temperature, and $S_{2}(x)=x^{2}/2-(d+2)x/2+d(d+2)/8$ is the
second Sonine polynomial where $d$ is space dimension. Within this
approximation, the evolution equations of $T$ and $a_{2}$ constitute a system
of two coupled ODES, in which the control function is the intensity of the
thermostat $\chi(t)$. The problem of finding the protocol that minimizes the
connection time between two NESSs corresponding to different values of the
driving intensity, $\chi_{\scriptsize\textnormal{i}}$ and
$\chi_{\scriptsize\textnormal{f}}$, has been analyzed in Ref. [62]. This
control problem is, at first, non-trivial, since the evolution equations in
the Sonine approximation are non-linear in $(T,a_{2})$ and then not
analytically solvable. Still, the control function $\chi(t)$ enters linearly
in the evolution equations, and this simplifies the mathematical problem: the
optimal controls are of bang-bang type. That is, the optimal control comprises
two time-windows, inside each of them $\chi(t)$ is constant and equal to
either its maximum possible value $\chi_{\max}$ or its minimum possible value
$\chi_{\min}$, with one switching between these extreme values at a certain
intermediate time.666For evolution equations that are linear both in the
controls and in the variables, there exist rigorous theorems that ensure that
the optimal control is of bang-bang type, with $n-1$ switchings for a system
with $n$ variables. For evolution equations that are non-linear in the
variables but linear in the controls, Pontryagin’s maximum principle implies
that a similar situation is expected and, since here $n=2$, there is one
switching [62]. For a full-power thermostat, $\chi_{\min}=0$ and
$\chi_{\max}=\infty$, the optimal control problem can be exactly solved—the
bangs with $\chi_{\max}=\infty$ are instantaneous whereas the bangs with
$\chi_{\min}=0$ correspond to time windows where the granular fluid freely
cools. Both the thermodynamic length and the information geometry cost can be
evaluated over the optimal protocols, being consistent with the recently
derived general inequalities that impose the existence of a speed limit for
arbitrary dynamics, not necessarily Markovian [148, 149]—see also Sec. 4.3.
The case of a more realistic thermostat, where $\chi_{\min}>0$ and
$\chi_{\max}<\infty$, has been addressed in the linear response regime [163].

##  6 Applications: heat engines and beyond

###  6.1 Heat engines

SST have been employed in the last decade to design irreversible heat-engines
[103, 164, 165, 84, 30, 166, 167, 75, 168, 169, 60, 170, 171, 172, 173, 174,
80, 175, 86, 176, 177]. Loosely speaking, these heat engines can be considered
as the mesoscopic counterparts of the classical, macroscopic, heat engines, in
which the branches of the corresponding cycle last for a finite time. Note
that we focus on classical heat engines, described at the mesoscopic level by
nonequilibrium classical statistical mechanics—their quantum counterparts are
thus not addressed in this review.

Let us go back to the colloidal particle trapped in a harmonic potential of
stiffness $\kappa$, and immersed in a fluid at equilibrium with temperature
$T$, thus described by the Smoluchowski equation (11lbg). Recall that both the
stiffness of the trap and the bath temperature can be time-dependent, with
their time-dependence being externally controlled. This system is
experimentally realizable, see for example Refs. [165] and [60] for
experimental implementations of Stirling-like and Carnot-like cycles,
respectively. Moreover, since it has been shown that active baths can in some
situations be mapped into regular “passive” baths with an effective
temperature, the approach below may be also relevant within the context of
active Brownian heat engines [178, 179].

A difficulty arises in the designing of these mesoscopic heat engines when
they involve, as is the case of the Otto or the Carnot cycles, adiabatic—in
the sense of zero heat, not in the sense of infinitely slow employed before
[3]—branches. Complete decoupling of the system from the heat bath is not
possible, since the interaction between the mesoscopic object—i.e., the
Brownian particle—and the heat bath cannot be switched off. In addition, zero-
heat (in average, fluctuations are unavoidable) and isoentropic processes are
not equivalent for finite-time protocols like those in SST. This makes the
definition of adiabatic processes a subtle issue in the mesoscopic world [103,
84, 60, 92, 175, 86].

In a pioneering work [103], Schmiedl and Seifert analyzed an irreversible
Carnot-like heat engine, built on an overdamped Brownian particle confined in
a harmonic trap. This heat engine works cyclically, with two isothermal and
two pseudoadiabatic branches [103]—like the original Carnot engine—that
connects equilibrium states. However, unlike with the original Carnot engine,
all the branches are irreversible. More specifically, during the hot (cold)
isotherm, the system is in contact with a heat bath at temperature $T_{h}$
($T_{c}$) for a certain finite time $t_{h}$ ($t_{c}$). In the instantaneous
pseudoadiabats, the temperature of the system suddenly jumps from $T_{h}$ to
$T_{c}$ (or vice versa) while keeping the probability distribution for the
position of the Brownian particle unchanged. Therefore, what remains constant
over the adiabats is the configurational contribution to the entropy or,
equivalenty, the ratio $T/\kappa$. As already noted in Ref. [103], this means
that there appears a nonzero heat—and a nonzero entropy increment—along the
pseudoadiabatic branches associated with the instantaneous change of the
kinetic contribution to the energy. This is the reason why we term these
branches pseudoadiabatic, to differentiate them from the actual adiabats, over
which the heat vanishes in average, as described below.

This heat engine is optimized in the following way: keeping the isotherm times
$t_{h}$ and $t_{c}$ fixed, one can minimize the irreversible work, i.e.,
maximize the output work $-W$ (which is a function of $t_{h}$ and $t_{c}$).
Afterwards, one can maximize the output power $P=-W/(t_{h}+t_{c})$ and obtain
the corresponding efficiency at maximum power $\eta^{*}$, which reads

$\eta_{\scriptsize\textnormal{SS}}=\frac{2\eta_{\scriptsize\textnormal{C}}}{4-\eta_{\scriptsize\textnormal{C}}},$
(11lbtbzcdcx)

where $\eta_{\scriptsize\textnormal{C}}$ is Carnot’s efficiency,

$\eta_{C}=1-\frac{T_{c}}{T_{h}}.$ (11lbtbzcdcy)

The study of efficiency at maximum power is a classic problem in the field of
finite-time thermodynamics [87, 180, 181, 182, 183, 184]. The so-called
Curzon-Ahlborn efficiency at maximum power is

$\eta_{\scriptsize\textnormal{CA}}=1-\sqrt{\frac{T_{c}}{T_{h}}},$
(11lbtbzcdcz)

which was obtained for an endoreversible heat engine [87]—an engine that
operates reversibly but for the irreversible coupling to the heat-baths.777A
microscopic theory for the Curzon-Ahlborn cycle has been proposed [185]. It
must be noted that efficiency at maximum power is a rather delicate concept,
which in general depends on the parameters with respect to which the
maximisation is carried out. Still, even if not completely general, the
Curzon-Ahlborn value is a quite frequent bound for the efficiency at maximum
power of a wide range of heat engines. In fact,
$\eta_{\scriptsize\textnormal{SS}}<\eta_{\scriptsize\textnormal{CA}}$,
although in the limit of small temperature difference, for which $\eta_{C}\ll
1$, both efficiencies coincide up to second order in $\eta_{C}$,

$\eta_{\scriptsize\textnormal{CA}}=\frac{\eta}{2}+\frac{\eta^{2}}{8}+O(\eta^{3}),\quad\eta_{\scriptsize\textnormal{SS}}-\eta_{\scriptsize\textnormal{CA}}=O(\eta^{3}).$
(11lbtbzcdda)

Nevertheless, the universality of the efficiency at maximum power up to
quadratic order in $\eta_{\scriptsize\textnormal{C}}$ is quite a general
feature [186, 164, 167, 166, 187, 169, 175, 86], which can be shown to hold
for heat engines with left-right symmetry (switching of the baths entail an
inversion of the fluxes) [188]. Less strict conditions for this universality
have been derived in Refs. [167, 187], which also apply to heat engines
without left-right symmetry such as the Feynman ratchet [189] or the Curzon-
Ahlborn endoreversible heat engine [87]. Besides, this kind of universality
has been extended to other figures of merit, beyond the maximum power regime
[190].

Different variants of this Carnot-like heat engine have been investigated in
the literature. Holubec considered an overdamped Brownian particle moving in a
log-harmonic potential [166]. The cycle is exactly the same as in Ref. [103],
with two optimal isotherms and two instantaneous pseudoadiabats, adapted to
the different binding potential. The efficiency at maximum power also agrees
with $\eta_{\scriptsize\textnormal{CA}}$ up to second order in
$\eta_{\scriptsize\textnormal{C}}$. Bo and Celani analyzed a Brownian particle
immersed in a fluid with inhomogeneous temperature [84]. The linear
temperature profile makes it necessary to consider the underdamped
description, because of the so-called entropic anomaly stemming from the
temperature gradient [191]. In this work, the authors also derive the
condition to obtain quasistatic (reversible) adiabatic branches:
$T^{2}/\kappa$ is the quantity that must be kept constant, see Sec. 3.4.
Martinez et al. employed these quasistatic adiabatic branches to realize
experimentally an irreversible Carnot-like engine [60], with a charged
colloidal particle immersed in water and optically trapped. The charge of the
particle allows for adding a noisy electrostatic force: in this way, the
effective temperature felt by the particle can be varied from room temperature
to thousands of kelvins. Over the “adiabats”, there is a nonzero fluctuating
heat that must be taken into account when defining the efficiency of such an
engine. The authors analyzed the probability distribution function of the
efficiency as a function of the number of cycles and observed the appearance
of super-Carnot efficiencies, even far from the quasistatic regime. Tu also
investigated a Carnot-like engine built with a Brownian particle in the
underdamped description, but with the temperature of the thermal bath being
homogeneous [75]. Suitable shortcuts are introduced along the isothermal
branches, whereas the Brownian particle is decoupled from the thermal bath and
submitted to a velocity-dependent force during the “adiabatic” branches. Both
the decoupling from the thermal bath and the velocity-dependent force limit
the experimental feasibility of such an engine, but makes it possible to
exactly derive the Curzon-Ahlborn value for the efficiency at maximum power.

As the discussion above suggests, the construction of the adiabatic branches
of the Carnot-like engine is a subtle issue. Plata et al. showed how to build
finite-time actually adiabatic—in the sense of zero average heat—branches for
an overdamped Brownian particle trapped in an arbitrary nonlinear potential
[83]—see also Secs. 3.4 and 4.2. Figure 5 illustrates the difference between
the classical quasistatic Carnot cycle and its optimal finite-time
counterpart. Nakamura et al. also thoroughly studied the adiabatic
connection—again in the sense of zero average heat—for the specific case of
harmonic confinement within the underdamped description of the dynamics [175].
Extending ideas of the fast-forward protocol—already discussed in Sec. 3.3,
they built a Carnot-like heat engine with fast-forwarded isothermal and
adiabatic branches. On the one hand, similarly to the situation in Ref. [75],
the extra potential contains terms involving the velocity; the efficiency at
maximum power then equals the Curzon-Ahlborn bound for “weak dissipation” (in
the “strongly” underdamped regime). On the other hand, for strong dissipation
(in the overdamped limit), the efficiency at maximum power is smaller than
$\eta_{\scriptsize\textnormal{SS}}$, as given by (11lbtbzcdcx). Nakamura et
al. claimed that the overdamped approximation cannot be used to describe the
full cycle of a Carnot-like stochastic heat engine. Still, Plata et al. built
an irreversible Carnot-like engine with an overdamped colloidal particle
trapped in a harmonic potential [86]. The main novelty with regard to Ref.
[103] is the authors’ employing of the finite-time adiabatic branches derived
in Ref. [83], instead of the instantaneous pseudoadiabats that involve a
nonvanishing heat. In addition to minimum work (maximum work output)
isotherms, minimum time adiabats are employed to construct the Carnot-like
cycle. Further optimization of the cycle shows that the efficiency at maximum
power of this overdamped Carnot-like heat engine is very close to the Curzon-
Ahlborn value throughout the whole range of temperature ratios $T_{c}/T_{h}$.

Figure 5: Sketch of a Carnot cycle with a overdamped harmonic oscillator. The
cycle is made by two isothermal branches (red lines) and two adiabatic ones
(green lines). On the one hand, in the traditional Carnot cycle (top panel),
the states are swept quasistatically, and the system keeps at equilibrium at
all times. On the other hand, the optimal finite-time version (bottom panel)
involves non-equilibrium evolution between operation equilibrium points. See
Ref. [83] for further details.

It is worth mentioning that several works have addressed the problem of
attaining the Carnot efficiency at finite power [192, 193, 194, 195, 171, 170,
196, 197, 198, 199]. Mostly, attaining—or even exceeding [198]—Carnot’s
efficiency is connected with the antiadiabatic limit of infinitely fast
transformations [194, 171, 170, 196, 199]. In Ref. [199], working in the small
temperature difference regime, this has been shown to be consistent with the
trade-off relation between efficiency and power recently derived on quite
general grounds [200].

Other cycles, of the Stirling or Otto types, have been studied in the
literature. In fact, the first experimental realization of a mesoscopic heat
engine was achieved with the Stirling cycle [165], which comprises two
isothermal and two isochoric branches. A Brownian particle (a single melamine
bead of diameter close to $3\mu$m) suspended in water was trapped in a two-
dimensional harmonic potential created with optical tweezers. The stiffness of
the trap linearly varies with time along the isothermal branches, whereas it
is kept constant in the almost instantaneous processes that connect the
isotherms—which are thus isochoric, the average work vanishes. The determined
efficiency at maximum power is, within experimental error, in agreement with
the Curzon-Ahlborn value. Optimal protocols for the overdamped Stirling engine
have been derived in Ref. [168] by a mapping to an optimal mass transport
problem, with an efficiency at maximum power given by Eq. (11lbtbzcdcx). Also,
Stirling engines with active baths have been investigated [201, 202]. Non-
Gaussianities seem to be responsible for the improved efficiency—as compared
to usual engine with non-active baths—of the active engine analyzed in Ref.
[201]. Still, these results are changed if a different definition of the
temperature, based on the diffusion constant of the particle without any
external potential, is employed [202].

SST for the Otto cycle have been considered in Ref. [30]. In this paper, it is
shown that the accelerated connection intrinsic to SST does not only increase
power output of the engine but also may increase its efficiency, both in the
quantum and the classical case. Abah et al. further investigated the quantum
Otto engine, establishing the energetic cost of driving with SST and finding
that the efficiency at maximum power is very close to the Curzon-Ahlborn bound
[173].

Recently, the optimization of quite general heat engines has also been
considered [111, 203, 204, 205, 206, 207]. In Ref. [111], the cycle comprises
two isothermal branches connected by the instantaneous pseudoadiabats
described above. The driving potential that minimizes irreversible work or
heat dissipation is obtained, which in relevant cases leads to results
previously derived in the framework of optimal mass transport [95, 96,
208]—see also Sec. 4 of this review. In Ref. [203], a very general class of
microscopic heat engines driven by arbitrary periodic protocols of temperature
and mechanical control parameters is considered. For slow driving, a universal
trade-off relation between efficiency and power has been derived by using
thermodynamic geometry arguments. In Ref. [204], a system in contact with a
heat bath with an arbitrary time-varying peri- odic temperature profile is
studied. Optimal control protocols that maximize power or efficiency are
derived. In Ref. [205], a linear response theory for generic Gaussian heat
engines obeying Fokker-Planck dynamics is developed. In this context, the
structure of the Onsager coefficients is investigated in detail. In Ref.
[206], the full space—within the linear response regime—of non-equilibrium
thermodynamic cycles is exploredI . Therein, the authors also employ
information geometry ideas for deriving an upper bound for the efficiency and
building finite-time heat engines that outperform—in terms of efficiency—other
recent proposals. In Ref. [207], the simultaneous optimization of the average
and the fluctuation of the efficiency in a finite-time Carnot cycle has been
investigated. The approach is also based on information geometry ideas and
holds for slow driving and over a coarse-grained timescale.

###  6.2 Other Applications

The idea to control the time required to reach a given transformation, or
perform a certain task has a respectable history in the field of engineering
[209], where a common situation is that pertaining to crane driving (trolley
displacement and rope length). Yet, the similarities between such a
macroscopic system and the dynamical equations ruling the transport of
ultracold ions or neutral atoms in effectively one-dimensional traps with
time-dependent controllable parameters (as used in advances towards scalable
quantum-information processing), opens the way for a transfer of method
between the two fields [210].

Besides, SST allow to circumvent some shortcomings of existing manipulation
techniques. In particular, optical manipulations of colloids are plagued by
the impracticality of creating time-dependent expulsive confinements, that are
required when seeking to deconfine a colloidal state [17, 160]. While feedback
protocols offer a way out [80], we seek a purely feedforward technique,
without applying any retroaction on the system. Taking advantage of
diffusiophoresis, i.e. the migration of colloids induced by solute gradients,
one can drive the system by a proper time-dependent control of the salt
concentration in a buffer in contact with the solution. A fast decompression
can thereby be achieved [81], from a joint optical and diffusiophoretic
driving. To remain in the field of soft matter, a related proposal of driving
a bulk system from its boundaries was put forward in [211]: by monitoring the
potential (or the charge) difference between the two plates of a
nanocapacitor, one can accelerate significantly electrolyte relaxation
following a charging of the system. We also report applications of SST to
systems driven by Lévy noise [212], as well as in the field of active matter
for active Brownian particles, to control the system’s state by a joint
monitoring of activity and the strength of trap confinement.

A rather unexpected application of SST has emerged in the context of
biological evolution [213]. The setting is provided by the Wright-Fisher model
[214], describing the evolution of a population of organisms through a space
of $M$ possible genotypes. The state of the population is characterized by a
time-dependent probability distribution $\rho(\mathbf{x},t)$, where
$\mathbf{x}\equiv(x_{1},\cdots x_{M})$ is a frequency vector whose component
$x_{m}$ gives the fraction of the population found in genotype $m\in[1,M]$ at
time $t$. The Wright-Fisher model describes the diffusive evolution of
$\rho(\mathbf{x},t)$ due to mutations that induce a random walk among
genotypes. It is additionally assumed that there exists an externally
controlled environmental parameter $\lambda$, corresponding for example to a
drug concentration that affects the fitness of a pathogen. If $\lambda$ is
held fixed, then under the Wright-Fisher model the population evolves to an
evolutionary equilibrium state $\rho^{\rm eq}(\mathbf{x},\lambda)$. If the
parameter is varied with time, then the actual state of the population
$\rho(\mathbf{x},t)$ lags behind the instantaneous equilibrium state
$\rho^{\rm eq}(\mathbf{x},\lambda(t))$. Applying the tools of SST, Ref. [213]
shows how to construct a counterdiabatic control protocol that eliminates this
lag. This opens the possibility of controlling biological evolution to drive a
population from an initial equilibrium state to a desired final equilibrium
state in finite time. This kind of counterdiabatic driving has been
generalized to discrete-state continuous-time Markov networks and applied to
control chaperon-assisted protein folding. Interestingly, the obtained control
protocols resemble natural strategies observed in actual experiments with E.
coli and yeast [215].

SST have also been used to develop strategies for implementing computational
tasks rapidly and accurately, and to study the inherent thermodynamic costs
associated with these tasks [129, 130]. Specifically, protocols were designed
to create, erase and transfer a single bit of information stored in a double-
well potential. These tasks dissipate energy into the bit’s thermal
surroundings. Ref. [129] explored how this dissipated work scales with the
rate of computation, the characteristic length scale of the system, and the
robustness of the (inherently metastable) information storage. A key finding
of Ref. [129] was that although Landauer’s principle [122, 121] establishes
the minimum dissipated energy needed to erase a bit of information, there is
no similar bound on the amount of time needed to erase the bit. Using SST, a
bit can be erased arbitrarily rapidly and with arbitrarily high fidelity,
provided one is willing to pay the requisite thermodynamic cost in dissipated
energy. A tight bound on this irreversible cost has been derived [130]. The
results of Refs. [129, 130] illustrate, in a computational thermodynamic
setting, the refined second law of thermodynamics obtained in Ref. [96].
Generalizing these approaches to underdamped systems is challenging [123].

To conclude, we emphasize that the key ideas behind SST, and inverse
engineering, can be illustrated in an early undergraduate course, with RC, LC
or RLC circuits. An experimental demonstration was proposed in [216], that
furthermore provides an original approach to the venerable teaching of
differential equations.

##  7 Perspectives and conclusions

Shortcuts to adiabaticity (STA) are finite-time protocols that produce the
same state as an infinitely slow (so-called adiabatic) driving. We have here
generalized this approach to finite-time protocols connecting arbitrary
states, referred to under the terminology of SST (Swift State-to-state
Transformations). For both Hamiltonian and stochastic dynamics, we have
classified SST into different types, inverse-engineering, counterdiabatic
(CD), and fast-forward (FF), mirroring the usual categorization of quantum
STA. This categorization is indeed useful for presentation purposes but not so
clear-cut as it may seem: in the stochastic case, CD protocols can be thought
both as a special case of inverse-engineering ones and as a certain limit of
FF protocols—which somehow smears the boundaries between the different
categories. One of the goals of our review is to highlight the common
fundamentals behind all of them. To conclude the paper, we offer below some
perspectives that emerge.

For isolated Hamiltonian systems (Sec. 2) we foresee three topics of potential
future investigation. In systems with one degree of freedom, one could drop
the assumption that the steering potential $U(x,t)$ is a confining potential
with a single minimum, and instead consider a confining potential with a
double-well structure, with a local maximum separating the two wells. If
$U_{\rm max}$ gives the value of the potential at this local maximum, then
each energy shell with $E>U_{\rm max}$ forms a single closed loop in phase
space, while each energy shell $E<U_{\rm max}$ forms a pair of non-
intersecting closed loops. The shell $E=U_{\rm max}$ is a separatrix, with the
characteristic shape of a “figure-eight” closed loop. The adiabatic invariance
of the action $I(E,t)$ breaks down in the vicinity of the separatrix [25] and
this in turn leads to subtle topological effects [217], which may have
interesting implications for the design of shortcuts for such systems. Next,
as already noted in Sec. 2.2.5, it is plausible that Hamiltonian shortcuts can
be generalized to integrable systems with $N>1$ degrees of freedom, as such
systems effectively decompose into $N$ independent one-degree-of-freedom
systems. Finally, as also suggested in Sec. 2.2.5, it would be worthwhile to
investigate the design of SST for ergodic and chaotic Hamiltonian systems that
constitute canonical families. These are systems with the (non-generic)
property that $H(\mathbf{x},\mathbf{p},t)$ and
$H(\mathbf{x},\mathbf{p},t^{\prime})$ are related by a canonical
transformation, for any $t,t^{\prime}\in[0,t_{\scriptsize\textnormal{f}}]$.
Progress on any of these three questions may in turn provide insight into
quantum shortcuts for corresponding systems.

Most SST have been developed for one-dimensional systems. As a rule, the
generalization of these protocols—either inverse engineering, counterdiabatic,
fast-forward, or thermal engineering—to higher-dimensional systems is not
straightforward. Still, inverse engineering protocols, akin to those
originally devised for a colloidal particle in a one-dimensional harmonic trap
[17], have been worked out for an arbitrary curved configuration space [218].
This shows the feasibility of extending SST to arbitrary geometries, but
engineering optimal protocols—in the sense of minimizing work, heat, entropy
production, connection time or any other relevant physical property over
them—remains a challenge in this case. On a related note, the protocols
detailed in our review are mostly designed in the overdamped regime. Future
theoretical developments in SST are expected in the underdamped regime to
accompany the development of new experimental setups having a tunable damping
rate [219].

The irreversible work unavoidably appearing in SST has been linked to
information geometry ideas in linear response [104]. In the information
geometry approach, a key role is played by the Fisher information, which at
equilibrium is related to derivatives of the distribution function and, in
certain specific situations, to fluctuations of relevant quantities [157, 89,
114, 220]. This theoretical framework also makes it possible to establish a
speed-limit inequality for SST, Eq. (11lbtbzcdcn), which resembles the quantum
speed limit (the time-energy uncertainty relation [139, 141]). Recently, the
information geometry approach has been employed beyond linear response. This
has led to the emergence of several versions of classical speed-limits
inequalities—also resembling the quantum time-energy uncertainty relation—for
systems described by stochastic dynamics [144, 143, 145, 147, 149, 148, 107].
Nevertheless, the tightness of the obtained classical bounds, for instance
that given by Eq. (11lbtbzcdcs), is not well established—unlike in the quantum
case [142, 140]. Clarifying this specific issue and, on a more general note,
linking information geometry concepts and physical quantities beyond linear
response are open questions. Answering them constitutes an enticing
perspective for future work, and may provide a novel way to tackle and move
forward non-equilibrium statistical mechanics and, more specifically, the
field of SST.

We have exposed here several strategies for the control of systems in contact
with a thermostat. Closed-loop control protocols, i.e. those with feedback,
have not been discussed since they are intrinsically subject to the system’s
own time scales. However, such approaches have the advantage of not requiring
a perfect modeling of the system under study. This is to be contrasted with
open-loop control protocols detailed in this review article, that are of the
feed-forward type. They provide a wide range of strategies to accelerate the
transition from a given state to another over an a priori arbitrarily short
amount of time. The methods presented here provide such solutions, despite the
intrinsic randomness associated to the coupling with a reservoir, and even
propose to engineer the randomness.

##  Appendix A Summary of acronyms

CD | Counterdiabatic
---|---
FF | Fast-forward
FP | Fokker-Planck
GCD | Global Counterdiabatic
LCD | Lobal Counterdiabatic
NESS | Non-equilibrium Steady State
PDF | Probability density function
STA | Shortcut(s) to adiabaticity
SST | Swift State-to-state Transformation(s)

##  Appendix B Stochastic processes framework

###  B.1 Langevin and Fokker-Planck descriptions

For the sake of concreteness, let us consider a colloidal particle immersed in
a fluid in equilibrium at temperature $T$ in the overdamped regime. For one-
dimensional motion, the evolution equation for the position $x$ of the
Brownian particle is

$\gamma\dot{x}=-\partial_{x}U(x,t)+\sqrt{2\gamma k_{B}T(t)}\,\eta(t),$
(11lbtbzcddb)

where $\gamma$ is the friction coefficient, $U(x,t)$ is the driving potential,
$T(t)$ is the temperature, $k_{B}$ is the Boltzmann constant, and $\eta(t)$ is
a Gaussian white noise of unit variance,

$\langle\eta(t)\rangle=0,\quad\langle\eta(t)\eta(t^{\prime})\rangle=\delta(t-t^{\prime}).$
(11lbtbzcddc)

Note that, in general, both the potential and the temperature may depend on
time—as explicitly stated in our notation. The diffusion coefficient and the
temperature verify Einstein’s relation $D=k_{B}T/\gamma$, so the diffusion
coefficient is in general also time-dependent. Usually, the friction
coefficient $\gamma$ is assumed to be constant, including in the situation
where colloid temperature is artificially controlled, as discussed below Eq.
(11lbtbw), by randomly shaking the center-of-force of the trapping
888Temperature control is done by adding an independent stochastic white-noise
force as detailed in Refs. [79, 59], which increases the variance of the noise
acting on the Brownian particle and effectively heats it, with the temperature
of the background fluid remaining unchanged.. Different realizations of the
noise $\eta(t)$ give rise to different trajectories for the colloidal
particle. In the Langevin description, $\langle\cdots\rangle$ thus mean
averaging over the different trajectories of the stochastic process.

It is also possible to analyze the stochastic motion of the colloid by using
the ensemble picture. Therein, we introduce the PDF $\rho(x,t)$, which obeys
the FP equation

$\gamma\partial_{t}\rho(x,t)=\partial_{x}\left\\{[\partial_{x}U(x,t)]\rho(x,t)+k_{B}T(t)\partial_{x}\rho(x,t)\right\\}.$
(11lbtbzcddd)

For time-independent potential and temperature, the FP equation has only one
stationary solution that is the equilibrium one,
$\rho_{eq}(x)\propto\exp[-\beta U(x)]$, with $\beta=(k_{B}T)^{-1}$.

It is interesting to note that the FP equation can be cast in the form of a
continuity equation,

$\partial_{t}\rho(x,t)=-\partial_{x}J(x,t),$ (11lbtbzcdde)

where the probability current $J(x,t)$ is defined as

$J(x,t)\equiv-\gamma^{-1}\left\\{\left[\partial_{x}U(x,t)\right]\rho(x,t)+k_{B}T(t)\partial_{x}\rho(x,t)\right\\}.$
(11lbtbzcddf)

Writing the probability current as $J(x,t)=\rho(x,t)v(x,t)$, a velocity field
$v(x,t)$ is introduced as

$v(x,t)=-\gamma^{-1}\left[\partial_{x}U(x,t)+k_{B}T(t)\partial_{x}\ln\rho(x,t)\right].$
(11lbtbzcddg)

Note that, despite our assumption of constant $\gamma$, Eqs.
(11lbtbzcddd)-(11lbtbzcddg) remain valid for time-dependent friction.

###  B.2 Stochastic thermodynamics

Following Sekimoto [4], we can write down the first principle at the mesoscale
by defining stochastic work $W$ and heat $Q$. As usual, work is the potential
energy change associated with the variation of the external parameters
$\boldsymbol{\lambda}(t)$ in the potential (volume, stiffness of the trap,
etc.). In other words, we consider

$\displaystyle U(x,t)$ $\displaystyle=U(x,\boldsymbol{\lambda}(t)),$
(11lbtbzcddh) $\displaystyle\partial_{t}U(x,t)$
$\displaystyle=\boldsymbol{\dot{\lambda}}(t)\cdot\left[\partial_{\boldsymbol{\lambda}}U(x,\boldsymbol{\lambda})\right]_{\boldsymbol{\lambda}=\boldsymbol{\lambda}(t)}.$
(11lbtbzcddi)

For each trajectory of the stochastic process, i.e. for each realization of
the white noise in the Langevin equation, work is thus calculated as

$\displaystyle W$
$\displaystyle=\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,\left[\partial_{t}U(x,t)\right]_{x=x(t)}$
(11lbtbzcddj)
$\displaystyle=\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,\boldsymbol{\dot{\lambda}}(t)\cdot\left[\partial_{\boldsymbol{\lambda}}U(x,\boldsymbol{\lambda}(t))\right]_{x=x(t)}.$

Note that $W$ changes from trajectory to trajectory, i.e. it is stochastic. We
can also introduce the stochastic instantaneous power,

$\dot{W}=\boldsymbol{\dot{\lambda}}(t)\cdot\left[\partial_{\boldsymbol{\lambda}}U(x,\boldsymbol{\lambda}(t))\right]_{x=x(t)},$
(11lbtbzcddk)

such that $W=\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\dot{W}$. For each
trajectory of the stochastic process, the configurational contribution to heat
is thus the potential energy change due to the time evolution of the
particle’s position $x(t)$, i.e.

$Q_{\scriptsize\textnormal{conf}}=\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,\dot{x}(t)\left[\partial_{x}U(x,t)\right]_{x=x(t)}.$
(11lbtbzcddl)

For mathematical consistency, the above integral must be understood in the
Stratonovich sense [95]. Physically, this entails

$Q_{\scriptsize\textnormal{conf}}+W=\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,\frac{d}{dt}U(x(t),t)=\Delta
U,$ (11lbtbzcddm)

where $\Delta U$ is the potential energy increment over the considered
trajectory, i.e. $\Delta U$ is also stochastic.

In the Langevin picture, mean work and heat are obtained by averaging over
noise realizations. For example, the average work reads

$\left\langle
W\right\rangle=\left\langle\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,\boldsymbol{\dot{\lambda}}(t)\cdot\left[\partial_{\boldsymbol{\lambda}}U(x,\boldsymbol{\lambda}(t))\right]_{x=x(t)}\right\rangle,$
(11lbtbzcddn)

where $\left\langle\cdots\right\rangle$ on the rhs means average over the
noise $\eta(t)$. In the equivalent ensemble picture, the PDF $\rho(x,t)$ obeys
the FP equation (11lbtbzcddd) and one has

$\left\langle
W\right\rangle=\int_{-\infty}^{+\infty}\\!\\!dx\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,\boldsymbol{\dot{\lambda}}(t)\cdot\partial_{\boldsymbol{\lambda}}U(x,\boldsymbol{\lambda}(t))\,\rho(x,t).$
(11lbtbzcddo)

Consistently, the average configurational heat ensures that Eq. (11lbtbzcddm)
holds in average,

$\displaystyle\left\langle Q_{\scriptsize\textnormal{conf}}\right\rangle$
$\displaystyle=\left\langle\Delta U\right\rangle-\left\langle
W\right\rangle=\int_{-\infty}^{+\infty}\\!\\!dx\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,U(x,t)\partial_{t}\rho(x,t)$
(11lbtbzcddp)
$\displaystyle=\int_{-\infty}^{+\infty}\\!\\!dx\int_{0}^{t_{\scriptsize\textnormal{f}}}\\!\\!dt\,[\partial_{x}U(x,t)]v(x,t)\rho(x,t).$

In order to have energy conservation—i.e. the first principle—we must
incorporate the kinetic contribution to the energy. In the overdamped regime,
the velocity degree of freedom instantaneously relaxes to equilibrium, so that
the average kinetic energy $\left\langle K\right\rangle=k_{B}T(t)/2$ and
therefore average heat has a kinetic contribution,

$\left\langle Q\right\rangle=\underbrace{\frac{1}{2}k_{B}\Delta
T}_{\left\langle Q_{\scriptsize\textnormal{kin}}\right\rangle}+\left\langle
Q_{\scriptsize\textnormal{conf}}\right\rangle.$ (11lbtbzcddq)

The first principle states

$\left\langle Q\right\rangle+\left\langle W\right\rangle=\left\langle\Delta
E\right\rangle,$ (11lbtbzcddr)

where $E$ is the total energy, $E=K+U$. For isothermal processes, $T$ is
constant and then there is no kinetic contribution to the heat. Conversely,
$\left\langle Q_{\scriptsize\textnormal{kin}}\right\rangle\neq 0$ for non-
isothermal processes: this is important, for example, to correctly define
finite-time adiabatic (in the sense of zero average heat) processes [83]—see
also Sec. 3.4.

##  Appendix C Conditions for the existence of Boltzmann’s breathers under
static confinement

We work here in the framework of the Boltzmann equation for dilute gases, as
described in Sec. 2.4. We have reported that when the external confining
potential $U$ is time-independent and harmonic, permanent (undamped) undriven
breathing modes do exist, with a time-dependent and periodic temperature. We
address here the reciprocal question: under which conditions on $U$ do such
solutions exist? We restrict to a one-dimensional dynamics for simplicity.

We start from the potential given by (11las), where the contribution $\bar{U}$
stems from Eq. (11laq). The confining potential then reads

$U(x,t)\,=\,\frac{1}{\sigma^{2}}U_{0}\left(\frac{x}{\sigma}\right)-\frac{m}{2}\frac{\ddot{\sigma}}{\sigma}x^{2}\,+\,\frac{b}{x^{2}},$
(11lbtbzcdds)

in which we require that $\sigma$ be time dependent, with nevertheless a time
independent $U$ on the left hand side. Taking the third derivative with
respect to $x$, we arrive at

$\partial_{x}^{3}U\,=\,\frac{1}{\sigma^{5}}U_{0}^{{}^{\prime\prime\prime}}\left(\frac{x}{\sigma}\right)-\frac{24\,b}{x^{5}},$
(11lbtbzcddt)

which should be independent of time, and thus independent of $\sigma$. This
implies that
$\widetilde{x}\,U_{0}^{IV}(\widetilde{x})=-5\,U_{0}^{{}^{\prime\prime\prime}}(\widetilde{x})=0$,
i.e. $U_{0}^{{}^{\prime\prime\prime}}(\widetilde{x})$ must be a power-law
function in $1/\widetilde{x}^{5}$. Subsequent integration yields

$U_{0}(x)\,=\,\frac{A}{x^{2}}+Bx^{2}+Cx,$ (11lbtbzcddu)

where $A$, $B$ and $C$ are constants. Plugging this back into Eq.
(11lbtbzcdds) and requiring once more the time-independence of $U$, we get
$C=0$. This indicates

  * •

what differential equation $\sigma$ should fulfil:

$\frac{B}{\sigma^{4}}-\frac{m}{2}\frac{\ddot{\sigma}}{\sigma}\,=\,{\cal
C}\\!\hbox{onstant}.$ (11lbtbzcddv)

This is an Ermakov-like relation for which
$d^{3}\beta/dt^{3}+4\Omega^{2}\dot{\beta}=0$ with $\Omega^{2}=2\,{\cal
C}\\!\hbox{onstant}/m$, see Sec. 2.4 and in particular Eqs. (11laz) and
(11lba). We recover the breathing modes found out by Boltzmann himself, and
described in [53, 52].

  * •

that apart from the $1/r^{2}$ contribution already mentioned in Sec. 2.4 (here
$1/x^{2}$), $U$ should be a harmonic potential.

While the present argument transposes directly to isotropic potentials of the
type

$U(\mathbf{r},t)\,=\,\frac{1}{\sigma^{2}}U_{0}\left(\frac{r}{\sigma}\right)-\frac{m}{2}\frac{\ddot{\sigma}}{\sigma}r^{2}\,+\,\frac{b}{r^{2}},$
(11lbtbzcddw)

generalization to higher dimensions does not seem straightforward.

##  Appendix D Derivation of the expression for the minimum irreversible work

In this appendix, we give more details of the derivation of the expression for
the minimum irreversible work. The calculation below somehow puts on a common
ground the approaches developed in Refs. [95, 110, 111].

Our starting point is the Burgers equation (11lbtbzcc), which must be jointly
solved with the FP equation (11lbta) to find the optimal profiles for
$\rho(x,t)$ and $v(x,t)$. On the one hand, the characteristic system of ODEs
(Lagrange-Charpit equations) for the Burgers equation is

$\frac{dt}{ds}=1,\quad\frac{dx}{ds}=v,\quad\frac{dv}{ds}=0.$ (11lbtbzcddx)

Therefore, $s=t$, $v=v_{\scriptsize\textnormal{i}}$, and
$x=v_{\scriptsize\textnormal{i}}s+x_{\scriptsize\textnormal{i}}$. At the
initial time $t=t_{\scriptsize\textnormal{i}}=0$, we have that
$v_{\scriptsize\textnormal{i}}=\varphi(x_{\scriptsize\textnormal{i}})$, where
$\varphi$ is an an arbitrary smooth function, and thus
$v=\varphi(x_{\scriptsize\textnormal{i}})$, with
$x_{\scriptsize\textnormal{i}}=x-vt$. The solution of the Burgers equation can
be given in implicit form,

$v=\varphi(x-vt).$ (11lbtbzcddy)

On the other hand, the characteristic system of ODEs for the FP equation is

$\frac{dt}{ds}=1,\quad\frac{dx}{ds}=v,\quad\frac{d\rho}{ds}=-\rho\,\partial_{x}v.$
(11lbtbzcddz)

The two first equations are the same as for the Burgers equation, so we focus
on the third one,

$\frac{d\rho}{ds}=-\frac{\varphi^{\prime}(x_{\scriptsize\textnormal{i}})}{1+s\varphi^{\prime}(x_{\scriptsize\textnormal{i}})}\rho,$
(11lbtbzcdea)

which can be readily integrated to give

$\rho(s)=\frac{\rho_{\scriptsize\textnormal{i}}(x_{\scriptsize\textnormal{i}})}{1+s\varphi^{\prime}(x_{\scriptsize\textnormal{i}})}.$
(11lbtbzcdeb)

We have taken into account that
$\rho(s=0)=\rho_{\scriptsize\textnormal{i}}(x_{\scriptsize\textnormal{i}})$.
Going back to the original variables $(x,t)$,

$\rho(x,t)=\frac{\rho_{\scriptsize\textnormal{i}}(x-vt)}{1+t\varphi^{\prime}(x-vt)},$
(11lbtbzcdec)

in which $v(x,t)$ is in turn implicitly given by Eq. (11lbtbzcddy). Equation
(11lbtbzcdec) can also be written in an equivalent form: making use of Eq.
(11lbtbzcddy), $v_{x}=\left(1-tv_{x}\right)\varphi^{\prime}(x-vt)$ and

$v_{x}=\frac{\varphi^{\prime}(x-vt)}{1+t\varphi^{\prime}(x-vt)}\Rightarrow
1-tv_{x}=\frac{1}{1+t\varphi^{\prime}(x-vt)}.$ (11lbtbzcded)

Therefore, one has that

$\rho(x,t)=(1-tv_{x})\rho_{\scriptsize\textnormal{i}}(x-vt),$ (11lbtbzcdee)

and

$dx\rho(x,t)=dx_{\scriptsize\textnormal{i}}\rho_{\scriptsize\textnormal{i}}(x_{\scriptsize\textnormal{i}},t),\quad
x_{\scriptsize\textnormal{i}}\equiv x-vt.$ (11lbtbzcdef)

Now, the function $\varphi$ is determined by imposing that the system must
reach the target distribution $\rho_{\scriptsize\textnormal{f}}(x)$ at the
final time $t=t_{\scriptsize\textnormal{f}}$, i.e.

$\displaystyle\rho_{\scriptsize\textnormal{f}}(x)$
$\displaystyle=\frac{\rho_{\scriptsize\textnormal{i}}(x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}})}{1+t_{\scriptsize\textnormal{f}}\,\varphi^{\prime}(x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}})},$
(11lbtbzcdega) $\displaystyle v_{\scriptsize\textnormal{f}}(x)$
$\displaystyle\equiv
v(x,t_{\scriptsize\textnormal{f}})=\varphi(x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}}).$
(11lbtbzcdegb)

Taking spatial derivative in the last equation and inserting the result into
the previous one, we get

$\rho_{\scriptsize\textnormal{f}}(x)=\rho_{\scriptsize\textnormal{i}}(x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}})\left[1-v^{\prime}_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}}\right],$
(11lbtbzcdegeh)

which determines $v_{\scriptsize\textnormal{f}}(x)$ or, equivalently, the (up
to now) unknown function $\varphi$.

Zhang’s solution to the optimization problem [110, 111] is recovered by going
to the cumulative distribution function defined in Eq. (11lbp). Integrating
Eq. (11lbtbzcdegeh) from $-\infty$ to some arbitrary point $x$, one gets

$\mathcal{F}_{\scriptsize\textnormal{f}}(x)=\mathcal{F}_{\scriptsize\textnormal{i}}(x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}}),$
(11lbtbzcdegei)

after assuming that
$\lim_{x\to-\infty}\left[x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}}\right]=-\infty$;
$\mathcal{F}_{\scriptsize\textnormal{i}}$ and
$\mathcal{F}_{\scriptsize\textnormal{f}}$ are the cumulative distributions
corresponding to $\rho_{\scriptsize\textnormal{i}}$ and
$\rho_{\scriptsize\textnormal{f}}$, respectively. Since both
$\rho_{\scriptsize\textnormal{i}}$ and $\rho_{\scriptsize\textnormal{f}}$ are
positive definite for equilibrium states,
$\mathcal{F}_{\scriptsize\textnormal{i}}$ and
$\mathcal{F}_{\scriptsize\textnormal{f}}$ are strictly increasing functions
and we can explicitly solve the above equation for
$v_{\scriptsize\textnormal{f}}(x)$,

$x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}}=\mathcal{F}_{\scriptsize\textnormal{i}}^{-1}\left(\mathcal{F}_{\scriptsize\textnormal{f}}(x)\right),$
(11lbtbzcdegej)

where $\mathcal{F}_{\scriptsize\textnormal{i}}^{-1}$ is the inverse function
of $\mathcal{F}_{\scriptsize\textnormal{i}}$.999Note that Eq. (11lbtbzcdegej)
ensures that
$\lim_{x\to-\infty}\left[x-v_{\scriptsize\textnormal{f}}(x)t_{\scriptsize\textnormal{f}}\right]=-\infty$,
since $\mathcal{F}_{\scriptsize\textnormal{f}}(-\infty)=0$ and
$\mathcal{F}_{\scriptsize\textnormal{i}}^{-1}(0)=-\infty$. By defining the
function

$\Xi(x)\equiv\mathcal{F}_{\scriptsize\textnormal{i}}^{-1}\left(\mathcal{F}_{\scriptsize\textnormal{f}}(x)\right),$
(11lbtbzcdegek)

we get that

$v_{\scriptsize\textnormal{f}}(x)=\frac{x-\Xi(x)}{t_{\scriptsize\textnormal{f}}}.$
(11lbtbzcdegel)

Note that determining $v_{\scriptsize\textnormal{f}}(x)$ is utterly equivalent
to determining $\varphi(x)$, because Eq. (11lbtbzcdegb) tells us that

$v_{\scriptsize\textnormal{f}}(x)=\varphi(\Xi(x))\Rightarrow\varphi(x)=v_{\scriptsize\textnormal{f}}(\Xi^{-1}(x))=\frac{\Xi^{-1}(x)-x}{t_{\scriptsize\textnormal{f}}}.$
(11lbtbzcdegem)

It is useful to recall that, in the connection problem we are addressing, both
$\mathcal{F}_{\scriptsize\textnormal{i}}$ and
$\mathcal{F}_{\scriptsize\textnormal{f}}$ are given, so that the function
$\Xi(x)$ in (11lbtbzcdegek) is perfectly defined. Then, the function
$\varphi(x)$ follows from Eq. (11lbtbzcdegem) and $v(x,t)$ is thus implicitly
provided by (11lbtbzcddy), which completes the solution to the optimization
problem.

It is important to remark that shocks cannot appear in the obtained solution
when connecting equilibrium states,

$1+t\varphi^{\prime}(x)=1+\frac{t}{t_{\scriptsize\textnormal{f}}}\left(\frac{d}{dx}\Xi^{-1}(x)-1\right)>0$
(11lbtbzcdegen)

for all times $t\in[0,t_{\scriptsize\textnormal{f}}]$. Proving the above
inequality is equivalent to prove that the term inside the parenthesis in Eq.
(11lbtbzcdegen) is always larger than $-1$ or, equivalently, that
$\frac{d}{dx}\Xi^{-1}(x)>0$ for all $x$. This is indeed true, since Eq.
(11lbtbzcdegek) can be rewritten as

$\mathcal{F}_{\scriptsize\textnormal{i}}(x)=\mathcal{F}_{\scriptsize\textnormal{f}}(\Xi^{-1}(x))\Rightarrow\frac{d}{dx}\Xi^{-1}(x)=\frac{\rho_{\scriptsize\textnormal{i}}(x)}{\rho_{\scriptsize\textnormal{f}}(\Xi^{-1}(x))}>0.$
(11lbtbzcdegeo)

Bringing to bear Eqs. (11lbtbzcddy), (11lbtbzcdec), (11lbtbzcdef), and
(11lbtbzcdegem), the irreversible power over the optimal protocol is

$\displaystyle P_{\scriptsize\textnormal{irr}}^{*}$
$\displaystyle=\gamma\int_{-\infty}^{+\infty}\\!\\!\\!dx\,\rho(x,t)v^{2}(x,t)$
(11lbtbzcdegep)
$\displaystyle=\gamma\int_{-\infty}^{+\infty}\\!\\!\\!dx\,(1-tv_{x})\rho_{\scriptsize\textnormal{i}}(x-vt)\varphi^{2}(x-vt)$
$\displaystyle=\gamma\int_{-\infty}^{+\infty}\\!\\!\\!dx_{\scriptsize\textnormal{i}}\,\rho_{\scriptsize\textnormal{i}}(x_{\scriptsize\textnormal{i}})\left[\frac{\Xi^{-1}(x_{\scriptsize\textnormal{i}})-x_{\scriptsize\textnormal{i}}}{t_{\scriptsize\textnormal{f}}}\right]^{2}$
$\displaystyle=\frac{\gamma}{t_{\scriptsize\textnormal{f}}^{2}}\int_{-\infty}^{+\infty}\\!\\!\\!dx_{\scriptsize\textnormal{i}}\,\rho_{\scriptsize\textnormal{i}}(x_{\scriptsize\textnormal{i}})\left[\Xi^{-1}(x_{\scriptsize\textnormal{i}})-x_{\scriptsize\textnormal{i}}\right]^{2}.$

where Eq. (11lbtbzcdee) has been used and $x_{\scriptsize\textnormal{i}}$ was
defined in Eq. (11lbtbzcdef). The expression (11lbtbzcdegep) for the optimal
irreversible power is identical to Eq. (11lbtbzcdci) in the main text.
Therefrom, the minimum irreversible work is readily obtained,

$\displaystyle
W_{\scriptsize\textnormal{irr}}^{*}=t_{\scriptsize\textnormal{f}}P_{\scriptsize\textnormal{irr}}^{*}=\frac{\gamma}{t_{\scriptsize\textnormal{f}}}\int_{-\infty}^{+\infty}\\!\\!\\!dx_{\scriptsize\textnormal{i}}\,\rho_{\scriptsize\textnormal{i}}(x_{\scriptsize\textnormal{i}})\left[\Xi^{-1}(x_{\scriptsize\textnormal{i}})-x_{\scriptsize\textnormal{i}}\right]^{2},$
(11lbtbzcdegeq)

which is again identical to Eq. (11lbtbzcdcj)—we repeat it here to keep the
appendix self-contained.

##  Appendix E Comparison of speed limit inequalities for the harmonic case

It is illuminating to compare inequalities (11lbtbzcdcn) and (11lbtbzcdcs) for
the harmonic case, for which the probability distribution is Gaussian for all
times,
$\rho(x,t)=(2\pi\sigma^{2})^{-1/2}\exp\left(-\frac{x^{2}}{2\sigma^{2}}\right)$.
A simple calculation leads to

$P_{\scriptsize\textnormal{irr}}(t)=\gamma\dot{\sigma}^{2},\quad
I(t)=2\left(\frac{\dot{\sigma}}{\sigma}\right)^{2}.$ (11lbtbzcdeger)

Making use of these expressions, we have that

$\displaystyle\mathcal{L}_{\scriptsize\textnormal{lin}}=\sqrt{\gamma}\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\left|{\dot{\sigma}}\right|\geq\sqrt{\gamma}\left|{\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\dot{\sigma}}\right|=\sqrt{\gamma}\left|{\sigma_{\scriptsize\textnormal{f}}-\sigma_{\scriptsize\textnormal{i}}}\right|,$
(11lbtbzcdegesa)
$\displaystyle\mathcal{L}=\sqrt{2}\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\left|{\frac{d}{dt}\ln\sigma}\right|\geq\sqrt{2}\left|{\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\frac{d}{dt}\ln\sigma}\right|=\sqrt{2}\,\ln\\!\left|{\frac{\sigma_{\scriptsize\textnormal{f}}}{\sigma_{\scriptsize\textnormal{i}}}}\right|,$
(11lbtbzcdegesb)

and thus

$t_{\scriptsize\textnormal{f}}\,W_{\scriptsize\textnormal{irr}}\geq\gamma\left|{\sigma_{\scriptsize\textnormal{f}}-\sigma_{\scriptsize\textnormal{i}}}\right|^{2},\quad
t_{\scriptsize\textnormal{f}}\,\mathcal{C}\geq\left(\ln\left|{\frac{\sigma_{\scriptsize\textnormal{f}}}{\sigma_{\scriptsize\textnormal{i}}}}\right|\right)^{2}.$
(11lbtbzcdegeset)

On the one hand, these inequalities do not immediately translate to bounds for
the connecting time, because both $W_{\scriptsize\textnormal{irr}}$ and
$\mathcal{C}$ depend on $t_{\scriptsize\textnormal{f}}$—and on the path swept
by the system to go from the initial to the target state. In fact, in the
harmonic case we are analysing

$W_{\scriptsize\textnormal{irr}}=\gamma\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\dot{\sigma}^{2},\quad\mathcal{C}=\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\left(\frac{d}{dt}\ln\sigma\right)^{2}.$
(11lbtbzcdegeseu)

If one assumes that all the time dependence occurs through a reduced time
$s=t/t_{\scriptsize\textnormal{f}}$, i.e. $\sigma=\sigma(s)$,

$W_{\scriptsize\textnormal{irr}}=\frac{\gamma}{t_{\scriptsize\textnormal{f}}}\int_{0}^{1}ds\,\left(\frac{d\sigma}{ds}\right)^{2},\quad\mathcal{C}=\frac{1}{t_{\scriptsize\textnormal{f}}}\int_{0}^{t_{1}}ds\,\left(\frac{d}{ds}\ln\sigma\right)^{2},$
(11lbtbzcdegesev)

and $t_{\scriptsize\textnormal{f}}$ disappears from Eq. (11lbtbzcdegeset). On
the other hand, both inequalities are equivalent in the linear response
regime—a property that is somehow expected, since the Fisher information
$I(t)$ is directly related to the friction tensor
$\boldsymbol{\zeta}(\boldsymbol{\lambda})$ in Eq. (11lbtbzcdcl) [104]. Within
linear response, $\sigma=\sigma_{\scriptsize\textnormal{f}}+\delta\sigma$ and
non-linearities in $\delta\sigma$ are neglected. Therefore, we have that

$\mathcal{L}_{\scriptsize\textnormal{lin}}\sim\sqrt{\gamma}\left|{\delta\sigma_{\scriptsize\textnormal{i}}}\right|,\quad\mathcal{L}\sim\frac{\sqrt{2}}{\sigma_{\scriptsize\textnormal{f}}}\left|{\delta\sigma_{\scriptsize\textnormal{i}}}\right|,$
(11lbtbzcdegesew)

and

$W_{\scriptsize\textnormal{irr}}\sim\gamma\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\left(\frac{d}{dt}\delta\sigma\right)^{2},\quad\mathcal{L}\sim\frac{1}{\sigma_{\scriptsize\textnormal{f}}^{2}}\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\left(\frac{d}{dt}\delta\sigma\right)^{2}.$
(11lbtbzcdegesex)

Finally, both inequalities can be cast in the same form,

$t_{\scriptsize\textnormal{f}}\int_{0}^{t_{\scriptsize\textnormal{f}}}dt\,\left(\frac{d}{dt}\delta\sigma\right)^{2}\geq\left|{\delta\sigma_{\scriptsize\textnormal{i}}}\right|^{2}.$
(11lbtbzcdegesey)

Consistently with our notation,
$\delta\sigma_{\scriptsize\textnormal{i}}=\sigma_{\scriptsize\textnormal{i}}-\sigma_{\scriptsize\textnormal{f}}$.

##  Appendix F A shortcut to the fluctuation relation

The idea of counterdiabatic driving, in the framework of the Langevin/Fokker-
Planck equation used in Sec. 3.2, offers an economical derivation of the work
fluctuation relation [77]. In a potential $U(x,{\boldsymbol{\lambda}}(t))$
where $\boldsymbol{\lambda}(t)$ stands for the collection of external
parameters controlled by an operator (say, the experimentalist), the particle
density $\rho$ obeys Eq. (11lbf) with fixed temperature (i.e. fixed $\beta$).
Here, it is understood that a given protocol is chosen from the outset,
through the time dependence of the steering parameter
${\boldsymbol{\lambda}}$. What would be the counterdiabatic force to be
applied, such that the particle density would be forced to follow the
equilibrium distribution? In other words, we seek to enforce

$\rho(x,t)=\rho_{eq}(x,{\boldsymbol{\lambda}}(t))=\frac{1}{Z({\boldsymbol{\lambda}}(t))}e^{-\beta
U(x,{\boldsymbol{\lambda}}(t))}$ (11lbtbzcdegesez)

with the partition function

$Z({\boldsymbol{\lambda}})=\int_{-\infty}^{\infty}e^{-\beta
U(x,{\boldsymbol{\lambda}})}\,dx,$ (11lbtbzcdegesfa)

assumed well defined for all ${\boldsymbol{\lambda}}$ (i.e. $U$ is supposed to
be confining enough). The required counterdiabatic term is given by Eq.
(11lbo). Introducing the Fokker-Planck operator ${\cal L}_{u}$ associated to
$U(x,{\boldsymbol{\lambda}}(t))$, such that

${\cal
L}_{u}\rho\,\equiv\,\partial_{x}[\rho\,\partial_{x}U(x,{\boldsymbol{\lambda}}(t))]+\beta^{-1}\partial^{2}_{x}\rho$
(11lbtbzcdegesfb)

the evolution equation for $\rho_{eq}$ can be written

$\displaystyle\partial_{t}\rho_{eq}(x,{\boldsymbol{\lambda}}(t))$
$\displaystyle={\cal
L}_{u}\rho_{eq}-\beta\left(\dot{W}-\left\langle\dot{W}\right\rangle\right)\rho_{eq}$
(11lbtbzcdegesfc)
$\displaystyle=-\beta\left(\dot{W}-\left\langle\dot{W}\right\rangle\right)\rho_{eq}.$
(11lbtbzcdegesfd)

Here, we have introduced the instantaneous power

$\dot{W}(x,t)\,=\,\dot{\boldsymbol{\lambda}}\cdot\partial_{\boldsymbol{\lambda}}U(x,{\boldsymbol{\lambda}}(t)),$
(11lbtbzcdegesfe)

the dot denotes a time derivative, and the bracket is for an average with
weight $\rho_{eq}$:

$\left\langle\dot{W}\right\rangle\,=\,\int_{-\infty}^{\infty}\dot{W}\,\rho_{eq}(x,{\boldsymbol{\lambda}})\,dx.$
(11lbtbzcdegesff)

Note that integrating $\dot{W}$ over a trajectory, one obtains the
corresponding stochastic work $W$.

While relation (11lbtbzcdegesfd) is straightforwardly obtained from the very
definition of the equilibrium density in Eq. (11lbtbzcdegesez), we are here
more interested in the seemingly more complex form in Eq. (11lbtbzcdegesfc).
Consider indeed the auxiliary dynamics defined by the evolution equation

$\partial_{t}{\cal Q}(x,t)\,=\,{\cal L}_{u}{\cal Q}-\beta\dot{W}{\cal Q}$
(11lbtbzcdegesfg)

and initial condition

${\cal
Q}(x,t=0)\,=\,\rho_{eq}(x,{\boldsymbol{\lambda}}(0))=\frac{1}{Z({\boldsymbol{\lambda}}(0))}e^{-\beta
U(x,{\boldsymbol{\lambda}}(0))}.$ (11lbtbzcdegesfh)

Compared to Eq. (11lbtbzcdegesfc), the term in $\beta\langle\dot{W}\rangle Q$
is missing in (11lbtbzcdegesfg). To proceed, it is instructive to reinterpret
the dynamics encoded in (11lbtbzcdegesfc) as follows. A collection of
independent random walkers, diffusing in the force field $-\partial_{x}U$
(hence the presence of the operator ${\cal L}_{u}$), is subject to a
population/depopulation mechanism, such that walkers are added at rate
$\beta\dot{W}$. This leads to an overall population change, such that when $U$
increases locally, $\dot{W}>0$ and particles are removed, to match a lesser
probability of presence. Conversely, when $\dot{W}<0$, particles are added
locally. The counter-term in $\langle\dot{W}\rangle\rho_{eq}$ has the effect
to correct for this gain/loss, and conserve globally the total number of
walkers. The resulting integral $\int_{-\infty}^{+\infty}\rho_{eq}dx$ is
thereby conserved. The counter-term does not change the $x$ dependence; it
only affects normalization, and can be viewed as the action of a global
rescaling $\rho_{eq}\to\Lambda\rho_{eq}$. If this counter-term is absent, as
in the case in Eq. (11lbtbzcdegesfg), we obtain the unnormalized solution

${\cal Q}(x,t)\,=\,\frac{1}{Z({\boldsymbol{\lambda}}(0))}e^{-\beta
U(x,{\boldsymbol{\lambda}}(t))}.$ (11lbtbzcdegesfi)

The final step in the argument is the realization that $\cal Q$ can be
reinterpreted as [221]

${\cal Q}(x,t)=\left\langle e^{-\beta W}\delta(x-x(t))\right\rangle.$
(11lbtbzcdegesfj)

Here, the brackets correspond to averaging over all trajectories starting from
a point $x_{0}$ drawn according to the distribution
$\rho_{eq}(x_{0},{\boldsymbol{\lambda}}(0))$, and that end up at position $x$
at time $t$. The stochastic work is

$W\,=\,\int_{0}^{t}\dot{\boldsymbol{\lambda}}(\tau)\cdot\partial_{\boldsymbol{\lambda}}U(x(\tau),{\boldsymbol{\lambda}}(\tau))\,d\tau.$
(11lbtbzcdegesfk)

This is a facet of the Feynman-Kac correspondence. Since

$\left\langle e^{-\beta
W}\delta(x-x(t))\right\rangle\,=\,\frac{1}{Z({\boldsymbol{\lambda}}(0))}e^{-\beta
U(x,{\boldsymbol{\lambda}}(t))}$ (11lbtbzcdegesfl)

we can integrate over $x$ on both sides to get

$\left\langle e^{-\beta
W}\right\rangle\,=\,\frac{Z({\boldsymbol{\lambda}}(t))}{Z({\boldsymbol{\lambda}}(0))}\equiv
e^{-\beta\Delta F}$ (11lbtbzcdegesfm)

where the last equality serves to define the free-energy difference $\Delta
F$. The above relation, valid at all time $t$ for an arbitrary protocol
${\boldsymbol{\lambda}}(t)$, is the work fluctuation relation [77].

## Acknowledgments

We would like to thank A. Baldassarri, M. Baldovin, L. Bellon, B. Besga, A.
Chepelianskii, S. Ciliberto, S. Dago, T. de Guillebon, S.N. Majumdar, I.
Martínez, J.G. Muga, I. Palaia, A. Patrón, A. Petrossian, A. Puglisi, D.
Raynal, L. Rondin, N. Ruiz-Pino. C.A.P. and A.P. acknowledge financial support
from Grant PGC2018-093998-B-I00 funded by MCIN/AEI/10.13039/501100011033/ and
by ERDF “A way of making Europe.” C.A.P. also acknowledges financial support
from Junta de Andalucía and European Social Fund through the program PAIDI-
DOCTOR. This work was supported by the ‘Agence Nationale de la Recherche’
grant No. ANR-18-CE30-0013. C.J. acknowledges financial support from the
Simons Foundation through Award No. 681566, and from the U.S. National Science
Foundation under Grant No. 2127900.

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