Geodesic path for the minimal energy cost in shortcuts to isothermality

Geng Li,1 Jin-Fu Chen,2, 1 C. P. Sun,1, 2 and Hui Dong1, ∗

1Graduate School of China Academy of Engineering Physics, Beijing 100193, China2Beijing Computational Science Research Center, Beijing 100193, China

Shortcut to isothermality is a driving strategy to steer the system to its equilibrium states withinfinite time, and enables evaluating the impact of a control promptly. Finding optimal scheme tominimize the energy cost is of critical importance in applications of this strategy in pharmaceuticaldrug test, biological selection, and quantum computation. We prove the equivalence between de-signing the optimal scheme and finding the geodesic path in the space of control parameters. Suchequivalence allows a systematic and universal approach to find the optimal control to reduce theenergy cost. We demonstrate the current method with examples of a Brownian particle trapped incontrollable harmonic potentials.

Introduction.– Boosting system to its steady state iscritical to promptly evaluate the impact of a control [1–8]. In biological systems, the quest to timely evaluatethe impact of therapy or genotypes posts a requirementto steer the system to reach its steady state with a con-siderable tunable rate [1–5]. In adiabatic quantum com-putation, the task of solving the optimization problemis converted to the problem of driving systems from atrivial ground state to another nontrivial ground state.The speedup of the computational process needs to steerthe system to the target ground state in finite time [6–8].These quests to tune the system within finite time whilekeep it in equilibrium is eagerly needed.

Shortcut to isothermality was proposed as a finite-time driving strategy to steer the system evolving alongthe path of instantaneous equilibrium states [9]. Thestrategy has been applied in reducing transition timebetween equilibrium states [10–12], improving the effi-ciency of free-energy estimation [13], constructing finite-time heat engines [14–16], and controlling biological evo-lutions [4, 5]. The cost of the finite-time operation is theadditional energy cost due to irreversibility posted bythe fundamental thermodynamic law. Minimizing suchcost is in turn relevant to optimize the heat engine [17–19] and reconstruct the energy landscape of biologicalmacromolecules [20–22]. A question arises naturally, howto find the optimal control protocol to minimize the ir-reversible energy cost in shortcuts to isothermality.

In this Letter, we present a systematic approach forfinding the optimal protocol to minimize the energy cost.In Fig. 1, we show the equivalence of designing theoptimal control to finding the geodesic path on a Rie-mannian manifold, spanned by the control parameters[23–27]. In turn, the powerful tools developed in geome-try are adapted for solving the optimal control protocol.Our scheme is exemplified with a single Brownian parti-cle in the harmonic potential with controllable stiffnessand central position.

Geometric approach – The system is described by theHamiltonian Ho(~x, ~p,~λ) =

∑i p

2i /2 +Uo(~x, ~p,~λ) with the

∗ hdong@gscaep.ac.cn

Minimize energy cost

Geodesic path

Geometric space

Find geodesic path

(a) (b)Phase space

FIG. 1. (Color online) The equivalence between designingthe optimal control protocol and finding the geodesic path inthe parametric space. (a) The evolution of the system con-trolled by the shortcut scheme. An auxiliary HamiltonianHa = ~λ · ~f is added to steer the evolution along the instanta-neous equilibrium state ρeq(~λ(t)) of the original HamiltonianHo. Designing the optimal control protocol normally requiresto minimize the energy cost in the shortcut scheme. (b) Thegeodesic path in the equivalent geometric space. We can con-vert the designing task into finding the geodesic path in thegeometric space with the metric gµν = γ

∑i〈∂pifµ∂pifν〉eq.

coordinate ~x ≡ (x1, x2, · · · , xN ) and the momentum ~p ≡(p1, p2, · · · , pN ). It is immersed in a thermal reservoirwith a constant temperature T . ~λ(t) ≡ (λ1, λ2, · · · , λM )are time-dependent control parameters. For simplicity,we have set the mass of the system as a unit. In theshortcut scheme, an auxiliary Hamiltonian Ha(~x, ~p, t) isadded to steer the system evolving along the instanta-neous equilibrium states of the original Hamiltonian Ho

in the finite-time interval τ with boundary conditionsHa(0) = Ha(τ) = 0. The dynamical evolution underthe total Hamiltonian H = Ho +Ha is described by theLangevin equation as

xi =∂H

∂pi,

pi = −∂H∂xi− γxi + ξi(t), (1)

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where γ is the dissipation rate and ~ξ ≡ (ξ1, ξ2, · · · , ξN )are random variables of the Gaussian white noise.The evolution equation of the system distribu-tion ρ(~x, ~p, t) = δ(~x − ~x(t))δ(~p − ~p(t)) for a tra-jectory [~x(t), ~p(t)] is described by the Liouvilleequation as ∂tρ = −∑i[∂xi(xiρ) + ∂pi(piρ)]. Byaveraging over different noise realizations [~ξ(t)],we obtain the evolution of the observable prob-ability distribution P (~x, ~p, t) ≡ 〈ρ(~x, ~p, t)〉~ξ =sD[~x(t)]D[~p(t)]T [~x(t), ~p(t)]δ(~x − ~x(t))δ(~p − ~p(t))

as [28]

∂P

∂t=∑

i

[− ∂

∂xi(∂H

∂piP )+

∂

∂pi(∂H

∂xiP+γ

∂H

∂piP )+

γ

β

∂2P

∂p2i],

(2)where β ≡ 1/(kBT ) is the inverse temperature with theBoltzmann constant kB. Here T [~x(t), ~p(t)] is the proba-bility of the trajectory [~x(t), ~p(t)] associated with a noiserealization [~ξ(t)] [29]. To ensure the instantaneous equi-librium distribution

P (~x, ~p, t) = Peq(~x, ~p,~λ) = eβ[F (~λ)−Ho(~x,~p,~λ)], (3)

the auxiliary Hamiltonian is proved [9] to have the formHa(~x, ~p, t) = ~λ · ~f(~x, ~p,~λ) with ~f(~x, ~p,~λ) satisfying

∑

i

[γ

β

∂2fµ∂p2i

−γpi∂fµ∂pi

+∂fµ∂pi

∂Uo

∂xi−pi

∂fµ∂xi

] =dF

dλµ− ∂Uo

∂λµ,

(4)where F ≡ −β−1 ln[

sd~xd~p exp(−βHo)] is the free en-

ergy. The boundary conditions are presented explicitlyas ~λ(0) = ~λ(τ) = 0.

The cost of the energy in the shortcut scheme is eval-uated by the average work W ≡

⟨∫ τ0dt∂tH

⟩~ξ[30–33],

explicitly as

W = ∆F + γ∑

i

τw

0

dtx

d~xd~p

(∂Ha

∂pi

)2

Peq, (5)

where ∆F = F (~λ(τ))− F (~λ(0)) is the free energy differ-ence. Detailed derivation of Eq. (5) is presented in thesupplementary materials [28]. To consider the finite-timeeffect, we define the irreversible work Wirr ≡ W − ∆F,which follows

Wirr = γ∑

µνi

τw

0

dtλµλν

⟨∂fµ∂pi

∂fν∂pi

⟩

eq

, (6)

with 〈·〉eq =sd~xd~p [·]Peq. It follows from Eq. (6) that

the integrand scales as τ−2 through reducing the time s ≡t/τ , which results in the 1/τ scaling [23] of the irreversiblework, i.e, Wirr ∝ 1/τ . Such a 1/τ scaling, predictedin various finite-time studies [18, 34–43], was recentlyverified for the ideal gas system [44] at the long-timelimit. It is worth noting that in the shortcut scheme thecurrent scaling is valid for any duration time τ with norequirement of the long-time limit [23, 25, 41, 45, 46].

In the space of the control parameters ~λ, we define apositive semi-definite metric

gµν = γ∑

i

⟨∂fµ∂pi

∂fν∂pi

⟩

eq

, (7)

whose positive semi-definiteness is proved in the supple-mentary materials [28]. With this metric, the lengthof a curve in the current geometric space is character-ized via the thermodynamic length [23, 25–27, 45] as

L =∫ τ0dt∑µν

√λµλνgµν , which provides a lower bound

of the irreversible work Wirr as

Wirr ≥L2

τ. (8)

The lower bound is reached with the optimal controlscheme ~λ(t) (0 < t < τ), determined by the geodesicequation

λµ +∑

νκ

Γµνκλν λκ = 0, (9)

with the given boundary conditions ~λ(0) and ~λ(τ).Here the Christoffel symbol is defined as Γµνκ ≡12

∑ι(g−1)ιµ(∂λκgιν + ∂λνgικ − ∂λιgνκ). For the case

with the single control parameter λ(t), the analytical so-lution [26] for Eq. (9) is obtained as λ(t) = (λ(τ) −λ(0))g(λ(t))−1/

∫ τ0dt′g(λ(t′))−1, with g = γ〈(∂pf)2〉.

For the case with multiple parameters, the shootingmethod is an available option which treats the two-pointboundary-value problem as an initial-value problem [47].See the supplementary materials for details about theshooting method to our problems [28].

The strategy of current formalism is shown in Fig. 1.Firstly, we obtain the control operators ~f(~x, ~p,~λ) in Fig.1(a) by solving Eq. (4). Secondly, the metric gµν in Fig.1(b) for the parametric space is calculated via Eq. (7).Finally, the optimal control is obtained by solving thegeodesic equation in Eq. (9). The current strategy pro-vides an effective approach to find the optimal control tominimize the energy cost, i.e., the total work done dur-ing the shortcut-to-isothermal process. The strategy isillustrated through two examples with one or two controlparameters as follows.

Brownian motion in the harmonic potential– TheBrownian particle is trapped by the one-dimensionalbreathing harmonic potential with tunable stiffness λ(t)under the Hamiltonian Ho(x, p, λ) = p2/2 + λ(t)x2/2.Its auxiliary Hamiltonian was derived in Ref. [9] asHa(x, p, t) = λf(x, p, λ) with f = 1/(4γλ)[(p − γx)2 +λx2]. The metric in Eq. (7) in this case reduces to [28]

g =λ+ γ2

4γβλ3. (10)

And the lower bound of the irreversible work is reachedby the protocol satisfying the geodesic equation λ +

3

λ2∂λg/(2g) = 0. The solution

λgp(t) =

√1 + 2γ2(ms − nst/τ) + 1

2(ms − nst/τ), (11)

offers an optimal protocol to minimize the energy costin the shortcut scheme. Here ms = 1/λ(0) + γ2/(2λ(0))and ns = 1/λ(0) + γ2/(2λ(0))− 1/λ(τ)− γ2/(2λ(τ)) areconstants for single control-parameter case. And the ir-reversible work of the geodesic protocol reaches its min-imum Wmin

irr =∫ τ0λ2gdt = n2s/τ , which is consistent

with the lower bound given by the thermodynamic length

L =∫ τ0

√λ2gdt = ns through the relationWmin

irr = L2/τ .Underdamped Brownian motion with two control pa-

rameters –We consider a Brownian particle moving inthe one-dimensional harmonic potential with Hamilto-nian Ho(x, p, λ) = p2/2 + λ1x

2/2 − λ2x. The auxiliaryHamiltonian for the shortcut scheme takes the form [28]Ha(x, p, t) =

∑2µ=1 λµfµ(x, p, λ1, λ2) with

f1 =(p− γx)

2+ λ1x

2

4γλ1− λ2p

2λ21+ (

γλ22λ21− λ2

2γλ1)x,

f2 =p

λ1− γx

λ1. (12)

The metric in Eq. (7) for the control parameters ~λ isobtained as

g =

(1

4βγλ21

+ γ4βλ3

1+

γλ22

λ41−γλ2

λ31

−γλ2

λ31

γλ21

). (13)

The geodesic equation follows

λ1 −λ21(3γ2 + 2λ1)

2λ1(γ2 + λ1)= 0,

λ2 −2λ1λ2λ1

+λ21λ2(γ2 + 2λ1)

2λ21(γ2 + λ1)= 0, (14)

with the boundary conditions ~λ(0) and ~λ(τ) .The optimal scheme can be obtained by solving equa-

tions above using a general numerical method, i.e., theshooting method [47]. Here we firstly solve these equa-tions numerically to provide a general perspective on ourscheme. With the initial point ~λ(0), we choose an ini-tial rate ~λ(0+) and solve the geodesic equation with theEular algorithm to obtain a trial solution ~λtri(τ). New-ton’s method is utilized for updating the rate ~λ(0+) toreduce the distance between the trial solution ~λtri(τ) andthe target point ~λ(τ). In the simulation, we have chosenthe parameters ~λ(0) = (1, 1), ~λ(τ) = (16, 2), kBT = 1,and γ = 1. The geodesic path for the optimal control isillustrated as ~λgp,n(t) (triangles) in Fig. 2.

Fortunately, the analytical geodesic protocol for

Eq. (14) can be obtained as

λ1 =wb

τ

√λ31

λ1 + γ2,

λ2λ1

= mbt/τ + nb, (15)

where wb = −[2√

1 + γ2/λ1 + ln(√

1 + γ2/λ1 −1) − ln(

√1 + γ2/λ1 + 1)]|λ1(τ)

λ1(0), mb = (λ2(τ)λ1(0) −

λ2(0)λ1(τ))/(λ1(τ)λ1(0)), and nb = λ2(0)/λ1(0) are con-stants. In Fig. 2, we show the match between the optimalcontrol obtained from the numerical calculation ~λgp,n(t)

(triangles) and the analytical solution ~λgp,a(t) (solidlines). For the comparison, we also show the protocol ofthe simple linear control ~λlin(t) = (~λ(τ)−~λ(0))t/τ+~λ(0).

0 0.2 0.4 0.6 0.8 10

5

10

15

FIG. 2. (Color online) Geodesic protocols for the control withtwo parameters. In the simulation, we have set the temper-ature and the dissipation rate as kBT = 1 and γ = 1. Theparameters change from the initial point ~λ(0)=(1, 1) to the fi-nal point ~λ(τ)=(16, 2). The triangles represent the numericalgeodesic protocol ~λgp,n(t) while the solid lines represent theanalytical geodesic protocol ~λgp,a(t). The dash lines representthe linear protocol ~λlin(t). The numerical geodesic protocol(triangles) coincides well with the analytical geodesic protocol(solid lines).

To validate our results of optimization, we calculatethe irreversible work for the single Brownian particle inthe controllable harmonic potential with two control pa-rameters by solving the Langevin equation (1) throughthe Euler algorithm [28, 48]. The average work is cal-culated by the ensemble average of the stochastic workover 105 stochastic trajectories. Details of the simula-tion are presented in the supplementary material [28]. InFig. 3, we plot the irreversible work Wirr as a functionof duration τ ∈ {0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0} for boththe geodesic path (red circles) and the simple linear con-trol (blue squares). The geodesic protocol results in alower irreversible work than that from the linear proto-col. The black line shows the analytical results Wmin

irr =L2/τ , where the thermodynamic length L is calculated

as L =∫ τ0dt∑µν

√λµλνgµν =

√w2

b/(4βγ) + γm2b. The

4

10-1 100 101100

102

LinearGeodesic

FIG. 3. (Color online) The irreversible work of the geodesicprotocol (red circles) and the linear protocol (blue squares).The black line represents the theoretical lower bound givenby the thermodynamic length, i.e., Eq. (8). We per-form the simulation for different control duration τ ∈{0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0}. The irreversible cost given bythe geodesic protocol is lower than that from the linear pro-tocol. And the lower bound given by the geodesic protocolmatches the one given by the thermodynamic length.

simulation results match the lower bound presented bythe thermodynamic length, illustrated by the coincide ofthe simulated results (red circles) with the theoreticalline (black line). Figure 3 shows that the geodesic proto-col can largely reduce the irreversible work, which there-fore proves our findings about the geometric propertyof the control-parameter space in the shortcut scheme.Our findings simplify the procedure of finding the opti-mal control protocol in the shortcut scheme by applyingthe tools of Riemannian geometry.

Conclusions.– In summary, we have provided a geo-metric approach to find the optimal control scheme tosteer the evolution of the system along the path of in-stantaneous equilibrium states to reduce the energy cost.The proven equivalence between designing the optimalcontrol and finding the geodesic path in the parametricspace allows the application of the methods developed inRiemannian geometry to solve the optimization problemin thermodynamics. We have applied our approach intothe Brownian particle system tuned by both one and twocontrol parameters to find the optimal control for reduc-ing energy cost. Analytical and numerical results haveverified that the geodesic protocol can largely reduce theirreversible work in the shortcut scheme. Our strategyshall provide an effective tool to design the optimal finite-time control with the lowest energy cost.

Our results demonstrate that the optimal control withthe minimal energy cost to transfer the system betweenequilibrium states is to steer the system evolving alongthe geodesic path. Once the initial and final equilibriumstates are given, the geodesic path is determined by thegeodesic equation (9) for the given system. The dynamicsof the system is covered by the metric gµν in Eq. (7)without the need to treat the system on a case-by-casebasis. An intuitive determination of the performance ofthe controls is allowed with the proportional relation inEq. (8) between the minimal energy cost and the squareof the length of the geodesic path.

Acknowledgement.–This work is supported by the Na-tional Natural Science Foundation of China (NSFC)(Grants No. 12088101, No. 11534002, No. 11875049,No. U1930402, No. U1930403 and No. 12047549) andthe National Basic Research Program of China (GrantNo. 2016YFA0301201).

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Supplementary materials: Geodesic path for the minimal energy cost in shortcuts toisothermality

Geng Li,1 Jin-Fu Chen,2, 1 C. P. Sun,1, 2 and Hui Dong1, ∗

1Graduate School of China Academy of Engineering Physics, Beijing 100193, China2Beijing Computational Science Research Center, Beijing 100193, China

The supplementary materials are devoted to provide detailed derivations in the main context.

CONTENTS

I. The modified Kramers equation 1

II. Shortcuts to isothermality with multiple control parameters 4

III. General formalism: The mean work done in the process driven by shortcuts to isothermality 4A. Geometric approach to the irreversible work 4B. Positive semi-definteness of the metric gµν 7C. The 1/τ scaling of the irreversible work 7

IV. General formalism: Solving the geodesic equation with shooting method 7

V. Example: Underdamped Brownian motion 8A. The auxiliary Hamiltonian for an one-dimensional system 8B. The geodesic protocol for an underdamped Brownian particle system 9C. The stochastic simulations 10

References 11

I. THE MODIFIED KRAMERS EQUATION

The Kramers equation [1] was developed for describing systems with the form of Hamiltonian Ho = ~p2/2+Uo(~x,~λ).In the main text, we consider systems controlled by shortcuts to isothermality with the form of Hamiltonian H =Ho + Ha, with boundary conditions Ha(0) = Ha(τ) = 0 at the initial time t = 0 and the final time t = τ . In thissection, we derive a modified Kramers equation [2] for the total Hamiltonian H.

The evolution equation of the system probability distribution ρ(~x, ~p, t) = δ(~x− ~x(t))δ(~p− ~p(t)) for a trajectory[~x(t), ~p(t)] is governed by the Liouville equation as

∂ρ

∂t= −

∑

i

[∂

∂xi(xiρ) +

∂

∂pi(piρ)], (1)

where the evolution of variables is described by the Langevin equation as

xi =∂H

∂pi,

pi = −∂H∂xi− γxi + ξi(t). (2)

With Eqs. (1) and (2), we obtain

∂ρ

∂t=∑

i

[− ∂

∂xi(∂H

∂piρ) +

∂

∂pi(∂H

∂xiρ+ γ

∂H

∂piρ)− ∂

∂pi(ξiρ)]. (3)

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An observable probability to describe the average effect of the probability distribution over different realizations of[~ξ(t)] can be defined as

P (~x, ~p, t) ≡ 〈ρ(~x, ~p, t)〉~ξ=

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]ρ(~x, ~p, t)

=

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]δ(~x− ~x(t))δ(~p− ~p(t)), (4)

where T [~x(t), ~p(t)] is the probability of the trajectory [~x(t), ~p(t)] associated with a noise realization [~ξ(t)] [1]. In thefollowing, we derive the evolution equation for the observable probability P (~x, ~p, t).

Equation (3) can be rewritten as

∂ρ

∂t= −Ldρ− Lsρ, (5)

with the deterministic operator

Ld(t) ≡∑

i

[∂

∂xi(∂H

∂pi)− ∂

∂pi(∂H

∂xi+ γ

∂H

∂pi)],

and the stochastic operator

Ls(t) ≡∑

i

ξi∂

∂pi.

Introduce a new probability φ(~x, ~p, t) that satisfies

ρ(~x, ~p, t) = T e−∫ t0Ld(t

′)dt′φ(~x, ~p, t) = R(t)φ(~x, ~p, t), (6)

where R(t) ≡ T e−∫ t0Ld(t

′)dt′ with T representing the time-order operator. Substituting Eq. (6) into Eq. (5), weobtain

∂φ

∂t= −Oφ, (7)

where O(t) ≡ R−1(t)Ls(t)R(t) with R−1(t) ≡ (T e−∫ t0Ld(t

′)dt′)−1. Equation (7) has a formal solution as

φ(~x, ~p, t) = T e−∫ t0O(t′)dt′φ(~x, ~p, 0)

= [∑

n

(−1)n∫ t

0

dtn

∫ tn

0

dtn−1 · · ·∫ t2

0

dt1O(tn)O(tn−1) · · · O(t1)]φ(~x, ~p, 0). (8)

Averaging it over different realizations of [~ξ], we derive that

〈φ(~x, ~p, t)〉~ξ = [∑

n

(−1)n∫ t

0

dtn

∫ tn

0

dtn−1 · · ·∫ t2

0

dt1〈O(tn)O(tn−1) · · · O(t1)〉~ξ]φ(~x, ~p, 0)

= [∑

n

(−1)2n∫ t

0

dt2n

∫ t2n

0

dt2n−1 · · ·∫ t2

0

dt1〈O(t2n)O(t2n−1) · · · O(t1)〉~ξ]φ(~x, ~p, 0). (9)

In the second step, we have considered the fact that the noise ~ξ is Gaussian satisfying 〈ξi(t)〉~ξ = 0 and 〈ξi(t)ξj(t′)〉~ξ =

2γkBTδijδ(t−t′). The higher-order moments that contain odd number of ξi are zero. The remaining terms containingeven number of ξi can be decomposed into a sum of products of the second order moment 〈ξi(t)ξj(t′)〉~ξ. For example,the fourth order moment follows

〈ξi(t1)ξj(t2)ξk(t3)ξl(t4)〉~ξ = 〈ξi(t1)ξj(t2)〉~ξ〈ξk(t3)ξl(t4)〉~ξ + 〈ξi(t1)ξk(t3)〉~ξ〈ξj(t2)ξl(t4)〉~ξ+〈ξi(t1)ξl(t4)〉~ξ〈ξj(t2)ξk(t3)〉~ξ. (10)

3

In the right hand side of Eq. (10), the second and third terms vanish because of the time order t ≥ t2n · · · ≥ t1.Therefore, Eq. (9) reduces to the form

〈φ(~x, ~p, t)〉~ξ = [∑

n

(

∫ t

0

dt2n

∫ t2n

0

dt2n−1〈O(t2n)O(t2n−1)〉~ξ)(∫ t2n−1

0

dt2n−2

∫ t2n−2

0

dt2n−3〈O(t2n−2)O(t2n−3)〉~ξ)

× · · · (∫ t3

0

dt2

∫ t2

0

dt1〈O(t2)O(t1)〉)]φ(~x, ~p, 0). (11)

One of the integral in Eq. (11) is calculated as∫ t3

0

dt2

∫ t2

0

dt1〈O(t2)O(t1)〉

=∑

ij

∫ t3

0

dt2

∫ t2

0

dt1〈ξi(t2)ξj(t1)〉~ξR−1(t2)∂

∂piR(t2)R−1(t1)

∂

∂pjR(t1)

= 2γkBT∑

ij

∫ t3

0

dt2

∫ t2

0

dt1δijδ(t2 − t1)R−1(t2)∂

∂piR(t2)R−1(t1)

∂

∂pjR(t1)

= γkBT∑

i

∫ t3

0

dt2R−1(t2)

∂2

∂p2iR(t2). (12)

Then Eq. (11) proceeds as

〈φ(~x, ~p, t)〉~ξ = [∑

n

(γkBT )n(∑

i

∫ t

0

dt2nR−1(t2n)

∂2

∂p2iR(t2n))(

∑

i

∫ t2n

0

dt2n−2R−1(t2n−2)

∂2

∂p2iR(t2n−2))

× · · · (∑

i

∫ t4

0

dt2R−1(t2)

∂2

∂p2iR(t2))]φ(~x, ~p, 0)

= [∑

n

(γkBT )n(∑

i

∫ t

0

dtnR−1(tn)

∂2

∂p2iR(tn))(

∑

i

∫ tn

0

dtn−1R−1(tn−1)

∂2

∂p2iR(tn−1))

× · · · (∑

i

∫ t2

0

dt1R−1(t1)

∂2

∂p2iR(t1))]φ(~x, ~p, 0). (13)

Taking derivative of Eq. (13) over time t, we obtain

∂

∂t〈φ(~x, ~p, t)〉~ξ = [

∞∑

n=0

(γkBT )n(∑

i

R−1(t)∂2

∂p2iR(t))(

∑

i

∫ t

0

dtn−1R−1(tn−1)

∂2

∂p2iR(tn−1))

× · · · (∑

i

∫ t2

0

dt1R−1(t1)

∂2

∂p2iR(t1))]φ(~x, ~p, 0)

= γkBT (∑

i

R−1(t)∂2

∂p2iR(t))[

∞∑

n=1

(γkBT )n−1(∑

i

∫ t

0

dtn−1R−1(tn−1)

∂2

∂p2iR(tn−1))

× · · · (∑

i

∫ t2

0

dt1R−1(t1)

∂2

∂p2iR(t1))]φ(~x, ~p, 0)

= γkBT (∑

i

R−1(t)∂2

∂p2iR(t))〈φ(~x, ~p, t)〉~ξ. (14)

Therefore, the observable probability P (~x, ~p, t) = 〈ρ(~x, ~p, t)〉~ξ = R(t)〈φ(~x, ~p, t)〉~ξ follows the evolution equation

∂P

∂t= −LdR(t)〈φ(~x, ~p, t)〉~ξ + R(t)

∂

∂t〈φ(~x, ~p, t)〉~ξ

= (−Ld + γkBT∑

i

∂2

∂p2i)R(t)〈φ(~x, ~p, t)〉~ξ

=∑

i

[− ∂

∂xi(∂H

∂piP ) +

∂

∂pi(∂H

∂xiP + γ

∂H

∂piP ) + γkBT

∂2P

∂p2i], (15)

4

which is equivalent to Eq. (4) in the main text.

II. SHORTCUTS TO ISOTHERMALITY WITH MULTIPLE CONTROL PARAMETERS

The framework of the shortcut scheme was originally developed for systems with single control parameter [2–4]. Toestablish a general fromalism, we extend this framework for systems with multiple control parameters.

In the shortcut scheme, the system evolves according to Eq. (15) with H = Ho+Ha. Substituting the instantaneousequilibrium distribution

Peq(~x, ~p,~λ) = eβ[F (~λ)−Ho(~x,~p,~λ)] (16)

into Eq. (15), we obtain the requirement for the auxillary Hamiltonian Ha as

∑

µ

(dF

dλµ− ∂Uo

∂λµ)λµ =

∑

i

(γ

β

∂2Ha

∂p2i− γpi

∂Ha

∂pi+∂Ha

∂pi

∂Uo

∂xi− pi

∂Ha

∂xi). (17)

The solution for the auxillary Hamiltonian is Ha(~x, ~p, t) =∑µ λµfµ(~x, ~p,~λ) with fµ(~x, ~p,~λ) satisfying

dF

dλµ− ∂Uo

∂λµ=∑

i

(γ

β

∂2fµ∂p2i

− γpi∂fµ∂pi

+∂fµ∂pi

∂Uo

∂xi− pi

∂fµ∂xi

). (18)

Once the form of the original potential Uo is given, we can solve Eq. (18) for the function fµ(~x, ~p,~λ). With theboundary condition

~λ(0) = ~λ(τ) = 0, (19)

we can realize the shortcut scheme for systems with multi-parameters.

III. GENERAL FORMALISM: THE MEAN WORK DONE IN THE PROCESS DRIVEN BYSHORTCUTS TO ISOTHERMALITY

In this section, we will derive the work done in the shortcut scheme, and show its geometric expression.

A. Geometric approach to the irreversible work

The work done in an individual stochastic trajectory reads [2]

w[~x(t), ~p(t)] ≡∫ τ

0

dt∂Ho(~x(t), ~p(t), ~λ)

∂t+

∫ τ

0

dt∂Ha(~x(t), ~p(t), t)

∂t

=

∫ τ

0

dt∂Ho(~x(t), ~p(t), ~λ)

∂t+∑

i

∫ τ

0

dt(dHa

dt− xi(t)

∂Ha

∂xi− pi(t)

∂Ha

∂pi)

=

∫ τ

0

dt∂Ho(~x(t), ~p(t), ~λ)

∂t−∑

i

∫ τ

0

dt(xi(t)∂Ha(~x(t), ~p(t), t)

∂xi+ pi(t)

∂Ha(~x(t), ~p(t), t)

∂pi). (20)

5

In the above derivations, we have used integration by part and considered the boundary conditions in Eq. (19). Takingan ensemble average over the trajectory work w in Eq. (20), we obtain the mean work as

W ≡ 〈w〉~ξ =

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]w[~x(t), ~p(t)]

=

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]

∫ τ

0

dt

∫∫d~xd~pδ(~x− ~x(t))δ(~p− ~p(t))

×[∂Ho(~x(t), ~p(t), ~λ)

∂t−∑

i

(xi(t)∂Ha(~x(t), ~p(t), t)

∂xi+ pi(t)

∂Ha(~x(t), ~p(t), t)

∂pi)]

=

∫ τ

0

dt

∫∫d~xd~p[

∂Ho(~x, ~p,~λ)

∂t〈δ(~x− ~x(t))δ(~p− ~p(t))〉~ξ

−∑

i

(∂Ha(~x, ~p, t)

∂xi〈xi(t)δ(~x− ~x(t))δ(~p− ~p(t))〉~ξ +

∂Ha(~x, ~p, t)

∂pi〈pi(t)δ(~x− ~x(t))δ(~p− ~p(t))〉~ξ)]

=

∫ τ

0

dt

∫∫d~xd~p[

∂Ho(~x, ~p,~λ)

∂tP (~x, ~p, t)

−∑

i

(∂Ha(~x, ~p, t)

∂xi〈xi(t)ρ(~x, ~p, t)〉~ξ +

∂Ha(~x, ~p, t)

∂pi〈pi(t)ρ(~x, ~p, t)〉~ξ)]. (21)

In the shortcut scheme, the system keeps in the instantaneous equilibrium state, P = Peq = exp[β(F −Ho)], whichleads to [2] ∆F =

∫ τ0dt∫∫

d~xd~pPeq∂tHo. The irreversible work Wirr ≡W −∆F then follows as

Wirr = W −∫∫

d~xd~p

∫ τ

0

dt∂Ho(~x, ~p,~λ)

∂tPeq(~x, ~p, t)

= −∫∫

d~xd~p∑

i

∫ τ

0

dt(∂Ha(~x, ~p, t)

∂xi〈xi(t)ρ(~x, ~p, t)〉~ξ +

∂Ha(~x, ~p, t)

∂pi〈pi(t)ρ(~x, ~p, t)〉~ξ). (22)

With Eq. (2), we can calculate 〈xi(t)ρ(~x, ~p, t)〉~ξ and 〈pi(t)ρ(~x, ~p, t)〉~ξ as

〈xi(t)ρ(~x, ~p, t)〉~ξ =

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]xi(t)δ(~x− ~x(t))δ(~p− ~p(t))

=

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]δ(~x− ~x(t))δ(~p− ~p(t))∂H(~x(t), ~p(t), t)

∂pi

=∂H(~x, ~p, t)

∂pi

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]δ(~x− ~x(t))δ(~p− ~p(t))

=∂H(~x, ~p, t)

∂piP (~x, ~p, t), (23)

and

〈pi(t)ρ(~x, ~p, t)〉~ξ =

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]pi(t)δ(~x− ~x(t))δ(~p− ~p(t))

=

∫∫D[~x(t)]D[~p(t)]T [~x(t), ~p(t)]δ(~x− ~x(t))δ(~p− ~p(t))(−∂H(~x(t), ~p(t), t)

∂xi− γxi(t) + ξi(t))

= −(∂H(~x, ~p, t)

∂xi+ γ

∂H(~x, ~p, t)

∂pi)P (~x, ~p, t) + 〈ξi(t)ρ(~x, ~p, t)〉~ξ. (24)

6

Since the stochastic force ~ξ(t) commutes with the deterministic operator Ld(t), we have

〈ξi(t)ρ(~x, ~p, t)〉~ξ = 〈ξi(t)R(t)φ(~x, ~p, 0)〉~ξ= R(t)〈ξi(t)φ(~x, ~p, t)〉~ξ

= R(t)〈ξi(t)[∞∑

n=0

(−1)n∫ t

0

dtn

∫ tn

0

dtn−1 · · ·∫ t2

0

dt1O(tn)O(tn−1) · · · O(t1)]φ(~x, ~p, 0)〉~ξ

= R(t)[∞∑

n=0

(−1)n∫ t

0

dtn〈ξi(t)O(tn)〉~ξ∫ tn

0

dtn−1 · · ·∫ t2

0

dt1〈O(tn−1) · · · O(t1)〉~ξ]φ(~x, ~p, 0)

= −R(t)(γkBT )R−1(t)∂

∂piR(t)[

∞∑

n=1

(−1)n−1∫ t

0

dtn−1 · · ·∫ t2

0

dt1〈O(tn−1) · · · O(t1)〉~ξ]φ(~x, ~p, 0)

= −γkBT∂

∂piR(t)〈φ(~x, ~p, t)〉~ξ

= −γkBT∂

∂piP (~x, ~p, t). (25)

Combining Eqs. (23), (24), and (25), we obtain

Wirr = −∑

i

∫ τ

0

dt

∫∫d~xd~p[

∂Ha

∂xi

∂H

∂piPeq −

∂Ha

∂pi(∂H

∂xiPeq + γ

∂H

∂piPeq + γkBT

∂Peq

∂pi)]

= −∑

i

∫ τ

0

dt

∫∫d~xd~p[

∂Ha

∂xi

∂Ho

∂pi− ∂Ha

∂pi

∂Ho

∂xi− γ(

∂Ha

∂pi)2]Peq

= −∑

i

∫ τ

0

dt

∫∫d~xd~p[− 1

β

∂Ha

∂xi

∂Peq

∂pi+

1

β

∂Ha

∂pi

∂Peq

∂xi− γ(

∂Ha

∂pi)2Peq]

= −∑

i

∫ τ

0

dt

∫∫d~xd~p[− 1

β

∂

∂pi(∂Ha

∂xiPeq) +

1

β

∂

∂xi(∂Ha

∂piPeq)− γ(

∂Ha

∂pi)2Peq]

= γ∑

i

∫ τ

0

dt

∫∫d~xd~p(

∂Ha

∂pi)2Peq

= γ∑

µνi

∫ τ

0

dtλµλν

∫∫d~xd~p

∂fµ∂pi

∂fν∂pi

Peq

=∑

µν

∫ τ

0

dtλµλνgµν , (26)

with the metric

gµν = γ∑

i

∫∫d~xd~p

∂fµ∂pi

∂fν∂pi

Peq ≡ γ∑

i

⟨∂fµ∂pi

∂fν∂pi

⟩

eq

. (27)

In the derivations of Eq. (26), we have used integration by part and assumed that the boundary term

∑

i

(∂Ha

∂xiPeq)|pi=+∞ =

∑

i

(∂Ha

∂xiPeq)|pi=−∞ = 0,

and

∑

i

(∂Ha

∂piPeq)|xi=+∞ =

∑

i

(∂Ha

∂piPeq)|xi=−∞ = 0.

Equation (26) is equivlant to Eq. (7) in the main text.

7

B. Positive semi-definteness of the metric gµν

The positive semi-definiteness of the metric in Eq. (27) are guaranteed by the structure, gµν = γ∑i〈∂pifµ∂pifν〉eq.

For any vector ~v ≡ (v1, v2, · · · , vM ), we have

~vT g~v = γ∑

µν

vµvν∑

i

⟨∂fµ∂pi

∂fν∂pi

⟩

eq

= γ∑

i

⟨(∑

µ

vµ∂fµ∂pi

)(∑

ν

vν∂fν∂pi

)

⟩

eq

= γ∑

i

⟨(∑

µ

vµ∂fµ∂pi

)2

⟩

eq

= γ∑

i

∫d~xd~pPeq(

∑

µ

vµ∂fµ∂pi

)2. (28)

The integrand in Eq. (28) is non-negative, which ensures the non-negativity of ~vT g~v. Therefore, we prove that themetric in Eq. (27) is positive semi-definite.

C. The 1/τ scaling of the irreversible work

It is natural to choose the function form of the protocol ~λ(t) = ~Λ(t/τ). After a change of variable s ≡ t/τ , thecontrol protocol ~Λ(s) is independent of the protocol duration τ . The irreversible work in Eq. (26) follows

Wirr = γ∑

µνi

∫ τ

0

dtλµ(t)λν(t)

∫∫d~xd~p

∂fµ(~x, ~p,~λ(t))

∂pi

∂fν(~x, ~p,~λ(t))

∂piPeq(~x, ~p,~λ(t))

=γ

τ

∑

µνi

∫ 1

0

dsΛ′µ(s)Λ

′ν(s)

∫∫d~xd~p

∂fµ(~x, ~p, ~Λ(s))

∂pi

∂fν(~x, ~p, ~Λ(s))

∂piPeq(~x, ~p, ~Λ(s)),

where the prime in Λ′µ(s) represents the derivative of Λµ(s) with respective to s. The irriversible workWirr is inversely

proportional to the protocol duration τ .

IV. GENERAL FORMALISM: SOLVING THE GEODESIC EQUATION WITH SHOOTING METHOD

According to Eq. (26), the task of designing the optimal protocol in the shortcut scheme is converted to findingthe geodesic path in the parameter space. The geodesic path is obtained by solving the geodesic equation

λµ +1

2

∑

νκι

(g−1)ιµ(∂gιν∂λκ

+∂gικ∂λν

− ∂gνκ∂λι

)λν λκ = 0, (29)

with boundary conditions ~λ(0) = ~λ0, ~λ(τ) = ~λτ , and ~λ(0) = ~λ(τ) = 0. Shooting method is one of the popular toolsthat treats the above two-point boundary values problem as an initial value problem [5]. Specifically, the shootingmethod solves the initial valve problem

λµ = yµ(t, ~λ, ~λ) ≡ 1

2

∑

νκι

(g−1)ιµ(∂gνκ∂λι

− ∂gιν∂λκ

− ∂gικ∂λν

)λν λκ, (30)

with the initial conditions ~λ(0) = ~λ0 and ~λ(0+) = ~d. We remark here that the first order derivation ~λ(0) is notcontinuous at t = 0, noticing ~λ(0) = 0. The initial rate ~d is updated until the solution of Eq. (30) satisfies theboundary condition ~λ(τ) = ~λτ . The shooting method can be realized by using the Eular algorithm to solve Eq. (30)and Newton’s method [6] to approach the final condition ~λ(τ) = ~λτ . To update the initial rate ~d, we treat the protocol

8

as a function of the initial rate, i.e., ~λ(t, ~d), and define zµν(t, ~d) ≡ ∂dνλµ(t, ~d). At the final time τ , it follows that inNewton’s method, the solution of the equation ~λτ = ~λ(τ, ~d) is approximated as the solution of the equation

~λτ ≈ ~λ(τ, ~d(k)) + z(τ, ~d(k))(~d(k+1) − ~d(k)), (31)

where ~d(k) represents the current iteration and ~d(k+1) represents the next iteration. Rearranging Eq. (31) yields

~d(k+1) = ~d(k) + z−1(τ, ~d(k))(~λτ − ~λ(τ, ~d(k))), (32)

which gives the process to obtain each new iteration ~d(k+1) from the previous iteration ~d(k). Here, z−1(τ, ~d(k)) in Eq.(32) is obtained by solving the evolution equation as

zµν(t, ~d) =∂λµ∂dν

=∂yµ∂dν

=∑

κ

(∂yµ∂λκ

∂λκ∂dν

+∂yµ

∂λκ

∂λκ∂dν

)

=∑

κ

(∂yµ∂λκ

zκν +∂yµ

∂λκzκν), (33)

which is derived by taking derivative of Eq. (30) over ~d. The accompanied initial conditions follow as zµν(0, ~d) = 0

and zµν(0+, ~d) = δµν . The shooting method to solve the geodesic equation (29) is summarized as follows. Firstly,choosing a proper initial rate ~d(1), we solve Eqs. (30) and (33) to obtain the first iteration ~λ(τ, ~d(1)) and z(τ, ~d(1)).Secondly, we get the updated rate ~d(2) by using Eq. (32) and repeat the first step to solve for the next iteration~λ(τ, ~d(2)) and z(τ, ~d(2)). The iterator finally stops in the kth iteration when |~λτ − ~λ(τ, ~d(k))| < ε with ε representingthe termination precision. Then, the solution of the the geodesic equation (29) is ~λ(t, ~d(k)).

V. EXAMPLE: UNDERDAMPED BROWNIAN MOTION

A. The auxiliary Hamiltonian for an one-dimensional system

We consider an underdamped Brownian particle system in an one-dimensional harmonic potential with the Hamil-tonian

Ho(x, p, ~λ) =p2

2+ Uo(x,~λ), (34)

where Uo(x,~λ) = λ1(t)x2/2− λ2(t)x is a controllable potential. The auxiliary Hamiltonian Ha = Ha(x, p, t) follows

(dF

dλ1− ∂Uo

∂λ1)λ1 + (

dF

dλ2− ∂Uo

∂λ2)λ2 =

γ

β

∂2Ha

∂p2− γp∂Ha

∂p+∂Uo

∂x

∂Ha

∂p− p∂Ha

∂x. (35)

The function f1 and f2 in the auxiliary Hamiltonian Ua(x, p, t) = λ1f1(x, p, λ1, λ2) + λ2f2(x, p, λ1, λ2) satisfy thefollowing equations

γ

β

∂2f1∂p2

− γp∂f1∂p

+∂Uo

∂x

∂f1∂p− p∂f1

∂x=dF

dλ1− ∂Uo

∂λ1, (36)

and

γ

β

∂2f2∂p2

− γp∂f2∂p

+∂Uo

∂x

∂f2∂p− p∂f2

∂x=dF

dλ2− ∂Uo

∂λ2. (37)

By assuming that f1 = a1(t)p2 + a2(t)xp+ a3(t)p+ a4(t)x2 + a5(t)x, we can exactly derive the form

f1(x, p, λ1, λ2) =1

4γλ1[(p− γx)2 + λ1x

2]− λ2p

2λ21+ (

γλ22λ21− λ2

2γλ1)x. (38)

9

With similar derivations, we can obtain

f2(x, p, λ1, λ2) =p

λ1− γx

λ1. (39)

Therefore, the auxiliary Hamiltonian takes the form

Ha(x, p, t) = λ1{1

4γλ1[(p− γx)2 + λ1x

2]− λ2p

2λ21+ (

γλ22λ21− λ2

2γλ1)x}+ λ2(

p

λ1− γx

λ1). (40)

B. The geodesic protocol for an underdamped Brownian particle system

The metric in Eq. (27) for the current underdamped Brownian motion takes the form

g =

(1

4βγλ21

+ γ4βλ3

1+

γλ22

λ41−γλ2

λ31

−γλ2

λ31

γλ21

),

which results in the geodesic equations for the minimal work as

λ1 −λ21(3γ2 + 2λ1)

2λ1(γ2 + λ1)= 0,

λ2 −2λ1λ2λ1

+λ21λ2(γ2 + 2λ1)

2λ21(γ2 + λ1)= 0. (41)

Two boundary conditions ~λ(0) = ~λ0 and ~λ(τ) = ~λτ are accompanied with the geodesic equation. We first solveEq. (41) using the shooting method mentioned above. The geodesic equation (41) can be rewritten as

λ1 = y1(t, ~λ, ~λ) ≡ λ21(2λ1 + 3γ2)

2λ1(λ1 + γ2),

λ2 = y2(t, ~λ, ~λ) ≡ 2λ1λ2λ1

− λ21λ2(2λ1 + γ2)

2λ21(λ1 + γ2). (42)

As shown in Sec. IV, the algorithm to solve Eq. (42) proceeds as follows:In the simulation, we set the parameters as ~λ(0) = (1, 1), ~λ(τ) = (16, 2), kBT = 1, and γ = 1. The initial rate is

chosen as ~d = (1, 1) , the operation time is τ = 1 with the time step δt = 10−3, and the termination precision is setas ε = 10−4. The simulation results are presented in Fig. 2 of the main text.

Fortunately, the geodesic equation (41) can also be solved analytically. Substituting the auxiliary Hamiltonian inEq. (40) into the irreversible work in Eq. (26), we obtain

Wirr =λ21

4βγλ21+

γλ214βλ31

+γλ21λ

22

λ41− 2γλ1λ2λ2

λ31+

2γλ22λ21

=λ21(λ1 + γ2)

4βγλ31+ γ(

λ1λ2λ21− λ2λ1

)2

=λ21(λ1 + γ2)

4βγλ31+ γ[

d

dt(λ2λ1

)]2. (43)

We can simplify the expression in Eq. (43) with a new set of parameters,

y ≡ λ1√λ1 + γ2

4βγλ31, z ≡ d

dt(

√γλ2

λ1), (44)

and the irreversible work follows Wirr = y2+z2, indicating a flatten manifold in the geometric space. The correspondinggeodesic equation follows y = 0 and z = 0 which gives

λ1 =wb

τ

√λ31

λ1 + γ2,

λ2λ1

= mbt/τ + nb, (45)

10

Algorithm 1: Shooting method

Data: Choose ~λ(0+) = ~d(k), with k = 0; Choose δt such that Nδt = τ where N is the number of steps;Result: Optimal control protocol ~λ(τ);

1 while (|~λτ − ~λ(τ)| > ε) do2 k = k + 1;

3 ~λ(0) = ~λ0, ~λ(0) = ~d(k), z(0) = 0, z11(0) = z22(0) = 1,z12(0) = z21(0) = 0;4 for m = 0, 1, 2, · · · , N − 1 do5 for µ = 1, 2 do6 λµ((m+ 1)δt) = λµ(mδt) + λµ(mδt)δt;7 λµ((m+ 1)δt) = λµ(mδt) + yµ(mδt)δt;8 end9 for µ = 1, 2 do

10 for ν = 1, 2 do11 zµν((m+ 1)δt) = zµν(mδt) + zµν(mδt)δt;12 zµν((m+ 1)δt) = zµν(mδt) +

∑κ[∂yµ∂λκ

zκν +∂yµ

∂λκzκν ]δt;

13 end14 end15 end16 d1(k + 1) = d1(k) +

(λτ1−λ1(τ))z22−(λτ2−λ2(τ))z12z11z22−z21z12 ;

17 d2(k + 1) = d2(k) +(λτ2−λ2(τ))z21−(λτ1−λ1(τ))z11

z21z12−z11z22 ;18 end

where the parameters wb = −[2√

1 + γ2/λ1 + ln(√

1 + γ2/λ1 − 1)− ln(√

1 + γ2/λ1 + 1)]|λ1(τ)λ1(0)

, mb = (λ2(τ)λ1(0)−λ2(0)λ1(τ))/(λ1(τ)λ1(0)), and nb = λ2(0)/λ1(0). The final geodesic protocol is shown as Fig. 2 in the main text.

C. The stochastic simulations

The motion of the Brownian particle is governed by the Langevin equation as

x =∂Ho

∂p+∂Ha

∂p,

p = −∂Ho

∂x− ∂Ha

∂x− γx+ ξ(t), (46)

where ξ represents the standard Gaussian white noise satisfying 〈ξ(t)〉 = 0 and 〈ξ(t)ξ(t′)〉 = 2γkBTδ(t − t′). Weintroduce the characteristic length lc ≡ (kBT/λ1(0))1/2, the characteristic times τ1 = m/γ and τ2 = γ/λ1(0) to definethe dimensionless coordinate x ≡ x/lc, momentum p ≡ pτ/(mlc), time s ≡ t/τ , and the control protocol λ ≡ λ/(kl2c).The dimensionless Langevin equation follows

x′ = p+ ατ2∂Ha

∂p,

p′ = −ατ2 ∂Ho

∂x− ατ2 ∂Ha

∂x− τ x′ + τ

√2ατζ(s), (47)

where τ ≡ τ/τ1 and α ≡ τ1/τ2. The prime represents the derivative with respective to dimensionless time s. ζ(s) is aGaussian white noise satisfying 〈ζ(s)〉 = 0 and 〈ζ(s1)ζ(s2)〉 = δ(s1 − s2). The Hamiltonian Ho and Ha are rewrittenwith the dimensionless parameters as

Ho(x, s) =1

ατ2p2

2+

1

2λ1x

2 − λ2x (48)

and

Ha(x, p, s) =λ′1

4τ λ1[

1

ατ2(p− τ x)2 + λ1x

2]

− λ′1λ22ατ2λ21

(p+ τ x− ατ λ1x) + λ′2(p

ατ2λ1− x

ατ λ1). (49)

11

We solve the Langevin equation (47) by using the Euler algorithm as

x(s+ δs) = x(s) + pδs+ ατ2∂Ha

∂pδs,

p(s+ δs) = p(s)− ατ2 ∂Ho

∂xδs− ατ2 ∂Ha

∂xδs− τ(p+ ατ2

∂Ha

∂p)δs+ τ

√2ατδsθ(s), (50)

where δs is the time step and θ(s) is a random number sampled from Gaussian distribution with zero mean and unitvariance. The trajectory work of the system takes

w ≡ w

kBT=

∫ 1

0

(∂Ho

∂s+∂Ha

∂s)ds

≈∑

(∂Ho

∂s+∂Ha

∂s)δs. (51)

In the simulation, we have chosen the parameters ~λ(0) = (1, 1), ~λ(τ) = (16, 2), kBT = 1, γ = 1, and m = 1. Themean work is obtained as the ensemble average over the trajectory work of 105 stochastic trajectories. We performthe simulation for different duration τ ∈ {0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0}.

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96(1):012144, jul 2017.[3] John A. C. Albay, Sarah R. Wulaningrum, Chulan Kwon, Pik-Yin Lai, and Yonggun Jun. Thermodynamic cost of a

shortcuts-to-isothermal transport of a brownian particle. Phys. Rev. Research, 1(3):033122, nov 2019.[4] Geng Li and Z. C. Tu. Equilibrium free-energy differences from a linear nonequilibrium equality. Phys. Rev. E, 103(3):032146,

mar 2021.[5] Marcel Berger. A Panoramic View of Riemannian Geometry. Springer Berlin Heidelberg, June 2007.[6] Daan Frenkel and Berend Smit. Understanding Molecular Simulation. Elsevier Science Publishing Co Inc, October 2001.