Geodesic path for the optimal nonequilibrium transition Momentum-independent protocol Geng Li1C. P. Sun1 2and Hui Dong1

2025-05-06 0 0 1.22MB 10 页 10玖币

侵权投诉

Geodesic path for the optimal nonequilibrium transition: Momentum-independent

protocol

Geng Li,1C. P. Sun,1, 2 and Hui Dong1, ∗

1Graduate School of China Academy of Engineering Physics, Beijing 100193, China

2Beijing Computational Science Research Center, Beijing 100193, China

Accelerating controlled thermodynamic processes requires an auxiliary Hamiltonian to steer the

system into instantaneous equilibrium states. An extra energy cost is inevitably needed in such

ﬁnite-time operation. We recently develop a geodesic approach to minimize such energy cost for

the shortcut to isothermal process. The auxiliary control typically contains momentum-dependent

terms, which are hard to be experimentally implemented due to the requirement of constantly mon-

itoring the speed. In this work, we employ a variational auxiliary control without the momentum-

dependent force to approximate the exact control. Following the geometric approach, we obtain the

optimal control protocol with variational minimum energy cost. We demonstrate the construction

of such protocol via an example of Brownian motion with a controllable harmonic potential.

I. INTRODUCTION

The quest to accelerate a system evolving toward a

target equilibrium state is ubiquitous in various applica-

tions [1–8]. In the biological pharmacy, pathogens are ex-

pected to evolve to an optimum state with maximal drug

sensitivity [1–4]. Controlling the evolution of pathogens

towards the target state with a considerable rate is crit-

ically relevant to confronting the threat of increasing

antibiotic resistance and determining optimal therapies

for infectious disease and cancer. In adiabatic quantum

computation, the solution of the optimization problem is

transformed to the ground state of the problem Hamil-

tonian [5–8]. Speeding up the computation requires to

steer the system evolving from a trivial ground state to

another nontrivial ground state within ﬁnite time. These

examples require to tune the system from one equilibrium

state to another one within ﬁnite time.

The scheme of shortcuts to isothermality was devel-

oped as such a control strategy to maintain the system

in instantaneous equilibrium states during evolution pro-

cesses [9–11]. Relevant results have been applied in accel-

erating state-to-state transformations [12–15], raising the

eﬃciency of free-energy landscape reconstruction [16, 17],

designing the nano-sized heat engine [18–21], and steer-

ing biological evolutions [3, 4]. Additional energy cost

is required due to the irreversibility in the ﬁnite-time

driving processes. Much eﬀort has been devoted to ﬁnd

the minimum energy requirement in the driving processes

[22–28]. We recently proved that the optimal path for

the shortcut scheme is equivalent to the geodesic path in

the geometric space spanned by control parameters [29].

Such an equivalence allows us to ﬁnd the optimal path

through methods developed in geometry.

Implementing such shortcut scheme remains a chal-

lenge task since the driving force required in the short-

cut scheme is typically momentum-dependent [7, 8, 15].

One solution is to use an approximate scheme [17, 30]

∗hdong@gscaep.ac.cn

to replace the exact one. Such scheme has been applied

in the underdamped case to obtain a driving force with-

out any momentum terms. The key idea is to use an

approximate auxiliary control without the momentum-

dependent terms.

In this work, we employ the variational method and

the geometric approach to ﬁnd an experimental proto-

col with minimum dissipation for realizing the shortcut

scheme. In Sec. II, we brieﬂy introduce the shortcut

scheme and the geometric approach for ﬁnding the opti-

mal control protocol with minimum energy cost. In Sec.

III, we apply a variational method to overcome the dif-

ﬁculty of the momentum-dependent terms in the driving

force. As illustrated in Fig. 1, the variational method is

separated into two steps. In step I, a variational shortcut

scheme is used to obtain an approximate auxiliary con-

trol without high-order momentum-dependent terms. In

step II, a gauge transformation scheme is used to remove

the linear momentum-dependent terms and an experi-

mentally testable protocol is obtained. In Sec. IV, we

demonstrate our protocol through a Brownian particle

moving in the harmonic potential with two controllable

parameters. In Sec. V, we conclude the paper with ad-

ditional discussions.

II. GEOMETRIC APPROACH AND THE

AUXILIARY HAMILTONIAN

In this section, we brieﬂy review our geodesic approach

of the shortcut to isothermality and show the possible

experimental diﬃculties to apply the obtained auxiliary

Hamiltonian.

Consider a system with the Hamiltonian Ho(~x, ~p, ~

λ) =

Pip2

i/(2m) + Uo(~x, ~

λ)immersed in a thermal reser-

voir with a constant temperature T. Here ~x ≡

(x1, x2,··· , xN)are coordinates, ~p ≡(p1, p2,··· , pN)

are momentum, mis mass, and ~

λ(t)≡(λ1, λ2,··· , λM)

are time-dependent control parameters. In the shortcut

scheme, an auxiliary Hamiltonian Ha(~x, ~p, t)is added to

steer the evolution of the system along the instantaneous

arXiv:2210.10986v1 [cond-mat.stat-mech] 20 Oct 2022

Step I: Variational shortcut Step II: Gauge transformationExact shortcut

Equivalent

process

Approximate

process

Schematic example of a Brownian particle with the Hamiltonian

Evolution equation:

Auxiliary control:

Instantaneous distribution:

Auxiliary control:

Evolution equation:

Instantaneous distribution:

Auxiliary control:

Evolution equation:

Instantaneous distribution:

(momentum-dependent) (linearly momentum-dependent) (momentum-independent)

FIG. 1. (Color online) Schematic example of a Brownian particle with the Hamiltonian Ho=p2/(2m) + λx2/2. The dynamical

evolution of the system is governed by the Langevin equation, where γis the dissipation coeﬃcient. In the shortcut scheme,

a momentum-dependent auxiliary control Hais added to escort the system distribution in the instantaneous equilibrium

distribution Peq. In step I, a variational shortcut scheme is used to obtain an approximate auxiliary control H∗

awhich can

keep the system in the approximate equilibrium distribution P∗

eq. In step II, a gauge transformation scheme is used to obtain

a momentum-independent auxiliary control Uawhich can maintain the system distribution in PID.

equilibrium states Peq = exp[β(F−Ho)] in the ﬁnite-time

interval t∈[0, τ]with the boundary conditions Ha(0) =

Ha(τ) = 0.Here F≡ −β−1ln[RRd~xd~p exp(−βHo)] is

the free energy and β= 1/(kBT)is the inverse temper-

ature with the Boltzmann constant kB. The probability

distribution of the system’s microstate P(~x, ~p, t)evolves

according to the Kramers equation

∂P

∂t =X

[−∂

∂xi

(∂H

∂pi

P)+ ∂

∂pi

(∂H

∂xi

P+γ∂H

∂pi

P)+ γ

∂2P

∂p2

(1)

where H≡Ho+Hais the total Hamiltonian, and γis

the dissipation coeﬃcient. The auxiliary Hamiltonian is

proved to have the form Ha(~x, ~p, t) = ˙

λ·~

f(~x, ~p, ~

λ)with

f(~x, ~p, ~

λ)depending on

[γ

∂2Ha

∂p2

i−γpi

∂Ha

∂pi

+∂Ha

∂pi

∂Ho

∂xi−pi

∂Ha

∂xi

] = dF

dt −∂Ho

∂t .

(2)

The boundary conditions for the auxiliary Hamiltonian

Ha(t)are presented explicitly as ˙

λ(0) = ˙

λ(τ)=0.The

irreversible energy cost Wirr ≡W−∆Fin the shortcut

scheme follows as [29]

Wirr =X

µν Zτ

dt ˙

λµ˙

λνgµν ,(3)

where the positive semi-deﬁnite metric is gµν =

γPih(∂fµ/∂pi)(∂fν/∂pi)ieq with h·ieq =RRd~xd~p[·]Peq.

Here W≡ hRτ

0dt∂tHiis the mean work with h·i repre-

senting the ensemble average over stochastic trajectories

and ∆F≡F(~

λ(τ)) −F(~

λ(0)) is the free energy diﬀer-

ence. The metric gµν endows a Riemannian manifold in

the space of thermodynamic equilibrium states marked

by the control parameters ~

λ. Minimizing the irreversible

work in Eq. (3) is equivalent to ﬁnding the geodesic path

in the geometric space with the metric gµν . This prop-

erty allows us to obtain the optimal control protocol in

the shortcut scheme by using methods developed in ge-

ometry [31].

Generally, the auxiliary Hamiltonian Haare

momentum-dependent that are hard to be imple-

mented. For example, the auxiliary Hamiltonian for a

one-dimensional harmonic system Ho=p2/(2m)+λx2/2

is obtained as [9, 29] Ha=˙

λ[(p−γx)2+mλx2]/(4γλ).

The quadratic momentum-dependent term p2and the

linear momentum-dependent term xp in the auxiliary

Hamiltonian are hard to be realized in experiment

due to the requirement of constantly monitoring the

momentum [8].

III. APPROXIMATE SHORTCUT SCHEME

The variational shortcut scheme is an approximation

of the exact shortcut scheme. The auxiliary Hamiltonian

Hain the exact shortcut scheme is replaced by the ap-

proximate auxiliary Hamiltonian H∗

a. We deﬁne a semi-

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Geodesicpathfortheoptimalnonequilibriumtransition:Momentum-independentprotocolGengLi,1C.P.Sun,1,2andHuiDong1,1GraduateSchoolofChinaAcademyofEngineeringPhysics,Beijing100193,China2BeijingComputationalScienceResearchCenter,Beijing100193,ChinaAcceleratingcontrolledthermodynamicprocessesrequiresanauxil...

展开>> 收起<<

Geodesic path for the optimal nonequilibrium transition Momentum-independent protocol Geng Li1C. P. Sun1 2and Hui Dong1.pdf

共10页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Geodesic path for the optimal nonequilibrium transition Momentum-independent protocol Geng Li1C. P. Sun1 2and Hui Dong1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: