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arXiv:1105.5456v3 [cond-mat.stat-mech] 20 Jul 2011

Effects of error on fluctuations under feedback control

Sosuke Ito and Masaki Sano

Department of Physics, The University of Tokyo - Hongo, Bunkyo-ku, Tokyo, Japan

(Dated: July 21, 2011)

We consider a one-dimensional Brownian motion under nonequilibrium feedback control. Gener-

ally, the fluctuation-dissipation theorem (FDT) is violated in driven systems under nonequilibrium

conditions. We find that the degree of the FDT violation is bounded by the mutual information

obtained by the feedback system when the feedback protocol includes measurement errors. We

introduce two simple models to illustrate cooling processes by feedback control and demonstrate

analytical results for the cooling limit in those systems. Especially in a steady state, lower bounds

to the effective temperature are given by an inequality similar to the Carnot efficiency.

INTRODUCTION

Discussions on the Maxwell’s demon provided a bet-

ter understanding of the relation between information

entropy and entropy production [1–3]. As a generaliza-

tion of the relation between information entropy and en-

tropy production, the second law of thermodynamics is

extended to an open system under feedback control [4, 5].

The generalization of the second law is denoted by

β (?W? − ∆F) ≥ −?I?,(1)

where ?W? is the ensemble average of work W exerted on

the open system, ∆F is the free energy difference gained

in the open system, and ?I? is the mutual information

obtained by the feedback protocol.

is in contact with a thermal reservoir at temperature

T = (kBβ)−1, where kBis the Boltzmann constant. The

difference ?W?−∆F amounts to a dissipated work in the

open system. When the dissipated work becomes nega-

tive, the feedback can extract work from a heat reservoir.

The amount of work is bounded by the mutual informa-

tion ?I?, owing to the generalized second law, Eq.(1).

Feedback control in Brownian systems has important

applications in noise cancellation, namely cold damping

or entropy pumping [6, 7]. For instance in the cold damp-

ing, thermal noise of the cantilever in atomic force mi-

croscope (AFM) was canceled through a measurement of

velocity and feedback control with a force proportional

to the velocity of the cantilever [7]. Similarly in the en-

tropy pumping, the reduction of thermal fluctuations by

optical tweezers under velocity-dependent feedback con-

trol was proposed [6]. These discussions did not take into

account noise effects in the feedback system which is un-

avoidable in a real experiment. An ideal condition that

the effective temperature reaches 0 K was only discussed

in Ref. [7]. The fundamental limit of the cooling by feed-

back in the presence of measurement errors has not been

discussed.

To discuss the noise effects, we study the generalized

second law for the one-dimensional Langevin system and

derive the relation between fluctuations and mutual in-

formation.In our derivation, we can apply following

The open system

remarkable progresses in nonequilibrum statistical me-

chanics. The fluctuation theorem (FT) [8–10] and the

Jarzynski equality [11] are remarkable progresses which

are connected to the second law. The premise of the FT,

so-called the detailed fluctuation theorem, which is the

FT for specific trajectory, is also the premise of the gen-

eralized second law [5]. The detailed fluctuation theorem

can be derived for many systems including the Langevin

system [12–14]. The Maxwell’s demon can be discussed

using the FT for the Langevin system [6]. Moreover there

are relations between fluctuations and the entropy change

for the Langevin system. The Harada-Sasa equality or

the generalized fluctuation-dissipation theorem [15–17]

clarifies the relations between the rate of energy dissipa-

tion and the violation of the fluctuation dissipation the-

orem (FDT). The FDT is the relations between thermal

fluctuations and the dissipation in equilibrium and con-

nected to the FT and the Jarzynski equality [11, 12]. The

generalizations of the FDT for nonequilibrium processes

can be generally obtained by a perturbation dependence

of a path probability [17, 18].

In our discussion, we derive the Harada-Sasa equal-

ity and the generalized second law for a nonequilibrium

transition performed by feedback control. Since these two

equalities are connected in terms of the entropy change in

the heat reservoir, we can obtain the bounds to the FDT

violation. The FDT violation is bounded by the mutual

information characterizing as measurement errors of the

feedback system. Hence, the expression of the bounds

quantifies effects of error on the FDT violation. Here

we show that effects of error are dominant especially in

the cold damping system. We construct two cold damp-

ing models under velocity-dependent feedback control in-

cluding measurement errors and discuss the effects of er-

ror on the FDT violation. Furthermore, in view of the

effective temperature, the bounds to the FDT violation

give the cooling limit of the effective temperature in a

steady state. The lower bound to the effective tempera-

ture is determined by the balance between the informa-

tion obtained by the measurement for feedback control

and the information lost as a result of the relaxation.

The inequality giving the lower bound to the effective

temperature has a similar form to Carnot efficiency.

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SYSTEM AND FEEDBACK PROTOCOL

We study an underdamped Langevin equation includ-

ing the feedback described as

m¨ x(t) + γ ˙ x(t) = Fλ(t,y)(x(t)) + ǫfp(t) + ξ(t), (2)

where m is the mass of a Brownian particle and γ is the

friction coefficient. We assume that the friction coeffi-

cient γ does not depend on time t. The feedback force

Fλ(t,y)(x(t)) is the external force that generally includes

a potential force −∂U/∂x, and a constant driving force

fex as in Ref. [16]. λ(t,y) is a control parameter for a

nonequilibrium transition which depend on time t and

measurement outcomes y = {y1,...,yn}. ǫfp(t) is the

perturbation force which is introduced for the discus-

sion of the response function. The thermal noise ξ(t)

is zero-mean white Gaussian noise with variance 2γkBT.

Throughout our paper, multiplications of stochastic vari-

ables are assumed to be interpreted as the Stratonovich-

type integral without explicit remarks.

We consider a nonequilibrium transition performed by

the feedback force Fλ(t,y)(x(t)) from time t = 0 to t = τ.

We note the phase space point of the Langevin system at

time t as Γ(t) = (x(t), ˙ x(t)) and the trajectory of a tran-

sition asˆΓ = {Γ(t)|0 ≤ t ≤ τ}. We assume that mea-

surements for the feedback control are performed at time

t = tMi(i = 1,...,n), where 0 ≤ tM1≤ ... ≤ tMn≤ τ,

and the measurement outcomes yiare obtained at time

t = tMi. The probability of obtaining the measure-

ment outcome yi is depend on the phase space point

ΓMi= Γ(tMi). Therefore the stochastic process of mea-

surement outcomes y is determined by conditional prob-

abilities

pi(yi|ΓMi) = gi(yi,ΓMi), (3)

where gi(yi,ΓMi) is a function which characterizes the

measurement error of the feedback system. The condi-

tional probabilities are normalized as?dyipi(yi|ΓMi) =

1.

In the system, the measurement outcomes y determine

the time evolution of the feedback force Fλ(t,y)(x(t)).

Due to the causality of feedback control, the time evo-

lution of the feedback force Fλ(t,y)(x(t)) depends on the

i-th measurement outcome yifor t ≥ tMi(see Fig. 1).

When the measurement outcomes y are fixed, the time

evolution of the control parameter λ(t,y) is uniquely

determined. Then the path probability Pǫ

for the Langevin equation Eq.(2), is given by the

Stratonovich-type path-integral expression as

λ(t,y)[ˆΓ|Γ(0)]

Pǫ

λ(t,y)[ˆΓ|Γ(0)] =

1

Ne−β

4γ

?τ

0dt(m¨ x+γ ˙ x−Fλ−ǫfp)2, (4)

where N is a normalization constant independent of ǫ.

The path probability Pǫ

sity is defined as Pǫ

λ(t,y)[ˆΓ] including an initial den-

λ(t,y)[ˆΓ] = ρ0(Γ(0))Pǫ

λ(t,y)[ˆΓ|Γ(0)],

???????

?????????

????????

??????

??????

???????????

FIG. 1. An open system subjected to thermal noise is cou-

pled to a feedback system. The measurement outcome yi is

obtained as a function of the phase space ΓMi.

trol parameter λ depends on the measurement outcomes y

under feedback control. Ii is the mutual information which

is characterized as the dependence between the measurement

outcome yi and the phase space ΓMi. 2γkBT is the value of

the thermal fluctuation.

The con-

where ρ0(Γ) is an initial probability density at time t = 0

for a transition . This path probability is assumed to be

normalized by the path integral as?[DˆΓ]Pǫ

A probability density at time t is defined as ρt(Γ) =

?[DˆΓ]δ(Γ(t)−Γ)Pǫ

ensemble average of arbitrary path function A[ˆΓ] and ar-

bitrary phase function B(Γ) are defined as

λ(t,y)[ˆΓ] = 1.

λ(t,y)[ˆΓ]. For the feedback system, the

?A?ǫ=

?

Πidyi

?

[DˆΓ]A[ˆΓ]Pǫ

λ(t,y)[ˆΓ]pi(yi|ΓMi), (5)

and

?B(t)?ǫ=

?

Πidyi

?

[DˆΓ]B(Γ(t))Pǫ

λ(t,y)[ˆΓ]pi(yi|ΓMi).

(6)

These ensemble averages are the averages for all paths

and all measurement outcomes. To discuss the FDT vio-

lation, we define the response function R(t;s) of the sys-

tem for t > s using ǫ dependence of the ensemble average

as

?˙ x(t)?ǫ= ?˙ x(t)?0+ ǫ

?t

0

dsR(t;s)fp(s) + O(ǫ2), (7)

where ?...?0is an ensemble average when the perturba-

tion force ǫfpis zero. Due to the causality, R(t,s) = 0 is

satisfied for t < s. Moreover, the same time response

is defined as R(t,t) = 1/2[R(t;t − 0) + R(t;t + 0)] =

1/2R(t;t − 0).

To consider a steady state, we generalize Eq.(1) for sev-

eral measurements and feedbacks. We define the i-th mu-

tual information Iibetween the system’s state ΓMiand

the measurement outcome yias Ii≡ lnpi(yi|ΓMi)/pi(yi)

where pi(yi) is the probability of obtaining the outcome

yi in the i-th measurement. The probability pi(yi) is

calculated as

pi(yi) =

?

Πj?=idyj

?

[DˆΓ]Pǫ

λ(t,y)[ˆΓ]Πkpk(yk|ΓMk). (8)

Here, the normalization?dyipi(yi) = 1 is satisfied.

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3

MAIN RESULT

For the Langevin system including feedback effects, we

prove the following inequality

β

?τ

0

dtγ

??˙ x(t)2?

0−2

βR(t;t)

?

≥ ?∆φ?0−

?

i

?Ii?0,

(9)

where ?∆φ?0= ?lnρ0(Γ(0)) − lnρτ(Γ(τ))?0is the en-

tropy change of the system. This inequality shows that

the time integral of the FDT violation is bounded by

the sum of the mutual information. On the other hand,

when the measurement outcome is obtained without er-

ror, the mutual information ?Ii?0goes to infinity. In this

limit, the bounds to the violation of FDT are vanished.

When the measurement outcome is obtained with error,

the mutual information has finite values.

is interpreted as effects of error on the FDT violation.

The bounds are crucial especially in the condition that

the correlation term?˙ x2(t)?is smaller than the response

term 2R(t;t)/β. Therefore, the bounds can be important

for a problem where the dynamical feedback makes the

effective temperature of the system lower than the tem-

perature of the heat reservoir because the effective tem-

perature is defined as the ratio of the correlation term to

the response term in a steady state,

This result

Teff=

?˙ x2(t)?

0

2kBR(t;t).(10)

The relation between the generalized FDT and the ef-

fective temperature is discussed in Ref. [19]. When the

state of the system is considered to be in a steady state

approximately by a time-independent feedback protocol

as a result of coarse graining over time, we can prove the

following relation for the effective temperature Teff

Teff− T

T

≥ −

?

i?Ii?0

τ

tr. (11)

where tr = m/γ is the relaxation time,?

sum of the mutual information obtained within the time

duration τ. Then?

information rate obtained by the measurement. The left

hand side of Eq.(11) is similar to the Carnot efficiency

and the right hand side is considered as the information

obtained in the relaxation time. It is worth indicating

that tris the characteristic time of the relaxation to an

equilibrium state in terms of the velocity without exter-

nal forces (Fλ+ ǫfp = 0). In other words, the system

practically forgets the information of the velocity after

time m/γ. In order to cool the system down to a lower

temperature, we should obtain the information of the

velocity of a particle before the system looses the infor-

mation of the velocity and apply feedback control. If the

system is considered to be an over-damped Langevin sys-

tem (m/γ → 0), the right hand side of Eq.(11) becomes

i?Ii?0is the

i?Ii?0/τ is considered to be a mutual

zero. Then this inequality indicates an inability to cool

the over-damped Langevin system by the feedback force.

In addition, if the relaxation time is smaller than the

measurement interval, cooling the system is difficult and

the lower bounds to the effective temperature are decided

by this inequality. We prove these inequalities in the next

section.

PROOF

For the discussion of the detailed fluctuation theorem,

a reversal process is introduced. Then we define a time-

reversal map as (x, ˙ x)∗= (x,−˙ x). WhenˆΓ is considered

to be a trajectory of a forward process, a trajectory of the

reverse process is described asˆΓ†= {Γ∗(τ−t)|0 ≤ t ≤ τ}.

In the reversal process, a control parameter is introduced

as λ(τ − t,y) using a protocol of the forward process

λ(t,y). We assume that the initial probability density

of the trajectory of the reversed process is equal to the

final probability density of one of the forward process

(ρ0(Γ∗(τ)) = ρτ(Γ(τ))). According to Eq.(4), the local

detailed balance for the Langevin system is derived as

Pǫ

λ(τ−t,y)[ˆΓ†]

λ(t,y)[ˆΓ]

Pǫ

= exp

??τ

0

dtω(t) − ∆φ

?

,(12)

where ω(t) is the entropy production rate defined as

ω(t) = β ˙ x(t)?Fλ(t,y)(x(t)) + ǫfp(t) − m¨ x(t)?.

The entropy production rate ω(t) = β ˙ x(t)[γ ˙ x(t) − ξ(t)]

is consistent with the definition of the energy dissipation

rate in Ref. [20]. The generalized Jarzynski equality [5]

for the system can be derived using the definition of the

ensemble average including the feedback, Eq.(5), as

(13)

?

?

e−?τ

0dtω(t)+∆φ−?

iIi

?

ǫ

=Πidyipi(yi)

?

[DˆΓ]Pǫ

λ(τ−t,y)[ˆΓ†]

= 1. (14)

Due to a concavity of the exponential function, the

Jensen’s inequality for Eq.(14) is obtained.

Jensen’s inequality for ǫ = 0 is equal to the generaliza-

tion of the second law for the feedback Langevin system

as

?τ

0

Then the

β dt?˙ x(t)?Fλ(t,y)(x(t)) − m¨ x(t)??

−?∆φ?0≥ −

0

?

i

?Ii?0,(15)

because the left hand side of Eq.(15) is the entropy pro-

duction from time t = 0 to t = τ and the entropy produc-

tion is bounded by the sum of mutual information from

time t = 0 to t = τ.

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To discuss the violation of FDT, we start with the

identity

∂

∂ǫ

?

˙ x(t)e−ǫβ?τ

????

0dt′˙ x(t′)fp(t′)?

?τ

0

ǫ

????

ǫ=0

=∂ ?˙ x(t)?ǫ

∂ǫ

ǫ=0

− β

dt′fp(t′)?˙ x(t)˙ x(t′)?0. (16)

The definition of the response function, Eq.(7), give us

the relation

∂ ?˙ x(t)?ǫ

∂ǫ

????

ǫ=0

=

?t

0

dt′R(t;t′)fp(t′). (17)

Moreover, we can calculate the identity, Eq.(16), exactly

using the path probability, Eq.(4), as

∂

∂ǫ

?

?τ

0

˙ x(t)e−ǫβ?τ

0dt′˙ x(t′)fp(t′)?

ǫ

????

ǫ=0

=

β

2γ

+m¨ x(t′)]?0.

dt′fp(t′)?˙ x(t)?−γ ˙ x(t′) − Fλ(t′,y)(x(t′))

(18)

A small impulse force fp(t′) = δ(t′−t+s) is substituted

for Eqs.(16), (17) and (18) for s ?= t then the generalized

FDT for the feedback system can be derived as

γ

?

?˙ x(t)˙ x(t − s)?0−2

=?˙ x(t)?Fλ(t−s,y)(x(t − s)) − m¨ x(t − s)??

using the causality R(t;t + s) = 0 for s > 0. Owing to

the definition of the Stratonovich-integral and the same

time response R(t;t), the relation between the same time

response and correlation can be obtained as

βR(t;t − s)

?

0. (19)

γ

??˙ x2(t)?

=?˙ x(t)?Fλ(t,y)(x(t)) − m¨ x(t)??

This equality is the Harada-Sasa equality for the

Langevin system with feedback.

of Eq.(20) is the degree of the violation of FDT

and the right hand side of Eq.(20) represents the en-

ergy dissipation rate. In an equilibrium state, the

FDT violation is vanished because the feedback force

Fλ(t,y)(x(t)) is considered to be a time-independent po-

tential force −∂U(x)/∂x. Therefore both the correlations

?˙ x(t)Fλ(t,y)(x(t))?

Therefore we obtain the first main result Eq.(9) from

Eqs.(15) and (20). This result is valid for the Langevin

dynamics driven by the feedback force.

To discuss the effective temperature, we assume that

the system is considered to be in a nonequilibrium steady

state approximately as a result of coarse graining over

time. A steady state can be introduced when a feed-

back protocol is independent in time. In our protocol,

0−2

βR(t;t)

?

0.(20)

The left hand side

0and ?˙ x(t)¨ x(t)?0are zero.

the i-th measurement outcome dependence of the feed-

back force is independent of i. In a steady state, the

correlation term?˙ x2(t)?and the response term R(t;t) do

not depend on time t. The effective temperature Teff

is defined by the ratio of the correlation term to the re-

sponse term in a steady state as Eq.(10). In an equilib-

rium state, the effective temperature is equal to the tem-

perature of the heat reservoir because the degree of the

FDT violation is zero?˙ x2(t)?

in the nonequilibrium steady state, the response func-

tion R(t;t) is calculated by using the Furutsu-Novikov-

Donsker formula as in Refs. [21, 22] when the noise term

ξ(t) is a zero-mean white Gaussian noise.

lation ?˙ x(t)ξ(t)?0becomes 2R(t;t)/β. Moreover, we can

calculate ?˙ x(t)ξ(t)?0by the definition of the Stratonovich

integral. When ǫ = 0, the correlation ?˙ x(t)ξ(t)?0is cal-

culated as γ/(mβ). The same time response R(t;t) in

a steady state is obtained exactly as R(t;t) = 1/(2m).

This fact shows that the effective temperature fulfills

0− 2R(t;t)/β = 0. While

The corre-

?1

2m˙ x2

?

0

=1

2kBTeff.(21)

If the probability of particle’s velocity is a zero-mean

Gaussian distribution, Eq.(21) means that the distribu-

tion of a steady state is considered to be the Maxwell-

Boltzmann distribution with the temperature Teff. Let

the value R(t;t) = 1/(2m), a steady-state condition

?∆φ?0= 0 and Eq.(21) substitute the first main result,

Eq.(9), then we can obtain the second main result, Eq.

(11).

MODELS FOR COLD DAMPING

At first we consider the cold damping process [7] or

entropy pumping [6] generally given by the following

Langevin equation

m¨ x(t) + γx(t) = −γ′˙ x(t) + ξ(t).

In this model, γ′is positive. This cold damping pro-

cess was proposed in the experiment of cooling a Brow-

nian particle by applying a velocity-dependent feedback

−γ′˙ x(t). In a realistic experimental setup, this feedback

can be realized by using optical tweezers [6, 23]. In a

steady state, the effective temperature of this system

Tγ/(γ + γ′) was found to be lower than the temperature

of the heat reservoir T. Thus this model is considered as

the noise cancellation. The feedback of this model in-

cludes the velocity of the Brownian particle ˙ x(t) without

a measurement error.

We substitute Fλ

= −γ′˙ x(t) into Eq.(20), then

the FDT violation of the system is calculated as

−γ′?˙ x2(t)?

condition d/dt?(m/2) ˙ x2(t)?

term?(m/2) ˙ x2(t)?

(22)

0−d/dt?(m/2) ˙ x2(t)?

0. In a steady state, the

0= 0 is derived because the

0does not depend on time t. Then the

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FDT violation −γ′?˙ x2(t)?

state. The effective temperature of the system is calcu-

lated by the definition Eq.(10) as Teff = Tγ/(γ + γ′).

In the limit of γ′→ ∞, the effective temperature Teff

reaches 0 K. This model does not have the cooling bounds

to the effective temperature by the mutual information

because the feedback protocol is free from measurement

errors, thus the mutual information goes to infinity. In

terms of the measurement error, this model cannot de-

scribe the actual setup because the feedback protocol has

measurement errors in the actual experimental setup. If

the feedback protocol of the cold damping has measure-

ment errors, the bounds to the FDT violation given by

Eq.(9) are dominant and therefore the effective tempera-

ture cannot reach 0 K. To discuss the effects of errors on

the FDT violation, we consider the following two models

including measurement errors. We show the validity of

the bounds to the FDT violation given by Eq.(9).

0is always negative in a steady

Case 1

A model for cold damping with continuous output feed-

back can be described by the Langevin equation

m¨ x(t) + γ ˙ x(t) = Fλ(t,y)(x(t)) + ξ(t). (23)

We consider the following feedback protocol for one cy-

cle. Firstly, a measurement about the velocity ˙ x(0) = ˙ x0

is performed at time t = 0. Secondly, a measurement

outcome y about the velocity ˙ x0is obtained. In order to

introduce the measurement error, we consider the condi-

tional probability is Gaussian with variance σ2

erras

p(y|˙ x0) =

1

?2πσ2

err

exp

?

−(˙ x0− y)2

2σ2

err

?

. (24)

Thirdly, a constant force Fλ(t,y)(x(t)) = −γ′y is applied

to the system from time t = 0 to t = τ.

back sequence defines one cycle. In repeating this cycle,

we assume that the system has the same Gaussian dis-

tribution about the velocity at time t = 0 and t = τ,

instead of the assumption of a steady state, described as

p(˙ x0) = p(˙ x(τ)) = 1/√2πσ2exp?−˙ x2

the noise cancellation, the variance of the steady state

density becomes smaller than the one of the original

Maxwell-Boltzmann distribution with temperature T as

1/(mβ) ≥ σ2.

In this model, we can show the validity of Eq.(9) for

one cycle. Let the left hand side of Eq.(9) be defined as

Ωτ = β?τ

tion, Ωτ, can be calculated using Eq.(20) as

This feed-

0/?2σ2??. Due to

0dtγ??˙ x(t)2?

0− 2R(t;t)/β?. The FDT viola-

Ωτ= β

?τ

0

?

dt?˙ x(t)Fλ(t,y)(x(t))?

βm

2

0

−

?˙ x2(0) − ˙ x2(τ)??

0,(25)

In this condition,

?m/2?˙ x2(0) − ˙ x2(τ)??

probability distribution is the same at t = 0 and t = τ.

Then we compare the value of the FDT violation Ωτ

and the mutual information ?I? to discuss the validity

of Eq.(9). We can exactly calculate the violation of FDT

as

the relations ?∆φ?0

0= 0 is calculated because the

= 0 and

Ωτ= −β

?τ

0

dt

?∞

−∞

dy

?∞

−∞

d˙ x0p(˙ x0)p(y|˙ x0)γ′y¯˙ x(t),

(26)

where¯˙ x(t) is the average of the velocity in terms of the

thermal noise ξ(t). ¯˙ x(t) obeys the equation of motion

m(d/dt)¯˙ x(t) = −γ¯˙ x(t) − γ′y, then the solution of the

equation of motion is calculated as

¯˙ x(t) = −γ′y

γ

+

?

˙ x0+γ′y

γ

?

e−γ

mt.(27)

Then we substitute Eqs.(27) and (24) into Eq.(26) to

obtain the value of the FDT violation as

Ωτ= βγ′2

γ(σ2+ σ2

−βγ′m

γ

err)τ

?

σ2+γ′

γ

?σ2+ σ2

err

???

1 − e−γ

mτ?

.(28)

When (dΩτ/dτ) |τ=τmin= 0, the FDT violation has min-

imum value in terms of τ. The value of τminis calculated

as τmin = m/γ ln?1 + (γ/γ′)?σ2/?σ2+ σ2

fore, the minimum value of the FDT violation Ωτminis

obtained as

err

???. There-

Ωτmin=βmγ′2(σ2+ σ2

err)

γ2

ln

?

1 +γ

γ′

σ2

σ2+ σ2

err

?

−βmγ′σ2

≃ −mβσ2

γ

2

1

1 + σ2

r

.(29)

where σr = σerr/σ. In this calculation, the logarithmic

term is expanded in terms of σ2/?σ2+ σ2

On the other hand, the mutual information ?I?0can

be calculated. The probability of obtaining the measure-

ment outcome p(y) is calculated as

err

?(≤ 1).

p(y) =

?∞

−∞

d˙ x0p(y|˙ x0)p(˙ x0)

1

?2π(σ2+ σ2

=

err)exp

?

−

y2

2(σ2+ σ2

err)

?

. (30)

Then, the mutual information ?I?0is obtained as

?∞

−∞ −∞

=1

2lnσ2

r

?I?0= dy

?∞

d˙ x0p(˙ x0)p(y|˙ x0)lnp(y|˙ x0)

?

p(y)

?

1 +

1

. (31)

Page 6

6

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5

FDT violation Ω

Normalized error σr

-<I>0

Ωτmin (mβσ2=1)

Ωτmin (mβσ2=1/2)

FIG. 2. Minimum values of FDT violation (dashed lines) and

mutual information (solid line) in Case 1. Mutual information

is less than FDT violation. Then the main result, Eq.(9), is

valid in Case 1.

The results of Eqs.(29), (31) and the condition of the

variance 1/(mβ) ≥ σ2give us the inequality

Ωτmin≥ −1

2 1 + σ2

1

r

≥ −?I?0. (32)

Thus the bounds to the FDT violation given by Eq.(9)

are valid in this model. According to Fig. 2, the bounds

to the FDT violation is effective when the measurement

error cannot be negligible (σ2

depend on any of the parameters τ, m, γ, γ′, σ and σerr.

In other words, the validity of Eq.(9) does not depend on

the feedback parameter in this model.

r≫ 1). This result does not

Case 2

Here we consider the case that the output of feedback

control system takes only discrete values. We assume

that the system has only binary states for the measure-

ment outcome. In such a case, without loss of generality,

the measurement outcome y can be simply represented

by y = 0 for negative values of ˙ x0observations, or y = 1

otherwise. The measurement error rate q (0 ≤ q ≤ 1/2)

is introduced by the conditional probability as

p(0|˙ x0) =

?q (˙ x0≥ 0)

1 − q (˙ x0< 0), (33)

p(1|˙ x0) =

?1 − q (˙ x0≥ 0)

q (˙ x0< 0). (34)

Here, a constant force Fλ(t,0)(x(t)) = γ′or Fλ(t,1)(x(t)) =

−γ′(γ′> 0) is applied to the system from time t = 0

to t = τ, depending on the value of y.

back sequence is considered as one cycle.

ing this cycle, we also assume that the system has the

This feed-

In repeat-

same Gaussian distribution about the velocity at time

t = 0 and t = τ described as p(˙ x0) = p(˙ x(τ)) =

1/√2πσ2exp?−˙ x2

condition 1/(mβ) ≥ σ2. In this case, the violation of

FDT Ωτ can be also calculated as

0/?2σ2??. Moreoverwe also assume the

Ωτ= β

?

y

?τ

0

dt

?∞

−∞

d˙ x0p(˙ x0)p(y|˙ x0)Fλ(t,y)(x(t))¯˙ x(t),

(35)

where¯˙ x(t) is calculated as

¯˙ x(t) = ±γ′

γ−

?

−˙ x0±γ′

γ

?

e−γ

mt. (36)

The plus and minus signs of Eq.(36) correspond to y = 0

and y = 1, respectively. By substituting Eqs.(33), (34)

and (36) into Eq.(35), we obtain the value of the FDT

violation as

Ωτ= βγ′2

γτ − βm

γ

?γ′2

γ

− 2(1 − 2q)γ′σ

√2π

??

1 − e−γ

mτ?

(37)

.

When (dΩτ/dτ) |τ=τmin

hasminimum

case,the value of τmin is calculated as τmin

m/γ ln?1 + (2q − 1)?2σγ/?√2πγ′???.

minimum value of the FDT violation Ωτminis obtained

as

= 0, the FDT violation

ofτ.value intermsInthis

=

Therefore, the

Ωτmin=βmγ′

γ

?γ′

γln

?

1 + (1 − 2q)

?

2γσ

√2πγ′

?

−(1 − 2q)2σ

≃ −mβσ2

π

√2π

(1 − 2q)2. (38)

In this calculation, the logarithmic term is expanded in

terms of (1 − 2q) (≤ 1).

On the other hand, the probability of obtaining the

measurement outcome p(y) is calculated as p(0) = 1/2

and p(1) = 1/2. Then, the mutual information ?I?0is

obtained as

?I?0=

?

y

?∞

−∞

d˙ x0p(˙ x0)p(y|˙ x0)lnp(y|˙ x0)

p(y)

= ln2 + q lnq + (1 − q)ln(1 − q)

The results Eqs.(38), (39) and the condition of the vari-

ance 1/(mβ) ≥ σ2give us the inequality (see Fig. 3)

(39)

Ωτmin≥ −(1 − 2q)2

π

≥ −?I?0. (40)

Therefore we demonstrated that Eq.(9) is valid regardless

of different feedback protocols.

Page 7

7

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-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.1 0.2 0.3 0.4 0.5

FDT violation Ω

Measurement error rate q

-<I>0

Ωτmin (mβσ2=1)

Ωτmin (mβσ2=1/2)

FIG. 3. Minimum values of FDT violation (dashed lines) and

mutual information (solid line) in Case 2. Mutual information

is less than FDT violation. Then the main result, Eq.(9), is

valid in Case 2.

DISCUSSION

In this paper, we have discussed the effects of error on

the FDT violation and the effective temperature using

the Langevin dynamics under feedback with error. The

bounds to the FDT violation and the effective temper-

ature as a function of the mutual information are de-

rived. Then we presented two simple models to demon-

strate analytical calculations for the validity of the gen-

eralized second law Eq.(1) for Langevin system including

the velocity-dependent feedback with error. Moreover,

the result about the effective temperature is considered

to be the relation between the information obtained by

the measurement and the relaxation. We believe this re-

sult is a valuable approach to the nonequilibrium steady

state dynamics when contents of the information play a

significant role in the feedback control systems.

As a possible experimental realization of the proposed

results, cooling of the Brownian particle by applying a

feedback force with laser tweezers might be a good can-

didate, since the velocity of the Brownian particle is

measurable in the present technology [23]. In vacuum,

millikelvin cooling of the Brownian particle was recently

archived and the lowest temperature would be limited

by the noise [24]. For the generalized second law, the

inequality Eq.(1) has been tested by our group in the

feedback system of a Brownian particle [25]. Therefore

experimental verification may be technically feasible. A

more important extension of the present result will be

the generalization to quantum system in which measure-

ment error comes from quantum fluctuations, or general-

ization to many particle systems. It is worth noting that

the stochastic cooling in particle acceleration technology

uses a periodic feedback control. It would be interesting

to look for a theoretical relation with mutual information

in many particle systems as in Ref.[26].

ACKNOWLEDGMENTS

The authors would like to thank Dr. T. Sagawa and

Prof. S. Sasa for their valuable comments.

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