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.
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 . Similarly in the en-
tropy pumping, the reduction of thermal fluctuations by
optical tweezers under velocity-dependent feedback con-
trol was proposed . 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. . The fundamental limit of the cooling by feed-
back in the presence of measurement errors has not been
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  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 . 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 . 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.
0 0.1 0.2 0.3 0.4 0.5
FDT violation Ω
Measurement error rate q
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.
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 . In vacuum,
millikelvin cooling of the Brownian particle was recently
archived and the lowest temperature would be limited
by the noise . For the generalized second law, the
inequality Eq.(1) has been tested by our group in the
feedback system of a Brownian particle . 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..
The authors would like to thank Dr. T. Sagawa and
Prof. S. Sasa for their valuable comments.
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