One-Dimensional Shift of a Brownian Particle under the Feedback Control

Suzuki, Hiroyuki; Fujitani, Youhei

doi:10.1143/jpsj.78.074007

Cited by 16 publications

(37 citation statements)

References 23 publications

(23 reference statements)

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“…A corresponding inequality in a classical system was studied numerically. 6) Another inequality involving the time-reversal of the process was also studied. 7,8) Sagawa and Ueda recently considered a stochastic process of a classical system, whose output is measured at a single time.…”

Section: Introductionmentioning

confidence: 99%

Jarzynski Equality Modified in the Linear Feedback System

Fujitani

Suzuki

2010

J. Phys. Soc. Jpn.

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It is known that one can derive the Jarzynski equality in a stochastic process of a classical system by assuming the local detailed balance. We study how the equality is modified in the linear feedback system. There, measurement is performed on the state following a linear Langevin equation, the measured variable is linear to the state variable with white sensor noise, the Kalman filter estimates the state by utilizing measured values in the past, and a linear regulator controls the state dynamics. Although a stochastic process produced by this dynamics is non-Markovian because of the feedback loop, it is known in the control theory that a Markov process for the estimation can be separated from the whole process. To the exponent in the Jarzynski equality, we find an additional term, whose average gives the mutual information between state variables and measured variables in the Markov process for the estimation. The resultant equality holds whether the gain is optimal or not.

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Section: Introductionmentioning

confidence: 99%

Jarzynski Equality Modified in the Linear Feedback System

Fujitani

Suzuki

2010

J. Phys. Soc. Jpn.

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“…This behavior was also observed in the model studied in ref. 7. The device finally appears to reach the origin with a sufficiently low speed in Fig.…”

Section: Resultsmentioning

confidence: 88%

“…These two kinds of independence were also found in the model discussed in ref. 7. Below we use R 1 ¼ 10 À1 and R 2 ¼ 10 4 .…”

Section: Resultsmentioning

confidence: 99%

“…[1][2][3][4][5][6] We concentrate on the linear feedback system as in our previous works. 3,7) The dynamics of the plant state is described by a linear Langevin equation having a term representing the external force. At many times, with white sensor noise, we measure some variables, which are linear to the state variables.…”

Section: Introductionmentioning

confidence: 99%

“…(1.1) is convenient in numerical studies on the Gaussian process under the linear feedback system. 16,17) We once studied a simple linear feedback system, 7) where a charged Brownian particle is shifted in one dimension from a position to another within a time interval by means of an electric field. Because the field is switched off after the shift process, the initial equilibrium state is the same as the final equilibrium state, which leads to ÁF ¼ 0.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Study on the Generalized Second Law for a Brownian Particle under the Linear Feedback Control

Suzuki

Fujitani

2012

J. Phys. Soc. Jpn.

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The second law of thermodynamics can be written by means of an inequality between the work done to the system and the change in its Helmholtz free energy during an isothermal process. The work is minimized and the inequality becomes the equality when the process is quasistatic. It is known that, under a feedback control, the inequality is modified to contain a quantity representing information content in the measurement associated with the control. In this study, we numerically investigate a model, where a Brownian particle is shifted in one dimension by means of the harmonic potential under the linear feedback loop designed so that the average work is decreased as much as possible. The average work under the optimal control depends on the sensor noise, the process duration time, and other factors. It is shown that, when we decrease the average work by changing a factor, other than the particle friction, the inequality does not approach the equality significantly.

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Stochastic thermodynamics, fluctuation theorems and molecular machines

2012

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Stochastic thermodynamics as reviewed here systematically provides a framework for extending the notions of classical thermodynamics such as work, heat and entropy production to the level of individual trajectories of well-defined non-equilibrium ensembles. It applies whenever a non-equilibrium process is still coupled to one (or several) heat bath(s) of constant temperature. Paradigmatic systems are single colloidal particles in time-dependent laser traps, polymers in external flow, enzymes and molecular motors in single molecule assays, small biochemical networks and thermoelectric devices involving single electron transport. For such systems, a first-law like energy balance can be identified along fluctuating trajectories. For a basic Markovian dynamics implemented either on the continuum level with Langevin equations or on a discrete set of states as a master equation, thermodynamic consistency imposes a local-detailed balance constraint on noise and rates, respectively. Various integral and detailed fluctuation theorems, which are derived here in a unifying approach from one master theorem, constrain the probability distributions for work, heat and entropy production depending on the nature of the system and the choice of non-equilibrium conditions. For non-equilibrium steady states, particularly strong results hold like a generalized fluctuation-dissipation theorem involving entropy production. Ramifications and applications of these concepts include optimal driving between specified states in finite time, the role of measurement-based feedback processes and the relation between dissipation and irreversibility. Efficiency and, in particular, efficiency at maximum power can be discussed systematically beyond the linear response regime for two classes of molecular machines, isothermal ones such as molecular motors, and heat engines such as thermoelectric devices, using a common framework based on a cycle decomposition of entropy production.

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One-Dimensional Shift of a Brownian Particle under the Feedback Control

Cited by 16 publications

References 23 publications

Jarzynski Equality Modified in the Linear Feedback System

Jarzynski Equality Modified in the Linear Feedback System

Numerical Study on the Generalized Second Law for a Brownian Particle under the Linear Feedback Control

Stochastic thermodynamics, fluctuation theorems and molecular machines

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