### Abstract

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.

Original language | English |
---|---|

Article number | 104003 |

Journal | Journal of the Physical Society of Japan |

Volume | 79 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2010 Oct |

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### Keywords

- Helmholtz free-energy
- Separation theorem
- The second law
- Time-reversal process

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Journal of the Physical Society of Japan*,

*79*(10), [104003]. https://doi.org/10.1143/JPSJ.79.104003

**Jarzynski equality modified in the linear feedback system.** / Fujitani, Youhei; Suzuki, Hiroyuki.

Research output: Contribution to journal › Article

*Journal of the Physical Society of Japan*, vol. 79, no. 10, 104003. https://doi.org/10.1143/JPSJ.79.104003

}

TY - JOUR

T1 - Jarzynski equality modified in the linear feedback system

AU - Fujitani, Youhei

AU - Suzuki, Hiroyuki

PY - 2010/10

Y1 - 2010/10

N2 - 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.

AB - 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.

KW - Helmholtz free-energy

KW - Separation theorem

KW - The second law

KW - Time-reversal process

UR - http://www.scopus.com/inward/record.url?scp=77958061402&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958061402&partnerID=8YFLogxK

U2 - 10.1143/JPSJ.79.104003

DO - 10.1143/JPSJ.79.104003

M3 - Article

AN - SCOPUS:77958061402

VL - 79

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 10

M1 - 104003

ER -