Authors

Jean-Charles Delvenne, Henrik Sandberg

Abstract

In this paper, we identify a class of time-varying port-Hamiltonian systems that is suitable for studying problems at the intersection of statistical mechanics and control of physical systems. Those port-Hamiltonian systems are able to modify their internal structure as well as their interconnection with the environment over time. The framework allows us to prove the First and Second Laws of thermodynamics, but also lets us apply results from optimal and stochastic control theory to physical systems. In particular, we show how to use linear control theory to optimally extract work from a single heat source over a finite time interval in the manner of Maxwell’s demon. Furthermore, the optimal controller is a time-varying port-Hamiltonian system, which can be physically implemented as a variable linear capacitor and transformer. We also use the theory to design a heat engine operating between two heat sources in finite-time Carnot-like cycles of maximum power, and we compare those two heat engines.

Keywords

Hamiltonian systems; Statistical mechanics; Thermodynamics; Optimal control theory; Stochastic control theory

Citation

  • Journal: Physica D: Nonlinear Phenomena
  • Year: 2014
  • Volume: 267
  • Issue:
  • Pages: 123–132
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.physd.2013.07.017
  • Note: Evolving Dynamical Networks

BibTeX

@article{Delvenne_2014,
  title={{Finite-time thermodynamics of port-Hamiltonian systems}},
  volume={267},
  ISSN={0167-2789},
  DOI={10.1016/j.physd.2013.07.017},
  journal={Physica D: Nonlinear Phenomena},
  publisher={Elsevier BV},
  author={Delvenne, Jean-Charles and Sandberg, Henrik},
  year={2014},
  pages={123--132}
}

Download the bib file

References

  • Kubo, R. The fluctuation-dissipation theorem. Reports on Progress in Physics vol. 29 255–284 (1966) – 10.1088/0034-4885/29/1/306
  • Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
  • Willems, J. C. Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis vol. 45 352–393 (1972) – 10.1007/bf00276494
  • Borkar, A note on stochastic dissipativeness. (2002)
  • Brockett, R. & Willems, J. Stochastic control and the second law of thermodynamics. 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes 1007–1011 (1978) doi:10.1109/cdc.1978.268083 – 10.1109/cdc.1978.268083
  • Brockett, Control of stochastic ensembles. (1999)
  • Salamon, P., Nulton, J. D., Siragusa, G., Andersen, T. R. & Limon, A. Principles of control thermodynamics. Energy vol. 26 307–319 (2001) – 10.1016/s0360-5442(00)00059-1
  • Mitter, S. K. & Newton, N. J. Information and Entropy Flow in the Kalman?Bucy Filter. Journal of Statistical Physics vol. 118 145–176 (2005) – 10.1007/s10955-004-8781-9
  • Eberard, D., Maschke, B. M. & van der Schaft, A. J. An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes. Reports on Mathematical Physics vol. 60 175–198 (2007) – 10.1016/s0034-4877(07)00024-9
  • Haddad, (2005)
  • Gromov, On the stability of interconnected thermodynamic systems with heat and work exchange. (2012)
  • Warden, R. B., Aris, R. & Amundson, N. R. An analysis of chemical reactor stability and control—VIII. Chemical Engineering Science vol. 19 149–172 (1964) – 10.1016/0009-2509(64)85027-2
  • Favache, A. & Dochain, D. Thermodynamics and chemical systems stability: The CSTR case study revisited. Journal of Process Control vol. 19 371–379 (2009) – 10.1016/j.jprocont.2008.07.007
  • Sandberg, H., Delvenne, J.-C. & Doyle, J. C. On Lossless Approximations, the Fluctuation- Dissipation Theorem, and Limitations of Measurements. IEEE Transactions on Automatic Control vol. 56 293–308 (2011) – 10.1109/tac.2010.2056450
  • Sandberg, H. & Delvenne, J.-C. The Observer Effect in Estimation with Physical Communication Constraints*. IFAC Proceedings Volumes vol. 44 12483–12489 (2011) – 10.3182/20110828-6-it-1002.02312
  • Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
  • Cervera, J., van der Schaft, A. J. & Baños, A. Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica vol. 43 212–225 (2007) – 10.1016/j.automatica.2006.08.014
  • (2009)
  • Maxwell, (1897)
  • Leff, (2010)
  • Andresen, (1983)
  • Orlov, V. N. & Berry, R. S. Power output from an irreversible heat engine with a nonuniform working fluid. Physical Review A vol. 42 7230–7235 (1990) – 10.1103/physreva.42.7230
  • Hoffmann, An introduction to endoreversible thermodynamics. AAPP—Phys., Math., and Nat. Sci. (2008)
  • Delvenne, J.-C., Sandberg, H. & Doyle, J. C. Thermodynamics of linear systems. 2007 European Control Conference (ECC) 840–847 (2007) doi:10.23919/ecc.2007.7068722 – 10.23919/ecc.2007.7068722
  • Sandberg, Linear-quadratic-gaussian heat engines. (2007)
  • Mandal, D. & Jarzynski, C. Work and information processing in a solvable model of Maxwell’s demon. Proceedings of the National Academy of Sciences vol. 109 11641–11645 (2012) – 10.1073/pnas.1204263109
  • Strasberg, P., Schaller, G., Brandes, T. & Esposito, M. Thermodynamics of a Physical Model Implementing a Maxwell Demon. Physical Review Letters vol. 110 (2013) – 10.1103/physrevlett.110.040601
  • Schöberl, M. & Schlacher, K. On an intrinsic formulation of time-variant Port Hamiltonian systems. Automatica vol. 48 2194–2200 (2012) – 10.1016/j.automatica.2012.06.014
  • Willems, J. Terminals and Ports. IEEE Circuits and Systems Magazine vol. 10 8–26 (2010) – 10.1109/mcas.2010.938635
  • Duindam, (2009)
  • Nyquist, H. Thermal Agitation of Electric Charge in Conductors. Physical Review vol. 32 110–113 (1928) – 10.1103/physrev.32.110
  • Caldeira, A. O. & Leggett, A. J. Path integral approach to quantum Brownian motion. Physica A: Statistical Mechanics and its Applications vol. 121 587–616 (1983) – 10.1016/0378-4371(83)90013-4
  • Gupta, M. S. Thermal fluctuations in driven nonlinear resistive systems. Physical Review A vol. 18 2725–2731 (1978) – 10.1103/physreva.18.2725
  • Anderson, B. D., Spaulding, D. A. & Newcomb, R. W. The time-variable transformer. Proceedings of the IEEE vol. 53 634–634 (1965) – 10.1109/proc.1965.3948
  • Cover, (1991)
  • Chambadal, (1957)
  • Novikov, I. I. The efficiency of atomic power stations (a review). Journal of Nuclear Energy (1954) vol. 7 125–128 (1958) – 10.1016/0891-3919(58)90244-4
  • Åström, (2006)