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The Dissipation Function: Its Relationship to Entropy Production, Theorems for Nonequilibrium Systems and Observations on Its Extrema

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Beyond the Second Law

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this chapter we introduce the dissipation function, and discuss the behaviour of its extrema. The dissipation function allows the reversibility of a nonequilibrium process to be quantified for systems arbitrarily close to or far from equilibrium. For a system out of equilibrium, the average dissipation over a period, t, will be positive. For field driven flow in the thermodynamic and small field limits, the dissipation function becomes proportional to the rate of entropy production from linear irreversible thermodynamics. It can therefore be considered as an entropy-like quantity that remains useful far from equilibrium and for relaxation processes. The dissipation function also appears in three important theorems in nonequilibrium statistical mechanics: the fluctuation theorem, the dissipation theorem and the relaxation theorem. In this chapter we introduce the dissipation function and the theorems, and show how they quantify the emergence of irreversible behaviour in perturbed, steady state, and relaxing nonequilibrium systems. We also examine the behaviour of the dissipation function in terms of the extrema of the function using numerical and analytical approaches.

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Notes

  1. 1.

    Note that this is often trivially satisfied in a stochastic system.

  2. 2.

    Outside of equilibrium, microscopic temperature expressions are ill defined. Often expressions such as the kinetic temperature (the equipartition expression in momenta) or configurational temperature (a similar expression in position) are used, however these expressions only correspond to the temperature of the system, and each other, at equilibrium [15].

  3. 3.

    Equation (2.7) applies to systems with no field or a constant field. For the case of a time-dependent field see [19].

  4. 4.

    Note that a special case of this relation was derived much earlier, see [21] for details.

  5. 5.

    We note that \( - \int_{0}^{ - t} \varOmega (\varvec{\varGamma }(s))ds = \int_{0}^{t} \varOmega (\varvec{\varGamma }^{*} (s))ds \) if the dynamics are time reversible and that \( \int \varOmega (\varvec{\varGamma }^{*} ,s)f(\varvec{\varGamma })d\varvec{\varGamma } = \int \varOmega (\varvec{\varGamma }^{*} ,s)f(\varvec{\varGamma }^{*} )d\varvec{\varGamma }^{*} = \left\langle {\varOmega (s)} \right\rangle \) since the probability of observing ensemble members is constant (\( f(\varvec{\varGamma })d\varvec{\varGamma } = f(\varvec{\varGamma }^{*} )d\varvec{\varGamma }^{*} \), see [1]). We assume that the system eventually relaxes to an equilibrium state, \( f_{2} (\varvec{\varGamma }) = \mathop {\lim }\nolimits_{t \to \infty } \ln f(\varvec{\varGamma },t) \). From Eq. (2.7), \( \mathop {\lim }\nolimits_{t \to \infty } \ln f(\varvec{\Upgamma},t) = \mathop {\lim }\nolimits_{t \to \infty } ( - \int_{0}^{ - t} \Upomega (\varvec{\varGamma }(s))ds + \ln f(\varvec{\varGamma },0)) \), which can be expressed \( \ln f_{2} (\varvec{\varGamma }) = \int_{0}^{\infty } \varOmega (\varvec{\varGamma }^{*} (s))ds + \ln f_{1} (\varvec{\varGamma }) \). Then taking the ensemble average with respect to the initial distribution function we obtain \( \left\langle {\ln f_{2} } \right\rangle_{{f_{1} }} = \left\langle {\varOmega_{\infty } } \right\rangle_{{f_{1} }} + \left\langle {\ln f_{1} } \right\rangle_{{f_{1} }}. \)

  6. 6.

    A conformal system relaxes such that the nonequilibrium distribution is of the form (\( f(\varvec{\varGamma },t) = \exp ( - \beta H(\varvec{\varGamma }) + \lambda (t)g(\varvec{\varGamma }))/Z,\forall t \)) and the deviation function, \( g \), is a constant over the relaxation.

  7. 7.

    Strictly a system relaxes as time tends towards infinity, but in practice at \( t_{eq} \) the system has relaxed.

  8. 8.

    Simulation parameters: 50 Fluid particles, 22 Wall particles, T = 1, \( \rho = 0.3 \), 100,000 trajectories.

  9. 9.

    Simulation parameters: 64 Fluid particles, 64 Wall particles, T = 1, \( \rho = 0.8 \), 100,000 Trajectories.

References

  1. Evans, D.J., Williams, S.R., Searles, D.J.: J. Chem. Phys. 135, 194107 (2011)

    Article  Google Scholar 

  2. de Groot, S., Mazur, P.: Non-Equilibrium Thermodynamics. Dover Books on Physics. Dover Publications, NY (1984)

    Google Scholar 

  3. Evans, D.J., Searles, D.J.: Adv. Phys. 51, 1529 (2002)

    Article  Google Scholar 

  4. Reid, J.C., Sevick, E.M., Evans, D.J.: Europhys. Lett. 72, 726 (2005)

    Article  MathSciNet  Google Scholar 

  5. Kurchan, J.: J. Phys. A: Math. Gen. 31, 3719 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lebowitz, J.L., Spohn, H.: J. Stat. Phys. 95, 333 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Monnai, T.: Phys. Rev. E 72, 027102 (2005)

    Article  Google Scholar 

  8. Evans, D.J., Searles, D.J.: Phys. Rev. E 50, 1645 (1994)

    Article  Google Scholar 

  9. Maxwell, J.: Nature 17, 278 (1878)

    Article  Google Scholar 

  10. Broda, E., Gay, L.: Ludwig Boltzmann: man, physicist, philosopher. Ox Bow Press, Woodbridge (1983). Translating: L. Boltzmann, Re-joinder to the Heat Theoretical Considerations of Mr E. Zermelo (1896)

    Google Scholar 

  11. Searles, D.J., Evans, D.J.: Aust. J. Chem. 57, 1119 (2004)

    Article  Google Scholar 

  12. Carberry, D.M., Reid, J.C., Wang, G.M., Sevick, E.M., Searles, D.J., Evans, D.J.: Phys. Rev. Lett. 92, 140601 (2004)

    Article  Google Scholar 

  13. Joubaud, S., Garnier, N.B., Ciliberto, S.: J. Stat. Mech: Theory Exp. 2007, P09018 (2007)

    Article  Google Scholar 

  14. Garnier, N., Ciliberto, S.: Phys. Rev. E 71, 060101 (2005)

    Article  Google Scholar 

  15. Jepps, O.G., Ayton, G., Evans, D.J.: Phys. Rev. E 62, 4757 (2000)

    Article  Google Scholar 

  16. Evans, D.J., Searles, D.J., Williams, S.R.: J. Chem. Phys. 128, 014504 (2008)

    Article  Google Scholar 

  17. Evans, D.J., Searles, D.J., Williams, S.R.: J. Stat. Mech: Theory Exp. 2009, P07029 (2009)

    Article  Google Scholar 

  18. Evans, D.J., Searles, D.J., Williams, S.R.: Diffusion fundamentals III. In: Chmelik, C., Kanellopoulos, N., Karger, J., Theodorou, D. (eds.), pp. 367–374. Leipziger Universitats Verlag, Leipzig (2009)

    Google Scholar 

  19. Williams, S.R., Evans, D.J.: Phys. Rev. E 78, 021119 (2008)

    Article  Google Scholar 

  20. Evans, D.J., Williams, S.R., Searles, D,J.: Nonlinear dynamics of nanosystems. In: Radons, G., Rumpf, B., Schuster, H. (eds.), pp. 84–86. Wiley-VCH, NJ (2010)

    Google Scholar 

  21. Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Cambridge Unviersity Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  22. Todd, B.D.: Phys. Rev. E 56, 6723 (1997)

    Article  Google Scholar 

  23. Desgranges, C., Delhommelle, J.: Mol. Simul. 35(5), 405 (2009)

    Article  Google Scholar 

  24. Brookes, S.J., Reid, J.C., Evans, D.J., Searles, D.J.: J. Phys: Conf. Ser. 297, 012017 (2011)

    Article  Google Scholar 

  25. Hartkamp, R., Bernardi, S., Todd, B.D.: J. Chem. Phys. 136, 064105 (2012)

    Article  Google Scholar 

  26. Evans, D.J., Searles, D.J., Williams, S.R.: J. Chem. Phys. 132, 024501 (2010)

    Article  Google Scholar 

  27. Reid, J.C., Evans, D.J., Searles, D.J.: J. Chem. Phys. 136, 021101 (2012)

    Article  Google Scholar 

  28. Dewar, R.: J. Phys. A: Math. Gen. 36, 631 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dewar, R.C.: Entropy 11, 931 (2009)

    Article  Google Scholar 

  30. Dewar, R.C.: J. Phys. A: Math. Gen. 38, L371 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hoover, W.G.: J. Stat. Phys. 42, 587 (1986)

    Article  MathSciNet  Google Scholar 

  32. Evans, D.J.: Phys. Rev. A 32, 2923 (1985)

    Article  Google Scholar 

  33. Lorenz, R.: Science 299, 837 (2003)

    Article  Google Scholar 

  34. Weeks, J.D., Chandler, D., Andersen, H.C.: J. Chem. Phys. 54, 5237 (1971)

    Article  Google Scholar 

  35. Kröger, M.: Phys. Rep. 390, 453 (2004)

    Article  MathSciNet  Google Scholar 

  36. Williams, S.R., Evans, D.J., Mittag, E.: C.R. Phys. 8, 620 (2007)

    Article  Google Scholar 

  37. Zhang, F., Isbister, D.J., Evans, D.J.: Phys. Rev. E 64, 021102 (2001)

    Article  Google Scholar 

  38. Cui, S.T., Evans, D.J.: Mol. Simul. 9, 179 (1992)

    Article  Google Scholar 

  39. Meiburg, E.: Phys. Fluids 29, 3107 (1986)

    Article  Google Scholar 

  40. Dzwinel, W., Alda, W., Pogoda, M., Yuen, D.: Physica D 137, 157 (2000)

    Article  Google Scholar 

  41. Kadau, K., Germann, T.C., Hadjiconstantinou, N.G., Lomdahl, P.S., Dimonte, G., Holian, B.L., Alder, B.J.: Proc. Natl. Acad. Sci. U.S.A. 101, 5851 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hoover, W.G.: Lecture Notes in Physics 258, Molecular Dynamics. Springer, London (1986)

    Google Scholar 

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Authors and Affiliations

  1. Australian Institute of Bioengineering and Nanotechnology and School of Chemistry and Molecular Biosciences, The University of Queensland, Brisbane, Qld 4072, Australia

    James C. Reid & Debra J. Searles

  2. Queensland Micro- and Nanotechnology Centre, School of Biomolecular and Physical Sciences, Griffith University, Brisbane, Qld 4072, Australia

    Sarah J. Brookes & Debra J. Searles

  3. Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia

    Denis J. Evans

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Correspondence to James C. Reid .

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Editors and Affiliations

  1. Australian National University, Research School of Biology, Canberra, Aust Capital Terr, Australia

    Roderick C. Dewar

  2. Australian National University, Research School of Astronomy and Astrophysics and the Research School of Earth Sciences, Canberra, Aust Capital Terr, Australia

    Charles H. Lineweaver

  3. School of Engineering and Information Technology, University of New South Wales at ADFA, Canberra, Aust Capital Terr, Australia

    Robert K. Niven

  4. School of Earth and Environment, University of Western Australia, CSIRO Earth Science and Resource Engineering, Crawley, West Australia, Australia

    Klaus Regenauer-Lieb

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Reid, J.C., Brookes, S.J., Evans, D.J., Searles, D.J. (2014). The Dissipation Function: Its Relationship to Entropy Production, Theorems for Nonequilibrium Systems and Observations on Its Extrema. In: Dewar, R., Lineweaver, C., Niven, R., Regenauer-Lieb, K. (eds) Beyond the Second Law. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40154-1_2

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