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The Time Evolution of Entropy Production in Nonlinear Dynamic Systems

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Beyond the Second Law

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

Abstract

General characteristics of entropy production in a fluid system are investigated from a thermodynamic viewpoint. A basic expression for entropy production due to irreversible transport of heat or momentum is formulated together with balance equations of energy and momentum in a fluid system. It is shown that entropy production always decreases with time when the system is of a pure diffusion type without advection of heat or momentum. The minimum entropy production (MinEP) property is thus intrinsic to a pure diffusion-type system. However, this MinEP property disappears when the system is subject to advection of heat or momentum due to dynamic motion. When the rate of advection exceeds the rate of diffusion of heat or momentum, entropy production tends to increase over time. The maximum entropy production (MaxEP), suggested as a selection principle for steady states of nonlinear non-equilibrium systems, can therefore be understood as a characteristic feature of systems with dynamic instability. The observed mean state of vertical convection of the atmosphere is consistent with the condition for MaxEP presented in this study.

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Notes

  1. 1.

    One can include a reversible flux due to advection in the balance equation of entropy, but it results in no contribution to entropy production after the integration over the whole volume of a fluid system (see, e.g., [21], Sec. 49; [12], Sec. 2.4).

  2. 2.

    Assuming linearity, \( {\boldsymbol{\Uppi}}:\nabla {\mathbf{v}} = \, [ 2\mu (\nabla {\mathbf{v}})^{\text{s}} - \left( { 2/ 3} \right)\mu (\nabla \cdot {\mathbf{v}}){\boldsymbol{\delta}} \left]: \right[(\nabla {\mathbf{v}})^{\text{s}} \, + \, \, (\nabla {\mathbf{v}} )^{\text{a}} ] \, = { 2}\mu (\nabla {\mathbf{v}})^{\text{s}} :(\nabla {\mathbf{v}})^{\text{s}} - \left( { 2/ 3} \right)\mu (\nabla \cdot {\mathbf{v}})^{ 2},\) with δ denoting the unit tensor, and T s and T a denoting symmetric and asymmetric parts of a tensor T. Then, \( \partial ({\boldsymbol{\Uppi}}:\nabla {\mathbf{v}})/\partial t = { 2}[ 2\mu (\nabla {\mathbf{v}})^{\text{s}} - \left( { 2/ 3} \right)\mu (\nabla \cdot {\mathbf{v}}){\boldsymbol{\delta}}\left]: \right[\nabla (\partial {\mathbf{v}}/\partial {\text{t}})]^{\text{s}} = 2{\boldsymbol{\Uppi}}:\nabla (\partial {\mathbf{v}}/\partial {t}) \).

  3. 3.

    In a general case, the forth term in the right-hand side of Eq. (6.12) should be expressed as a sum of the viscosity μ and the second viscosity λ. Using Stokes’ relation (λ = −2μ/3), μ + λ = μ/3.

  4. 4.

    There are a few exceptions. Laminar (or non-turbulent) flow can be realized even with advection of momentum, e.g., in a converging nozzle. However, the flow direction is not parallel in this case, and it may not be regarded as “laminar” in the strict sense of the word.

  5. 5.

    The exact correspondence between the time evolution of a system and the probability of states requires an additional assumption, which is related to profound and not yet fully solved problems of the ergodic hypothesis.

  6. 6.

    There is also generation of mechanical energy by volume expansion of the air at the surface. The rate of energy conversion is about 2 W m−2, which thereafter dissipates into heat in the atmosphere. We have included this rate in the convective energy transport considered here.

Abbreviations

A :

Surface of a system or the Earth (m2)

c v :

Specific heat at constant volume (J K−1 kg−1)

e :

Unit vector (–)

F c :

Convective heat flux density (sensible and latent heat) (J m−2 s−1)

F r :

Radiation flux density (J m−2 s−1)

F LW :

Longwave radiation flux density (J m−2 s−1)

F SW :

Shortwave radiation flux density (J m−2 s−1)

J i :

Diffusive flux density of i-th component

J h :

Diffusive flux density of heat (J m−2 s−1)

J m :

Diffusive flux density of momentum (kg m−2 s−1)

k :

Thermal conductivity (J m−1 K−1 s−1)

L h :

Kinetic coefficient for heat diffusion (J m−1 K s−1)

n :

Unit vector normal to system’s surface (–)

p :

Pressure (Pa)

t :

Time (s)

T :

Temperature (K)

T e :

Effective radiation temperature at the top of the atmosphere (K)

T r :

Effective radiation temperature (K)

T s :

Surface temperature (K)

T sun :

Emission temperature of the sun (K)

V :

Volume of a system (m3)

v :

Velocity (m s−1)

X i :

Gradient of intensive variable for i-th diffusive flux

δ :

Unit tensor (–)

\( \kappa \) :

Thermal diffusivity (m2 s−1)

\( \lambda \) :

Second viscosity (kg m−1 s−1)

\( \mu \) :

Viscosity (kg s−1 m−1)

\( \nu \) :

Kinematic viscosity (m2 s−1)

\( {\boldsymbol{\Uppi}} \) :

Viscous stress tensor (Pa)

\( \rho \) :

Density (kg m−3)

σB :

Stefan–Boltzmann constant ≈ 5.67 × 10−8 (J m−2 K−4 s−1)

\( \dot{\sigma } \) :

Rate of entropy production (J K−1 s−1)

\( \dot{\sigma }_{\text{conv}} \) :

Rate of entropy production due to convective heat flux (J K−1 s−1)

\( \dot{\sigma }_{\text{h}} \) :

Rate of entropy production due to heat diffusion (J K−1 s−1)

\( \dot{\sigma }_{\text{m}} \) :

Rate of entropy production due to momentum diffusion (J K−1 s−1)

\( \dot{\sigma }_{\text{rad}} \) :

Rate of entropy production due to absorption of radiation (J K−1 s−1)

\( \dot{\sigma }_{\text{tot}} \) :

Total rate of entropy production in the atmosphere (J K−1 s−1)

stat:

Static state with no motion

lam:

Laminar flow state

adv:

State with advection

References

  1. Ziegler, H.: Zwei Extremalprinzipien der irreversiblen Thermodynamik. Ing. Arch. 30, 410–416 (1961)

    Article  MathSciNet  Google Scholar 

  2. Paltridge, G.W.: Global dynamics and climate—a system of minimum entropy exchange. Q. J. Roy. Meteorol. Soc. 101, 475–484 (1975)

    Article  Google Scholar 

  3. Paltridge, G.W.: The steady-state format of global climate. Q. J. Roy. Meteorol. Soc. 104, 927–945 (1978)

    Article  Google Scholar 

  4. Schneider, E.D., Kay, J.J.: Life as a manifestation of the second law of thermodynamics. Math. Comput. Model. 19, 25–48 (1994)

    Article  Google Scholar 

  5. Ozawa, H., Shimokawa, S., Sakuma, H.: Thermodynamics of fluid turbulence: a unified approach to the maximum transport properties. Phys. Rev. E 64, 026303 (2001)

    Article  Google Scholar 

  6. Lorenz, R.D., Lunine, J.I., Withers, P.G., McKay, C.P.: Titan, Mars and Earth: entropy production by latitudinal heat transport. Geophys. Res. Lett. 28, 415–418 (2001)

    Article  Google Scholar 

  7. Shimokawa, S., Ozawa, H.: On the thermodynamics of the oceanic general circulation: irreversible transition to a state with higher rate of entropy production. Q. J. Roy. Meteorol. Soc. 128, 2115–2128 (2002)

    Article  Google Scholar 

  8. Shimokawa, S., Ozawa, H.: Thermodynamics of irreversible transitions in the oceanic general circulation. Geophys. Res. Lett. 34, L12606 (2007). doi:10.1029/2007GL030208

    Article  Google Scholar 

  9. Hill, A.: Entropy production as the selection rule between different growth morphologies. Nature 348, 426–428 (1990)

    Article  Google Scholar 

  10. Nohguchi, Y., Ozawa, H.: On the vortex formation at the moving front of lightweight granular particles. Physica D 238, 20–26 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lorenz, E.N.: Generation of available potential energy and the intensity of the general circulation. In: Pfeffer, R.L. (ed.) Dynamics of Climate, pp. 86–92. Pergamon, Oxford (1960)

    Chapter  Google Scholar 

  12. Ozawa, H., Ohmura, A., Lorenz, R.D., Pujol, T.: The second law of thermodynamics and the global climate system: a review of the maximum entropy production principle. Rev. Geophys. 41, 1018 (2003). doi:10.1029/2002RG000113

    Article  Google Scholar 

  13. Dewar, R.: Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states. J. Phys. A 36, 631–641 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Niven, R.K.: Steady state of a dissipative flow-controlled system and the maximum entropy production principle. Phys. Rev. E 80, 021113 (2009)

    Article  Google Scholar 

  15. Prigogine, I.: Modération et transformations irréversibles des systémes ouverts. Bulletin de la Classe des Sciences. Academie Royale de Belgique 31, 600–606 (1945)

    Google Scholar 

  16. Glansdorff, P., Prigogine, I.: Sur les propriétés différentielles de la production d’entropie. Physica 20, 773–780 (1954)

    Article  MATH  Google Scholar 

  17. De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. North-Holland, Amsterdam (1962)

    Google Scholar 

  18. Gyarmati, I.: Non-equilibrium Thermodynamics. Field Theory and Variational Principles. Springer, Berlin (1970). [First published in Hungarian, 1967]

    Google Scholar 

  19. Prigogine, I.: From Being to Becoming. Time and Complexity in the Physical Sciences. Freeman, San Fransisco (1980)

    Google Scholar 

  20. Sawada, Y.: A thermodynamic variational principle in nonlinear non-equilibrium phenomena. Prog. Theor. Phys. 66, 68–76 (1981)

    Article  Google Scholar 

  21. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Pergamon Press, Oxford (1987). [First published in Russian (1944)]

    MATH  Google Scholar 

  22. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)

    MATH  Google Scholar 

  23. Helmholtz, H.: Zur Theorie der stationären Ströme in reibenden Flüssigkeiten. Verhandlungen des naturhistorisch-medicinischen Vereins zu Heidelberg 5, 1–7 (1869)

    Google Scholar 

  24. Rayleigh, Lord.: On the motion of a viscous fluid. Phil. Mag. 26, 776–786 (1913)

    Article  MATH  Google Scholar 

  25. Suzuki, M., Sawada, Y.: Relative stabilities of metastable states of convecting charged-fluid systems by computer simulation. Phys. Rev. A 27, 478–489 (1983)

    Article  Google Scholar 

  26. Kawazura, Y., Yoshida, Z.: Entropy production rate in a flux-driven self-organizing system. Phys. Rev. E 82, 066403 (2010)

    Article  MathSciNet  Google Scholar 

  27. Malkus, W.V.R.: Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  28. Busse, F.H.: Bounds for turbulent shear flow. J. Fluid Mech. 41, 219–240 (1970)

    Article  MATH  Google Scholar 

  29. Malkus, W.V.R.: Borders of disorder: in turbulent channel flow. J. Fluid Mech. 489, 185–198 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Paulus Jr, D.M., Gaggioli, R.A.: Some observations of entropy extrema in fluid flow. Energy 29, 2487–2500 (2004)

    Article  Google Scholar 

  31. Niven, R.K.: Simultaneous extrema in the entropy production for steady-state fluid flow in parallel pipes. J. Non-Equilib. Thermodyn. 35, 347–378 (2010)

    Article  MATH  Google Scholar 

  32. Ozawa, H., Ohmura, A.: Thermodynamics of a global-mean state of the atmosphere —a state of maximum entropy increase. J. Clim. 10, 441–445 (1997)

    Article  Google Scholar 

  33. Heisenberg, W.: Nonlinear problems in physics. Phys. Today 20(5), 27–33 (1967)

    Article  Google Scholar 

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Acknowledgments

The authors wish to express their cordial thanks to the organizers of the MaxEP 2011 Workshop in Canberra where the authors’ interest in this subject has been stimulated. Valuable comments from two anonymous reviewers are also gratefully acknowledged.

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Authors and Affiliations

  1. Hiroshima University, Higashi-Hiroshima, 739-8521, Japan

    Hisashi Ozawa

  2. National Research Institute for Earth Science and Disaster Prevention, Tsukuba, 305-0006, Japan

    Shinya Shimokawa

Authors
  1. Hisashi Ozawa
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  2. Shinya Shimokawa
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Corresponding author

Correspondence to Hisashi Ozawa .

Editor information

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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Ozawa, H., Shimokawa, S. (2014). The Time Evolution of Entropy Production in Nonlinear Dynamic Systems. 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_6

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  • DOI: https://doi.org/10.1007/978-3-642-40154-1_6

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