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Thermodynamics pp 239–279Cite as

Non-Equilibrium Thermodynamics

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Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

Abstract

In the preceding chapters with few exceptions we studied systems at equilibrium. This means that the systems do not change or evolve over time.

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Notes

  1. 1.

    Here * means that time is in units of \(\sqrt{m \sigma ^2/\epsilon }\), where \(m\) is the particle mass and \(\epsilon \) and \(\sigma \) are the parameters in the LJ-potential.

  2. 2.

    Ludwig Boltzmann, austrian physicist, *Wien 20.2.1844, †Duino 5.9.1906; fundamental contributions to Statistical Mechanics. His tombstone bears the inscription \(S= k \log W\) (cf. Eq. (6.16)).

  3. 3.

    For the sake of simplicity we treat the \( L_{ij}\) as scalar quantities.

  4. 4.

    An explicit example in the case of energy fluctuations was worked out on p. 191.

  5. 5.

    We discuss entropy production in more detail in the next section.

  6. 6.

    Ilya Prigogine, Nobel Prize in chemistry for his contributions to non-equilibrium thermodynamics, 1977.

  7. 7.

    which we discuss in more detail below.

  8. 8.

    A particular importance of autocatalytic reactions is their key role in models of prebiotic evolution—an idea that was developed quite a long time ago (G. Allen Reflexive catalysis, a possible mechanism of molecular duplication in prebiological evolution. Amer. Natur. 91, 65 (1957)). We return to this aspect in the next section.

  9. 9.

    This is an early representative of the coupled reactions schemes discussed in the context of chemical oscillations. Perhaps the most famous experimental representative is the Belousov-Zhabotinsky reaction.

  10. 10.

    We assume constant volume.

  11. 11.

    We use the square because each particular fixed point is visited every second iteration.

  12. 12.

    Not visible at this resolution are the self-similar copies of the original graph inside the “white gaps”.

  13. 13.

    Notice that this is the intercept of two lines of research. One objective is the understanding of the transition from order to chaos, whereas another group of researchers, Prigogine et al., peruse the opposite direction.

  14. 14.

    Molecular pattern formation due to gradient induced gene transcription is part of the early (Drosophila) embryo development (Ch. Nüsslein-Volhard (2006) Coming to Life. How Genes Drive Development. Yale University Press).

  15. 15.

    There may be more than one steady state.

  16. 16.

    The principle may even be applied to the optimization of technical systems or material properties. So called genetic algorithms consist of a set of operators simulating reproduction, combination, and mutation applied to linear parameter sets defining the technical system (e.g. D. E. Goldberg (1989) Genetic Algorithms in Search, Optimization & Machine Learning. Addison-Wesley).

  17. 17.

    Manfred Eigen, Nobel prize in chemistry for his work on the kinetics of fast reactions, 1967; he is perhaps better known for his work on prebiotic evolution.

  18. 18.

    The principle is analogous to a high jump competition. If a jumper clears the bar, the others must also clear this height in order to remain in the competition.

  19. 19.

    The logarithm in the numerator does ensure that the latter will not be large. In addition, replication without additional mechanisms enhancing its precision limits the approach of \(1-\bar{q}_i\) towards zero.

  20. 20.

    Equilibrium structures—are formed and maintained through reversible transformations implying no appreciable deviations from equilibrium; dissipative structures—are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions.

  21. 21.

    On an infinite lattice the result will be a step function dropping to zero at \(p_c=0.5927\).

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Authors and Affiliations

  1. Department of Mathematics and Natural Sciences, Bergische Universität, Gaußstr.20, 42097, Wuppertal, Germany

    Reinhard Hentschke

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  1. Reinhard Hentschke
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Correspondence to Reinhard Hentschke .

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Hentschke, R. (2014). Non-Equilibrium Thermodynamics. In: Thermodynamics. Undergraduate Lecture Notes in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36711-3_7

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  • DOI: https://doi.org/10.1007/978-3-642-36711-3_7

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  • Print ISBN: 978-3-642-36710-6

  • Online ISBN: 978-3-642-36711-3

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