Skip to main content
SpringerLink
Log in

Entropy of chemical processes versus numerical representability of orderings

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

Leaning on the mathematical concept of an interval order, we show that intransitivities that appear in several chemical processes, mainly related to mixing and competition, can actually be located and handled within a thermodynamical setting whose basis is the classical axiomatics due to Carathéodory, now using two intertwined entropy functions. Interdisciplinary comparisons to other similar theories (e.g., Utility Theory) are also made, pointing out the common mathematical background based on the numerical representability of total preorders and interval orders.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. To put an example concerning this setting, in [25] it is textually said that: (\(\ldots \)) “if \(t_1\), \(t_2\) and \(t_3\) are equilibrium states of three systems such as \(t_1\) is in thermal equilibrium with \(t_2\), and \(t_2\) is in thermal equilibrium with \(t_3\), then \(t_3\) is also in thermal equilibrium with \(t_1\)”. This law strongly resembles the first axiom of Euclidean geometry (ca. 300 BC), that is, “things equal to the same thing are equal to one another” (\(\ldots \)).

  2. Following [25], the second axiom in Carathéodory’s setting can be better understood when it is read together with Kelvin’s formulation of the second law, that “no cycle can exist whose net effect is a total conversion of heat into work”.

  3. A well-known topological condition of this kind is asking the order topology—on a nonempty set X endowed with a total preorder \(\precsim \)—to satisfy the second countability axiom. (For further details see e.g., [38, p. 243], or else Proposition 1.6.11 on p. 23 in [39].)

  4. The necessary definitions and concepts related to General Topology may be seen in the classical reference [40] among many others.

  5. This is quite important because, as commented in [27], “Intransitivities are possible in chemical reactions. The Belousov–Zhabotinsky reaction is known to display a cyclic chemical behavior in a homogeneous mixture, which is unusual since most chemical kinetics tend to monotonically converge to an equilibrium or steady state. (\(\ldots \)) The simplest chemical kinetic scheme that can adequately simulate the Belousov-Zhabotinsky reaction is Oregonator [41]. This scheme involves an essential intransitivity”. Obviously, if the accessibility relation is a total preorder, this kind of intransitive schemes would never occur. Therefore, it is crucial to say which are the possible phase spaces that we will deal with, in order to avoid intransitivities.

  6. The symbol \(\lnot \) is also commonly used to mean “negation”, so that \(\lnot (x \mathcal {R}y)\) is interpreted as “\(y \mathcal {R} x\) never occurs”.

  7. Needless to say, this is only one election among many possible others. In particular, the newest contributions [28, 34] as well as other previous ones as e.g., [5, 9, 20, 22, 24, 26] undoubtedly could also have been used to identical purposes.

  8. In Klimenko’s words (see [27]): “In non-equilibrium phenomena, the production of physical entropy is typically high, in perfect argument with the laws of thermodynamic. Although no direct violation of the laws of thermodynamics is known, thermodynamics struggles to explain complexity, which is often observed in essentially non-equilibrium phenomena: turbulent mixing and combustion as well as evolution of life forms may serve as typical examples. The entropy of turbulent fluctuations does not seem to be maximal and the same applies to entropies characterizing distributions in other complex non-equilibrium processes. These entropies have similarities with but are not the same as the molecular entropy, which characterises disorder of molecular movements and is subject to the laws of thermodynamics. We use the term apparent entropy to distinguish entropy-like quantities from the molecular entropy”.

  9. We have kept here Cooper’s notation introduced in [19].

  10. Warning! In this Sect. 3 we will only interpret those intransitivities that come from an interval order. In fact, such kind of intransitivities could be dealt with by means of two different entropy functions. However, when the intransitivity is of a totally different nature (obviously, not all the intransitive binary relations on a set are interval orders!) the corresponding study will remain open.

  11. This property is also known as the translation-invariance of the total order \(\precsim \) as regards the binary operation \(\circ \). Notice that, in particular, a totally ordered semigroup is always cancellative, namely \(s \circ u = t \circ u \Leftrightarrow s = t \Leftrightarrow u \circ s = u \circ t\quad (s,t,u \in S).\)

  12. Despite we are working with totally ordered semigroups, it can be proved that we could actually be working with a totally preordered semigroup, where \(\precsim \) is a total preorder but not necessarily a linear order (i.e., the binary relation \(\precsim \) could fail to be antisymmetric). When this happens, we might pass to be working with a quotient space \(S{/}\sim \) whose elements are the indifference classes of the elements of S with respect to \(\sim \). That is, given \(s \in S\), its corresponding class is the set \(\lbrace t \in S: t \sim s \rbrace \). Provided that there is a compatibility between the total preorder \(\precsim \) and the binary operation \(\circ \) such that \(s \precsim t \Leftrightarrow s \circ u \precsim t \circ u \Leftrightarrow u \circ s \precsim u \circ t\) holds true for every \(s,t,u \in S\), it is easy to see that \(S/\sim \) inherits a structure of totally ordered semigroup by considering in a natural way that the binary operation \(\circ \) as well as \( \precsim \) directly act. on the indifference classes that \(\sim \) induces on S. By this reason, in what follows we will be working with totally ordered semigroups, instead of just totally preordered semigroups, unless otherwise stated.

  13. In this setting, a mapping f with these properties is said to be an additive utility function.

  14. In our opinion, it seems more plausible to interpret the phase system as a semigroup rather than a group, because there are many chemical reactions that cannot be reverted, so that the concept of a “converse element” that we need to deal with the mathematical structure of an algebraic group could make no sense in this setting. However, the idea of the mixing (composition, reaction) being associative, seems to be more suitable.

  15. As a matter of fact, Cooper’s setting (see [19]) is more complicated, since he consideres n-ary operations (instead of just binary ones) to achieve composition of systems. However, perhaps we might still assume, at least implicitly, that the special kind of n-ary operations considered in Cooper’s approach could actually be decomposed as a suitable iteration of binary compositions.

  16. Here a jump is understood as a pair \(a,b \in X\) for which \(a \prec _L b\), but there is no \(c \in X\) such that \(a \prec _L c \prec _L b\).

  17. Cooper textually says in [19] that: “The physical model for a thermodynamic system is a system isolated from the external world by barriers impassible to heat but through which mechanical, electromagnetic, gravitational or other interactions with the external world are possible: these interactions will be summed up under the term interactions of the ground theories. The ground theories are the parts of physics established independently of thermodynamics such as mechanics, electromagnetic theory. Within the thermodynamic system, subsystems capable of being isolated by barriers impassible to heat may exist: but it must be assumed that these internal barriers can be removed”.

  18. Here we should recall and bear in mind the famous Newton’s sentence: “Natura non fact saltum”.

  19. If we just start with a totally ordered semigroup \((S, \circ , \precsim )\) we should take into account that no topology is given a priori on S, except maybe the order topology.

  20. Here on \(S \times S\) we will consider the product topology coming from \(\tau \) on S.

  21. Notice that this is, so-to-say, a theorem about “automatic continuity”. It actually states that on a totally ordered group \((G,\circ ,\precsim )\), both the operation \(\circ \) and the unary operation of taking an inverse are, a fortiori, continuous as regards the order topology \(\tau _{\precsim }\).

  22. The continuity is here understood with respect to the given topology \(\tau \) on X and the usual Euclidean topology on the real line.

  23. Observe that next result (Theorem 5) is only a partial answer to the problem of characterizing the continuous representability of an interval order, because nothing is said about the possibility of the existence of a pair (u, v) of continuous functions representing \(\prec \) but such that either u does not represent \(\precsim ^{**}\) or v does not represent \(\precsim ^{*}\).

References

  1. J.C. Candeal, J.R. De Miguel, E. Induráin, G.B. Mehta, Utility and entropy. Econ. Theory 17, 233–238 (2001)

    Article  Google Scholar 

  2. J.C. Candeal, J.R. De Miguel, E. Induráin, G.B. Mehta, On a theorem of Cooper. J. Math. Anal. Appl. 258, 701–710 (2001)

    Article  Google Scholar 

  3. J.C. Candeal, J.R. De Miguel, E. Induráin, G.B. Mehta, Representations of ordered semigroups and the physical concept of entropy. Appl. Gen. Topol. 5(1), 11–23 (2004)

    Article  Google Scholar 

  4. T. Sousa, T. Domingos, Is neoclassical microeconomcs formally valid? An approach based on an analogy with equilibrium thermodynamics. Ecol. Econ. 58(1), 160–169 (2006)

    Article  Google Scholar 

  5. T. Sousa, T. Domingos, Equilibrium econophysics: a unified formalism for neoclassical economics and equilibrium thermodynamics. Phys. A 371(2), 492–612 (2006)

    Article  Google Scholar 

  6. J. Bryant, A thermodynamic theory of economics. Int. J. Exergy 4(3), 302–337 (2007)

    Article  Google Scholar 

  7. E. Smith, D.K. Foley, Classical thermodynamics and economic general equilibrium. J. Econ. Dyn. Control 32(1), 7–65 (2008)

    Article  Google Scholar 

  8. M. Shubik, E. Smith, Building theories of economic process. Complexity 14(3), 77–92 (2009)

    Article  Google Scholar 

  9. A. Dionisio, A.H. Reis, L. Coelho, Utility function estimation: the entropy approach. Phys. A 387(15), 3862–3867 (2008)

    Article  Google Scholar 

  10. V.N. Pokrovskii, Econodynamics: The Theory of Social Production, New Economic Windows 12 (Springer, Berlin, 2012)

    Book  Google Scholar 

  11. F.S. Roberts, R.D. Luce, Axiomatic thermodynamics and extensive measurement. Synthese 18, 311–326 (1968)

    Article  Google Scholar 

  12. P.J. Davis, Entropy and society: can the physical/mathematical notions of entropy be usefully imported into the social sphere? J. Humanist. Math. 1(1), 119–136 (2011)

    Article  Google Scholar 

  13. C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)

    Article  Google Scholar 

  14. A. Meucci, Fully flexible views: theory and practice. Risk 21(10), 97–102 (2008)

    Google Scholar 

  15. R. Zhou, R. Cai, G. Tong, Applications of entropy in finance: a review. Entropy 15, 4909–4931 (2013)

    Article  Google Scholar 

  16. B. Kosko, Fuzzy entropy and conditioning. Inf. Sci. 40, 165–174 (1986)

    Article  Google Scholar 

  17. G. Bosi, M.J. Campión, J.C. Candeal, E. Induráin, Interval-valued representability of qualitative data: the continuous case. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 15(3), 299–319 (2007)

    Article  Google Scholar 

  18. C. Carathéodory, Untersuchung über die Grundlagen der Thermodynamik. Math. Ann. 67, 355–386 (1909)

    Article  Google Scholar 

  19. J.L.B. Cooper, The foundations of thermodynamics. J. Math. Anal. Appl. 17, 172–193 (1967)

    Article  Google Scholar 

  20. O. Redlich, Fundamental thermodynamics since Carathéodory. Rev. Mod. Phys. 49(1), 556–563 (1968)

    Article  Google Scholar 

  21. E.C. Zachmanoglou, Carathéodory theorem on the second law of thermodynamics. SIAM J. Appl. Math. 25(4), 592–596 (1973)

    Article  Google Scholar 

  22. P.T. Landsberg, Main ideas in the axiomatics of thermodynamics. Pure Appl. Chem. 22(3–4), 215–228 (1970)

    CAS  Google Scholar 

  23. E.H. Lieb, J. Yngvason, A guide to entropy and the second law of thermodinamics. Not. Am. Math. Soc. 45(5), 571–581 (1998)

    Google Scholar 

  24. P.T. Landsberg, Entropies galore!. Braz. J. Phys. 29(1), 46–49 (1999)

    Article  Google Scholar 

  25. L. Pogliani, M.N. Berberan-Santos, Constantin Carathéodory and the axiomatic thermodynamics. J. Math. Chem. 28(1–3), 313–324 (2000)

    Article  CAS  Google Scholar 

  26. J. Duda, J. Milewska-Duda, A theoretical model for evaluation of configurational entropy of mixing with respect to shape and size of particles. J. Math. Chem. 17(1), 69–109 (1995)

    Article  CAS  Google Scholar 

  27. A.Y. Klimenko, Mixing, entropy and competition. Phys. Scr. 85(6), 068201 (2012)

    Article  CAS  Google Scholar 

  28. A.Y. Klimenko, Complex competitive systems and competitive thermodynamics. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 271(1982), 20120244 (2013)

    Google Scholar 

  29. A.Y. Klimenko, Complexity and intransitivity in technological development. J. Syst. Sci. Syst. Eng. 23(2), 128–152 (2014)

    Article  Google Scholar 

  30. R.F. Nalewajski, Entropy/information bond indices of molecular fragments. J. Math. Chem. 38(1), 43–66 (2005)

    Article  CAS  Google Scholar 

  31. R.F. Nalewajski, Communication-theory perspective on valence-bond theory. J. Math. Chem. 45(3), 709–724 (2009)

    Article  CAS  Google Scholar 

  32. R.F. Nalewajski, Entropy/information coupling between orbital-communications in molecular subsystems. J. Math. Chem. 47(2), 808–832 (2010)

    Article  CAS  Google Scholar 

  33. L. Tarko, A new manner to use application of Shannon entropy in similarity computation. J. Math. Chem. 49(10), 2330–2344 (2011)

    Article  CAS  Google Scholar 

  34. A.Y. Klimenko, Entropy and equilibria in competitive systems. Entropy 16, 1–22 (2014)

    Article  Google Scholar 

  35. G. Arzamendi, R. Ferrero, A.R. Pierna, L.M. Gandía, Kinetics of methyl ethyl ketone combustion in air at low concentrations over a commercial Pt/Al\(_2\)O\(_3\) catalyst. Ind. Eng. Chem. Res. 46, 9037–9044 (2007)

    Article  CAS  Google Scholar 

  36. I. Reyero, G. Arzamendi, L.M. Gandía, Heterogeneization of the biodiesel synthesis catalysis: CaO and novel calcium compounds as transesterification catalysts. Chem. Eng. Res. Des. 92, 1519–1530 (2014)

    Article  CAS  Google Scholar 

  37. G. Bosi, J.C. Candeal, E. Induráin, E. Olóriz, M. Zudaire, Numerical representations of interval orders. Order 18, 171–190 (2001)

    Article  Google Scholar 

  38. G. Birkhoff, Lattice Theory, 3rd edn. (American Mathematical Society, Providence, 1967)

    Google Scholar 

  39. D.S. Bridges, G.B. Mehta, Representations of Preference Orderings (Springer, Berlin, 1995)

    Book  Google Scholar 

  40. J. Dugundji, Topology (Allyn and Bacon, Boston, 1966)

    Google Scholar 

  41. R.J. Field, Limit cycle oscillations in the reversible oregonator. J. Chem. Phys. 63(6), 2289 (1975)

    Article  CAS  Google Scholar 

  42. R.D. Luce, Semiorders and a theory of utility discrimination. Econometrica 24, 178–191 (1956)

    Article  Google Scholar 

  43. P.C. Fishburn, Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)

    Article  Google Scholar 

  44. P.C. Fishburn, Intransitive indifference in preference theory: a survey. Oper. Res. 18(2), 207–228 (1970)

    Article  Google Scholar 

  45. A.Y. Klimenko, Intransitivity in theory and in the real world. Entropy 17, 4364–4412 (2015)

    Article  Google Scholar 

  46. N. Wiener, Contribution to the theory of relative position. Math. Proc. Camb. Philos. Soc. 17, 441–449 (1914)

    Google Scholar 

  47. N. Wiener: A new theory of measurement. Proc. Lond. Math. Soc. 19, 181–205 (1919–1920)

  48. D. Scott, P. Suppes, Foundational aspects of theories of measurement. J. Symb. Log. 23, 113–128 (1958)

    Article  Google Scholar 

  49. A. Tversky, Intransitivity of preferences. Psychol. Rev. 76(1), 31–48 (1969)

    Article  Google Scholar 

  50. O. Hölder, Der Axiome der Quantität und die Lehre vom Mass, Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig. Math. Phys. Kl 53, 1–64 (1901)

    Google Scholar 

  51. N.G. Alimov, On ordered semigroups. Izv. Akad. Nauk SSSR Ser. Mat. 14, 569–576 (1950). (in Russian)

    Google Scholar 

  52. L. Fuchs, Partially Ordered Algebraical Systems (Pergamon Press, Oxford, 1963)

    Google Scholar 

  53. J.R. De Miguel, J.C. Candeal, E. Induráin, Archimedeaness and additive utility on totally ordered semigroups. Semigroup Forum 52, 335–347 (1996)

    Article  Google Scholar 

  54. G. Debreu, Representation of a preference ordering by a numerical function, in Decision Processes, ed. by R.M. Thrall, et al. (Wiley, New York, 1954)

    Google Scholar 

  55. G. Debreu, Continuous properties of Paretian utility. Int. Econ. Rev. 5, 285–293 (1964)

    Article  Google Scholar 

  56. S. Eilenberg, Ordered topological spaces. Am. J. Math. 63, 39–45 (1941)

    Article  Google Scholar 

  57. J.C. Candeal, J.R. De Miguel, E. Induráin, Extensive measurement: continuous additive utility functions on semigroups. J. Math. Psychol. 40(4), 281–286 (1996)

    Article  Google Scholar 

  58. J.C. Candeal, J.R. De Miguel, E. Induráin, Topological additively representable semigroups. J. Math. Anal. Appl. 210, 385–389 (1997)

    Article  Google Scholar 

  59. P.J. Nyikos, H.C. Reichel, Topologically orderable groups. Gen. Topol. Appl. 45(5), 571–581 (1998)

    Google Scholar 

  60. J.C. Candeal, E. Induráin, Semiorders and thresholds of utility discrimination: solving the Scott–Suppes representability problem. J. Math. Psychol. 54, 485–490 (2010)

    Article  Google Scholar 

  61. M.A. Nielsen, Conditions for a class of entanglement transformations. Phys. Rev. Lett. 83, 436 (1999)

    Article  CAS  Google Scholar 

  62. R. Gielerak, M. Sawerwain: Sorting of quantum states with respect to amount of entanglement included, in Computer Networks. 16th Conference, CN 2009, Wisła, Poland, June 16–20, 2009. ed. by A. Kwiecień et al. Proc. Commun. Comput. Inf. Sci. 39, 11–18 (2009)

  63. J.C. Candeal-Haro, E. Induráin-Eraso, Utility representations from the concept of measure. Math. Soc. Sci. 26, 51–62 (1993)

    Article  Google Scholar 

  64. B. Subiza, J.E. Peris, A note on numerical representations for weak-continuous acyclic preferences. Rev. Esp. Econ. 15(19), 15–21 (1998)

    Google Scholar 

Download references

Acknowledgments

Thanks are given to the editor and three anonymous referees for their valuable suggestions and comments on a previous version of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics and Institute for Advanced Materials (InaMat), Public University of Navarre, Campus Arrosadía (Las Encinas), 31006, Pamplona, Spain

    M. J. Campión & E. Induráin

  2. Department of Applied Chemistry and Institute for Advanced Materials (InaMat), Public University of Navarre, Campus Arrosadía (Los Acebos), 31006, Pamplona, Spain

    G. Arzamendi & L. M. Gandía

Authors
  1. M. J. Campión
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Arzamendi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. M. Gandía
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Induráin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. J. Campión.

Additional information

This work has been partially supported by the research Projects ENE2012-37431-C03-03, MTM2012-37894-C02-02 and TIN2013-40765-P (Spain).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Campión, M.J., Arzamendi, G., Gandía, L.M. et al. Entropy of chemical processes versus numerical representability of orderings. J Math Chem 54, 503–526 (2016). https://doi.org/10.1007/s10910-015-0565-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-015-0565-8

Keywords

Mathematics Subject Classification

Search

Navigation