Abstract
The paper rebuts a currently popular criticism against a certain take on the referential role of discontinuities and singularities in the physics of first-order phase transitions. It also elaborates on a proposal I made previously on how to understand this role within the framework provided by the distinction between data and phenomena.
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Notes
Although irrelevant for the actual content of the discussion here, authors and specific papers can be assigned to each side of the debate (under the usual caveat that they may not agree with this categorization). Thus, Batterman’s (2001, 2005, 2006, 2010, 2016) are illustrative for Referentialism, and similar sympathies can be detected in Morrison’s (2012, 2014) and Bangu (2009, 2015, forthcoming). As for the other camp, I will be discussing mainly Norton (2012), and Menon and Callender (2013), as well as gesturing in passing to Butterfield (2011) and Shech (2013). Ardourel (in press) and P. Palacios can perhaps be added to this group, too.
Except for emphasizing certain aspects, the presentation and the diagrams follow the textbook treatment (e.g., Zemansky and Dittman 1997).
The graph is a projection of the complete three-dimensional phase diagram, also including a volume axis V, onto the P–T plane.
The current classification originates in, but refines, P. Ehrenfest’s: if the first-order derivative of the free energy is discontinuous, the transition is first-order (discontinuous). A transition is continuous when a second (or higher)-order derivative (and the parameter it represents) is discontinuous, or when certain parameters diverge to infinity.
A number of technicalities had to be left out of this presentation. Z is typically evaluated for a homogeneous system at fixed volume, but the discussion here involved a heterogeneous system, i.e., a system where liquid and gas coexist (at point B), and pressure instead of volume is the control parameter. Thus, a calculation of F from Z gives an unphysical function p (V, T) with an unstable region (see Fig. 8.6.2 in Reif (1965, p. 308)). In fact, the evaluation of the full partition function in these conditions does not yield the spontaneous symmetry-breaking characteristic for most phase transitions (as the analysis of the Ising model reveals for a magnetic system). So, to work with the appropriate partition function, one needs to confine the sum to only part of the state space (as well as make a few other further assumptions).
More conditions have to be satisfied for the existence of the thermodynamic limit; see Liu (1999, p. S96); Goldenfeld (1992, pp. 25–28) discusses examples where the limit does not exist. Sklar (1993, pp. 78–81) lists several reasons why the limit is generally useful (e.g., it makes the effect of fluctuations vanish). The idea of such a limit was first proposed by H. Kramers, in the late 1930 s. L. Onsager (1944) was able to show that an infinite 2-dimensional Ising model undergoes a continuous phase transition (near criticality, the specific heat diverges logarithmically). Yang and Lee (1952) proved that the limit yields discontinuous transitions too (it generalizes to other types of transitions).
I replaced Reif’s notation VB, VA with V2 and V1, respectively.
It is because of this ‘gap’, I suspect, that one may (legitimately) begin to wonder how the very idea of an inconsistency between the existence of discontinuities and the finiteness of the statistical mechanical system may arise—if the systems are finite, which they are, their volumes (for instance) can’t have discontinuities. We will see below that this inconsistency is only apparent.
As Liu (1999, p. S103) correctly observed: “the transition is neither ‘smooth’ nor ‘singular’”.
Interestingly, and perhaps not accidentally, one central example used in the 1988 paper to introduce the distinction was that of a phase transition: the melting point of lead. Although quite old, the distinction still draws philosophers’ of science attention, eliciting both acceptance and criticism. See Woodward (2011) for, as far as I know, his latest restatement and defense of it.
The specific inference procedures vary with the area of scientific research, but a common feature is that they rely on statistical techniques (1988, p. 311).
This alludes to Norton’s insistence on approximations, to be discussed in a moment.
This way of speaking is meant to emphasize that (weak) Referentialism squares remarkably well with Kadanoff’s remark that “in some sense, phase transitions are not exactly embedded in the finite world but, rather, are products of the human imagination.” (2009, p. 778). I submit that the sense in question here is captured precisely by understanding a phase transition as a phenomenon—whose shaping from data requires, as we saw, ‘human imagination’, although not in an unrestrained way, but, as we also saw, in the form of mathematical modelling. This is perhaps an unnecessary clarification, since this ‘unrestrained way’ was surely not how Kadanoff intended the remark. (I should add the caveat that this is my reading of this remark, and that as far as I know neither Batterman, nor Kadanoff ever endorsed its interpretation along these lines.).
The case for realism about phenomena is made in section VIII of their 1988 paper.
In connection to this, let me note that, as a referee pointed out (and I am almost quoting the suggestion here), it is possible that in the above quote Norton did not mean “rapid” in a temporal sense. So, the proposal then is to interpret him more broadly as saying that the rate of change of a thermodynamic quantity, with respect to temperature, pressure, etc. is so large as to be effectively infinite. If so, there would still be wiggle room for his view, since after all, the very concept of a gas’ volume is well-defined only up to a certain degree of precision associated for example with the atomic granularity of the container walls. Then the more general lesson here would be that observability is problematic: physical concepts are only well-defined up to a finite margin of precision, and that would then seem to support the view that the strict mathematical discontinuities of the thermodynamic representation of the system cannot completely reflect the behavior of the real physical quantities, which are defined only up to a limited margin of precision. I do not discount the possibility that Norton meant something else by what he wrote, but I believe this rendering is a bit too charitable; it is pretty clear from the context that the claim should be understood in a temporal sense. And, after all, the principal sense of the word is, according to the Oxford dictionary, the temporal one: ‘happening in a short time’. See https://en.oxforddictionaries.com/definition/rapid. Hence the remarks above don’t change the situation too much in my view.
Among physicists, Norton (2012, p. 219) singles out Langford (1975) and Ruelle (2004; the passage I also cited above). He also points out that in Le Bellac et al. (2004) “any condition on an actual infinity of components and the corresponding behavior of that infinite system” is “conspicuously absent”. As an aside comment, I am not confident that this absence is the result of these authors’ careful deliberate choice; a perusal of their monograph did not convince me that they were concerned with conceptual clarity at the level Norton suggests they were. In fact, it is Ruelle’s treatise that seemed to me to explicitly aspire to this.
I deliberately refrained from engaging with what has recently been dubbed the ‘paradox’ of phase transitions (Shech 2013), originating in Callender (2001). See Bangu (2009, 2015) for a detailed take on it, including the comments on the several attempts of finite statistical mechanics to deal with (first-order) phase transitions. For very recent work on this, see Ardourel (in press).
References
Ardourel, V. (in press). The infinite limit as an eliminable approximation for phase transitions. Studies in History and Philosophy of Modern Physics. https://doi.org/10.1016/j.shpsb.2017.06.002.
Bangu, S. (2009). Understanding thermodynamic singularities, phase transitions, data and phenomena. Philosophy of Science, 76(4), 488–505.
Bangu, S. (2015). Why does water boil? Fictions in scientific explanation. In U. Mäki et al. (Eds.), Recent Developments in the philosophy of science. EPSA 13 Helsinki (pp. 319–330). Cham: Springer.
Bangu, S. (forthcoming). Phase transitions. In E. Knox and A. Wilson (Eds.) The Routledge companion to the philosophy of physics. Routledge.
Batterman, R. (2001). The devil in the details. Oxford: Oxford University Press.
Batterman, R. (2005). Critical phenomena and breaking drops: Infinite idealizations in physics. Studies in History and Philosophy of Modern Physics, 36, 225–244.
Batterman, R. (2006). Hydrodynamic vs. molecular dynamics: Intertheory relations in condensed matters physics. Philosophy of Science, 73, 888–904.
Batterman, R. (2010). On the explanatory role of mathematics in empirical science. British Journal for Philosophy of Science, 61, 1–25.
Batterman, R. (2016). Philosophical implications of Kadanoff’s work on the renormalization group. Journal of Statistical Physics. https://doi.org/10.1007/s10955-016-1659-9.
Bogen, J., & Woodward, J. (1988). Saving the phenomena. Philosophical Review, 97, 303–352.
Butterfield, J. (2011). Less is different: Emergence and reduction reconciled. Foundations of Physics, 41, 1065–1135.
Callender, C. (2001). Taking thermodynamics too seriously. Studies in History and Philosophy of Modern Physics, 32, 539–553.
Carpenter, K., Bahadur V. (2015). Saltwater icephobicity: Influence of surface chemistry on saltwater icing. Scientific Reports 5. Article number: 17563.
Goldenfeld, N. (1992). Lectures on phase transitions and the renormalization group. New York: Perseus Books.
Kadanoff, L. (2009). More is the same; mean field theory and phase transitions. Journal of Statistical Physics, 137, 777–797.
Lanford, O. (1975). Time evolution of large classical systems. In J. Moser (Ed.), Dynamical systems, theory and applications: Lecture notes in theoretical physics (Vol. 38, pp. 1–111). Berlin, Heidelberg: Springer.
Le Bellac, M., Mortessagne, F., & Batrouni, G. (2004). Equilibrium and non-equilibrium statistical thermodynamics. Cambridge: Cambridge University Press.
Liu, C. (1999). Explaining the emergence of cooperative phenomena. Philosophy of Science, 66, S92–S106.
Liu, C. (2001). Infinite systems in SM explanations: Thermodynamic limit, renormalization (semi-) groups, and irreversibility. Philosophy of Science, 68, S325–S344.
Menon, T., & Callender, C. (2013). Turn and face the strange. Ch–ch-changes: Philosophical questions raised by phase transitions. In Batterman (Ed.), The Oxford handbook of philosophy of physics. Oxford: Oxford University Press.
Morrison, M. (2012). Emergent physics and micro-ontology. Philosophy of Science, 79, 141–166.
Morrison, M. (2014). Reconstructing reality: Mathematics, models and simulation. Oxford: Oxford Univ Press.
Norton, J. (2012). Approximation and idealization: Why the difference matters. Philosophy of Science, 79(2), 207–232.
Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order-disorder transition. Physical Review, 65, 11749.
Reif, F. (1965). Fundamentals of statistical and thermal physics. New York: McGraw-Hill.
Ruelle, D. (2004). Thermodynamic formalism (2nd ed.). Cambridge: Cambridge University Press.
Shech, E. (2013). What is the paradox of phase transitions? Philosophy of Science, 80(5), 1170–1181.
Sklar, L. (1993). Physics and chance. Cambridge: Cambridge Univ. Press.
Woodward, J. (2000). Data, phenomena, and reliability. Philosophy of Science, 67(Proceedings), S163–S179.
Woodward, J. (2011). Data and phenomena: A restatement and defense. Synthese, 182, 165–179.
Yang, C. N., & Lee, T. D. (1952). Statistical theory of equations of state and phase transitions. I. Theory of condensation. Physical Review, 87, 404–409.
Zemansky, M., & Dittman, R. (1997). Heat and thermodynamics (7th ed.). New York: McGraw-Hill.
Acknowledgements
I thank the editors for the invitation to contribute to this collection, as well as to the two anonymous referees, for detailed and constructive criticisms, and suggestions. Special thanks to Professor Lapo Casetti, from Dipartimento di Fisica, Università di Firenze, for several illuminating points he made during a conference organized in Louvain-La-Neuve by Vincent Ardourel and Alexandre Guay. The final version of the paper is entirely my responsibility.
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Bangu, S. Discontinuities and singularities, data and phenomena: for Referentialism. Synthese 196, 1919–1937 (2019). https://doi.org/10.1007/s11229-018-1747-2
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DOI: https://doi.org/10.1007/s11229-018-1747-2