The arguments presented in this paper give us good reason to think that the appeal to the infinite limits in the theory of phase transitions does not represent a challenge for reduction, at least not for limiting reduction. In fact, contra what has been argued by Batterman (2001, 2009) and Morrison (2012), these arguments suggest that the infinities and divergences character-istic of the physics of phase transitions are not essential for giving an account of the phenomena since from finite statistical mechanics one can recover the

16This is also pointed out by Butterfield (Butterfield, 2011, p. 69).

thermodynamic behavior of phase transitions even in the case of continuous phase transitions, as it was shown in section 5.

Nevertheless, this does not mean that phase transitions are not incon-sistent with other notions of reduction that have also been discussed in the philosophical literature. Norton (2013), for instance, correctly points out that the case of continuous phase transitions does not satisfy what he calls

“few-many reduction”, according to which there will be a reduction if the behavior of a system with a few components can be used to explain the be-havior of a system with a large number of them. The reason for this is that continuous phase transitions are intrinsically fluctuation phenomena that can only arise when N is sufficiently large.

Likewise, continuous phase transitions also seem to be at odds with the kind of reductive explanation that requires the explanans to give us accu-rate and detailed information about the microscopic causal mechanisms that produce the phenomenon (e.g. Kaplan (2011)). As it has been pointed out by Batterman (2002), Batterman and Rice (2014) and Morrison (2012), the impossibility of giving such an account is related with the robustness of the fixed point solutions under different choices of the initial conditions. This im-plies that the critical behavior is largely independent of specific microscopic details characterizing the different models and that the statistical mechani-cal account of phase transitions does not give us complete information about the microscopic mechanisms underlying the transitions. However, as it was shown in the paper, these senses in which reduction ”fails” do not threat the project of inter-theory reductionism in any relevant sense.

## Chapter 3

## Had We But World Enough, and Time... But We Don’t!

### Justifying the Thermodynamic and Infinite-time Limits in Statistical Mechanics

“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever” [N. H.

Abel 1828]

### 3.1 Introduction

“Had we but world enough, and time” are the words with which Andrew Marvell begins his passionate poem in which he tells his lover that things would be different if they had infinite space and time. While neither the number of particles in real systems nor the time of measurements are infinite, it is common in statistical mechanics to take the number of particles and time to infinity in order to recover the values of thermodynamic observables. These are called the thermodynamic limit and the infinite-time limit, respectively.

This raises the following questions: What justifies the empirical adequacy of

scientific models that involve infinite limits? And what is the justification that we have for applying such a theory to finite systems? Sure enough, there would be a straightforward justification for the limits if one could show that, at least for the purpose of inferring the values of the thermodynamic observables, the infinite case is rather similar to the finite case (contrary to the situation described by Marvell!). But, is this so?

As it was seen in the previous chapter, there has been a fervent contro-versy around the use of the thermodynamic limit in the statistical mechanical treatment of phase transitions, in which has been claimed by some authors (e.g. Batterman 2005, Jones 2006, Batterman 2011, Bangu 2009, Bangu 2011) that the use of the thermodynamic limit – and so of an infinite system – is indispensable to give an account of phase transitions. As a consequence, it has been said that the behavior in the limit is physically real (Batterman 2005) or that phase transitions are not reducible to statistical mechanics (e.g. Batterman 2011, Bangu 2011, Morrison 2012). Others (e.g. Butterfield 2011, Butterfield and Buoatta 2011, Norton 2012) have argued against these conclusions saying that the thermodynamic limit can be justified straightfor-wardly, because the thermodynamic limit gives an approximate description of the behavior of real systems. They generally arrive at that conclusion by saying that the thermodynamic limit satisfies what Landsman (2013) calls Butterfield’s principle, according to which a limit is justified and can be re-garded as mathematically convenient and empirically adequate if the same behavior that arises in the limit also arises, at least approximately, “on the way to the limit”.

In this chapter, I will take the side of the ones that believe that there is a straightforward justification for the thermodynamic limit, but I will ar-gue against the idea that the so-called “Butterfield Principle” is sufficient to give a straightforward justification for the use of infinite limits in general. I arrive at that conclusion by comparing the use of the thermodynamic limit in the theory of phase transitions with the infinite-time limit in the

expla-nation of equilibrium states, which has generally been left aside from the recent philosophical debate around the use of infinite idealizations in statis-tical mechanics. In the case of phase transitions, I will argue (Section 3.2) that the thermodynamic limit can be justified pragmatically, since the limit behavior also arises before we get to the limit and for a number of particles N that is physically significant. However, I will contend (Section 3.3) that the justification of the infinite-time limit is less straightforward. In fact, I will point out that even in cases where one can recover the limit behavior for finite time t, i.e. before we get to the limit, one fails to recover this behavior for realistic time scales. In my view this leads us to reconsider the role that the rate of convergence plays in the justification of infinite limits in general and calls for a revision of the so-called Butterfield’s principle. I will end this paper (Section 4.4) by offering a criterion for the justification of infinite limits based on the notion of controllable approximations.