Forma de actuación durante la emergencia:

The main results of this section are contained in Theorems 3.7 and 3.10. Theorem 3.7 states that if the Legendre-Fenchel transform of the pressure is convex then we have equivalence of ensembles. This not only holds for the canonical and grand canonical en- sembles but also for the restricted versions. If we also know that the canonical ensembles concentrate on configurations with a particular bound on the maximum then this can lead to equivalence between the unrestricted canonical ensemble and the restricted grand canonical ensemble. This is the content of Corollary 3.12. Under the canonical mea- sures, the macroscopic maximum site occupation concentrates on the global minimum ofIρ(·).

measures, I(ρ, m), can be calculated following Theorem 3.10. Notice that conditioning the canonical measures at densityρ to have macroscopic maximum less than or equal to ρdoes not impose any restriction. This implies thatscan,ρ(ρ) =scan(ρ) so Theorem 3.10

allows us to calculate the canonical entropy by contracting on the maximum occupied site. The second part of Theorem 3.10 states that we need not know the entire rate function, only its value at the minimum, in order to perform the contraction. We may be required to iterate the contraction for fixed ρ to find the full rate function I(ρ, m) and finally scan(ρ). If there exists an ¯m such that Dpm¯ ⊃ Dp (that is there is a value

of the cut-off that provides a non-trivial extension of the pressure) then for any ρ >0 we have to perform this iterative procedure only finitely many times to calculateI(ρ,·). This implies that scan(ρ) exists and we can calculate it by applying the contraction on

finitely many macroscopically occupied sites.

The results of this chapter do not rely on the random variable SL being positive.

They therefore also hold for the re-centred density, TL(η) = 1 bL X x∈ΛL (ηx−ν(η1)) (3.79)

which counts the ‘excess’ density in the system, above or below the expected number under ν on the scale bL. The moderate deviations of this random variable give rise to

the leading order finite-size effects in Chapter 4, and can be found using the results of this chapter.

The methods of this chapter are essentially equivalent to applying Laplace/steepest- descent methods to asymptomatically estimate the finite sums defining the canonical partition function Z(L, ρ) and the truncated version Q(L, ρ, m) (see Appendix D.1), which gives rise to the cumulative probability distribution for the largest mass, for details on this approach see [47].

In the following chapters the results are applied to three examples; finite size effects in a standard condensing zero-range process, a size-dependent zero-range process, and the recently introduced inclusion process. The methods apply in general to any system that conserves mass and exhibits stationary product measures. For instance they can be applied to study other size-dependent zero-range processes such as those analysed in [114]. There it was shown that, depending on the size-dependence of the jump rates, multiple condensates can be stabilised in the thermodynamic limit. This included the case of finitely many macroscopic condensates as well as infinitely many mesoscopic con- densates. The former case can be analysed in the context of the results in this chapter, however, to derive useful rigorous results on the latter case would require contracting over a number of mesoscopically occupied sites growing with L. This would require a slight adjustment of Theorem 3.10 to allow for contractions on the events that include the excess mass being distributed over a growing number of lattice sites.

It is also possible to apply similar methods to continuous mass versions of the systems discussed here, in which the local state space is replaced with R+. If the single site

one another) then, with some technical effort, one can extend the analysis in this chapter to these continuous mass systems. It is then possible to define canonical measures by conditioning the reference measures on a single real value of the mass (despite the set having measure zero). Care must be taken, since the canonical measures of a system of size L are defined on an L −1 dimensional simplex, and so the relative entropy must be taken with respect to νL−1. Interesting metastability phenomena have been observed for size-dependent continuous mass transport systems in [133]. Another family of continuous mass models that these results could apply to is given by the Brownian energy and Brownian momentum processes [56, 57], which are closely related to the inclusion process discussed in Chapter 6.

Necessary and sufficient conditions for the existence of product stationary measures for a large class of discrete and continuous mass transport systems have been found [49, 63, 132]. Also the accuracy of mean-field predictions derived from approximations of the stationary measures by product measures have been investigated for continuous mass systems [134]. It follows from these studies that the results of this chapter apply to a wide range of mass transport systems that may exhibit a condensation transition. It would be interesting to extend the results here to systems with weak correlations between sites. For such systems extended condensates could be formed, over more than a single lattice site. In this case the surface tension associated with the formation of a condensate may also contribute to the canonical entropy densities. There has been recent work in this area, considering mass transport models with pair-factorised steady states [126, 127, 128]. The models in this work represent a good starting point for generalising the methods in this chapter.

Another possible extension are processes with multiple particle species [65]. In this case the relevant thermodynamic quantities such as the pressure and entropy densities are defined on higher dimensional spaces, and an extension of the analysis requires some technical work.