Elsevier

Advances in Mathematics

Volume 385, 16 July 2021, 107768
Advances in Mathematics

Multiple phase transitions on compact symbolic systems

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Abstract

Let ϕ:XR be a continuous potential associated with a symbolic dynamical system T:XX over a finite alphabet. Introducing a parameter β>0 (interpreted as the inverse temperature) we study the regularity of the pressure function βPtop(βϕ) on an interval [α,) with α>0. We say that ϕ has a phase transition at β0 if the pressure function Ptop(βϕ) is not differentiable at β0. This is equivalent to the condition that the potential β0ϕ has two (ergodic) equilibrium states with distinct entropies. For any α>0 and any increasing sequence of real numbers (βn) contained in [α,), we construct a potential ϕ whose phase transitions in [α,) occur precisely at the βn's. In particular, we obtain a potential which has a countably infinite set of phase transitions.

Introduction

Broadly speaking a phase transition refers to a qualitative change of the statistical properties of a dynamical system. The precise definition of this notion differs depending on which settings and properties one studies. A phase transition might mean co-existence of several equilibrium states resulting from some optimization [9], [11], [13], lack of the Gateaux differentiability of the pressure functional [8], [10], [14], [20], or loss of analyticity of the pressure with respect to external physical parameters such as temperature [15], [16], [24]. The latter gives rise to further differentiation between the first-order, second-order, or higher order phase transitions.

We briefly note the relationship between the regularity of the pressure and coexistence of several equilibria. We refer to Section 2.1 for the definitions and a formal discussion. It was shown by Walters [27] that the Gateaux differentiability at ϕ of the pressure functional acting on the space of continuous potentials is equivalent to the uniqueness of the equilibrium state for ϕ. On the other hand, non-differentiability of the pressure in the direction of ϕ necessarily implies coexistence of several equilibrium states. Recently, Leplaideur [18] discovered the surprising fact that the converse to this statement is not true. He provided an example of a continuous potential ϕ defined on a mixing subshift of finite type such that the pressure is analytic in the direction of ϕ, but uniqueness of equilibrium states fails. Moreover, in Leplaideur's example the uniqueness of equilibrium states fails for two distinct inverse temperature values.

The existence of phase transitions has also been established for parabolic systems and the geometric potential, see, e.g., [1], [3], [6], [26]. Roughly speaking in these examples the degree of parabolicity near the parabolic point(s) determines whether the second equilibrium state is finite or σ-finite.

A number of related questions have been studied in the statistical physics literature. Miekisz [19] starts from the well-known Robinson two-dimensional shift of finite type that admits only aperiodic configurations. These configurations have a highly regular structure, and are considered to be a quasi-crystal. Miekisz's model allows “forbidden” configurations to appear with a local energy cost. There is a conjectural picture in the statistical physics literature which would imply that at finite temperatures, the equilibrium states exhibit global periodicity with local fluctuations, while at zero temperature the equilibrium measure is supported on quasi-crystal states. In this picture, there is a sequence of phase transitions as the temperature is reduced at which the global period increases. In support of this conjecture, Miekisz establishes an increasing sequence of lower bounds on the global period as the temperature is reduced to zero. It should be pointed out that the paper does not demonstrate that the quasi-crystal state is not attained at positive temperature, but rather establishes that conditional on the positive temperature states not being quasi-crystalline, there must be an infinite sequence of phase transitions.

Another related model due to van Enter and Shlosman [12] deals with a model in dimension 2 or higher with a continuous alphabet (the circle). They describe a nearest neighbor potential (the “Seuss model”, named after the Cat in the Hat) with an infinite sequence of first order phase transitions in this setting.

In this note we are concerned with the first-order phase transitions of the pressure function with respect to a parameter regarded as the inverse temperature. We consider a continuous potential ϕ:XR associated with a symbolic dynamical system (X,T) over a finite alphabet. Given a positive real number β, we study the regularity of the pressure function βPtop(βϕ). We say ϕ has a phase transition at β0 if the pressure function βPtop(βϕ) is not differentiable at β0. This is equivalent to the condition that the potential β0ϕ has two (ergodic) equilibrium states with distinct entropies (see Section 2.1 for details).

It is a classical result due to Ruelle [22], [23] that if X is a transitive subshift of finite type then the pressure functional Ptop acts real analytically on the space of Hölder continuous potentials, that is, for all Hölder continuous ϕ,ψ:XR we have that βPtop(ϕ+βψ) is analytic in a neighborhood of 0. This immediately implies the uniqueness of equilibrium states for Hölder continuous potential, which is referred to as “lack of phase transitions” [11]. Therefore, in order to allow the possibility of phase transitions (i.e. the occurrence of distinct equilibrium states) one needs to consider potential functions that are merely continuous.

Although phase transitions have been studied for many classes of dynamical systems, to the best of our knowledge this is the first family of examples in the one-dimensional compact symbolic setting with more than two phase transitions. In this paper we develop a method to explicitly construct a continuous potential with any finite number of first order phase transitions in any given interval [α,) occurring at any sequence of predetermined points. We are able to go even further. Note that the convexity of the pressure implies that a continuous potential ϕ has at most countably many phase transitions. We show that the case of infinitely many phase transitions can indeed be realized. In the following statement we summarize our results given by Theorem 1, Corollary 1, Corollary 2, Corollary 3 and Remark 1.

Main Theorem

Let T:XX be the two-sided full shift, α be any positive number and (βn) be a strictly increasing (finite or infinite) sequence in [α,). Then there exists a continuous potential ϕ:XR such that the following holds:

  • (i)

    When βα the potential ϕ has a phase transition at β if and only if β=βn for some nN;

  • (ii)

    If limnβn=β<, then the family of equilibrium states of βϕ is constant for all ββ.

The idea behind the construction of the potential ϕ is to describe it first on a sequence of certain disjoint invariant subsets Xn. The objective here is to guarantee that the supports of the equilibria of βϕ pass through the Xn's as β increases. The values of ϕ at all other points of X are defined in terms of the distance to Xn in such a way that they do not change the behavior of the pressure function. There are other papers where the distance function to invariant subsets is considered to establish interesting dynamical phenomena. For example, Chazottes and Hochman [7] used this technique to show the possibility of the non-existence of the zero-temperature measure for Lipschitz potentials.

We note that our methods do not carry over to one-sided shifts. The fact that the shift map is a homeomorphism is essential to control the distances to subsets Xn and estimate the values of the potential along the drop off. A standard technique to transfer results from two-sided to one-sided shifts involves a construction of a cohomologous potential that does not depend on the backward history. In [28] Walters found a necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function. Since for subshifts of finite type this condition necessarily implies uniqueness of equilibrium states, our potential clearly cannot satisfy it. We speculate that infinitely many phase transitions can occur in the one-sided setting, but our present work does not exhibit such a phenomenon.

While we are able to completely control the function βP(βϕ) on the interval [α,), we do not have a full picture of the behavior for β values in the range [0,α). In the range [α,), the potential drops so sharply off the sets Xn that equilibrium states are forced to live on the Xn's. For β below α, the “cost” incurred by leaving Xn is not sufficiently high to prevent equilibrium states having support outside the union. We conjecture that there exists a β such that for β<β, the equilibrium state for βϕ is fully supported, and indeed in this region, we expect that the pressure depends analytically on β.

Of particular interest is the behavior of the pressure function βPtop(βϕ) as β. A simple argument shows that Ptop(βϕ) has an asymptote of the form aβ+b. Taking finitely many (βn)n=1N in part (i) of the Main Theorem we see that Ptop(βϕ) reaches its asymptote at β=βN. Hence, we have an ultimate phase transition at β=βN. Physically, this means that for some positive temperature 1/βN, the system reaches its ground state which is the unique measure of maximal entropy of a certain subshift of X in our construction, and then ceases to change. This phenomenon is often referred to as a freezing phase transition at which the system reaches a ground state. We refer to [4], [5] and the references therein for details about freezing phase transitions.

We point out that in the situation of the Main Theorem part (ii) with infinitely many βn values we also obtain a freezing phase transition. Namely, for all ββ there are precisely two ergodic equilibrium states both of which are fixed point measures and each equilibrium state is a convex combination of these fixed point measures. In particular, the set of equilibrium states of βϕ does not change anymore when the temperature 1/β is lowered.

Finally, we mention that the case of finite alphabet shift maps crucially differs from that of countable alphabet symbolic systems. Indeed, for countable Markov shifts Sarig established several new phenomena that are associated with the lack of analyticity of the pressure function. This includes positive Lebesgue measure non-analyticity points of the pressure function, which are associated with the existence of multiple equilibrium states and/or intervals of intermittent behavior, i.e. an interval of β's with an infinite conservative equilibrium state. We refer to [24], [25] for details.

We thank Aernout van Enter for helpful information about the statistical physics literature.

Section snippets

Thermodynamic formalism

Let T:XX be a homeomorphism on a compact metric space X, and denote by M=MT the set of all T-invariant probability measures on X endowed with the weak topology. This makes M a compact convex metrizable topological space. Further, let Me=MTe be the subset of ergodic measures. For μM the measure-theoretic entropy of μ, denoted by hμ(T), is defined as follows. Let P be a countable measurable partition of X. We define the entropy of the partition P with respect to μHμ(P)=PPμ(P)logμ(P). Here

Proof of the main theorem

We deal first with the case of a countably infinite number of phase transitions. The reduction to the case of finitely many phase transitions is done in Corollary 1.

Theorem 1

Let X be a two-sided full shift on two symbols. Then for any positive real number α and any strictly increasing sequence (βn)n=1 in [α,) there is a continuous potential ϕ:XR such that the pressure function βP(βϕ) on [α,) is not differentiable exactly at points βn,nN.

We build a potential ϕ on the full shift (X,T) with

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  • Cited by (0)

    1

    T.K. is supported by grants from the Simons Foundation #430032 and from the PSC-CUNY TRADA-48-19.

    2

    A.Q. is supported by a grant from NSERC.

    3

    C.W. was supported by grants from the Simons Foundation (#637594 to Christian Wolf) and from the PSC-CUNY (TRADB-51-63715).

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