Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality

Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality

Abstract

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume–Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter governing the speed at which the sequence approaches criticality is below a certain threshold . However, when exceeds , the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization. The asymptotic behavior of the finite-size magnetization is proved via a moderate deviation principle when and via a weak-convergence limit when . To the best of our knowledge, our results are the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model.

[
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10.1214/10-AAP679 \volume20 \issue6 2010 \firstpage2118 \lastpage2161

\runtitle

Asymptotic behavior of the finite-size magnetization

{aug}

A]\fnmsRichard S. \snmEllis\corref\thanksreft1label=e1]rsellis@math.umass.edu, B]\fnmsJonathan \snmMachtalabel=e2]machta@physics.umass.edu and C]\fnmsPeter Tak-Hun \snmOttolabel=e3]potto@willamette.edu

\thankstext

t1Supported by NSF Grant DMS-06-04071.

class=AMS] \kwd[Primary ]60F10 \kwd60F05 \kwd[; secondary ]82B20. Finite-size magnetization \kwdthermodynamic magnetization \kwdsecond-order phase transition \kwdfirst-order phase transition \kwdtricritical point \kwdmoderate deviation principle \kwdlarge deviation principle \kwdscaling limit \kwdBlume–Capel model \kwdfinite-size scaling.

1 Introduction

For the mean-field Blume–Capel model, as for other mean-field spin systems, the magnetization in the thermodynamic limit is well understood within the theory of large deviations. In this framework the thermodynamic magnetization arises as the unique, positive, global minimum point of the rate function in a large deviation principle. The question answered in this paper is whether, in a neighborhood of criticality, the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization, which is the expected value of the spin per site. A similar question is answered by the heuristic, statistical mechanical theory of finite-size scaling. This paper is both motivated by the theory of finite-size scaling and puts that theory on a firm foundation in the context of mean-field spin systems. It is hoped that our results suggest how this question can be addressed in the context of much more complicated, short-range spin systems.

Our approach is to evaluate the asymptotic behaviors of the thermodynamic magnetization and the physically relevant, finite-size magnetization along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume–Capel model. The thermodynamic magnetization is then considered to be a physically relevant estimator of the finite-size magnetization when these two quantities have the same asymptotic behavior. Our main finding is that the value of the parameter governing the speed at which the sequence approaches criticality determines whether or not the asymptotic behaviors of these two quantities are the same. Specifically, we show in Theorem 4.1 that the thermodynamic magnetization and the finite-size magnetization are asymptotic when is below a certain threshold and that therefore the thermodynamic magnetization is a physically relevant estimator when . However, when exceeds , then according to Theorem 4.2, the thermodynamic magnetization converges to 0 much faster than the finite-size magnetization, and therefore the thermodynamic magnetization is not a physically relevant estimator when . An advantage of using the thermodynamic magnetization as an estimator of the finite-state magnetization when is that the asymptotic behavior of the former quantity is much easier to derive than the asymptotic behavior of the latter quantity [see the discussion at the end of the paragraph after (5)].

The investigation is carried out for a mean-field version of an important lattice spin model due to Blume and Capel, to which we refer as the B–C model [4, 6, 7, 8]. This mean-field model is equivalent to the B–C model on the complete graph on vertices. It is one of the simplest models that exhibits the following intricate phase-transition structure: a curve of second-order points, a curve of first-order points and a tricritical point, which separates the two curves. A generalization of the B–C model is studied in [5].

The mean-field B–C model is defined by a canonical ensemble that we denote by ; equals the number of spins, is the inverse temperature and is the interaction strength. is defined in (7) in terms of the Hamiltonian

in which represents the spin at site and takes values in . The configuration space for the model is the set containing all sequences with each . Expectation with respect to is denoted by . The finite-size magnetization is defined by , where equals the total spin .

Before introducing the results in this paper, we summarize the phase-transition structure of the model. For and we denote by the set of equilibrium values of the magnetization. coincides with the set of global minimum points of the free-energy functional , which is defined in (11). It is known from heuristic arguments and is proved in [16] that there exists a critical inverse temperature and that for there exists a quantity and for there exists a quantity having the following properties. The positive quantity appearing in the following list is the thermodynamic magnetization.

  1. Fix . Then for , consists of the unique pure phase 0, and for , consists of two nonzero values .

  2. For , undergoes a continuous bifurcation at , changing continuously from for to for . This continuous bifurcation corresponds to a second-order phase transition.

  3. Fix . Then for , consists of the unique pure phase 0, for , consists of 0 and two nonzero values and for , consists of two nonzero values .

  4. For , undergoes a discontinuous bifurcation at , changing discontinuously from for to for to for . This discontinuous bifurcation corresponds to a first-order phase transition.

Because of items 2 and 4, we refer to the curve as the second-order curve and to the curve as the first-order curve. Points on the second-order curve are called second-order points, and points on the first-order curve first-order points. The point separates the second-order curve from the first-order curve and is called the tricritical point. The two-phase region consists of all points in the positive - quadrant for which consists of two values. Thus this region consists of all above the second-order curve, above the tricritical point and above the first-order curve; that is, all satisfying and and satisfying and . The sets that describe the phase-transition structure of the model are shown in Figure 1.

Figure 1: The sets that describe the phase-transition structure of the mean-field B–C model: the second-order curve , the first-order curve and the tricritical point . The phase-coexistence region consists of all above the second-order curve, above the tricritical point, on the first-order curve and above the first-order curve. The extension of the second-order curve to is called the spinodal curve.

For fixed lying in the two-phase region the finite-size magnetization converges to the thermodynamic magnetization as . In order to see this, we use the large deviation principle (LDP) for with respect to in [16], Theorem 3.3, and the fact that the set of global minimum points of the rate function in that LDP coincides with the set  [16], Proposition 3.4, the structure of which has just been described. Since for lying in the two-phase region , the LDP implies that the -distributions of put an exponentially small mass on the complement of any open set containing . Symmetry then yields the weak-convergence limit

(1)

This implies the desired result

(2)

The limit in the last display is closely related to the main focus of this paper. It shows that because the thermodynamic magnetization is the limit, as the number of spins goes to , of the finite-size magnetization, the thermodynamic magnetization is a physical relevant estimator of the finite-size magnetization, at least when evaluated at fixed in the two-phase region.

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization in a more general sense, namely, when evaluated along a class of sequences that converge to a second-order point or the tricritical point . The criterion for determining whether is a physically relevant estimator is that as , is asymptotic to the finite-size magnetization , both of which converge to 0. In this formulation we let in the finite-size magnetization; that is, we let the number of spins coincide with the index parametrizing the sequence . As summarized in Theorems 4.1 and 4.2, our main finding is that is a physically relevant estimator if the parameter governing the speed at which approaches criticality is below a certain threshold ; however, this is not true if . For the sequences under consideration the parameter determines the limits

which are assumed to exist and not to be both 0. The value of depends on the type of the phase transition—first-order, second-order or tricritical—that influences the sequence, an issue addressed in Section 5 of [13].

We illustrate the results contained in these two theorems by applying them to six types of sequences. In the case of second-order points two such sequences are considered in Theorems 5.1 and 5.2, and in the case of the tricritical point four such sequences are considered in Theorems 5.35.6. Possible paths followed by these sequences are shown in Figure 2. We believe that modulo uninteresting scale changes, irrelevant higher order terms and other inconsequential modifications, these are all the sequences of the form and equal to plus a polynomial in , where is either a second-order point or the tricritical point and for some and .

Figure 2: Possible paths for the six sequences converging to a second-order point and to the tricritical point. The asymptotic results for the sequences converging on the paths labeled 1, 2, 3, 4a–4d, 5 and 6 are discussed in the respective Theorems 5.15.6. The sequences on the paths labeled 4a–4d are defined in (19) and in the second paragraph after that equation.

We next summarize our main results on the asymptotic behaviors of the thermodynamic magnetization and the finite-size magnetization, first for small values of and then for large values of . The relevant information is given, respectively, in Theorems 3.1, 4.1 and 4.2. These theorems are valid for suitable positive sequences parametrized by , lying in the two-phase region for all sufficiently large , and converging either to a second-order point or to the tricritical point. The hypotheses of these three theorems overlap but do not coincide. The hypotheses of Theorem 3.1 are satisfied by all six sequences considered in Section 5 while the hypotheses of each of the Theorems 4.1 and 4.2 are satisfied by all six sequences with one exception. For each of the six sequences the quantities and appearing in these asymptotic results are specified in Table 1.

Seq. Defn. Thm.
1 (14) Theorem 5.1
2 (16) Theorem 5.2
3 (18) Theorem 5.3
4 (19) Theorem 5.4
5 (21) Theorem 5.5
6 (23) Theorem 5.6
Table 1: The equations where each of the six sequences is defined, the theorems where the asymptotic results in (3), (4) and (5) are stated for each sequence, and the values of , and [see (6)] for each sequence

The difference in the asymptotic behaviors of the thermodynamic magnetization and the finite-size magnetization for is described in item 3. As we discuss in Section 6, the difference is explained by the statistical mechanical theory of finite-size scaling.

  1. According to Theorem 3.1, there exists positive quantities and such that for all

    (3)
  2. (). According to Theorem 4.1, there exists a threshold value such that for all

    (4)

    Because is asymptotic to the finite-size magnetization, is a physically relevant estimator of the finite-size magnetization. In this case converges to criticality slowly, and we are in the two-phase region, where the system is effectively infinite. Formally the first index parametrizing the finite-size magnetization can be sent to before the index parametrizing the sequence is sent to , and so we have

  3. (). According to Theorem 4.2, there exists a positive quantity such that for all

    (5)

    Because converges to 0 much faster than the finite-size magnetization, is not a physically relevant estimator of the finite-size magnetization. In this case converges to criticality quickly, and we are in the critical regime, where finite-size scaling effects are important.

The asymptotic behavior of the thermodynamic magnetization stated in (3) holds for all . It is derived in Theorem 3.2 in [13] and is summarized in Theorem 3.1 in the present paper. In (4) we state the asymptotic behavior of the finite-size magnetization for . This result is proved in part (a) of Theorem 4.1 as a consequence of the moderate deviation principle (MDP) for the spin in Theorem 7.1, the weak-convergence limit in Corollary 7.3, and the uniform integrability estimate in Lemma 7.4. The asymptotic behavior of stated in (5) for is proved in part (a) of Theorem 4.2 as a consequence of the weak-convergence limit for the spin in Theorem 8.1 and the uniform-integrability-type estimate in Proposition 8.3. In part (a) of Theorem 4.3 we state the asymptotic behavior of for . That result is a consequence of a weak-convergence limit analogous to the limit in Theorem 8.1 and the uniform-integrability-type estimate in Proposition 8.3. With changes in notation only, Theorem 3.1 and Theorems 4.14.3 also apply to other mean-field models including the Curie–Weiss model [12] and the Curie–Weiss–Potts model [17]. The proof of the asymptotic behavior of the thermodynamic magnetization in [13], Theorem 3.2, is purely analytic and is much more straightforward than the probabilistic proofs of the asymptotic behaviors of the finite-size magnetization in Theorems 4.14.3.

Figure 3: and for (a) , (b) . Graph (b) is not shown to scale. In fact, for the global minimum points of are much closer to the origin and are much shallower than shown in graph (b).

Figure 3 gives a pictorial representation of the phenomena that are summarized in (4) for and in (5) for . As we discuss in Section 2, for the sequences under consideration the thermodynamic magnetization can be characterized as the unique, positive, global minimum point in an LDP or, equivalently, as the unique, positive, global minimum point of the dual, free-energy functional defined in (11). According to graph (a) in Figure 3, for , has two deep, global minimum points at . Graph (b) in Figure 3, which is not shown to scale, exhibits the contrasting situation for . In this case the global minimum points of at are shallow and close to the origin. In the two graphs we also show the form of the distribution . For this probability distribution is sharply peaked at as . In contrast, for the probability distribution is peaked at 0 and its standard deviation is much larger than .

In a work in progress we refine the asymptotic result in (4), which states that for , is asymptotic to as . Define , which exceeds since . We conjecture that for a class of suitable sequences that includes the first five sequences considered in Section 5, there exists a positive quantity such that for all

(6)

This refined asymptotic result would extend part (b) of Theorem 4.1. It is a consequence of the conjecture that when is conditioned to lie in a suitable neighborhood of , the -distributions of converge in distribution to a Gaussian.

For easy reference we list in Table 1 information about the six sequences considered in Section 5. The first two columns list, respectively, the equation in which each sequence is defined and the theorem in which the asymptotic results in equations (3), (4) and (5) are stated for each sequence. In these theorems the quantities and appearing in the three asymptotic results are defined. The three asymptotic results involve the quantities , , and , the values of the first two of which are listed in the next two columns of the table. In the last column of the table we list the values of . Through the factor , governs the conjectured asymptotics of stated in (6).

The contents of this paper are as follows. In Section 2 we summarize the phase-transition structure of the mean-field B–C model. Theorem 3.1 in Section 3 gives the asymptotic behavior of the thermodynamic magnetization for suitable sequences converging either to a second-order point or to the tricritical point. The heart of the paper is Section 4. In this section Theorems 4.1, 4.2 and 4.3 give the asymptotic behavior of the finite-size magnetization for three respective ranges of , and . The quantity is a threshold value that depends on the type of the phase transition—first-order, second-order or tricritical—that influences the associated sequence . These theorems also compare the asymptotic behaviors of the thermodynamic magnetization and the finite-size magnetization, showing that they are the same for but not the same for . In Section 5 the three theorems in the preceding section are applied to six specific sequences , the first two of which converge to a second-order point and the last four of which converge to the tricritical point. Section 6 gives an overview of the statistical mechanical theory of finite-size scaling, which gives insight into the physical phenomena underlying our mathematical results. Part (a) of Theorem 4.1 is derived in Section 7 from the MDP for the spin in Theorem 7.1, the weak-convergence limit for the spin in Corollary 7.3, and the uniform integrability estimate in Lemma 7.4. Finally, part (a) of Theorem 4.2 is derived in Section 8 from the weak-convergence limit for the spin in Theorem 8.1 and the uniform-integrability-type estimate in Proposition 8.3.

2 Phase-transition structure of the mean-field B–C model

After defining the mean-field B–C model, we introduce a function , called the free-energy functional. The global minimum points of this function define the equilibrium values of the magnetization. The phase-transition structure of the model is summarized in Theorems 2.1 and 2.2. The first theorem shows that the model exhibits a second-order phase transition for , where is the critical inverse temperature of the model. The second theorem shows that the model exhibits a first-order phase transition for .

For the mean-field B–C model is a lattice-spin model defined on the complete graph on vertices . The spin at site is denoted by , a quantity taking values in . The configuration space for the model is the set containing all sequences with each . In terms of a positive parameter representing the interaction strength, the Hamiltonian is defined by

for each . Let be the product measure on with identical one-dimensional marginals . Thus assigns the probability to each . For inverse temperature and for , the canonical ensemble for the mean-field B–C model is the sequence of probability measures that assign to each subset of the probability

(7)

In this formula is the partition function equal to

Expectation with respect to is denoted by .

The analysis of the canonical ensemble is facilitated by absorbing the noninteracting component of the Hamiltonian into the product measure , obtaining

(8)

In this formula equals the total spin , is the product measure on with identical one-dimensional marginals

(9)

is the normalization equal to and is the normalization equal to .

We denote by the set of equilibrium macrostates of the mean-field B–C model. In order to describe this set, we introduce the cumulant generating function of the measure defined in (9); for this function is defined by

(10)

For we define

(11)

As shown in Proposition 3.4 in [16], the set of equilibrium macrostates of the mean-field B–C model can be characterized as the set of global minimum points of :

(12)

In [16] the set was denoted by .

We also define the canonical free energy

where is the normalizing constant in (8). This limit exists and equals . Because of this property of , we call the free-energy functional of the mean-field B–C model.

The next two theorems use (12) to determine the structure of for and for . The positive quantity appearing in these theorems is called the thermodynamic magnetization. The first theorem, proved in Theorem 3.6 in [16], describes the continuous bifurcation in for as crosses a curve . This bifurcation corresponds to a second-order phase transition, and this curve is called the second-order curve. The quantity , defined in (13), is denoted by in [16].

Theorem 2.1

For , we define

(13)

For these values of , has the following structure:

(a) For , .

(b) For , there exists such that .

(c) is a positive, increasing, continuous function for , and as , . Therefore, exhibits a continuous bifurcation at .

The next theorem, proved in Theorem 3.8 in [16], describes the discontinuous bifurcation in for as crosses a curve . This bifurcation corresponds to a first-order phase transition, and this curve is called the first-order curve. As shown in Theorem 3.8 in [16], for all , . The quantity is denoted by in [16].

Theorem 2.2

For , has the following structure in terms of the quantity , denoted by in [16] and defined implicitly for on page 2231 of [16]:

(a) For , .

(b) For there exists such that .

(c) For there exists such that .

(d) is a positive, increasing, continuous function for , and as , . Therefore, exhibits a discontinuous bifurcation at .

The phase-coexistence region is defined as the set of all points in the positive - quadrant for which consists of more than one value. According to Theorems 2.1 and 2.2, the phase-coexistence region consists of all points above the second-order curve, above the tricritical point, on the first-order curve and above the first-order curve; that is,

Our derivation of the asymptotic behavior of the finite-size magnetization in this paper is valid for a class of sequences lying in the phase-coexistence region for all sufficiently large and converging either to a second-order point or to the tricritical point. In the next section we state an asymptotic formula for for a general class of such sequences. That asymptotic formula will be used later in the paper when we study the asymptotic behavior of the finite-size magnetization .

3 Asymptotic behavior of

The main result in this section is Theorem 3.1. It states the asymptotic behavior of the thermodynamic magnetization for sequences lying in the phase-coexistence region for all sufficiently large and converging either to a second-order point or to the tricritical point. The asymptotic behavior is expressed in terms of the unique positive, global minimum point of an associated polynomial that is introduced in hypothesis (iii) of the theorem. With several modifications the hypotheses of the next theorem are also the hypotheses under which we derive the rates at which later in the paper.

As shown in part (iii) of Theorem 3.1, the asymptotics of depend on the asymptotics of the scaled free-energy function . Because of Lemma 7.2, the asymptotics of the finite-size magnetization in Theorems 4.14.3 depend on precisely the same asymptotics. Lemma 7.2 coincides with Lemma 4.1 in [9]. In that paper the connections among the asymptotics of the scaled free-energy functional, the limit theorems underlying the asymptotics of the finite-size magnetization and Lemma 4.1 are described in detail. These limit theorems are analogues of the MDP in Theorem 7.1 and of the weak convergence limit in Theorem 8.1.

Theorem 3.1 restates the main theorem in [13], Theorem 3.2. Hypotheses (iii)(a) and (iv) in the next theorem coincide with hypotheses (iii)(a) and (iv) in Theorem 3.2 in [13] except that the latter hypotheses are expressed in terms of and while here we have substituted the formulas for and . Hence and no longer appear.

Theorem 3.1

Let be a positive sequence that converges either to a second-order point , , or to the tricritical point . We assume that satisfies the following four hypotheses: {longlist} lies in the phase-coexistence region for all sufficiently large . The sequence is parametrized by . This parameter regulates the speed of approach of to the second-order point or the tricritical point in the following sense: both exist, and and are not both ; if , then equals or . There exists an even polynomial of degree or satisfying as together with the following two properties; is called the Ginzburg–Landau polynomial. {longlist} [(a)  (iii)] (a) and such that for all uniformly for in compact subsets of . (b) has a unique, positive global minimum point ; thus the set of global minimum points of equals or . (iv) There exists a polynomial satisfying as together with the following property: such that sufficiently large and satisfying , . Under hypotheses (i)–(iv), for any If , then this becomes .

It is clear from the proof of the theorem that if hypotheses (iii) and (iv) are valid for a specific value of , then we obtain the asymptotic formula for that value of .

In the next section, we state the main results on the rates at which for small satisfying , for large satisfying , and for intermediate satisfying . We also compare these rates with the asymptotic behavior of the thermodynamic magnetization .

4 Main results on rates at which

Let be a positive sequence converging to 0. In stating the three results on the rates at which the finite-size magnetization , we write

and we write

Let be the quantity parametrizing the sequences as explained in hypothesis (ii) of Theorem 3.1. We begin with Theorem 4.1, which gives the rate at which for small satisfying . Theorem 4.2 gives the rate at which for large satisfying while Theorem 4.3 gives the rate at which for intermediate satisfying . In all three cases we compare these rates with the rate at which . In the next section we specialize these theorems to the six sequences mentioned in the Introduction.

Part (a) of the next theorem gives the rate at which for , and part (b) shows that for these values of , . It follows that for , is a physically relevant estimator of the finite-size magnetization because it has the same asymptotic behavior as that quantity.

The next theorem is valid under hypotheses (i) and (ii) of Theorem 3.1, hypotheses (iii)(a) and (iv) of that theorem for all , the inequality , and a new hypothesis (iii)(b). The inequality is satisfied by all six sequences considered in Section 5. The new hypothesis (iii)(b) restricts hypothesis (iii)(b) of Theorem 3.1 by assuming that the set of global minimum points of the Ginzburg–Landau polynomial equals for some . As we remark after the statement of the theorem, this restriction is needed in order to prove part (a). The proof does not cover the case where the set of global minimum points of equals for some . The conjecture is that in this case there exists such that (see the discussion before Corollary 7.3). An example of a sequence for which the set of global minimum points of contains three points is given in case (d) of sequence 4 in the next section. By contrast, all the other sequences considered in the next section satisfy the new hypothesis that the set of global minimum points of equals for some .

Theorem 4.1 (())

Let be a positive sequence parametrized by and converging either to a second-order point , , or to the tricritical point . We assume hypotheses (i) and (ii) of Theorem 3.1 together with hypotheses (iii)(a) and (iv) of that theorem for all . We also assume the inequality and the following hypothesis, which restricts hypothesis (iii)(b) of Theorem 3.1: {longlist}[(iii)(b)] (iii)(b) The set of global minimum points of the Ginzburg–Landau polynomial equals for some . The following conclusions hold: (a) For all

(b) For all , .

Part (a) of the theorem is proved from the moderate deviation principle (MDP) for the -distributions of in Theorem 7.1, which shows that the rate function equals . The inequality is used to control an error term in the proof of the MDP. According to hypothesis (iii)(b), the set of global minimum points of equals for some . It quickly follows from the MDP that the sequence of -distributions of converges weakly to