Contact geometric approach to Glauber dynamics near a cusp and its limitation Shin-itiro Goto Shai Lerer Leonid Polterovich

2025-04-27 0 0 368.59KB 18 页 10玖币

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Contact geometric approach to Glauber dynamics

near a cusp and its limitation

Shin-itiro Goto∗

, Shai Lerer†

, Leonid Polterovich†

March 8, 2023

Abstract

We study a nonequilibrium mean ﬁeld Ising model in the low temperature phase

regime, where metastable equilibrium states develop a cuspidal (spinodal) singularity.

We focus on celebrated Glauber dynamics, and design a contact Hamiltonian ﬂow

which captures some of its rough features in this regime. We prove, however, that

there is an inevitable discrepancy between the scaling laws for the relaxation time in

the Glauber and the contact Hamiltonian dynamical systems.

1 Introduction

Considerable activity is being devoted to establish a solid foundation of nonequilib-

rium thermodynamics and statistical mechanics [10, 18, 19]. Among various problems

in constructing a viable general theory, establishing a concise description of dynamical

properties of thermodynamic systems with phase transitions is one of the main points.

Recalling the success with the Ising model in developing equilibrium statistical me-

chanics, one recognizes that analyzing a canonical dynamical model is expected to be

the ﬁrst step towards the establishment of a nonequilibrium theory. One can choose

a spin kinetic model as a canonical model, where its dynamics is called Glauber dy-

namics. In particular the model with the mean ﬁeld type spin coupling enables one

to derive a simple dynamical system (see equation (5) below) for an expectation or

thermal average of magnetization with some approximation [5, 14]. This expectation

variable is the thermodynamic conjugate variable of the externally applied magnetic

ﬁeld, and shows relaxation processes. Here, roughly speaking, relaxation is a dynami-

cal process starting from a nonequilibrium state to a point of the equilibrium state set

in thermodynamic phase space. At equilibrium, for the Ising model with mean ﬁeld

type interactions, the equation of state is explicitly derived by calculating the partition

∗Center for Mathematical Science and Artiﬁcial Intelligence, Chubu University, 1200 Matsumoto-cho,

Kasugai, Aichi 487-8501, Japan

†School of Mathematical Sciences, Tel Aviv University, 6997801, Tel Aviv, Israel

arXiv:2210.00703v4 [math-ph] 7 Mar 2023

function in the thermodynamic limit, and the system at equilibrium exhibits a ﬁrst-

order phase transition together with metastable states (see [7] for this derivation). For

Glauber dynamics, various scaling relations have been proposed, and one of them is

the scaling of the relaxation time near the critical point [1, 11].

To advance our understanding of Glauber dynamics, one may employ reliable and

well-developed mathematical theories. One of them is contact geometry whose central

object is the Gibbs 1-form dz −pdq in the 3-dimensional thermodynamic phase space

equipped with coordinates z(minus free energy), p(magnetization), and q(exterior

magnetic ﬁeld) [4]. The equilibrium submanifold of the mean ﬁeld Ising model is rep-

resented by a smooth curve. The restriction of the Gibbs form to the equilibrium curve

vanishes, which manifests the fundamental thermodynamic relation. In the presence of

the 1st order phase transition, the equilibrium curves necessarily develop singularities

when projected to the (z, q)-plane. We are especially interested in so-called spinodal

points where such a singularity is a cusp. The interplay between scaling relations for

the relaxation time and the geometry of the equilibrium curve near a spinodal point is

the main theme of the present paper.

Another merit of contact geometry is that it provides a natural class of dynami-

cal systems, so called contact Hamiltonian ﬂows on the thermodynamic phase space,

which preserve the Gibbs form up to a conformal factor. Loosely speaking, contact

Hamiltonian ﬂows are odd-dimensional cousins of the standard Hamiltonian ﬂows of

classical mechanics. Contact Hamiltonian ﬂows model processes of nonequilibrium

thermodynamics (see e.g. [9, 8, 6, 3, 15, 4, 7]). In this paper, we focus on designing

a contact Hamiltonian system whose dynamics captures the equilibria and their sta-

bility patterns of the Glauber-Suzuki-Kubo ordinary diﬀerential equation (ODE) (5)

near critical points, show time-scales near critical points for the phase transition, and

compare this with other proposals in the literature. In addition to contact geometry,

we use some basics of singularity theory. Because of these, commonly used notations

employed in contact geometry are adopted in this paper.

2 Mean ﬁeld Ising model

Thermodynamics of the mean ﬁeld Ising model in the presence of a constant mag-

netic ﬁeld qis described by its free energy (taken with the opposite sign) z, and

magnetization p, where zand pare obtained by dividing by the number of total spins

N. In obtaining the thermodynamic quantities zand pfrom the microscopic model

with the standard statistical method, the limit N1 has been taken. In addition,

the statistical average over spin variables is assumed to yield a proper scaling of Nso

that the existence of the thermodynamic limit is guaranteed. These involved variables

can be written in terms of commonly used notations in physics as shown in Table 1.

In equilibrium, we have the relations

p=φ0(q+bp), z =φ(q+bp)−b

2p2,(1)

where φ(u) = β−1ln (2 cosh(βu)) and the ﬁrst equation represents the so-called self-

consistent equation (see e.g. [7]). The real parameter β > 0 is the inverse temperature,

Table 1: Notations of various quantities

Quantity Geometry oriented symbol Physics oriented symbol

Magnetization p m

Magnetic ﬁeld q h

Interaction b J

Free energy −z f

and b > 0 is determined by the strength of the interaction and the geometry of the

model. Equations (1) can be resolved as

q(p) = −bp +1

2βln 1 + p

1−p,

and

z(p) = 1

βln 2 −1

2βln(1 −p2)−bp2

Note that

−z(p) = −bp2

2−pq(p) + 1 + p

2βln(1 + p) + 1−p

2βln(1 −p)−1

βln 2 ,

which is another expression for the free energy of the mean ﬁeld Ising model [2, formula

(13.1.14)],[12, formula (1)].

Consider the curve L={(q(p), p)},p∈(−1,1) in the (q, p)-plane given by the ﬁrst

equation in (1). The point (q∗, p∗) on Lwith q∗=q(p∗), dq/dp(p∗) = 0 is called the

spinodal. Spinodal points exist when

bβ > 1,(2)

in which case the value of p∗is given by

p∗=±r1−1

bβ ,(3)

see Figure 2. In what follows, without loss of generality we choose the plus sign and

put q∗=q(p∗). The explicit expression of q∗in terms of band βis

q∗=−bsbβ −1

bβ +1

βarctanhsbβ −1

bβ .

Note that q∗<0. Since dq/dp = 1/(dp/dq), spinodal points are where dp/dq diverges,

and they are physically interpreted as the points where a response of p(magnetization)

due to a change of q(exterior magnetic ﬁeld) diverges.

In the present paper we study nonequilibrium thermodynamics of the mean ﬁeld

Ising model in a small neighbourhood of the spinodal point. We deal with relaxation

Figure 1: Lagrangian projection of Λ to the (q, p)-plane.

processes where the magnetic ﬁeld qis constant, while the magnetization p(t) converges

to a limit p∞as t→ ∞. For q6=q∗, we shall prove existence of the limit

−τ−1:= lim

t→+∞

ln |p(t)−p∞|

t.(4)

When τ > 0, we have an exponential convergence of p(t) to its equilibrium value. In

this case τis called the relaxation time. We shall focus on τas a function of the

magnetic ﬁeld in two models of nonequilibrium thermodynamics.

The ﬁrst model, a classical one, is Glauber dynamics. In [5] Glauber described a

Markov process which models relaxation of the Ising chain to the equilibrium. Fur-

thermore, Glauber [5] and later Suzuki and Kubo [14] proposed an ordinary diﬀerential

equation

˙p=−p+φ0(q+bp),(5)

where qis constant, see equation (4.3) in [14], which provides a molecular ﬁeld approx-

imation to the Markov evolution in the thermodynamic limit. One of the features of

this approximation is as follows. Consider the regime when bβ > 1, cf. (2). In this

regime, for qfrom the interval (q∗,−q∗) equation (5) has three equilibrium points: the

maximal and the minimal equilibria are stable, and the one in the middle is unstable.

The stable equilibrium having bigger free energy (i.e., the smaller value of z) corre-

sponds to the metastable equilibrium of the Markov process. Metastability, roughly

speaking, means that in the thermodynamic limit, i.e., as the size of the chain increases,

the chain spends larger and larger time in the metastable region, see Theorem 13.1 in

[2] for a precise formulation. Metastable equilibria are represented by a solid line on

Figure 2.

Convention: In this paper, by Glauber dynamics we mean the dynamics of ODE

(5). By the metastable equilibrium of (5) we mean the dynamically stable equilibrium

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ContactgeometricapproachtoGlauberdynamicsnearacuspanditslimitationShin-itiroGoto*,ShaiLerer,LeonidPolterovichMarch8,2023AbstractWestudyanonequilibriummeaneldIsingmodelinthelowtemperaturephaseregime,wheremetastableequilibriumstatesdevelopacuspidal(spinodal)singularity.WefocusoncelebratedGlauberdyn...

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Contact geometric approach to Glauber dynamics near a cusp and its limitation Shin-itiro Goto Shai Lerer Leonid Polterovich

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