Metastability and transfer-matrix finite-range scaling

Rikvold, Per Arne; Gorman, B. M.; Novotny, M. A.

doi:10.1063/1.42386

Cited by 4 publications

(9 citation statements)

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“…Recently one of us [30,31] introduced a transfer-matrix method, based on the concept of "constrained" joint probability densities, to obtain analogues of the free-energy density for constrained states. Preliminary applications [30][31][32] to the Q1DI model have shown qualitative agreement between the behavior of the free-energy-density analogue associated with the metastable eigenvalue branch of the transfer matrix and the analytically continued free-energy density. In this study we use this constrained-transfer-matrix (CTM) method with finite-range scaling [34] to obtain more quantitative results for the scaling of the imaginary part of the metastable "constrained" free-energy density.…”

Section: The Constrained-transfer-matrix Methodsmentioning

confidence: 85%

“…In a similar study [29] of an Ising model with N×∞ cylindrical geometry in which the interaction range is linear in the cross section, evidence from the transfer-matrix eigenvalue spectrum was found for the emergence of a classical spinodal in the limit of weak, longrange interactions. In the present work we apply a recently developed "constrained-transfermatrix" (CTM) method [30][31][32][33] to study the properties of the metastable phase of this quasione-dimensional Ising (Q1DI) model. Since the transfer matrix for the N×∞ Q1DI model has a rank of N+1 [29], instead of the typical value of 2 N for short-range Ising systems, it is relatively easy to study large systems by transfer-matrix techniques.…”

Section: Introductionmentioning

confidence: 99%

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Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

1994

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We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N ×∞ quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuationf of the free energy across the first-order transition, assuming that the system escapes the metastable state by nucleation of noninteracting droplets. We find that corrections to the field-dependence are substantial, and by solving the EulerLagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfermatrix method we associate with subdominant eigenvectors of the transfer matrix a complex-valued "constrained" free-energy density f α computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, f α exhibits finite-range scaling behavior in agreement withf over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.

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Section: The Constrained-transfer-matrix Methodsmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

1994

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“…Recently, complex-valued constrained free energies were numerically obtained for both the two-dimensional nearest-neighbor Ising ferromagnet [11,12] and for models with weak long-range forces [13][14][15] by a constrained-transfer-matrix method introduced by one of us [16]. Although no dynamical aspects were explicitly considered to obtain the constrained free energies, the average free-energy cost of a critical droplet was obtained over a wide range of fields and temperatures, in good agreement with the predictions of field-theoretical droplet models [8][9][10]14,15,17] and Monte Carlo (MC) simulations [1,[4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…Since it is well known that both the equilibrium and the metastable properties of equivalentneighbor models are exactly described by mean-field theory in the thermodynamic limit [2,[13][14][15], these models are often referred to as "mean-field models" [22]. Consistent with this usage, we call the class of dynamics that we define here "macroscopic mean-field dynamics.…”

Section: Introductionmentioning

confidence: 99%

Method to study relaxation of metastable phases: Macroscopic mean-field dynamics

1995

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We propose two different macroscopic dynamics to describe the decay of metastable phases in many-particle systems with local interactions. These dynamics depend on the macroscopic order parameter m through the restricted free energy F (m) and are designed to give the correct equilibrium distribution for m. The connection between macroscopic dynamics and the underlying microscopic dynamic are considered in the context of a projectionoperator formalism. Application to the square-lattice nearest-neighbor Ising ferromagnet gives good agreement with droplet theory and Monte Carlo simulations of the underlying microscopic dynamic. This includes quantitative agreement for the exponential dependence of the lifetime τ on the inverse of the applied field H, and the observation of distinct field regions in which Λ ≡ d ln τ /d|H| 1−d depends differently on |H|. In addition, at very low temperatures we observe oscillatory behavior of Λ with respect to |H|, due to the discreteness of the lattice and in agreement with rigorous results. Similarities and differences between this work and earlier works on finite Ising models in the fixed-magnetization ensemble are discussed.PACS Numbers: 64.60 My, 64.60 Qb, 02.70.Lq, 05.50 +q Typeset using REVT E XRecently, complex-valued constrained free energies were numerically obtained for both the two-dimensional nearest-neighbor Ising ferromagnet [11,12] and for models with weak

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“…A comparison of the CTM formalism applied to a short-range-force model, as presented here, and to two-and three-state models with weak long-range forces, presented elsewhere [45,46,61,62,63], clearly reveals the differences and similarities between these models [64]. The analytic continuation of the equilibrium free energy in long-range-force models has a branch-point singularity at a well-defined non-zero spinodal field [14,15,16,17,26,27,28,45,46,61,62,63], whereas it has an essential singularity on the coexistence line at zero magnetic field in short-range-force models [29,30,33,34,35,36,37,38,39,40,41,42,43,44]. Moreover, whereas long-range-force models exhibit infinitely long-lived metastable phases in the limit of infinite interaction range, short-range-force models display finite, albeit very long lifetimes, even in the thermodynamic limit [22,48,49,50,51,52,53,54,55,56,57,…”

Section: Introductionmentioning

confidence: 99%

Application of a constrained-transfer-matrix method to metastability in the d = 2 Ising ferromagnet

Günther

Rikvold

Novotny

1994

Physica A: Statistical Mechanics and its Applications

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Applying a numerical transfer-matrix formalism, we obtain complex-valued constrained free energies for the two-dimensional square-lattice nearest-neighbor Ising ferromagnet below its critical temperature and in an external magnetic field. In particular, we study the imaginary part of the constrained free-energy branch that corresponds to the metastable phase. Although droplets are not introduced explicitly, the metastable free energy is obtained in excellent agreement with field-theoretical droplet-model predictions. The finite-size scaling properties are different in the weak-field and intermediate-field regimes, and we identify the corresponding different criticaldroplet shapes. For intermediate fields, we show that the surface free energy of the critical droplet is given by a Wulff construction with the equilibrium surface tension. We also find a prefactor exponent in complete agreement with the field-theoretical droplet model. Our results extend the region of validity for known results of this field-theoretical droplet model, and they indicate that this transfer-matrix approach provides a nonperturbative numerical continuation of the equilibrium free energy into the metastable phase. * Submitted to Physica A.A The relationship between |Imf ms | and ξ N at H=0As may be seen from Fig. 3, the metastable phase for |H|

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Metastability and transfer-matrix finite-range scaling

Cited by 4 publications

References 0 publications

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

Method to study relaxation of metastable phases: Macroscopic mean-field dynamics

Application of a constrained-transfer-matrix method to metastability in the d = 2 Ising ferromagnet

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