Finite-size scaling of the three-state Potts model on a simple cubic lattice

Fukugita, M.; Mino, H.; Okawa, M.; Ukawa, A.

doi:10.1007/bf01334757

Cited by 41 publications

(39 citation statements)

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“…It is well-known that renormalization-group fixed point picture leads to a critical exponent dν = 1, α = 1 and γ = 1 for a first-order phase transition [19][20][21]. Our estimate dν = 0.93(5) for the correlation exponent deviates from unity and rather indicates that the 'helix-coiltransition" is a strong second order transition.…”

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confidence: 57%

Partition Function Zeros and Finite Size Scaling of Helix-Coil Transitions in a Polypeptide

Alves

Hansmann

2000

Phys. Rev. Lett.

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We report on multicanonical simulations of the helix-coil transition of a polypeptide. The nature of this transition was studied by calculating partition function zeros and the finite-size scaling of various quantities. New estimates for critical exponents are presented.87.15. He, 64.70Cn, 02.50.Ng A common, ordered structure in proteins is the α-helix and it is conjectured that formation of α-helices is a key factor in the early stages of protein-folding [1]. It is long known that α-helices undergo a sharp transition towards a random coil state when the temperature is increased. The characteristics of this so-called helix-coil transition have been studied extensively [2], most recently in Refs. [3,4]. They are usually described in the framework of Zimm-Bragg-type theories [5] in which the homopolymers are approximated by a one-dimensional Ising model with the residues as "spins" taking values "helix" or "coil", and solely local interactions. Hence, in such theories thermodynamic phase transitions are not possible. However, in preliminary work [4] it was shown that our all-atom model of poly-alanine exhibits a phase transition between the ordered helical state and the disordered random-coil state. It was conjectured that this transition is due to long range interactions in our model and the fact that it is not one-dimensional: it is known that the 1D Ising model with long-range interactions also exhibits a phase transition at finite T if the interactions decay like 1/r σ with 1 ≤ σ < 2 [6]. Our aim now is to investigate this transition in the frame work of a critical theory by means of the finite size scaling (FSS) analysis of partition function zeros. Analysis of partition function zeros is a well-known tool in the study of phase transitions, but was to our knowledge never used before to study biopolymers.For our project, the use of the multicanonical algorithm [7] was crucial. The various competing interactions within the polymer lead to an energy landscape characterized by a multitude of local minima. Hence, in the low-temperature region, canonical simulations will tend to get trapped in one of these minima and the simulation will not thermalize within the available CPU time. One standard way to overcome this problem is the application of the multicanonical algorithm [7] and other generalized-ensemble techniques [8] to the protein folding problem [9]. For poly-alanine, both the failure of standard Monte Carlo techniques and the superior performance of the multicanonical algorithm are extensively documented in earlier work [10].In the multicanonical algorithm [7] conformations with energy E are assigned a weight w mu (E) ∝ 1/n(E). Here, n(E) is the density of states. A simulation with this weight will lead to a uniform distribution of energy:This is because the simulation generates a 1D random walk in the energy, allowing itself to escape from any local minimum. Since a large range of energies are sampled, one can use the reweighting techniques [11] to calculate thermodynamic quantities over a wide range of...

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confidence: 57%

Partition Function Zeros and Finite Size Scaling of Helix-Coil Transitions in a Polypeptide

Alves

Hansmann

2000

Phys. Rev. Lett.

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“…[16] . The analysis should be insensitive to the actual fraction, b, of C MAX (χ MAX ) considered for defining T 1 and T 2 which was verified from our numerical data fitting of poly-alanine chains.The scaling exponents, α, ν, and γ, have their usual meaning in critical phenomena; however, the above scaling relations also hold formally for the case of a first-order transition, with effective scaling exponents dν = α = γ = 1 [16,17]. Note that d is the dimensionality of the system, and it always appears in the combination dν.…”

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confidence: 99%

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confidence: 99%

Is there a universality of the helix-coil transition in protein models?

2000

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The similarity in the thermodynamic properties of two completely different theoretical models for the helix-coil transition is examined critically. The first model is an all-atomic representation for a poly-alanine chain, while the second model is a minimal helix-forming model that contains no system specifics. Key characteristics of the helix-coil transition, in particular, the effective critical exponents of these two models agree with each other, within a finite-size scaling analysis.Pacs: 87.15. He, 64.70Cn, 02.50.Ng The importance of understanding the statistical physics of the protein-folding problem has been stressed recently [1, 2]. For instance, it is now often assumed that the energy landscape of a protein resembles a partially rough funnel. Folding occurs by a multi-pathway kinetics and the particulars of the folding funnel determine the transitions between the different thermodynamic states [1, 3]. This "new view" [1] of folding was derived from studies of minimal protein models which capture only a few, but probably dominant parameters (chain connectivity, excluded volume, etc.) in real proteins.An implicit yet fundamentally crucial assumption is that the basic mechanism of structural transitions in biological molecules depends solely on gross features of the energy function, not on their details, and that a law of corresponding states can be used to explain dynamics and structural properties of real proteins from studies of related minimal models. This assumption needs to be proven. An even stronger notion in statistical physics is the universality hypothesis for critical phenomena. The 1 critical exponents are identical for different theoretical models and realistic systems belonging to the same universality class. Many theoretical concepts in protein folding, such as coil-helix or coil-globular transitions involve phase transition or phase transition-like behavior. Thus, one wonders if physical measurements between two model systems for the same transition would have any "universal" properties.The purpose of this article is to examine these questions for the helix-coil transition in homopolymers of amino acids [4,5]. Traditionaly, the coil-helix transition is described by theories such as the Zimm-Bragg model [6] in which the homopolymers are regarded as one dimensional systems with only local interactions; as such a true thermodynamic phase transition is impossible. However, recently there have been [4,5] indications that the coil-helix transition near the transition temperature displays phase-transition like behavior. We use here finite-size scaling analysis, a common tool in statistical physics, to examine the question of universality of the helix-coil transition in two completely different, illuminating models. On one hand, we have a detailed, all-atomic representation of a homo poly-alanine chain [7]. On the other hand, we have a simple coarse-grained model describing the general features of helix-forming polymers [4]. In this article, our interest lies in finding out how fa...

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“…These turn out to be extremely close to the theoretical values 2Kc given in Eq. (11) almost independently of the lattice size l, for 2;;;;,q;;;;, 6. These values are listed in the row labelled by 2Kc(CL) in Table I.…”

Section: H>mentioning

confidence: 99%

“…These values 2Kc for 2;;;;, q;;;;, 7 are listed in the row labelled by 2Kc(SE) in Table I, and are compared to the theoretical values in Eq. (11). Figure 1 indicates the manner in which the pseudocritical coupling parameters 2Ko(l) from the specific heat anomaly approach the bulk values as the lattice size l increases.…”

Section: H>mentioning

confidence: 99%

Finite Size Effects and Pseudocritical Behavior. II: Potts Spin Model

Ohya

Katsumori

1996

Progress of Theoretical Physics

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851Using the transfer matrices constructed by Kramers and Wannier's successive screw method for the 2-d (! x oo) and 3-d (l x m x oo) q state Potts model (2;?. q ;;;> 7), we examine the extent to which the presently available supercomputers can provide us with finite size effects and pseudocritical properties of the systems.We estimate the bulk coupling parameters Kc from the specific heat anomaly in two and three dimensions and also from the correlation length anomaly in two dimensions. In two dimensions, our results Kc are in good agreement with those given by the Potts duality formula. In three dimensions, however, we need extended precision computations to obtain the numerical results with sufficient accuracy.We regard the behavior of the exponent f.l., which characterizes the relation between the specific heat peak Cvo(l) and the lattice size l, as an indicator to discriminate the order of the phase transition by investigating Cvo(l) for several small values of l. § 1. IntroductionIn a previous work,1l referred to as I hereafter, we numerically investigated pseudocritical properties of the semi-infinite Ising spin and lattice gauge models in two and three dimensions using the transfer matrices constructed by Kramers and Wannier's successive screw method.2 lIn the present paper, the successive screw method is applied to the q-state Potts modePHl in two and three dimensions, as a generalization of the Ising model. We would like to examine the extent to which the available supercomputers can provide us with finite size effects and pseudocritical properties of the Potts model by means of the long-established transfer matrix method.A great number of numerical studies on the Potts model have been published. Among others, recent Monte Carlo studies 6 H 5 l have given significant results with a high degree of precision. The precision of our estimates in the present work in general is inferior to that of the latest Monte Carlo results. 13 H 5 l However, we hope that an extended precision computation will considerably improve the present results in the near future_ We show that the specific heat peak exponent f-t can be regarded as an indicator for discriminating the order of the phase transition, even though the lattice size is small.In § 2 the transfer matrices are constructed from the eigenvalue equations given by the successive screw method. In § § 3 and 4 the finite size effects are calculated in two and three dimensions, respectively_ The transition order of the Potts model is

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Finite-size scaling of the three-state Potts model on a simple cubic lattice

Cited by 41 publications

References 30 publications

Partition Function Zeros and Finite Size Scaling of Helix-Coil Transitions in a Polypeptide

Partition Function Zeros and Finite Size Scaling of Helix-Coil Transitions in a Polypeptide

Is there a universality of the helix-coil transition in protein models?

Finite Size Effects and Pseudocritical Behavior. II: Potts Spin Model

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