Chai-Yu Lin

Parallel tempering simulations of HP-36.

We report results from all‐atom Monte Carlo simulations of the 36‐residue villin headpiece subdomain HP‐36. Protein‐solvent interactions are approximated by an implicit solvent model. The parallel tempering is used to overcome the problem of slow convergence in low‐temperature protein simulations. Our results show that this technique allows one to sample native‐like structures of small proteins and points out the need for improved energy functions. Proteins 2003;52:436–445.

Universal scaling functions and quantities in percolation models

We briefly review recent work on universal finite-size scaling functions (UFSSFs) and quantities in percolation models. The topics under discussion include: (a) UFSSFs for the existence probability (also called crossing probability) Ep, the percolation probability P, and the probability Wn of the appearance of n percolating clusters, (b) universal slope for average number of percolating clusters, (c) UFSSFs for a q-state bond-correlated percolation model corresponding to the q-state Potts model. We also briefly mention some very recent related developments and discuss implications of our results.

Universality of critical existence probability for percolation on three-dimensional lattices

Using a histogram Monte Carlo simulation method, we calculate the existence probability for bond percolation on simple cubic (sc) and body-centred cubic (bcc) lattices, and site percolation on sc lattices with free boundary conditions. The spanning rule considered by Reynolds, Stanley, and Klein is used to define percolating clusters. We find that for such systems has very good finite-size scaling behaviour and the value of at the critical point is universal and is about .

Renormalization-group approach to an Abelian sandpile model on planar lattices.

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Polydispersity Effect and Universality of Finite-Size Scaling Function

We derive an equation for the existence probability Ep for general percolation problem using an analytical argument based on exponential-decay behaviour of spatial correlation function. It is shown that the finite-size scaling function is well approximated by the error function. The present argument explain why it is universal. We use Monte Carlo simulation to calculate Ep for polydisperse continuum percolation and find that mono- and polydisperse system have the same finite-size scaling function.

Universality in critical exponents for toppling waves of the BTW sandpile model on two-dimensional lattices

Universality and scaling for systems driven to criticality by a tuning parameter has been well studied. However, there are very few corresponding studies for the models of self-organized criticality, e.g., the Bak, Tang, and Wiesenfeld (BTW) sandpile model. It is well known that every avalanche of the BTW sandpile model may be represented as a sequence of waves and the asymptotic probability distributions of all waves and last waves have critical exponents, 1 and 11/8, respectively. By an inversion symmetry, Hu, Ivashkevich, Lin, and Priezzhev showed that in the BTW sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1 (Phys. Rev. Lett. 85 (2000) 4048). Such predictions have been confirmed by extensive numerical simulations of the BTW sandpile model on square lattices. Very recently, we used Monte Carlo simulations to find that the waves of the BTW model on square, honeycomb, triangular, and random lattices have the same set of critical exponents.

Proteinlike behavior of a spin system near the transition between a ferromagnet and a spin glass.

A simple spin system is studied as an analog for proteins. We investigate how the introduction of randomness and frustration into the system affects the designability and stability of ground-state configurations. We observe that the spin system exhibits proteinlike behavior in the vicinity of the transition between a ferromagnet and a spin glass. Our results illuminate some guiding principles in protein evolution.

Inversion symmetry and exact critical exponents of dissipating waves in the sandpile model.

By an inversion symmetry, we show that in the Abelian sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1. Our extensive numerical simulations not only support these predictions, but also show that inversion symmetry is useful for the analysis of the two-wave probability distributions.

Monte Carlo Approaches to Universal Finite-Size Scaling Functions

The universality of critical exponents in critical phenomena has been well known for long time and it is generally believed that systems within a given universality class have different finite-size scaling functions. In 1984, Privman and Fisher proposed the idea of universal finite-size scaling functions (UFSSF) and nonuniversal metric factors for static critical phenomena. From 1984 to 1994, the progress of research in this direction was very slow. In this paper, we review recent developments relating to universal finite-size scaling functions in static and dynamic critical phenomena. The topics under discussion include: 1. UFSSF of the existence probability E p and the percolation probability P in lattice percolation models, 2. UFSSF of the probability for the appearance of n percolating clusters W n in lattice percolation models, 3. UFSSF of E p and W n in continuum percolation models, and 4. UFSSF in dynamic critical phenomena of the Ising model.