Biomedical subjects

Youjin Deng

Publications and source records attributed to Youjin Deng.

At least 19 recordsLinked to original sources

PubMed2006

Finite-size scaling of energylike quantities in percolation.

We study the bond-percolation model in two and three dimensions by Monte Carlo simulation, and investigate the finite-size scaling behavior of several quantities that account for fluctuations of the total numbers of clusters and occupied bonds, Nc and Nb, respectively. These quantities include C(2c) = ( - 2)/Ld and C(cb) = ( - )/Ld, where L is the linear system size and d is the spatial dimensionality. In statistical models with thermal fluctuations, C(2c) and C(cb) are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of C(2c) and C(cb) is described by the thermal critical exponent y(t)-d. We also measure quantity kappa b = 2 / - / - and an analogous quantity kappa c for Nc, where S2 and S4 are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that kappa b and kappa c diverge as L(y)t for L --> infinity. The analysis of the data of kappa b and kappa c yields y(t) = 1.145(2) for the three-dimensional percolation, in good agreement with existing results.

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PubMed2006

Bulk and surface phase transitions in the three-dimensional O4 spin model.

We investigate the O(4) spin model on the simple-cubic lattice by means of the Wolff cluster algorithm. Using the toroidal boundary condition, we locate the bulk critical point at coupling K(c) = 0.935 856(2), and determine the bulk thermal magnetic renormalization exponents as y(t) = 1.337 5(15) and y(h) = 2.482 0(2), respectively. The universal ratio Q=m(2)(2)/m(4) is also determined as 0.9142(1). The precision of these estimates significantly improves over that of the existing results. Then, we simulate the critical O(4) model with two open surfaces on which the coupling strength K(1) can be varied. At the ordinary transitions, the surface magnetic exponent is determined as y((o))(h1) = 1.020 2(12). Further, we find a so-called special surface transition at (k) = K(1)/K-1 = 1.258(20). At this point, the surface thermal exponent y(s)(t1) is rather close to zero, and we cannot exclude that the corresponding surface transition is Kosterlitz-Thouless-like. The surface magnetic exponent is y((s))/h1 = 1.816(2).

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PubMed2005

Dilute Potts model in two dimensions.

We study the two-dimensional dilute q-state Potts model by means of transfer-matrix and Monte Carlo methods. Using the random-cluster representation, we include noninteger values of q. We locate phase transitions in the three-dimensional parameter space of q, the Potts coupling K>>0, and the chemical potential of the vacancies. The critical plane is found to contain a line of fixed points that divides into a critical branch and a tricritical one, just as predicted by the renormalization scenario formulated by Nienhuis et al for the dilute Potts model. The universal properties along the line of fixed points agree with the theoretical predictions. We also determine the density of the vacancies along these branches. For q=2-squareroot of 2 we obtain the phase diagram in a three-dimensional parameter space that also includes a coupling V> or = 0 between the vacancies. For q=2, the latter space contains the Blume-Capel model as a special case. We include a determination of the tricritical point of this model, as well as an analysis of percolation clusters constructed on tricritical Potts configurations for noninteger q. This percolation study is based on Monte Carlo algorithms that include local updates flipping between Potts sites and vacancies. The bond updates are performed locally for and by means of a cluster algorithm for q>1. The updates for q>1 use a number of operations per site independent of the system size.

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PubMed2005

Surface and bulk transitions in three-dimensional O(n) models.

Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the O(n) models on the simple-cubic lattice with n=1 , 2, and 3, i.e., the Ising, XY , and Heisenberg models. For the critical couplings we find Kc (n=2) =0.454 1659 (10) and Kc (n=3) =0.693 003 (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be y (o)(h1) =0.7374 (15) , 0.781 (2), and 0.813 (2) for n=1 , 2, and 3, respectively. Then we vary the surface coupling K1 and locate the so-called special transition at kappa(c) (n=1) =0.502 14 (8) and kappa(c) (n=2) =0.6222 (3), where kappa= K1 /K-1 . The corresponding surface thermal and magnetic exponents are y (s)(t1) =0.715 (1) and y (s)(h1) =1.636 (1) for the Ising model, and y(s)(t1) =0.608 (4) and y(s)(h1) =1.675 (1) for the XY model. Finite-size corrections with an exponent close to -1/2 occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.

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PubMed2005

Monte Carlo study of the site-percolation model in two and three dimensions.

We investigate the site-percolation problem on the square and simple-cubic lattices by means of a Monte Carlo algorithm that in fact simulates systems with size L(d-1) x infinity, where L specifies the linear system size. This algorithm can be regarded either as an extension of the Hoshen-Kopelman method or as a special case of the transfer-matrix Monte Carlo technique. Various quantities, such as the magnetic correlation function, are sampled in the finite directions of the above geometry. Simulations are arranged such that both bulk and surface quantities can be sampled. On the square lattice, we locate the percolation threshold at p(c) =0.592 746 5 (4) , and determine two universal quantities as Q(gbc) =0.930 34 (1) and Q(gsc) =0.793 72 (3) , which are associated with bulk and surface correlations, respectively. These values agree well with the exact values 2(-5/48) and 2(-1/3) , respectively, which follow from conformal invariance. On the simple-cubic lattice, we locate the percolation threshold at p(c) =0.311 607 7 (4) . We further determine the bulk thermal and magnetic exponents as y(t) =1.1437 (6) and y(h) =2.5219 (2) , respectively, and the surface magnetic exponent at the ordinary phase transition as y (o)(hs) =1.0248 (3) .

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PubMed2005

Percolation between vacancies in the two-dimensional Blume-Capel model.

Using suitable Monte Carlo methods and finite-size scaling, we investigate the Blume-Capel model on the square lattice. We construct percolation clusters by placing nearest-neighbor bonds between vacancies with a variable bond probability p(b) . At the tricritical point, we locate the percolation threshold of these vacancy clusters at p(bc) =0.706 33 (6) . At this point, we determine the fractal dimension of the vacancy clusters as Xf =0.1308 (5) approximately equal to 21/160, and the exponent governing the renormalization flow in the p(b) direction as y(p) =0.426 (2) approximately equal to 17/40 . For bond probability p(b) > p(bc) , the vacancy clusters maintain strong critical correlations; the fractal dimension is Xf =0.0750 (2) approximately equal to 3/40 and the leading correction exponent is y(p) =-0.45 (2) approximately equal to -19/40 . The above values fit well in the Kac table for the tricritical Ising model. These vacancy clusters have much analogy with those consisting of Ising spins of the same sign, although the associated quantities rho and magnetization m are energylike and magnetic quantities, respectively. However, along the critical line of the Blume-Capel model, the vacancies are more or less uniformly distributed over the whole lattice. In this case, no critical percolation correlations are observed in the vacancy clusters, at least in the physical region p(b) < or = 1 .

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PubMed2005

Constrained tricritical phenomena in two dimensions.

We investigate several tricritical models on the square lattice by means of Monte Carlo simulations. These include the Blume-Capel model, Baxter's hard-square model, and the q=1 , 3, and 4 Potts models with vacancies. We use a combination of the Wolff and geometric cluster methods, which conserves the total number of vacancies or lattice-gas particles and suppresses critical slowing down. Several quantities are sampled, such as the specific heat C and the structure factor C(s) , which accounts for the large-scale spatial inhomogeneity of the energy fluctuations. We find that the constraint strongly modifies some of the critical singularities. For instance, the specific heat C reaches a finite value at tricriticality, while C(s) remains divergent as in the unconstrained system. We are able to explain these observed constrained phenomena on the basis of the Fisher renormalization mechanism generalized to include a subleading relevant thermal scaling field. In this context, we find that, under the constraint, the leading thermal exponent y(t1) is renormalized to 2- y(t1) , while the subleading exponent y(t2) remains unchanged.

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PubMed2005

Edge phase transitions of the tricritical Potts model in two dimensions.

Using Monte Carlo techniques and finite-size analysis, we investigate several two-dimensional lattice models with open edges, including the Blume-Capel model and the q=1 and 3 Potts models with vacancies. At bulk tricriticality, we find that the open edges are dominated by the vacancies when the surface coupling K(s) and the chemical potential D(s) of the vacancies assume the bulk values. When K(s) and/or D(s) is sufficiently enhanced, an edge phase transition takes place, beyond which spontaneous one-dimensional order occurs on the edges. Edge phase transitions can also be induced by a surface magnetic field H(s) . We numerically determine a number of edge critical exponents and derive phase diagrams in terms of K(s) , D(s) , and H(s) . In the low-temperature region, we observe first-order transitions when K(s) and D(s) are varied; the associated hysteresis loops of surface quantities are remarkably asymmetric. Some further insight into these edge transitions is provided by the exact equivalence of the tricritical q=1 Potts model and the Ising model.

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PubMed2005

Simulation algorithms for the random-cluster model.

We compare the performance of Monte Carlo algorithms for the simulation of the random-cluster representation of the q-state Potts model for continuous values of q. In particular we consider a local bond update method, a statistical reweighting method of percolation configurations, and a cluster algorithm, all of which generate Boltzmann statistics. The dynamic exponent z of the cluster algorithm appears to be quite small, and to assume the values of the Swendsen-Wang algorithm for q = 2 and 3. The cluster algorithm appears to be much more efficient than our versions of the other two methods for the simulation of the random-cluster model. The higher efficiency of the cluster method with respect to the local method is primarily due to the fact that the computer time usage of the local method increases more rapidly with system size; the difference between the dynamic exponents is less important.

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PubMed2005

Surface critical phenomena in three-dimensional percolation.

Using Monte Carlo methods and finite-size scaling, we investigate surface critical phenomena in the bond-percolation model on the simple-cubic lattice with two open surfaces in one direction. We decompose the whole lattice into percolation clusters and sample the surface and bulk dimensionless ratios Q1 and Qb, defined on the basis of the moments of the cluster-size distribution. These ratios are used to determine critical points. At the bulk percolation threshold pbc, we determine the surface bond-occupation probability at the special transition as p(s)1c = 0.418 17(2), and further obtain the corresponding surface thermal and magnetic exponents as y(s)t1 = 0.5387(2) and y(s)h1 = 1.8014(6), respectively. Next, from the pair correlation function on the surfaces, we determine y(o)h1 = 1.0246(4) and y(e)h1 = 1.25(6) for the ordinary and the extraordinary transition, respectively, of which the former is consistent with the existing result y(o)h1 = 1.024(4). We also numerically derive the line of surface phase transitions occurring at pb < pbc, and determine the pertinent asymptotic values of the universal ratios Q1 and Qb.

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PubMed2004

Red-bond exponents of the critical and the tricritical Ising model in three dimensions.

Using the Wolff and geometric cluster algorithms and finite-size scaling analysis, we investigate the critical Ising and the tricritical Blume-Capel models with nearest-neighbor interactions on the simple-cubic lattice. The sampling procedure involves the decomposition of the Ising configuration into geometric clusters, each of which consists of a set of nearest-neighboring spins of the same sign connected with bond probability p. These clusters include the well-known Kasteleyn-Fortuin clusters as a special case for p=1-exp(-2K) , where K is the Ising spin-spin coupling. Along the critical line K=Kc , the size distribution of geometric clusters is investigated as a function of p . We observe that, unlike in the case of two-dimensional tricriticality, the percolation threshold in both models lies at pc =1-exp(-2Kc) . Further, we determine the corresponding red-bond exponents as yr =0.757(2) and 0.501(5) for the critical Ising and the tricritical Blume-Capel models, respectively. On this basis, we conjecture yr =1/2 for the latter model.

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PubMed2004

Constrained tricritical Blume-Capel model in three dimensions.

Using the Wolff and geometric cluster Monte Carlo methods, we investigate the tricritical Blume-Capel model in three dimensions. Since these simulations conserve the number of vacancies and thus effectively introduce a constraint, we generalize the Fisher renormalization for constrained critical behavior to tricritical systems. We observe that, indeed, the tricritical behavior is significantly modified under this constraint. For instance, at tricriticality, the specific heat has only a finite cusp and the Binder ratio assumes a different value from that in unconstrained systems. Since 3 is the upper tricritical dimensionality of Ising systems, we expect that the mean-field theory correctly predicts a number of universal parameters in three dimensions. Therefore, we calculate the partition sum of the mean-field tricritical Blume-Capel model, and accordingly obtain the exact value of the Binder ratio. Under the constraint, we show that this mean-field tricritical system reduces to the mean-field critical Ising model. However, our three-dimensional data do not agree with this mean-field prediction. Instead, they are successfully explained by the generalized Fisher renormalization mechanism.

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PubMed2004

Magnetic and backbone exponents of the percolation and Ising models in three dimensions.

We investigate random-cluster representations of the q=1 - and 2-state Potts models in three dimensions, i.e., the bond-percolation and the Ising model, respectively. Using a recently developed sampling technique, we determine the probabilities C1 (r) and C2 (r) that a pair of lattice sites at a distance r are connected by at least one and two mutually independent paths, respectively. The scaling behavior of C1 and C2 at criticality is governed by the magnetic and the backbone scaling dimension X(h) and X(b) , respectively. From a finite-size analysis of the numerical data, we determine X(h) =0.4768 (7) and X(b) =1.125 (3) for the percolation and X(h) =0.5178 (7) and X(b) =0.829 (4) for the Ising model.

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PubMed2004

Spontaneous edge order and geometric aspects of two-dimensional Potts models.

Using suitable Monte Carlo methods and finite-size scaling, we investigate critical and tricritical surface phenomena of two-dimensional Potts models. For the critical two- and three-state models, we determine a surface scaling dimension describing percolation properties of the so-called Potts clusters near the edges. On this basis, we propose an exact expression describing this exponent for the whole critical branch. For tricritical Potts models we find that varying the surface coupling constant or the surface magnetic field can induce a continuous phase transition. At bulk tricriticality and sufficiently strong surface couplings, spontaneous one-dimensional order occurs on the edges. We determine several critical exponents describing these edge transitions. On the basis of these results and conformal field theory, we conjecture exact expressions for these exponents.

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PubMed2004

Anisotropic limit of the bond-percolation model and conformal invariance in curved geometries.

We investigate the anisotropic limit of the bond-percolation model in d dimensions, which is equivalent to a (d-1) -dimensional quantum q-->1 Potts model. We formulate an efficient Monte Carlo method for this model. Its application shows that the anisotropic model fits well with the percolation universality class in d dimensions. For three-dimensional rectangular geometry, we determine the critical point as t(c) =8.6429(4), and determine the length ratio as alpha(0) =1.5844(3), which relates the anisotropic limit of the percolation model and its isotropic version. On this basis, we simulate critical systems in several curved geometries including a spheroid and a spherocylinder. Using finite-size scaling and the assumption of conformal invariance, we determine the bulk and surface magnetic exponents in two and three dimensions. They are in good agreement with the existing results.

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PubMed2004

Geometric properties of two-dimensional critical and tricritical Potts models.

We investigate geometric properties of the general q-state Potts model in two dimensions, and define geometric clusters as sets of lattice sites in the same Potts state, connected by nearest-neighbor bonds with variable probability p. We find that, besides the random-cluster fixed point, both the critical and the tricritical Potts models have another fixed point in the p direction. For the critical model, the random-cluster fixed point p(r) is unstable and the other point p(g) > or =p(r) is stable; while p(r) is stable and p(g) < or =p(r) is unstable at tricriticality. Moreover, we show that the fixed point p(g) of a critical and tricritical q-state Potts models can be regarded to correspond to p(r) of a tricritical and critical q'-state Potts models, respectively. In terms of the coupling constant of the Coulomb gas g, these two models are related as gg'=16. By means of Monte Carlo simulations, we obtain p(g)=0.6227(2) and 0.6395(2) for the tricritical Blume-Capel and the q=3 Potts model, respectively, and confirm the predicted values of the magnetic and bond-dilution exponents near p(g).

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PubMed2004

Backbone exponents of the two-dimensional q-state Potts model: a Monte Carlo investigation.

We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formulate several efficient Monte Carlo methods and sample the probability P2 of a pair of points connected via at least two independent paths. Finite-size-scaling analysis of P2 yields X(b) as 0.3566(2), 0.2696(3), 0.2105(3), and 0.127(4) for the critical q=2-sqrt[3], 1,2, 3, and 4 state Potts model, respectively. At tricriticality, we obtain X(b)=0.0520(3) and 0.0753(6) for the q=1 and 2 Potts model, respectively. For the critical q-->0 Potts model it is derived that X(b)=3/4. From a scaling argument, we find that, at tricriticality, X(b) reduces to the magnetic exponent, as confirmed by the numerical results.

Algorithms↗

PubMed2003

Simultaneous analysis of several models in the three-dimensional Ising universality class.

We investigate several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling. These models include spin-1 / 2 models with nearest-neighbor interactions on the simple-cubic and on the diamond lattice. For the simple cubic lattice, we also include models with third-neighbor interactions of varying strength, and some "equivalent-neighbor" models. Also included are a spin-1 model and a hard-core lattice gas. Separate analyses of the numerical data confirm the Ising-like critical behavior of these systems. On this basis, we analyze all these data simultaneously such that the universal parameters occur only once. This leads to an improved accuracy. The thermal, magnetic, and irrelevant exponents are determined as y(t)=1.5868(3), y(h)=2.4816(1), and y(i)=-0.821(5), respectively. The Binder ratio is estimated as Q= (2)/ =0.62 341(4).

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