Monte Carlo study of self-avoiding surfaces

Two models of self-avoiding surfaces on the cubic lattice are studied by Monte Carlo simulations. Both the first model with fluctuating boundary and the second one with a fixed boundary are found to belong to the universality class of branched polymers. The algorithms as well as the methods used to extract the critical exponents are described in detail. The results are compared to other recent estimates in the literature.     

Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods

A Monte Carlo simulation was performed for loop-erased self-avoiding walks (LESAW) to ascertain the exponentv for the Z² and Z³ lattices. The estimated values were 2v=1.600±0.006 in two dimensions and 2v=1.232±0.008 in three dimensions, leading to the conjecturev=4/5 for the two-dimensional LESAW. These results add to existing evidence that the loop-erased self-avoiding walks are not in the same universality class as self-avoiding walks.     

The number and size of branched polymers in high dimensions

We consider two models of branched polymers (lattice trees) on thed-dimensional hypercubic lattice: (i)the nearest-neighbor model in sufficiently high dimensions, and (ii) a spread-out or long-range model ford>8, in which trees are constructed from bonds of length less than or equal to a large parameterL. We prove that for either model the critical exponent for the number of branched polymers exists and equals 5/2, and that the critical exponentv for the radius of gyration exists and equals 1/4. This improves our earlier results for the corresponding generating functions. The proof uses the lace expansion, together with an analysis involving fractional derivatives which has been applied previously to the self-avoiding walk in a similar context.     

Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

A Monte Carlo algorithm for lattice ribbons

A lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions. The ribbon can be closed to form an object which is topologically either a cylinder or a Möbius band, depending on whether its surface is orientable or nonorientable. We describe a grand canonical Monte Carlo algorithm for generating a sample of these ribbons, prove that the associated Markov chain is ergodic, and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons.     

New Monte Carlo method for the self-avoiding walk

We introduce a new Monte Carlo algorithm for the self-avoiding walk (SAW), and show that it is particularly efficient in the critical region (long chains). We also introduce new and more efficient statistical techniques. We employ these methods to extract numerical estimates for the critical parameters of the SAW on the square lattice. We find=2.63820 ± 0.00004 ± 0.00030=1.352 ± 0.006 ± 0.025v=0.7590 ± 0.0062 ± 0.0042 where the first error bar represents systematic error due to corrections to scaling (subjective 95% confidence limits) and the second bar represents statistical error (classical 95% confidence limits). These results are based on SAWs of average length 166, using 340 hours CPU time on a CDC Cyber 170–730. We compare our results to previous work and indicate some directions for future research.     

A new proposal for Monte Carlo simulation of fermions on a lattice

With F flavours of matter fields extrapolation to negative F effectively changes bosons into fermions and vice-versa. Extrapolatingresults of calculations performed with bosonic quarks we reproduce results obtained by evaluating the fermion determinant in some simple examples. We find an example of a long computer run apparently converging to the wrong answer when systematic lattice site access is used.     

Nonergodicity of local,length-conserving Monte Carlo algorithms for the self-avoiding walk

It is proved that every dynamic Monte Carlo algorithm for the self-avoiding walk based on a finite repertoire of local,N-conserving elementary moves is nonergodic (hereN is the number of bonds in the walk). Indeed, for largeN, each ergodic class forms an exponentially small fraction of the whole space. This invalidates (at least in principle) the use of the Verdier-Stockmayer algorithm and its generalizations for high-precision Monte Carlo studies of the self-avoiding walk.     

Monte carlo study of the interacting self-avoiding walk model in three dimensions

We consider self-avoiding walks on the simple cubic lattice in which neighboring pairs of vertices of the walk (not connected by an edge) have an associated pair-wise additive energy. If the associated force is attractive, then the walk can collapse from a coil to a compact ball. We describe two Monte Carlo algorithms which we used to investigate this collapse process, and the properties of the walk as a function of the energy or temperature. We report results about the thermodynamic and configurational properties of the walks and estimate the location of the collapse transition.     

Relaxation Monte Carlo for 3D branched polymers: The leading confluence exponent

A relaxation study of simple-cubic site trees taking 18 months of CDC Cyber computer time is used to estimate the radius-of-gyration dependence on the size, both as regards the correlation exponent and the confluent correction. The latter is found to be compatible with an analytic term within the uncertainty limits. The radius prefactor is estimated as 0.542±0.005 and the limiting site perimeter-to-size ratio as 2.905±0.005.     

Monte Carlo study of tricritical dynamics in two dimensions

The dynamical tricritical behavior for the spin-1 Ising model with single-ion interaction is investigated in two dimensions using Monte Carlo simulations. The nonlinear dynamical tricritical exponentz _t is determined from the asymptotic power-law relaxation of the magnetization. The valuez _t = 1.99 ± 0.04 reported here is the first estimate of the dynamical exponent at a multicritical point, in two dimensions.On leave from Departamento de Fisica, Universidade Federal de Pernambuco, Recife-PE, Brazil.     

On the upper critical dimension of lattice trees and lattice animals

We give a rigorous proof of mean-field critical behavior for the susceptibility (=1/2) and the correlation length (v=1/4) for models of lattice trees and lattice animals in two cases: (i) for the usual model with trees or animals constructed from nearest-neighbor bonds, in sufficiently high dimensions, and (ii) for a class of spread-out or long-range models in which trees and animals are constructed from bonds of various lengths, above eight dimensions. This provides further evidence that for these models the upper critical dimension is equal to eight. The proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point, and uses an expansion related to the lace expansion for the self-avoiding walk.     

Monte Carlo study of phase diagram for correlated site-bond percolation in two dimensions

At the critical point of the square Ising model, the percolation threshold for randomly active bonds between up spins is close top _Bc =0.60 and seems compatible with the predictionp _Bc =1-exp(–2J/k _B T _c )=0.586 of Coniglio and Klein. Longer simulations on larger lattices are necessary for a more precise clarification.     

Exact and Monte Carlo computations on a lattice model for change of conformation of a polymer

Conformational changes of linear polymers are studied by means of dynamic lattice models. The relaxation rates for the following four parameters describing the conformation of the polymer are studied for various polymer lengths: the square of the end-to-end distance, the square of the radius of gyration, thex component of the end-to-end vector, and the number of windings.In the most realistic models the relaxation rates for the first three of the above-mentioned properties decrease approximately proportional to the square of the number of monomers in agreement with the well-known Rouse model, while the relaxation of the winding number appears to be independent of the polymer length. The long-range interactions due to excluded volume restrictions are found to be of only minor importance compared to the rules presented for the local movements of the polymer segments.The results are obtained by diagonalizing the Markov matrix forn = 3, 4, 5, and 6 and by Monte Carlo simulation forn = 8, 16, 32, 64, and 128, wheren is the number of monomers.     

A new Monte Carlo code for absorption simulation of laser-skin tissue interaction

In laser clinical applications, the process of photon absorption and thermal energy diffusion in the target tissue and its surrounding tissue during laser irradiation are crucial. Such information allows the selection of proper operating parameters such as laser power, and exposure time for optimal therapeutic. The Monte Carlo method is a useful tool for studying laser-tissue interaction and simulation of energy absorption in tissue during laser irradiation. We use the principles of this technique and write a new code with MATLAB 6.5, and then validate it against Monte Carlo multi layer (MCML) code. The new code is proved to be with good accuracy. It can be used to calculate the total power bsorbed in the region of interest. This can be combined for heat modelling with other computerized programs.     

The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk

The pivot algorithm is a dynamic Monte Carlo algorithm, first invented by Lal, which generates self-avoiding walks (SAWs) in a canonical (fixed-N) ensemble with free endpoints (hereN is the number of steps in the walk). We find that the pivot algorithm is extraordinarily efficient: one effectively independent sample can be produced in a computer time of orderN. This paper is a comprehensive study of the pivot algorithm, including: a heuristic and numerical analysis of the acceptance fraction and autocorrelation time; an exact analysis of the pivot algorithm for ordinary random walk; a discussion of data structures and computational complexity; a rigorous proof of ergodicity; and numerical results on self-avoiding walks in two and three dimensions. Our estimates for critical exponents are=0.7496±0.0007 ind=2 and= 0.592±0.003 ind=3 (95% confidence limits), based on SAWs of lengths 200N10000 and 200N 3000, respectively.     

Monte Carlo study of the antiferromagnetical Ising model on a centred honeycomb lattice

We use the Monte Carlo method to study an antiferromagnetical Ising spin system on a centred honeycomb lattice, which is composed of two kinds of 1/2 spin particles A and B. There exist two different bond energies J_A-A and J_A-B in this lattice. Our study is focused on how the ratio of J_A-B to J_A-A influences the critical behaviour of this system by analysing the physical quantities, such as the energy, the order parameter, the specific heat, susceptibility, {etc} each as a function of temperature for a given ratio of J_A-B to J_A-A. Using these results together with the finite-size scaling method, we obtain a phase diagram for the ratio J_A-B / J_A-A. This work is helpful for studying the phase transition problem of crystals composed of compounds.     

A new multispin coding algorithm for Monte Carlo simulation of the Ising model

We present a new algorithm for Monte Carlo simulation of the Ising model. The usual serial architecture of a computer is exploited in a novel way, enabling parallel but independent calculations to be carried out on as many spins as there are bits in a computer word in each fundamental move. The algorithm enjoys a substantial increase in execution speed over more usual multispin coding algorithms. By its very nature, the algorithm constitutes a design for a special-purpose processor.