Limit Of Infinite Length Research Articles

Potts model partition functions for self-dual families of strip graphs

We consider the q-state Potts model on families of self-dual strip graphs G D of the square lattice of width L y and arbitrarily great length L x , with periodic longitudinal boundary conditions. The general partition function Z and the T=0 antiferromagnetic special case P (chromatic polynomial) have the respective forms ∑ j=1 N F, L y , λ c F, L y , j ( λ F, L y , j ) L x , with F= Z, P. For arbitrary L y , we determine (i) the general coefficient c F, L y , j in terms of Chebyshev polynomials, (ii) the number n F ( L y , d) of terms with each type of coefficient, and (iii) the total number of terms N F, L y , λ . We point out interesting connections between the n Z ( L y , d) and Temperley–Lieb algebras, and between the N F, L y , λ and enumerations of directed lattice animals. Exact calculations of P are presented for 2⩽ L y ⩽4. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), W( q). Generalizing q from Z + to C , we determine the continuous locus B in the complex q plane where W( q) is singular. We find the interesting result that for all L y values considered, the maximal point at which B crosses the real q-axis, denoted q c , is the same, and is equal to the value for the infinite square lattice, q c =3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of q c .

Exact T=0 partition functions for Potts antiferromagnets on sections of the simple cubic lattice.

We present exact solutions for the zero-temperature partition function of the q-state Potts antiferromagnet (equivalently, the chromatic polynomial P) on tube sections of the simple cubic lattice of fixed transverse size Lx x L(y) and arbitrarily great length L(z), for sizes Lx x L(y)=2 x 3 and 2 x 4 and boundary conditions (a) (FBC(x),FBC(y),FBC(z)) and (b) (PBC(x),FBC(y),FBC(z)), where FBC (PBC) denote free (periodic) boundary conditions. In the limit of infinite length, L(z)-->infinity, we calculate the resultant ground-state degeneracy per site W (=exponent of the ground-state entropy). Generalizing q from Z+ to C, we determine the analytic structure of W and the related singular locus Beta which is the continuous accumulation set of zeros of the chromatic polynomial. For the L(z)-->infinity limit of a given family of lattice sections, W is analytic for real q down to a value q(c). We determine the values of q(c) for the lattice sections considered and address the question of the value of q(c) for a d-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph K(m,m).

Exact Potts model partition functions on strips of the honeycomb lattice

We present exact calculations of the partition function of the q-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width L y =2 and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and W( q), the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) L y =3, cyclic, (v) L y =3, Möbius, (vi) L y =4, cylindrical, and (vii) L y =4, open. In the infinite-length limit we calculate W( q) and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the L y =4 strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in 10 5 for moderate q values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).

Exact Potts model partition functions on wider arbitrary-length strips of the square lattice

We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L y =3 vertices and arbitrary length L x with periodic longitudinal boundary conditions, of the following types: (i) (FBC y, PBC x) = cyclic, (ii) (FBC y,TPBC x) = Möbius, (iii) (PBC y,PBC x) = toroidal, and (iv) (PBC y,TPBC x) = Klein bottle, where FBC and ( T) PBC refer to free and (twisted) periodic boundary conditions. Results for the L y =2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the C 2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.

T=0 partition functions for Potts antiferromagnets on lattice strips with fully periodic boundary conditions

We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for families of arbitrarily long strip graphs of the square and triangular lattices with width L y =4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation (i.e., of torus or Klein bottle type). These boundary conditions have the advantage of removing edge effects. In the limit of infinite length, we calculate the exponent of the entropy, W( q) and determine the continuous locus B where it is singular. We also give results for toroidal strips involving “crossing subgraphs”; these make possible a unified treatment of torus and Klein bottle boundary conditions and enable us to prove that for a given strip, the locus B is the same for these boundary conditions.

EXACT PARTITION FUNCTION FOR THE POTTS MODEL WITH NEXT-NEAREST NEIGHBOR COUPLINGS ON ARBITRARY-LENGTH LADDERS

We present exact calculations of partition function Z of the q-state Potts model with next-nearest-neighbor spin–spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on n-vertex ladders with free, cyclic, and Möbius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus ℬ in the corresponding [Formula: see text] space, arising as the accumulation set of partition function zeros as n → ∞.

Exact Potts model partition function on strips of the triangular lattice

In this paper we present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex strip graphs, of width L y =2 and arbitrary length, of the triangular lattice with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus B in the corresponding C 2 space, arising as the accumulation set of partition function zeros as n→∞. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where B reduces to the accumulation set of chromatic zeros. Comparisons are made with our previous exact calculation of Potts model partition functions for the corresponding strips of the square lattice. Our present calculations yield, as special cases, several quantities of graph-theoretic interest.

Exact Potts model partition functions on ladder graphs

We present exact calculations of the partition function Z of the q-state Potts model and its generalization to real q, for arbitrary temperature on n-vertex ladder graphs, i.e., strips of the square lattice with width L y =2 and arbitrary length L x , with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of these ladder graphs and the thermodynamics is discussed. By comparison with strip graphs of other widths, we analyze how the singularities at the zero-temperature critical point of the ferromagnet on infinite-length, finite-width strips depend on the width. We point out and study the following noncommutativity at certain special values q s : lim n→∞ lim q→q s Z 1/n≠ lim q→q s lim n→∞ Z 1/n . It is shown that the Potts antiferromagnet on both the infinite-length line and ladder graphs with cyclic or Möbius boundary conditions exhibits a phase transition at finite temperature if 0< q<2, but with unphysical properties, including negative specific heat and non-existence, in the low-temperature phase, of an n→∞ limit for thermodynamic functions that is independent of boundary conditions. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus B in the corresponding C 2 space, arising as the accumulation set of partition function zeros as n→∞. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where B reduces to the accumulation set of chromatic zeros. Certain properties of the complex-temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice, showing that exact solutions on infinite-length strips provide a way of gaining insight into these complex-temperature phase diagrams.

Exact partition functions for Potts antiferromagnets on cyclic lattice strips

We present exact calculations of the zero-temperature partition function of the q-state Potts antiferromagnet on arbitrarily long strips of the square, triangular, and kagomé lattices with width L y =2 or 3 vertices and with periodic longitudinal boundary conditions. From these, in the limit of infinite length, we obtain the exact ground-state entropy S 0=k B ln W . These results are of interest since this model exhibits nonzero ground-state entropy S 0>0 for sufficiently large q and hence is an exception to the third law of thermodynamics. We also include results for homeomorphic expansions of the square lattice strip. The analytic properties of W( q) are determined and related to zeros of the chromatic polynomial for long finite strips.

T= 0 partition functions for Potts antiferromagnets on square lattice strips with (twisted) periodic boundary conditions

We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and (i) periodic (ii) twisted periodic boundary conditions in the longitudinal direction, so that the strip graphs are embedded on a (i) torus (ii) Klein bottle. In the limit of infinite length, we calculate the exponent of the entropy, W(q), show it to be the same for (i) and (ii), and determine its analytic structure.

Chromatic polynomials for J(∏ H) I strip graphs and their asymptotic limits

We calculate the chromatic polynomials P for n-vertex strip graphs of the form J(∏ l=1 m H) I, where J and I are various subgraphs on the left and right ends of the strip, whose bulk is comprised of m-fold repetitions of a subgraph H. The strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction. This extends our earlier calculations for strip graphs of the form (∏ l=1 m H) I. We use a generating function method. From these results we compute the asymptotic limiting function W=lim n→∞ P 1/ n ; for q∈ Z + this has physical significance as the ground-state degeneracy per site (exponent of the ground-state entropy) of the q-state Potts antiferromagnet on the given strip. In the complex q plane, W is an analytic function except on a certain continuous locus B . In contrast to the (∏ l=1 m H) I strip graphs, where B (i) is independent of I, and (ii) consists of arcs and possible line segments that do not enclose any regions in the q plane, we find that for some J(∏ l=1 m H) I strip graphs, B (i) does depend on I and J, and (ii) can enclose regions in the q plane. Our study elucidates the effects of different end subgraphs I and J and of boundary conditions on the infinite-length limit of the strip graphs.

FINITE ELEMENT MODELLING OF SPATIALLY CHAOTIC STRUCTURES

A simple non-linear mechanical system comprising a pin-jointed string of finite-length links, supported by elastic springs at the pins and compressed by an axial load, is viewed from two perspectives. When seen as an initial-value problem, equilibrium equations provide an iterative non-linear mapping. When seen as a boundary-value problem, it becomes a simple finite element model. At loads less than the critical buckling load, a preferred buckling configuration is found that is localized along the length. In the limit of infinite length this is described as a homoclinic connection in phase space, joining the flat equilibrium state to itself. The infinite sequence of homoclinic points thus defined embeds within the complex topological structure of a homoclinic tangle, within which also appear periodic, quasi-periodic, and chaotic spatial solutions. Implications in the finite element setting are discussed. © 1997 John Wiley & Sons, Ltd.

FINITE ELEMENT MODELLING OF SPATIALLY CHAOTIC STRUCTURES

A simple non-linear mechanical system comprising a pin-jointed string of finite-length links, supported by elastic springs at the pins and compressed by an axial load, is viewed from two perspectives. When seen as an initial-value problem, equilibrium equations provide an iterative non-linear mapping. When seen as a boundary-value problem, it becomes a simple finite element model. At loads less than the critical buckling load, a preferred buckling configuration is found that is localized along the length. In the limit of infinite length this is described as a homoclinic connection in phase space, joining the flat equilibrium state to itself. The infinite sequence of homoclinic points thus defined embeds within the complex topological structure of a homoclinic tangle, within which also appear periodic, quasi-periodic, and chaotic spatial solutions. Implications in the finite element setting are discussed. © 1997 John Wiley & Sons, Ltd.

Conformations of linear DNA

We examine the conformations of a model for underwound and overwound DNA. The molecule is represented as a cylindrically symmetric elastic string subjected to a stretching force and to constraints corresponding to a specification of the link number. We derive a fundamental relation between the Euler angles that describe the curve and the topological linking number. Analytical expressions for the spatial configurations of the molecule in the infinite-length limit are obtained. A unique configuration minimizes the energy for a given set of physical conditions. An elastic model incorporating thermal fluctuations provides excellent agreement with experimental results on the plectonemic transition.

Numerical Study of the String Order inS=1 Antiferromagnetic Heisenberg Chain

We study the spin-1 antiferromagnetic Heisenberg chain under the three boundary conditions, which are the free, periodic and antiperiodic boundary conditions, by numerical diagonalization up to 16 sites. We checked that the lowest excitation caused by the domain wall, which is imposed by the boundary condition and breaks the long-range string order, yields the same energy gap under all the boundary conditions in the infinite length limit. It is also found that the ground state and the first excited state have the same density (about 3%) of domain walls due to the quantum fluctuation which reduces the string order parameter about 10% smaller than that of the VBS model in the ground state. The analysis of the mean distance between two domain walls suggests that domain walls due to the quantum fluctuation form a bound state in pairs and does not break the long-range string order.

Elastic model of DNA supercoiling in the infinite-length limit

The energies of plectonemic and toroidal supercoiled DNA are calculated by treating DNA as an elastic rod with a finite radius. End effects are ignored and all extensive quantities (e.g., writhe, bend energy) are treated as linear densities (writhe per unit length, bend energy per unit length). Minimum energy configurations are found. For plectonemic DNA, the superhelical pitch angle α is in the range 45°&amp;lt;α≤90°. For low values of specific linking difference, most superhelicity is in writhe. As specific linking difference increases, a greater proportion of superhelicity is in twist. Under physiological conditions, roughly 88% of superhelicity is in writhe. Ionic strength effects are discussed, and it is found that variation of excluded volume with ionic strength has a large effect, resulting in significantly greater torsional stress in supercoiled DNA at low ionic strength. For biologically relevant values of specific linking difference, the plectonemic conformation is energetically favored over toroidal conformations. Results are compared with electron microscopy data. The application of the model to DNA conformational transitions is discussed.

Frequency response of a thermocouple wire: Effects of axial conduction

Theoretical expressions are derived for the steady-state frequency response of a thermocouple wire. In particular, the effects of axial heat conduction are demonstrated for a non-uniform wire with unequal material properties and wire diameters across the junction. The frequency response in the limit of equal wire diameters and material properties reduces to the expression derived by chomiak and Niedzialek [1]. In addition, the amplitude ratio at low frequency ω → 0 agrees with the results of Scadron and Warshawsky [2] for a steady-state temperature distribution. Moreover, the frequency response for a non-uniform wire in the limit of infinite length l → ∞ is shown to reduce to a simple expression that is analogous to the classic first order solution for a thermocouple wire with uniform properties.

Extrapolation to the infinite length limit of polymer energies. I. Hückel approximation

A method to extrapolate the energy of polymers (in the Hückel approximation) as a function of the number of monomers is presented. The method can be used to obtain the energy per monomer of the infinite chain, and test computations are presented. They show good results even in presence of a strongly oscillating behavior of the finite polymer energies.

Boundary condition effects on statistical mechanics properties of sine-Gordon systems

A functional approach to the statistical mechanics of the classic sine-Gordon field on a finite support is presented. The band spectrum determined by the periodic boundary conditions is discussed and the partition function is coherently calculated. The kink contr̀ibution to the free energy is derived and the “dilute gas approximation” is shown to be recovered in the infinite length limit.

Note on One-Dimensional Lattice Gas in External Fields

Abstract The fundamental equations of the scaled particle theory are derived for a one-dimensional lattice gas in an external potential. These equations relate the work required to add a particle, at a fixed point, to a N – 1 particle system to the activity and to a series of coordinate distribution functions. The equations hold in any dimension, and replacing sums by integrals, describe continuum fluids. The rigid lattice gas is solved by these means. When nearest neighbors interact, in a positive, increasing external potential, a formal solution is obtained by a matrix method. The grand partition function, in the infinite length limit, depends only on a single eigenvalue of an infinite product of matrices. The one-particle distribution, in this limit, is reduced to a series of terminating continued fractions, which is readily approximated in the high coordinate or low activity limit. Lastly, it is shown that the zeros of the grand partition function lie on the negative real axis of the complex activity ...