scholarly journals Solution of the random field XY magnet on a fully connected graph

2022 ◽  
Author(s):  
S Sumedha ◽  
Mustansir Barma
Keyword(s):  
Free Energy ◽  
Connected Graph ◽  
Large Deviation ◽  
Xy Model ◽  
Cubic Symmetry ◽  
Multicritical Points ◽  
Underlying Disorder ◽  
Function Of Two Variables ◽  
Fully Connected ◽  
Ordered State

Abstract We use large deviation theory to obtain the free energy of the XY model on a fully connected graph on each site of which there is a randomly oriented field of magnitude $h$. The phase diagram is obtained for two symmetric distributions of the random orientations: (a) a uniform distribution and (b) a distribution with cubic symmetry. In both cases, the ordered state reflects the symmetry of the underlying disorder distribution. The phase boundary has a multicritical point which separates a locus of continuous transitions (for small values of $h$) from a locus of first order transitions (for large $h$). The free energy is a function of a single variable in case (a) and a function of two variables in case (b), leading to different characters of the multicritical points in the two cases.

scholarly journals Cubic microlattices embedded in nematic liquid crystals: a Landau-de Gennes study

2021 ◽  
Author(s):  
Razvan-Dumitru Ceuca
Keyword(s):  
Free Energy ◽  
Additional Term ◽  
Energy Functional ◽  
Free Energies ◽  
Cubic Symmetry ◽  
Surface Anchoring ◽  
Free Energy Functional ◽  
Different Types ◽  
Embedded Particles ◽  
Anchoring Energy

We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material. In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature. In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain. Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.

scholarly journals THE HEIDER BALANCE: A CONTINUOUS APPROACH

2005 ◽  
Vol 16 (05) ◽  
pp. 707-716 ◽  
Author(s):  
KRZYSZTOF KUŁAKOWSKI ◽  
PRZEMYSŁAW GAWROŃSKI ◽  
PIOTR GRONEK
Keyword(s):  
Connected Graph ◽  
Social Group ◽  
Statistical Data ◽  
Structure Theorem ◽  
System Size ◽  
Random Numbers ◽  
Continuous Approach ◽  
Internal Relations ◽  
Group Members ◽  
Fully Connected

The Heider balance (HB) is investigated in a fully connected graph of N nodes. The links are described by a real symmetric array r (i, j), i, j =1, …, N. In a social group, nodes represent group members and links represent relations between them, positive (friendly) or negative (hostile). At the balanced state, r (i, j) r (j, k) r (k, i) > 0 for all the triads (i, j, k). As follows from the structure theorem of Cartwright and Harary, at this state the group is divided into two subgroups, with friendly internal relations and hostile relations between the subgroups. Here the system dynamics is proposed to be determined by a set of differential equations, [Formula: see text]. The form of equations guarantees that once HB is reached, it persists. Also, for N =3 the dynamics reproduces properly the tendency of the system to the balanced state. The equations are solved numerically. Initially, r (i, j) are random numbers distributed around zero with a symmetric uniform distribution of unit width. Calculations up to N =500 show that HB is always reached. Time τ(N) to get the balanced state varies with the system size N as N-1/2. The spectrum of relations, initially narrow, gets very wide near HB. This means that the relations are strongly polarized. In our calculations, the relations are limited to a given range around zero. With this limitation, our results can be helpful in an interpretation of some statistical data.

THE GROUNDSTATES AND PHASES OF THE TWO-DIMENSIONAL FULLY FRUSTRATED XY MODEL

2009 ◽  
Vol 23 (20n21) ◽  
pp. 3939-3950
Author(s):  
PETTER MINNHAGEN ◽  
SEBASTIAN BERNHARDSSON ◽  
BEOM JUN KIM
Keyword(s):  
Phase Transition ◽  
Phase Diagram ◽  
Phase Transitions ◽  
Xy Model ◽  
Two Dimensional ◽  
Spin Configuration ◽  
Multicritical Points ◽  
Symmetry Phase

The 2D Fully Frustrated XY(FFXY) class of models is shown to contain a new groundstate in addition to the checkerboard groundstate of the standard 2D XY model. The spin configuration of this additional groundstate is obtained and its connection to a broken Z2-symmetry explained. This means that the class of 2D FFXY models belongs within a U(1) ⊗ Z2 ⊗ Z2-symmetry phase-transition representation. The phase diagram is reviewed and the central charges of the four multicritical points described. The implications for the standard 2D FFXY-model are discussed and elucidated, in particular with respect to the long standing controversy concerning the phase transitions of the standard 2D FFXY-model.

SUPER SPANNING CONNECTIVITY OF THE FULLY CONNECTED CUBIC NETWORKS

2010 ◽  
Vol 11 (01n02) ◽  
pp. 61-70 ◽  
Author(s):  
CHERNG CHIN ◽  
HUAI-CHIH CHEN ◽  
LIH-HSING HSU ◽  
SHANG-CHIA CHIOU ◽  
KUO-TUNG LAI
Keyword(s):  
Connected Graph ◽  
Disjoint Paths ◽  
Fully Connected

A k-containerC(u, v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k*-container if it contains all vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices. The spanning connectivity of G, κ*(G), is defined to be the largest integer k such that G is w*-connected for all 1 ≤ w ≤ k if G is a 1*-connected graph and undefined otherwise. A graph G is super spanning connected if κ* (G) = κ(G). In this paper, we prove that the n-dimensional fully connected cubic network FCCNn is super spanning connected.

scholarly journals An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Muhammad Asif ◽  
Muhammad Hussain ◽  
Hamad Almohamedh ◽  
Khalid M. Alhamed ◽  
Sultan Almotairi
Keyword(s):  
Computer Program ◽  
Regular Graph ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Fixed Length ◽  
Inverse Degree ◽  
Fully Connected

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

scholarly journals Quantum Hopfield Model

Physics ◽  
2020 ◽  
Vol 2 (2) ◽  
pp. 184-196 ◽  
Author(s):  
Masha Shcherbina ◽  
Brunello Tirozzi ◽  
Camillo Tassi
Keyword(s):  
Free Energy ◽  
Long Range ◽  
Thermodynamic Limit ◽  
Random Variables ◽  
Hopfield Model ◽  
Xy Model ◽  
One Dimensional ◽  
Random Interaction ◽  
Independent Identically Distributed

We find the free-energy in the thermodynamic limit of a one-dimensional XY model associated to a system of N qubits. The coupling among the σ i z is a long range two-body random interaction. The randomness in the couplings is the typical interaction of the Hopfield model with p patterns ( p < N ), where the patterns are p sequences of N independent identically distributed random variables (i.i.d.r.v.), assuming values ± 1 with probability 1 / 2 . We show also that in the case p ≤ α N , α ≠ 0 , the free-energy is asymptotically independent from the choice of the patterns, i.e., it is self-averaging.

A Phase-Field Model for the Formation of Martensite and Bainite

2014 ◽  
Vol 922 ◽  
pp. 31-36 ◽  
Author(s):  
Tansel T. Arif ◽  
Rong Shan Qin
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Materials Science ◽  
Current Model ◽  
Phase Field Model ◽  
Field Method ◽  
Field Variable ◽  
Phase Field Method ◽  
Cubic Symmetry ◽  
Chemical Free Energy

The phase field method is rapidly becoming the method of choice for simulating the evolution of solid state phase transformations in materials science. Within this area there are transformations primarily concerned with diffusion and those that have a displacive nature. There has been extensive work focussed upon applying the phase field method to diffusive transformations leaving much desired for models that can incorporate displacive transformations. Using the current model, the formation of martensite, which is formed via a displacive transformation, is simulated. The existence of a transformation matrix in the free energy expression along with cubic symmetry operations enables the reproduction of the 24 grain variants of martensite. Furthermore, upon consideration of the chemical free energy term, the model is able to utilise both the displacive and diffusive aspects of bainite formation, reproducing the autocatalytic nucleation process for multiple sheaves using a single phase field variable. Transformation matrices are available for many steels, one of which is used within the model.

scholarly journals Influence of Dislocations on the Spatial Variation of Microstructure in Martensites

2011 ◽  
Vol 465 ◽  
pp. 77-80
Author(s):  
Roman Gröger ◽  
Turab Lookman
Keyword(s):  
Free Energy ◽  
Local Density ◽  
Parent Phase ◽  
Mean Field ◽  
Continuum Theory ◽  
Energy Functional ◽  
Cubic Symmetry ◽  
Macroscopic Models ◽  
Multiphase Materials ◽  
Continuum Theory Of Dislocations

The continuum theory of dislocations, as developed predominantly by Kröner and Kosevich, views each dislocation as a source of incompatibility of strains. We show that this concept can be employed efficiently in the Landau free energy functional to develop a mean-field mesoscopic model of materials with dislocations. The order parameters that represent the distortion of the parent phase (often of cubic symmetry) are written in terms of elastic strains which are themselves coupled by incompatibility constraints. Since the “strength” of the incompatibility depends on the local density of dislocations, the presence of dislocations affects the evolution of the microstructure and vice versa. An advantage of this formulation is that long range anisotropic interactions between dislocations appear naturally in the formulation of the free energy. Owing to the distortion of the crystal structure around dislocations, their presence in multiphase materials causes heterogeneous nucleation of the product phase and thus also shifts of the transformation temperature. This novel field-theoretical approach is very convenient as it allows to bridge the gap in studying the behavior of materials at the length and time scales that are not attainable by atomistic or macroscopic models.

Phase Diagram Features Associated with Multicritical Points in Alloy Systems

1982 ◽  
Vol 19 ◽  
Author(s):  
Samuel M. Allen ◽  
John W. Cahn
Keyword(s):  
Phase Diagram ◽  
Free Energy ◽  
Power Series ◽  
Phase Change ◽  
Critical Points ◽  
Binary Systems ◽  
Landau Theory ◽  
Magnetic Ordering ◽  
Simple Approach ◽  
Multicritical Points

ABSTRACTMany features in the vicinity of critical points in phase diagrams can be illustrated using a Landau type free energy expansion as a power series in one or more order parameters and composition. This simple approach can be used with any solution model. It also predicts limits to metastability, and is useful for understanding mechanisms of phase change. The theory is applied to all the critical points that can occur in binary systems according to a Landau theory: critical consolute points, order-disorder transitions, tricritical points, critical end points, as well as to systems in which two transitions such as chemical and magnetic ordering occur.

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