scholarly journals ARE THERE LOCAL MINIMA IN THE MAGNETIC MONOPOLE POTENTIAL IN COMPACT QED?

2004 ◽  
Vol 19 (29) ◽  
pp. 5017-5026 ◽  
Author(s):  
H. BOZKAYA ◽  
M. FABER ◽  
P. KOPPENSTEINER ◽  
M. PITSCHMANN
Keyword(s):  
Phase Transition ◽  
Magnetic Monopole ◽  
Local Minima ◽  
Local Extrema ◽  
Centers Of Mass ◽  
Definition Of

We investigate the influence of the granularity of the lattice on the potential between monopoles. Using the flux definition of monopoles we introduce their centers of mass and are able to realize continuous shifts of the monopole positions. We find periodic deviations from the 1/r-behavior of the monopole-antimonopole potential leading to local extrema. We suppose that these meta-stabilities may influence the order of the phase transition in compact QED.

scholarly journals Nucleation is more than critical: A case study of the electroweak phase transition in the NMSSM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Sebastian Baum ◽  
Marcela Carena ◽  
Nausheen R. Shah ◽  
Carlos E. M. Wagner ◽  
Yikun Wang
Keyword(s):  
Phase Transition ◽  
Parameter Space ◽  
Local Minima ◽  
Critical Temperatures ◽  
Electroweak Phase Transition ◽  
First Order ◽  
Vacuum Structure ◽  
The Universe ◽  
Scalar Sector

Abstract Electroweak baryogenesis is an attractive mechanism to generate the baryon asymmetry of the Universe via a strong first order electroweak phase transition. We compare the phase transition patterns suggested by the vacuum structure at the critical temperatures, at which local minima are degenerate, with those obtained from computing the probability for nucleation via tunneling through the barrier separating local minima. Heuristically, nucleation becomes difficult if the barrier between the local minima is too high, or if the distance (in field space) between the minima is too large. As an example of a model exhibiting such behavior, we study the Next-to-Minimal Supersymmetric Standard Model, whose scalar sector contains two SU(2) doublets and one gauge singlet. We find that the calculation of the nucleation probabilities prefers different regions of parameter space for a strong first order electroweak phase transition than the calculation based solely on the critical temperatures. Our results demonstrate that analyzing only the vacuum structure via the critical temperatures can provide a misleading picture of the phase transition patterns, and, in turn, of the parameter space suitable for electroweak baryogenesis.

The Hexagonal ↔ Orthorhombic Structural Phase Transition in Claringbullite, Cu4FCl(OH)6

2021 ◽  
Author(s):  
Mark D. Welch ◽  
Jens Najorka ◽  
Michael S. Rumsey ◽  
John Spratt
Keyword(s):  
Phase Transition ◽  
Data Storage ◽  
Structural Phase Transition ◽  
Structural Changes ◽  
Structural Phase ◽  
Magnetic Behavior ◽  
Orthorhombic Phase ◽  
New Mineral ◽  
Storage Devices ◽  
Definition Of

ABSTRACT Frustrated magnetic phases have been a perennial interest to theoreticians wishing to understand the energetics and behavior of quasi-chaotic systems at the quantum level. This behavior also has potentially wide applications to developing quantum data-storage devices. Several minerals are examples of such phases. Since the definition of herbertsmithite, Cu3ZnCl2(OH)6, as a new mineral in 2004 and the rapid realization of the significance of its structure as a frustrated antiferromagnetic phase with a triangular magnetic lattice, there has been intense study of its magnetic properties and those of synthetic compositional variants. In the past five years it has been recognized that the layered copper hydroxyhalides barlowite, Cu4BrF(OH)6, and claringbullite, Cu4FCl(OH)6, are also the parent structures of a family of kagome phases, as they also have triangular magnetic lattices. This paper concerns the structural behavior of claringbullite that is a precursor to the novel frustrated antiferromagnetic states that occur below 30 K in these minerals. The reversible hexagonal (P63/mmc) ↔ orthorhombic (Pnma or Cmcm) structural phase transition in barlowite at 200−270 K has been known for several years, but the details of the structural changes that occur through the transition have been largely unexplored, with the focus instead being on quantifying the low-temperature magnetic behavior of the orthorhombic phase. This paper reports the details of the structural phase transition in natural claringbullite at 100−293 K as studied by single-crystal X-ray diffraction. The transition temperature has been determined to lie between 270 and 293 K. The progressive disordering of Cu at the unusual trigonal prismatic Cu(OH)6 site on heating is quantified through the phase transition for the first time, and a methodology for refining this disorder is presented. Key changes in the behavior of Cu(OH)4Cl2 octahedra in claringbullite have been identified that suggest why the Pnma structure is likely stabilized over an alternative Cmcm structure. It is proposed that the presence of a non-centrosymmetric octahedron in the Pnma structure allows more effective structural relaxation during the phase transition than can be achieved by the Cmcm structure, which has only centrosymmetric octahedra.

scholarly journals Easy-hard phase transition in parameter estimation for optical waveguides

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Gunnar Claussen ◽  
Alexander K. Hartmann
Keyword(s):  
Phase Transition ◽  
Optical Waveguides ◽  
Energy Landscape ◽  
Global Optimum ◽  
Local Minima ◽  
Different Types ◽  
Two Phases ◽  
Parameter Values ◽  
Monochromatic Illumination

Abstract The determination of the parameters of cylindrical optical waveguides, e.g. the diameters $$\vec {d}=(d_1,\ldots ,d_r)$$ d → = ( d 1 , … , d r ) of r layers of (semi-) transparent optical fibres, can be executed by inverse evaluation of the scattering intensities that emerge under monochromatic illumination. The inverse problem can be solved by optimising the mismatch $$R(\vec {d})$$ R ( d → ) between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch $$R(\vec {d})$$ R ( d → ) can be seen as an energy landscape as a function of the diameters. In this work, we study the structure of the energy landscape for different values of the complex refractive indices $$\vec {n}$$ n → , for $$r=1$$ r = 1 and $$r=2$$ r = 2 layers. We find that for both values of r, depending on the values of $$\vec {n}$$ n → , two very different types of energy landscapes exist, respectively. One type is dominated by one global minimum and the other type exhibits a multitude of local minima. From an algorithmic viewpoint, this corresponds to easy and hard phases, respectively. Our results indicate that the two phases are separated by sharp phase-transition lines and that the shape of these lines can be described by one “critical” exponent b, which depends slightly on r. Interestingly, the same exponent also describes the dependence of the number of local minima on the diameters. Thus, our findings are comparable to previous theoretical studies on easy-hard transitions in basic combinatorial optimisation or decision problems like Travelling Salesperson and Satisfiability. To our knowledge our results are the first indicating the existence of easy-hard transitions for a real-world optimisation problem of technological relevance.

scholarly journals Local extrema in random trees

2005 ◽  
Vol 2005 (23) ◽  
pp. 3867-3882 ◽  
Author(s):  
Lane Clark
Keyword(s):  
Uniform Distribution ◽  
Rational Functions ◽  
Random Trees ◽  
Local Minima ◽  
Small Interval ◽  
Local Maxima ◽  
Local Extrema

The number of local maxima (resp., local minima) in a treeT∈𝒯nrooted atr∈[n]is denoted byMr(T)(resp., bymr(T)). We find exact formulas as rational functions ofnfor the expectation and variance ofM1(T)andmn(T)whenT∈𝒯nis chosen randomly according to a uniform distribution. As a consequence, a.a.s.M1(T)andmn(T)belong to a relatively small interval whenT∈𝒯n.

scholarly journals The Observable Representation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 310
Author(s):  
L. Schulman
Keyword(s):  
Phase Transition ◽  
Euclidean Space ◽  
Complex Systems ◽  
Random Walks ◽  
Stochastic Dynamics ◽  
Dimensional Euclidean Space ◽  
Significant Feature ◽  
Original Space ◽  
Definition Of ◽  
Low Dimensional

The observable representation (OR) is an embedding of the space on which a stochastic dynamics is taking place into a low dimensional Euclidean space. The most significant feature of the OR is that it respects the dynamics. Examples are given in several areas: the definition of a phase transition (including metastable phases), random walks in which the OR recovers the original space, complex systems, systems in which the number of extrema exceed convenient viewing capacity, and systems in which successful features are displayed, but without the support of known theorems.

scholarly journals Phase transition for the system of finite volume in the ϕ4 theory in the Tsallis nonextensive statistics

2018 ◽  
Vol 33 (12) ◽  
pp. 1850067 ◽  
Author(s):  
Masamichi Ishihara
Keyword(s):  
Phase Transition ◽  
Effective Mass ◽  
Finite Volume ◽  
Free Particle ◽  
Bare Mass ◽  
Expectation Value ◽  
Volume Formula ◽  
Nonextensive Statistics ◽  
Definition Of ◽  
Particle Approximation

We studied the effects of nonextensivity on the phase transition for the system of finite volume [Formula: see text] in the [Formula: see text] theory in the Tsallis nonextensive statistics of entropic parameter [Formula: see text] and temperature [Formula: see text], when the deviation from the Boltzmann–Gibbs (BG) statistics, [Formula: see text], is small. We calculated the condensate and the effective mass to the order [Formula: see text] with the normalized [Formula: see text]-expectation value under the free particle approximation with zero bare mass. The following facts were found. The condensate [Formula: see text] divided by [Formula: see text], [Formula: see text], at [Formula: see text] ([Formula: see text] is the value of the condensate at [Formula: see text]) is smaller than that at [Formula: see text] for [Formula: see text] as a function of [Formula: see text] which is the physical temperature [Formula: see text] divided by [Formula: see text]. The physical temperature [Formula: see text] is related to the variation of the Tsallis entropy and the variation of the internal energies, and [Formula: see text] at [Formula: see text] coincides with [Formula: see text]. The effective mass decreases, reaches minimum, and increases after that, as [Formula: see text] increases. The effective mass at [Formula: see text] is lighter than the effective mass at [Formula: see text] at low physical temperature and heavier than the effective mass at [Formula: see text] at high physical temperature. The effects of the nonextensivity on the physical quantity as a function of [Formula: see text] become strong as [Formula: see text] increases. The results indicate the significance of the definition of the expectation value, the definition of the physical temperature, and the constraints for the density operator, when the terms including the volume of the system are not negligible.

scholarly journals Transition pathways connecting crystals and quasicrystals

2021 ◽  
Vol 118 (49) ◽  
pp. e2106230118
Author(s):  
Jianyuan Yin ◽  
Kai Jiang ◽  
An-Chang Shi ◽  
Pingwen Zhang ◽  
Lei Zhang
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Saddle Points ◽  
Energy Functional ◽  
Computational Method ◽  
Local Minima ◽  
Free Energy Functional ◽  
Transition Pathways ◽  
Critical Nuclei ◽  
Quasicrystalline State

Due to structural incommensurability, the emergence of a quasicrystal from a crystalline phase represents a challenge to computational physics. Here, the nucleation of quasicrystals is investigated by using an efficient computational method applied to a Landau free-energy functional. Specifically, transition pathways connecting different local minima of the Lifshitz–Petrich model are obtained by using the high-index saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state, respectively.

scholarly journals Local extrema in random permutations and the structure of longest alternating subsequences

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Dan Romik
Keyword(s):  
Joint Distribution ◽  
Random Permutation ◽  
Random Permutations ◽  
Local Minima ◽  
New Approach ◽  
Successive Minimum ◽  
Local Extrema ◽  
Nous Montrons ◽  
Dimensional Distribution ◽  
International Audience

International audience Let $\textbf{as}_n$ denote the length of a longest alternating subsequence in a uniformly random permutation of order $n$. Stanley studied the distribution of $\textbf{as}_n$ using algebraic methods, and showed in particular that $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ and $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. From Stanley's result it can be shown that after rescaling, $\textbf{as}_n$ converges in the limit to the Gaussian distribution. In this extended abstract we present a new approach to the study of $\textbf{as}_n$ by relating it to the sequence of local extrema of a random permutation, which is shown to form a "canonical'' longest alternating subsequence. Using this connection we reprove the abovementioned results in a more probabilistic and transparent way. We also study the distribution of the values of the local minima and maxima, and prove that in the limit the joint distribution of successive minimum-maximum pairs converges to the two-dimensional distribution whose density function is given by $f(s,t) = 3(1-s)t e^{t-s}$. Pour une permutation aléatoire d'ordre $n$, on désigne par $\textbf{as}_n$ la longueur maximale d'une de ses sous-suites alternantes. Stanley a étudié la distribution de $\textbf{as}_n$ en utilisant des méthodes algébriques, et il a démontré en particulier que $\mathbb{E}(\textbf{as}_n) = (4n+1)/6$ et $\textrm{Var}(\textbf{as}_n) = (32n-13)/180$. A partir du résultat de Stanley on peut montrer qu'après changement d'échelle, $\textbf{as}_n$ converge vers la distribution normale. Nous présentons ici une approche nouvelle pour l'étude de $\textbf{as}_n$, en la reliant à la suite des extrema locaux d'une permutation aléatoire, dont nous montrons qu'elle constitue une sous-suite alternante maximale "canonique''. En utilisant cette relation, nous prouvons à nouveau les résultats mentionnés ci-dessus d'une façon plus probabiliste et transparente. En plus, nous prouvons un résultat asymptotique sur la distribution limite des paires formées d'un minimum et d'un maximum locaux consécutifs.

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