Metal–insulator transition in a ternary model with long range correlated disorder
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We study the metal–insulator transition of the one-dimensional diagonal Anderson ternary model with long range correlated disorder. The starting point of the model corresponds to a ternary alloy (i.e. with three possible on-site energies), and shows a metal–insulator transition when the random distribution of site energies is assumed to have a power spectrum . In this paper, we define a purity parameter for the ternary alloy which adjusts the occupancy probability of site potentials, and for any given α we calculate the critical purity parameter for which extended states are obtained. In this way, we show that the ternary alloy requires weaker correlations than the binary alloy to present a phase transition from localized to extended states. A phase diagram which separates the extended regime from the localized one for the ternary alloy is presented, obtained as the critical purity parameter in terms of the corresponding correlation exponent.Keywords:
Critical point (mathematics)
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Exponent
Let y(x) be a smooth sigmoidal curve, y (n) be its nth derivative and {x m,i } and {x a,i }, i = 1,2,…, be the set of points where respectively the derivatives of odd and even order reach their extreme values. We argue that if the sigmoidal curve y(x) represents a phase transition, then the sequences {x m,i } and {x a,i } are both convergent and they have a common limit x c that we characterize as the critical point of the phase transition. In this study, we examine the logistic growth curve and the Susceptible-Infected-Removed (SIR) epidemic model as typical examples of symmetrical and asymmetrical transition curves. Numerical computations indicate that the critical point of the logistic growth curve that is symmetrical about the point (x 0 , y 0 ) is always the point (x 0 , y 0 ) but the critical point of the asymmetrical SIR model depends on the system parameters. We use the description of the sol–gel phase transition of polyacrylamide-sodium alginate (SA) composite (with low SA concentrations) in terms of the SIR epidemic model, to compare the location of the critical point as described above with the "gel point" determined by independent experiments. We show that the critical point t c is located in between the zero of the third derivative t a and the inflection point t m of the transition curve and as the strength of activation (measured by the parameter k/η of the SIR model) increases, the phase transition occurs earlier in time and the critical point, t c , moves toward t a .
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The Phase diagram of QCD in a BAG+HRG based equation of state: appearance of a pseudo-critical point
Mapping the QCD phase boundary and locating critical end point still remains as an open problem in strong interaction physics. Predictions about the co-ordinates of the critical point in the $(T, \mu_B)$ plane, from different QCD motivated models show a wide variation. Lattice QCD calculations are also available, that give an estimation of the critical point for chiral phase transition, where the transition changes its nature from rapid cross over to first order transition. Recently co-ordinates of the critical point for deconfinement phase transition are claimed to be found as an endpoint of the first order phase transition line, in Bag model scenario. In the present paper we have shown that Bag model gives a complete first order phase transition line in the $(T, \mu_B)$ plane, and one can not have any point where the transition changes its nature.
Critical point (mathematics)
Deconfinement
Phase boundary
Transition point
Lattice (music)
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The phase diagram of QCD in a BAG+HRG based equation of state: appearance of a pseudo-critical point
Mapping the QCD phase boundary and locating critical end point still remains as an open problem in strong interaction physics. Predictions about the co-ordinates of the critical point in the $(T, \mu_B)$ plane, from different QCD motivated models show a wide variation. Lattice QCD calculations are also available, that give an estimation of the critical point for chiral phase transition, where the transition changes its nature from rapid cross over to first order transition. Recently co-ordinates of the critical point for deconfinement phase transition are claimed to be found as an endpoint of the first order phase transition line, in Bag model scenario. In the present paper we have shown that Bag model gives a complete first order phase transition line in the $(T, \mu_B)$ plane, and one can not have any point where the transition changes its nature.
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Deconfinement
Transition point
Phase boundary
Lattice (music)
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We introduce two notions for flows on quasi-diagonal C*-algebras, quasi-diagonal and pseudo-diagonal flows; the former being apparently stronger than the latter. We derive basic facts about these flows and give various examples. In addition we extend results of Voiculescu from quasi-diagonal C*-algebras to these flows.
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CALPHAD
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Dedicated to 60-th anniversary of E. Zelmanov Abstract. We show how to use generating exponent matrices to study the quivers of exponent matrices. We also describe the admissible quivers of 3 × 3 exponent matrices.
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We study the properties of the low-lying states at the critical point of the phase transition from U(3) to O(4) symmetry in the U(4) vibron model in detail. By analyzing the general characteristics and comparing the calculated results of the energy spectra and the E1, E2 transition rates in E(3) symmetry, in $r^4$ potential model and the finite boson number case in boson space, we find that the results in the $r^4$ potential demonstrates the characteristic of the classical limit at the critical point well and the E(3) symmetry over-predict the energy levels and under-predict the E1 and E2 transition rates of the states at the critical point. However, the E(3) symmetry may describe part of the properties of the system with boson number around 10 to 20. We also confirm that the 12C+12C system is an empirical evidence of the state at the critical point of the phase transition in the U(4) model when concerning the energies of the low-lying resonant states.
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Transition point
Interacting boson model
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Abstract We present precise measurements of the static dielectric constant near the smectic Ad-smectic A2 critical point in a binary system. Detailed analysis of the data obtained for several mixtures has been done to locate the critical point concentration. Results of a dielectric dispersion study near the Ad-A2 transition are also presented.
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The phase diagram of the pure substance ( is visually plotted on the P-T phase diagram based on pressure and temperature dependent experimental results. In the phase diagram, all three phases are present and the triple point where these phases are in equilibrium. In addition, a point on a liquid-gas equilibrium curve, commonly known as a critical point is also available on the phase diagram . In the classical P-T phase diagram used today, only shows the three phase and boundary curves that separate these phases from each other, the triple point where the three phases are at one point and the critical parameters where the liquid gas equilibrium curve ends (Tcr, Pcr, ρcr).
Triple point
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A dynamical model of traffic is proposed to take into account the effect of acceleration delay. In the limit of no delay, the model reproduces the optimal velocity model of traffic. When the delay is small, it is shown that the phase transition among the freely moving phase, the coexisting phase, and the uniform congested phase occurs below the critical point. Above the critical point, no phase transition occurs. The value ${a}_{c}$ of the critical point increases with increasing delay time $1/b,$ where $a$ is the friction coefficient (or sensitivity parameter). When the delay time is longer than $\frac{1}{2},$ the critical point disappears and the phase transition always occurs. The linear stability theory and nonlinear analysis are applied. The critical point predicted by the linear stability theory agrees with the simulation result. The modified Korteweg--de Vries (KdV) equation is obtained from the nonlinear analysis near the critical point. The phase separation line obtained from the modified KdV equation is consistent with the simulation result.
Critical point (mathematics)
Transition point
Critical ionization velocity
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