Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed

Antifreeze glycoproteins inhibit leakage from liposomes during thermotropic phase transitions.

Antifreeze glycoproteins (AFGPs), found in the blood of polar fish at concentrations as high as 35 g/liter, are known to prevent ice crystal growth and depress the freezing temperature of the blood. Previously, Rubinsky et al. [Rubinsky, B., Mattioli, M., Arav, A., Barboni, B. & Fletcher, G. L. (1992) Am. J. Physiol. 262, R542-R545] provided evidence that AFGPs block ion fluxes across membranes during cooling, an effect that they ascribed to interactions with ion channels. We investigated the effects of AFGPs on the leakage of a trapped marker from liposomes during chilling. As these liposomes are cooled through the transition temperature, they leak approximately 50% of their contents. Addition of less than 1 mg/ml of AFGP prevents up to 100% of this leakage, both during chilling and warming through the phase transition. This is a general effect that we show here applies to liposomes composed of phospholipids with transition temperatures ranging from 12 degrees C to 41 degrees C. Because these results were obtained with liposomes composed of phospholipids alone, we conclude that the stabilizing effects of AFGPs on intact cells during chilling reported by Rubinsky et al. may be due to a nonspecific effect on the lipid components of native membranes. There are other proteins that prevent leakage, but only under specialized conditions. For instance, antifreeze proteins, bovine serum albumin, and ovomucoid all either have no effect or actually induce leakage. Following precipitation with acetone, all three proteins inhibited leakage, although not to the extent seen with AFGPs. Alternatively, there are proteins such as ovotransferrin that have no effect on leakage, either before or after acetone precipitation.

Self-organized phase transitions in neural networks as a neural mechanism of information processing.

Transitions between dynamically stable activity patterns imposed on an associative neural network are shown to be induced by self-organized infinitesimal changes in synaptic connection strength and to be a kind of phase transition. A key event for the neural process of information processing in a population coding scheme is transition between the activity patterns encoding usual entities. We propose that the infinitesimal and short-term synaptic changes based on the Hebbian learning rule are the driving force for the transition. The phase transition between the following two dynamical stable states is studied in detail, the state where the firing pattern is changed temporally so as to itinerate among several patterns and the state where the firing pattern is fixed to one of several patterns. The phase transition from the pattern itinerant state to a pattern fixed state may be induced by the Hebbian learning process under a weak input relevant to the fixed pattern. The reverse transition may be induced by the Hebbian unlearning process without input. The former transition is considered as recognition of the input stimulus, while the latter is considered as clearing of the used input data to get ready for new input. To ensure that information processing based on the phase transition can be made by the infinitesimal and short-term synaptic changes, it is absolutely necessary that the network always stays near the critical state corresponding to the phase transition point.

Deficiency of retinoblastoma protein leads to inappropriate S-phase entry, activation of E2F-responsive genes, and apoptosis.

The retinoblastoma susceptibility gene (Rb) participates in controlling the G1/S-phase transition, presumably by binding and inactivating E2F transcription activator family members. Mouse embryonic fibroblasts (MEFs) with no, one, or two inactivated Rb genes were used to determine the specific contributions of Rb protein to cell cycle progression and gene expression. MEFs lacking both Rb alleles (Rb-/-) entered S phase in the presence of the dihydrofolate reductase inhibitor methotrexate. Two E2F target genes, dihydrofolate reductase and thymidylate synthase, displayed elevated mRNA and protein levels in Rb- MEFs. Since absence of functional Rb protein in MEFs is sufficient for S-phase entry under growth-limiting conditions, these data indicate that the E2F complexes containing Rb protein, and not the Rb-related proteins p107 and p130, may be rate limiting for the G1/S transition. Antineoplastic drugs caused accumulation of p53 in the nuclei of both Rb+/+ and Rb-/- MEFs. While p53 induction led to apoptosis in Rb-/- MEFs, Rb+/- and Rb+/+ MEFs underwent cell cycle arrest without apoptosis. These results reveal that diverse growth signals work through Rb to regulate entry into S phase, and they indicate that absence of Rb protein produces a constitutive DNA replication signal capable of activating a p53-associated apoptotic response.

Thermodynamic glass transition in a spin glass without time-reversal symmetry

Spin glasses are a longstanding model for the sluggish dynamics that appear at the glass transition. However, spin glasses differ from structural glasses in a crucial feature: they enjoy a time reversal symmetry. This symmetry can be broken by applying an external magnetic field, but embarrassingly little is known about the critical behavior of a spin glass in a field. In this context, the space dimension is crucial. Simulations are easier to interpret in a large number of dimensions, but one must work below the upper critical dimension (i.e., in d < 6) in order for results to have relevance for experiments. Here we show conclusive evidence for the presence of a phase transition in a four-dimensional spin glass in a field. Two ingredients were crucial for this achievement: massive numerical simulations were carried out on the Janus special-purpose computer, and a new and powerful finite-size scaling method.

First-order transition in a three-dimensional disordered system

We present the first detailed numerical study in three dimensions of a first-order phase transition that remains first order in the presence of quenched disorder (specifically, the ferromagnetic-paramagnetic transition of the site-diluted four states Potts model). A tricritical point, which lies surprisingly near the pure-system limit and is studied by means of finite-size scaling, separates the first-order and second-order parts of the critical line. This investigation has been made possible by a new definition of the disorder average that avoids the diverging-variance probability distributions that plague the standard approach. Entropy, rather than free energy, is the basic object in this approach that exploits a recently introduced microcanonical Monte Carlo method.

VO: a strongly correlated metal close to a Mott-Hubbard transition

Here, we present experimental and computational evidences to support that rocksalt cubic VO is a strongly correlated metal with non-Fermi-liquid thermodynamics and an unusually strong spin-lattice coupling. An unexpected change of sign of metallic thermopower with composition is tentatively ascribed to the presence of a pseudogap in the density of states. These properties are discussed as signatures of the proximity to a magnetic quantum phase transition. The results are summarized in an electronic phase diagram for the 3d monoxides, which resembles that of other strongly correlated systems. The structural and electronic simplicity of 3d monoxides makes them ideal candidates to progress in the understanding of highly correlated electron systems.

Glass transition and enthalpy relaxation of amorphous food saccharides: A review

Many food materials exist in a disordered amorphous solid state due to processing. Therefore, understanding the concept of amorphous state, its important phase transition (i.e., glass transition), and the related phenomena (e.g., enthalpy relaxation) is important to food scientists. Food saccharides, including mono-, di-, oligo-, and polysaccharides, are among the most important major components in food. Focusing on the food saccharides, this review covers important topics related to amorphous solids, including the concept and molecular arrangement of amorphous solid, the formation of amorphous food saccharides, the concept of glass transition and enthalpy relaxation, physical property changes and molecular mobility around the glass transition, measurement of the glass transition and enthalpy relaxation, their mathematical descriptions and models, and influences on food stability.

Threshold-induced phase transitions in perceptrons

Error rates of a Boolean perceptron with threshold and either spherical or Ising constraint on the weight vector are calculated for storing patterns from biased input and output distributions derived within a one-step replica symmetry breaking (RSB) treatment. For unbiased output distribution and non-zero stability of the patterns, we find a critical load, α p, above which two solutions to the saddlepoint equations appear; one with higher free energy and zero threshold and a dominant solution with non-zero threshold. We examine this second-order phase transition and the dependence of α p on the required pattern stability, κ, for both one-step RSB and replica symmetry (RS) in the spherical case and for one-step RSB in the Ising case.

Laminar-turbulent transition in Raman fiber lasers:a first passage statistics based analysis

Loss of coherence with increasing excitation amplitudes and spatial size modulation is a fundamental problem in designing Raman fiber lasers. While it is known that ramping up laser pump power increases the amplitude of stochastic excitations, such higher energy inputs can also lead to a transition from a linearly stable coherent laminar regime to a non-desirable disordered turbulent state. This report presents a new statistical methodology, based on first passage statistics, that classifies lasing regimes in Raman fiber lasers, thereby leading to a fast and highly accurate identification of a strong instability leading to a laminar-turbulent phase transition through a self-consistently defined order parameter. The results have been consistent across a wide range of pump power values, heralding a breakthrough in the non-invasive analysis of fiber laser dynamics.

The correlation of structure and electronic properties near the surface of transition metal oxides

The strong couplings between different degrees of freedom are believed to be responsible for novel and complex phenomena discovered in transition metal oxides (TMOs). The physical complexity is directly responsible for their tunability. Creating surfaces/interfaces add an additional ' man-made' twist, approaching the quantum phenomena of correlated materials. ^ The dissertation focused on the structural and electronic properties in proximity of surface of three prototype TMO compounds by using three complementary techniques: scanning tunneling microscopy, angle-resolved photoelectron spectroscopy and low energy electron diffraction, particularly emphasized the effects of broken symmetry and imperfections like defects on the coupling between charge and lattice degrees of freedom. ^ Ca1.5Sr0.5RuO4 is a layered ruthenate with square lattice and at the boundary of magnetic/orbital instability in Ca2-xSrxRuO4. That the substitution of Sr 2+ with Ca2+ causing RuO6 rotation narrows the dxy band width and changes the Fermi surface topology. Particularly, the γ(dxy) Fermi surface sheet exhibited hole-like in Ca1.5Sr0.5RuO4 in contrast to electron-like in Sr2RuO4, showing a strong charge-lattice coupling. ^ Na0.75CoO2 is a layered cobaltite with triangular lattice exhibiting extraordinary thermoelectric properties. The well-ordered CoO2-terminated surface with random Na distribution was observed. However, lattice constants of the surface are smaller than that in bulk. The surface density of states (DOS) showed strong temperature dependence. Especially, an unusual shift of the minimum DOS occurs below 230 K, clearly indicating a local charging effect on the surface. ^ Cd2Re2O7 is the first known pyrochlore oxide superconductor (Tc ∼ 1K). It exhibited an unusual second-order phase transition occurring at TS1 = 200 K and a controversial first-order transition at TS2 = 120 K. While bulk properties display large anomalies at TS1 but rather subtle and sample-dependent changes at TS2, the surface DOS near the EF show no change at T s1 but a substantial increase below TS2---a complete reversal as the signature for the transitions. We argued that crystal imperfections, mainly defects, which were considerably enhanced at the surface, resulted in the transition at TS2. ^

Exploring thermal and mechanical properties of selected transition elements under extreme conditions: Experiments at high pressures and high temperatures

Transition metals (Ti, Zr, Hf, Mo, W, V, Nb, Ta, Pd, Pt, Cu, Ag, and Au) are essential building units of many materials and have important industrial applications. Therefore, it is important to understand their thermal and physical behavior when they are subjected to extreme conditions of pressure and temperature. This dissertation presents: • An improved experimental technique to use lasers for the measurement of thermal conductivity of materials under conditions of very high pressure (P, up to 50 GPa) and temperature (T up to 2500 K). • An experimental study of the phase relationship and physical properties of selected transition metals, which revealed new and unexpected physical effects of thermal conductivity in Zr, and Hf under high P-T. • New phase diagrams created for Hf, Ti and Zr from experimental data. • P-T dependence of the lattice parameters in α-hafnium. Contrary to prior reports, the α-ω phase transition in hafnium has a negative dT/dP slope. • New data on thermodynamic and physical properties of several transition metals and their respective high P-T phase diagrams. • First complete thermodynamic database for solid phases of 13 common transition metals was created. This database has: All the thermochemical data on these elements in their standard state (mostly available and compiled); All the equations of state (EoS) formulated from pressure-volume-temperature data (measured as a part of this study and from literature); Complete thermodynamic data for selected elements from standard to extreme conditions. The thermodynamic database provided by this study can be used with available thermodynamic software to calculate all thermophysical properties and phase diagrams at high P-T conditions. For readers who do not have access to this software, tabulated values of all thermodynamic and volume data for the 13 metals at high P-T are included in the APPENDIX. In the APPENDIX, a description of several other high-pressure studies of selected oxide systems is also included. Thermophysical properties (Cp, H, S, G) of the high P-T ω-phase of Ti, Zr and Hf were determined during the optimization of the EoS parameters and are presented in this study for the first time. These results should have important implications in understanding hexagonal-close-packed to simple-hexagonal phase transitions in transition metals and other materials.

Phase diagram of the bosonic double-exchange model

The phase diagram of the simplest approximation to double-exchange systems, the bosonic double-exchange model with antiferromagnetic (AFM) superexchange coupling, is fully worked out by means of Monte Carlo simulations, large-N expansions, and variational mean-field calculations. We find a rich phase diagram, with no first-order phase transitions. The most surprising finding is the existence of a segmentlike ordered phase at low temperature for intermediate AFM coupling which cannot be detected in neutron-scattering experiments. This is signaled by a maximum (a cusp) in the specific heat. Below the phase transition, only short-range ordering would be found in neutron scattering. Researchers looking for a quantum critical point in manganites should be wary of this possibility. Finite-size scaling estimates of critical exponents are presented, although large scaling corrections are present in the reachable lattice sizes.

Numerical detection of the Gardner transition in a mean-field glass former.

Recent theoretical advances predict the existence, deep into the glass phase, of a novel phase transition, the so-called Gardner transition. This transition is associated with the emergence of a complex free energy landscape composed of many marginally stable sub-basins within a glass metabasin. In this study, we explore several methods to detect numerically the Gardner transition in a simple structural glass former, the infinite-range Mari-Kurchan model. The transition point is robustly located from three independent approaches: (i) the divergence of the characteristic relaxation time, (ii) the divergence of the caging susceptibility, and (iii) the abnormal tail in the probability distribution function of cage order parameters. We show that the numerical results are fully consistent with the theoretical expectation. The methods we propose may also be generalized to more realistic numerical models as well as to experimental systems.

410 - Phase transitions induced by coloured voltage noise in nerve membranes

Quantitative conditions are derived under which electrically excitable membranes can undergo a phase transition induced by an externally applied voltage noise. The results obtained for a non-cooperative and a cooperative form of the two-state model are compared. © 1981.