Thermalization in Harmonic Particle Chains with Velocity Flips

We propose a new mathematical tool for the study of transport properties of models for lattice vibrations in crystalline solids. By replication of dynamical degrees of freedom, we aim at a new dynamical system where the “local” dynamics can be isolated and solved independently from the “global” evolution. The replication procedure is very generic but not unique as it depends on how the original dynamics are split between the local and global dynamics. As an explicit example, we apply the scheme to study thermalization of the pinned harmonic chain with velocity flips. We improve on the previous results about this system by showing that after a relatively short time period the average kinetic temperature profile satisfies the dynamic Fourier’s law in a local microscopic sense without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the above thermalization period is at most of the order $L^{2/3}$ , where $L$ denotes the number of particles in the chain. In particular, even before the diffusive time scale Fourier’s law becomes a valid approximation of the evolution of the kinetic temperature profile. As a second application of the dynamic replica method, we also briefly consider replacing the velocity flips by an anharmonic onsite potential.     

Index Theory of One Dimensional Quantum Walks and Cellular Automata

If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much “quantum information” as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems — namely quantum walks and cellular automata — we make this intuition precise by defining an index, a quantity that measures the “net flow of quantum information” through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S ₁, S ₂ can be “pieced together”, in the sense that there is a system S which acts like S ₁ in one region and like S ₂ in some other region, if and only if S ₁ and S ₂ have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S ₁ into S ₂. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map \({S \mapsto {\rm ind} S}\) is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.     

Approximate Lifshitz Law for the Zero-Temperature Stochastic Ising Model in any Dimension

We study the Glauber dynamics for the zero-temperature stochastic Ising model in dimension d ≥ 4 with “plus” boundary condition. Let ${\mathcal{T}_+}$ be the time needed for an hypercube of size L entirely filled with “minus” spins to become entirely “plus”. We prove that ${\mathcal{T}_+}$ is O(L ²(log L) ^c ) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called “Lifshitz law” ${\mathcal{T}_{+} = O(L^{2})}$ (Fischer and Huse in Phys Rev B 35:6841–6848, 1987; Lifshitz in Sov Phys JETP 15:939–942, 1962) conjectured on heuristic grounds. The key point of our proof is to use the detailed knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from Caputo et al. (Comm Pure Appl Math 64:778–831, 2011) to get the result in higher dimension.     

More than six hundred new families of Newtonian periodic planar collisionless three-body orbits

The famous three-body problem can be traced back to Isaac Newton in the 1680 s. In the 300 years since this "three-body problem"was first recognized, only three families of periodic solutions had been found, until 2013 when ˇSuvakov and Dmitraˇsinovi′c [Phys.Rev. Lett. 110, 114301(2013)] made a breakthrough to numerically find 13 new distinct periodic orbits, which belong to 11 new families of Newtonian planar three-body problem with equal mass and zero angular momentum. In this paper, we numerically obtain 695 families of Newtonian periodic planar collisionless orbits of three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known figure-eight family found by Moore in 1993, the 11 families found by ˇSuvakov and Dmitraˇsinovi′c in 2013, and more than 600 new families that have never been reported, to the best of our knowledge. With the definition of the average period T = T=Lf, where Lf is the length of the so-called "free group element", these 695 families suggest that there should exist the quasi Kepler's third law T* ≈ 2:433 ± 0:075 for the considered case, where T*= T|E|~(3/2) is the scale-invariant average period and E is its total kinetic and potential energy,respectively. The movies of these 695 periodic orbits in the real space and the corresponding close curves on the "shape sphere"can be found via the website: http://numericaltank.sjtu.edu.cn/three-body/three-body.htm.     

Phase transitions in a holographic s $$$$ p model with back-reaction

In a previous paper (Nie et al. in JHEP 1311:087, arXiv:1309.2204 [hep-th], 2013), we presented a holographic s \(+\) p superconductor model with a scalar triplet charged under an SU(2) gauge field in the bulk. We also study the competition and coexistence of the s-wave and p-wave orders in the probe limit. In this work we continue to study the model by considering the full back-reaction. The model shows a rich phase structure and various condensate behaviors such as the “n-type” and “u-type” ones, which are also known as reentrant phase transitions in condensed matter physics. The phase transitions to the p-wave phase or s \(+\) p coexisting phase become first order in strong back-reaction cases. In these first order phase transitions, the free energy curve always forms a swallow tail shape, in which the unstable s \(+\) p solution can also play an important role. The phase diagrams of this model are given in terms of the dimension of the scalar order and the temperature in the cases of eight different values of the back-reaction parameter, which show that the region for the s \(+\) p coexisting phase is enlarged with a small or medium back-reaction parameter but is reduced in the strong back-reaction cases.     

High transverse momentum identified particle spectra in 200 GeV collisions from STAR

Significant baryon over meson enhancement was measured at RHIC in the intermediate transverse momentum range of p _T =2?4 GeV/c (“baryon-meson puzzle”). With STAR detector we were able to extend particle identification towards higher transverse momentum offering further insights into the particle production mechanisms at intermediate to high p _T . In this paper we present results on charged pion, proton and anti-proton spectra and ratios at intermediate to high p _T exploiting the relativistic rise of the specific ionization energy loss measured in the STAR Time Projection Chamber. These measurements provide valuable information about the production mechanisms of particles at intermediate p _T in relativistic heavy ion collisions, e.g. coalescence/recombination versus jet fragmentation.     

Nuclear spin structure in dark matter search: The finite momentum transfer limit

Spin-dependent elastic scattering of weakly interacting massive dark matter particles (WIMP) off nuclei is reviewed. All available, within different nuclear models, structure functions S(q) for finite momentum transfer (q > 0) are presented. These functions describe the recoil energy dependence of the differential event rate due to the spin-dependent WIMP-nucleon interactions. This paper, together with the previous paper “Nuclear Spin Structure in Dark Matter Search: The Zero Momentum Transfer Limit,” completes our review of the nuclear spin structure calculations involved in the problem of direct dark matter search.     

Ideal Gas with a Varying (Negative Absolute) Temperature: an Alternative to Dark Energy?

The present work is an attempt to investigate whether the evolutionary history of the Universe from the offset of inflation can be described by assuming the cosmic fluid to be an ideal gas with a specific gas constant but a varying negative absolute temperature (NAT). The motivation of this work is to search for an alternative to the “exotic” and “supernatural” dark energy (DE). In fact, the NAT works as an “effective quintessence” and there is need to deal neither with exotic matter like DE nor with modified gravity theories. For the sake of completeness, we release some clarifications on NATs in Section 3 of the paper.     

Metastability for the Ising Model on the Hypercube

We consider Glauber dynamics for the low-temperature, ferromagnetic Ising Model on the n-dimensional hypercube. We derive precise asymptotic results for the crossover time (the time it takes for the dynamics to go from the configuration with a “\(-1\)” at every vertex, to the configuration with a “\(+1\)” at each vertex) in the limit as the inverse temperature \(\beta \rightarrow \infty \).     

Constraints and entropy in a model of network evolution

Barabási–Albert’s “Scale Free” model is the starting point for much of the accepted theory of the evolution of real world communication networks. Careful comparison of the theory with a wide range of real world networks, however, indicates that the model is in some cases, only a rough approximation to the dynamical evolution of real networks. In particular, the exponent γ of the power law distribution of degree is predicted by the model to be exactly 3, whereas in a number of real world networks it has values between 1.2 and 2.9. In addition, the degree distributions of real networks exhibit cut offs at high node degree, which indicates the existence of maximal node degrees for these networks. In this paper we propose a simple extension to the “Scale Free” model, which offers better agreement with the experimental data. This improvement is satisfying, but the model still does not explain why the attachment probabilities should favor high degree nodes, or indeed how constraints arrive in non-physical networks. Using recent advances in the analysis of the entropy of graphs at the node level we propose a first principles derivation for the “Scale Free” and “constraints” model from thermodynamic principles, and demonstrate that both preferential attachment and constraints could arise as a natural consequence of the second law of thermodynamics.     

AQFT from <Emphasis Type="Italic">n</Emphasis>-Functorial QFT

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses n-functors which assign to each patch a “propagator of states”.In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor (“parallel surface transport”) naturally yields a local net, whose locality derives from the 2-categorical exchange law, and which is covariant if the 2-functor is equivariant. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general Lorentzian structure, bare lightcone structure and higher dimensions. It does not, however, by itself imply anything about the existence of a vacuum state or about positive energy representations.     

Theoretische Untersuchungen über die Lage des Zwischengitteratoms in Kupfer

The different stable atomic configurations, formation energies and changes in volume of the crystal for an interstitial in copper are calculated with the help of the electronic digital computer Z 22 using a general method developed byTewordt. For the interaction between a pair of ions the Born-Mayer potentialV ₁, given byHuntington, and the Morse potentialV _M, given byGirifalco-Weizer, are employed. Two equilibrium configurations for an interstitial are found. In the stable configuration“A” the interstitial and one next neighboured atom are symmetrically located relative to one of the elementary cube faces along a cubic axis passing through the cube center. In the stable configuration“B” the interstitial and one next neighboured atom are symmetrically located relative to a cube corner along a {111}-axis. The interstitial is found not to reside at the center of an elementary cube. Neglecting electronic contributions to the relaxation of the lattice due to the redistribution of the electrons the calculations showed that the interstitial moved along a cubic axis about 0.4a/2 away from the elementary cube center into a stable configuration“A”. Moreover the crowdion is found to be unstable. It is shown that the crowdion decays into an interstitial lying in a next neighboured configuration“A”. The configuration“B” is separated from surrounding“A” and unstable “body-centered” and “crowdion” configurations by energy barriers. The number of atoms around the mobile interstitial treated as movable discrete particles is about 150 for the configuration“A” and about 50 for the configuration“B”. Using the Born-Mayer potentialV ₁ the changes in volume of the crystal arising from the interstitial are found to be 1.126 atomic volumes for the configuration“A” and 1.432 atomic volumes for the configuration“B”. The contributions to the formation energy of an interstitial arising from the potentialV ₁ turn out to be 3.548 eV for the configuration“A” and 4.098 eV for the configuration“B”. The results of the theoretical calculations are discussed in connection with recent radiation damage experiments performed at low temperatures on copper.     

Theory of the Half-integer Quantum Hall Effect in Graphene

The unusual quantum Hall effect (QHE) in graphene is described in terms of the composite (c-) bosons, which move with a linear dispersion relation. The “electron” (wave packet) moves easier in the direction [1 1 0 c-axis] ≡ [1 1 0] of the honeycomb lattice than perpendicular to it, while the “hole” moves easier in [0 0 1]. Since “electrons” and “holes” move in different channels, the particle densities can be high especially when the Fermi surface has “necks”. The strong QHE arises from the phonon exchange attraction in the neighborhood of the “neck” surfaces. The plateau observed for the Hall conductivity and the accompanied resistivity drop is due to the superconducting energy gap caused by the Bose-Einstein condensation of the c-bosons, each forming from a pair of one-electron–two-fluxons c-fermions by phonon-exchange attraction. The half-integer quantization rule for the Hall conductivity: (1/2)(2P?1)(4e²/h), P=1,2,..., is derived.     

The Elusive p-air cross section

For the \(\bar pp\) and pp systems, we have used all of the extensive data of the Particle Data Group [K. Hagiwara et al. (Particle Data Group), Phys. Rev. D 66, 010001 (2002)]. We then subject these data to a screening process, the “Sieve” algorithm [M. M. Block, physics/0506010], in order to eliminate “ outliers” that can skew a χ² fit. With the “Sieve” algorithm, a robust fit using a Lorentzian distribution is first made to all of the data to sieve out abnormally high Δχ _i ² , the individual i^th point’s contribution to the total χ². The χ² fits are then made to the sieved data. We demonstrate that we cleanly discriminate between asymptotic ln s and ln² s behavior of total hadronic cross sections when we require that these amplitudes also describe, on average, low energy data dominated by resonances. We simultaneously fit real analytic amplitudes to the “sieved” high energy measurements of \(\bar pp\) and pp total cross sections and ρ-values for \(\sqrt s \) ≥ GeV, while requiring that their asymptotic fits smoothly join the the σ _pp and σ_pp total cross sections at \(\sqrt s \) = 4.0 GeV—again both in magnitude and slope. Our results strongly favor a high energy ln² s fit, basically excluding a ln s fit. Finally, we make a screened Glauber fit for the p-air cross section, using as input our precisely-determined pp cross sections at cosmic ray energies.     

Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization

Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N?1)-dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the “trust region subproblem” or “constraint least square problem”. When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number $\mathcal{N}_{tot}$ of critical (stationary) points in the cost function landscape. In the first regime $\mathcal{N}_{tot}$ remains of the order N and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor.