Entropy of the Monomer–Trimer System on a Hierarchical Lattice

Heat capacity, isothermal compressibility, isosteric heat of adsorption and thermal expansion of water confined in C-S-H

The problems of a freely rotating dipole interacting with a charge, and of a freely rotating dipole interacting with a polarizable surface (dielectric discontinuity), are solved here. Fully analytic expressions for angle-independent, i.e., radially symmetric, effective interaction potentials are derived and analyzed for their asymptotic forms. For the charge-dipole case the exact result takes a simple enough form to permit direct implementation in statistical mechanical calculations, and displays the appropriate limiting forms at large and small separations. The dipole-surface interaction potential is slightly more involved analytically but is nevertheless amenable for application in statistical thermodynamic study of bulk systems. Order parameter analyses demonstrate the orientational behavior of the dipole at small, large, and intermediate distances.

It is shown that generalized gradient approximations (GGAs) for exchange only, due to their very limited form, quite generally can not simultaneously reproduce both the asymptotic forms of the exchange energy density and the exchange potential of finite systems. Furthermore, mechanisms making GGAs formally approach at least one of these asymptotic forms do not improve the corresponding quantity in the relevant part of the asymptotic regime of atoms. By constructing a GGA which leads to superior atomic exchange energies compared to all GGAs heretofore but does not reproduce the asymptotic form of the exact exchange energy density it is demonstrated that this property is not important for obtaining extremely accurate atomic exchange energies. We conclude that GGAs by their very concept are not suited to reproduce these asymptotic properties of finite systems. As a byproduct of our discussion we present a particularly simple and direct proof of the well known asymptotic structure of the exchange potential of finite spherical systems.

We consider an Ising model on the triangular Apollonian network (AN), with a thermalized distribution of vacant sites. The statistical problem is formulated in a grand canonical ensemble, in terms of the temperature T and a chemical potential μ associated with the concentration of active magnetic sites. We use a well-known transfer-matrix method, with a number of adaptations, to write recursion relations between successive generations of this hierarchical structure. We also investigate the analogous model on the diamond hierarchical lattice (DHL). From the numerical analysis of the recursion relations, we obtain various thermodynamic quantities. In the μ→∞ limit, we reproduce the results for the uniform models: in the AN, the system is magnetically ordered at all temperatures, while in the DHL there is a ferromagnetic-paramagnetic transition at a finite value of T. Magnetic ordering, however, is shown to disappear for sufficiently large negative values of the chemical potential.

This paper studies the error correction problem for bit-shift channels with the so-called (d,k) input constraints (where successive 1’s are required to be separated by at least d and at most k zeros). Bounds on the size of optimal (d,k)-constrained codes correcting any given number of bit-shifts are derived, with a focus on their asymptotic form in the large block-length limit. The upper bound is obtained by a packing argument, while the lower bound follows from a construction based on a family of integer lattices. Several properties of (d,k)-constrained sequences that may be of independent interest are established as well; in particular, the exponential growth rate of the number of (d,k)-constrained constant-weight sequences is characterized.

We consider the structure of the operator product expansion (OPE) in conformal field theory by employing the OPE block formalism. The OPE block acted on the vacuum is promoted to an operator and its implications are examined on a non-vacuum state. We demonstrate that the OPE block is dominated by a light-ray operator in the Regge limit, which reproduces precisely the Regge behavior of conformal blocks when used inside scalar four-point functions. Motivated by this observation, we propose a new form of the OPE block, called the light-ray channel OPE block that has a well-behaved expansion dominated by a light-ray operator in the Regge limit. We also show that the two OPE blocks have the same asymptotic form in the Regge limit and confirm the assertion that the Regge limit of a pair of spacelike-separated operators in a Minkowski patch is equivalent to the OPE limit of a pair of timelike-separated operators associated with the original pair in a different Minkowski patch.

The plaquette expansion, a general non-perturbative method for calculating the properties of lattice Hamiltonian systems, is analytically investigated at the first non-trivial order for an arbitrary system. At this level the approximation describes systems with either a bounded or an unbounded spectrum, depending on simple inequalities formed from the first four cumulants. The analysis yields analytic forms for the ground state energy, mass gaps and density of states in the thermodynamic limit. An exact form for the resolvent oeprator is given for a finite number of sites, as well as the asymptotic form in the thermodynamic limit.

L-ensembles are a class of determinantal point processes which can be viewed as a statistical mechanical systems in the grand canonical ensemble. Circulant L-ensembles are the subclass which are locally translationally invariant and furthermore subject to periodic boundary conditions. Existing theory can very simply be specialised to this setting, allowing for the derivation of formulas for the system pressure, and the correlation kernel, in the thermodynamic limit. For a one-dimensional domain, this is possible when the circulant matrix is both real symmetric, or complex Hermitian. The special case of the former having a Gaussian functional form for the entries is shown to correspond to free fermions at finite temperature, and be generalisable to higher dimensions. A special case of the latter is shown to be the statistical mechanical model introduced by Gaudin to interpolate between Poisson and unitary symmetry statistics in random matrix theory. It is shown in all cases that the compressibility sum rule for the two-point correlation is obeyed, and the small and large distance asymptotics of the latter are considered. Also, a conjecture relating the asymptotic form of the hole probability to the pressure is verified.

We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Levy's distributions with a power-law decay at-∞, and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtain the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.

The asymptotic form of the density of states for a p-brane is computed (using the semi-classical approximation for the expression of the mass formula), and a value is found which only resembles the exponential growth of black holes in the formal limit p → ∞. Some physical consequences for the thermodynamics of these objects are also presented.

The asymptotic form of partition functions of linear chain molecules: An application of renewal theory

The leading- and higher-twist distribution amplitudes and light-cone wave functions of real and virtual photons are analyzed in chiral quark models. The calculations are performed in the nonlocal quark model based on the instanton picture of the QCD vacuum, as well as in the spectral quark model and the Nambu--Jona-Lasinio model with the Pauli-Villars regulator, which both treat interaction of quarks with external fields locally. We find that in all considered models the leading-twist distribution amplitudes of the real photon defined at the quark-model momentum scale are constant or remarkably close to the constant in the $x$ variable, thus are far from the asymptotic limit form. The QCD evolution to higher momentum scales is necessary and we carry it out at the leading order of the perturbative theory for the leading-twist amplitudes. We provide estimates for the magnetic susceptibility of the quark condensate ${\ensuremath{\chi}}_{\mathrm{m}}$ and the coupling ${f}_{3\ensuremath{\gamma}}$, which in the nonlocal model turn out to be close to the estimates from QCD sum rules. We find the higher-twist distribution amplitudes at the quark model scale and compare them to the Wandzura-Wilczek estimates. In addition, in the spectral model we evaluate the distribution amplitudes and light-cone wave functions of the $\ensuremath{\rho}$-meson.

To filter an image by an aperture function A(r), or in Fourier space by its transform A(f), requires that A(r) and A(f) are physically realizable and mathematically exist as transform pairs. This constrains both functions to being square-integrable to guarantee their uniform convergence both in spatial and frequency domains. This paper presents a general method for deriving filter forms which maximize image SNR and also satisfy the existence constraints. We show that asymptotic limiting forms of the constrained filters reduce to the "ideal observer" result, but demonstrate that the "ideal" filter form has a divergent square integral in some simple cases important in medical imaging. We apply these properly constrained filter forms to radiographic image and CT image models, and derive the range of arguable signal-to-noise advantage, which "more" optimum filters may exhibit with respect to the simple "matched" filter limit.

The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the general CFT 2 with c > 1. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit z → 1, which was unknown until now. In this study, we computed it in general by studying the pole structure of the fusion matrix (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value frac{c-1}{12} if the total Liouville momentum exceeds beyond the BTZ threshold. This might be interpreted as a black hole formation in AdS3.As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench.We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general n-th Renyi entropy after a local quench and out of time ordered correlators.

The many-body Green functions for the three-dimensional electron–phonon Holstein model are calculated via the Feynman diagram technique. The lowest-order contributions to the one-electron self-energy are considered (the Hartree–Fock approximation). The chemical potential is adjusted to yield the desired band-filling. The temperature dependence of the chemical potential has the correct asymptotic form in the non-interacting limit. The behavior of the one-electron occupancy factor is examined for various parameter values of the model. Tian's theorem is satisfied at half-filling. Luttinger liquid behavior is observed in the strong electron–phonon coupling regime.