Commensurability in one-dimensional lattices at finite temperature

The theory of commensurability transitions in one-dimensional atomic chains has been applied to charge density waves, mercury chain compounds, superionic conductors, etc. Previous numerical and analytical results on a model with chains of atoms with nearest neighbor interactions and periodic external potentials have dealt mainly with equilibrium and dynamical properties atT = 0 K. These studies are extended to nonzero temperatures. It is found that the reversible work per particle to slide the chain vanishes in the thermodynamic limit for any nonzero temperature. The mathematical pathologies associated with the commensurability transition atT= 0 (i.e., the devil's stair) are absent at finite temperature; only thermodynamic evidence of low-order Commensurability transitions remains.Supported by the Department of Energy, Contract number EG-77-S-02-4432.     

On the linear chain with arbitrary masses and force constants. II. Ergodic properties

The ergodic properties of linear and quadratic phase functions of the classical linear chain are studied for the uniform statistical distributions on the energy surface and in the manifold belonging to fixed values of the energy and the total momentum. This is done for the finite chain by using the time dependent correlation functions studied in a previous paper¹). The thermodynamic limit is also discussed. As an example, sufficient conditions on the masses and force constants are given to ensure that the kinetic energy of a certain particle remains nonergodic in the thermodynamic limit, the conditions defining a non-exceptional set of chains.     

The spin-1/2 Heisenberg Chains in Constant Magnetic Field and Coherent states1

In the ferromagnetic Heisenberg chains of XXX and XXZ types with the hidden symmetries of Lie bi-algebra su(2) and quantum bi-algebra su_q(2), we show that at thermodynamic limit the algebra contractions give the boson algebra h(4) and the q-deformed boion algebra h_q(2) as the hidden symmetries respectively. The chains in constant magnetic field are studied and the ground states and lowest excited states are given explicitly with energy spectra. The phonon (or angular momentum) excitations are shown to be bosonic for the isotropic case and q-bosonic for the anisotropic case, and the ground states and lowest excited states of the systems of the chains in field are given explicitly. We give the phonon coherent states in the isotropic Heisenberg chain and the q-coherent states of the anisotropic chain at the thermodynamic limit. The q-coherent states are shown to be a squeezed states of phonon excitations.     

Thermodynamic properties of extended Hubbard chains

In this contribution we discuss the thermodynamic behaviour of Hubbard chains with an additional site-diagonal interaction. It is shown rigorously that the strong coupling limit is equivalent to a generalized anisotropic Heisenberg system, which in some recently discussed limiting cases degenerates to an Ising orX-Y system. The whole procedure also works for a collection of chains with interchain interaction, thus it is not restricted to strictly one-dimensional systems. The effect of a magnetic field is discussed, and the results are compared with a recent perturbation approach by using numerical results for the anisotropic Heisenberg chain.     

Coefficient of restitution for one-dimensional harmonic solids

Using a numerical algorithm based on the time evolution of normal modes, we calculate the coefficient of restitution eta for various one-dimensional harmonic solids colliding with a hard wall. We find that, for a homogeneous chain, eta=1 in the thermodynamic limit. However, for a chain in which weaker springs are introduced in the colliding front half, eta remains significantly less than one even in the thermodynamic limit, and the "lost" energy goes mostly into low frequency normal modes. An understanding of these results is given in terms of how the energy is redistributed among the normal modes as the chain collides with the wall. We then contrast these results with those for collisions of one-dimensional harmonic solids with a soft wall. Using perturbation theory, we find that eta=1 for all harmonic chains in the extremely soft wall limit, but that inelasticity grows with increasing chain size in contrast to hard wall collisions.     

Van der Waals forces in electrolyte solutions

In this article we study van der Waals forces in electrolyte solutions from a local point of view. It is shown for arbitrary geometry that the classical limit of the force density exerted on the ions, as found in the macroscopic theory of van der Waals forces, is identical with the force density calculated from electrostatics and thermodynamic fluctuation theory. Thus neither retardation, nor the Lorentz force affect the average force density.     

横场中非束缚类准周期伊辛链的赝临界点

本文系统地研究了有限尺寸下非束缚类准周期量子伊辛链在横场中的赝临界点的行为. 首先, 通过计算平均磁矩和协作参量, 发现这两个量的导数随着横场的变化都会出现两个清晰的峰. 这与束缚类伊辛链和无序伊辛链的结果明显不同. 其次, 研究了横场中赝临界点的概率分布情况, 发现概率分布并不是高斯型的. 这也与无序的结果不同. 最后, 分析了赝临界点产生的原因, 发现赝临界点是由非束缚类准周期伊辛链中的集团结构造成的.     

On the microcanonical ensemble for a spin system interacting with one-mode electromagnetic field

A system described by the Dicke Hamiltonian is considered. It is known that for the canonical density operator describing such a system, the internal energyU is strictly negative. On the other hand, the microcanonical density operator can be defined for an arbitrary value ofU. The thermodynamic limit for the microcanonical density operator is investigated. In the caseU<0 the obtained results are completely equivalent to those of the canonical density operator. However, in the caseU>0 it turns out that the photon density is non-zero, and that the entropy and the mean spin are independent ofU. Moreover, forU>0 the coupling constant of the Hamiltonian does not appear in any thermodynamic formula after the thermodynamic limit is performed.     

Thermodynamic limit for classical systems with Coulomb interactions in a constant external field

We introduce a new method for studying the thermodynamic limit for systems of particles with Coulomb interactions. The method is based on calculating the potential energy of the Coulomb interactions from the electric or magnetic fields in the system rather than from the energy of the individual particle — particle interactions. We are able to include the effects of a constant external field being imposed at the boundary of the system. The difficulties associated with Coulomb potentials being not even weakly tempered are overcome by imposing the boundary condition that at the boundary of the region containing the particles, the electric or magnetic field has normal component equal to that of the applied field. We prove that the thermodynamic free energy density exists and is independent of the sequence of regions used to define the limit. We introduce sequences of regions all of the same shape and show that for these sequences of regions the thermodynamic free energy density is independent of shape. Finally, we prove that the thermodynamic free energy is a convex function of the density of particles and of the applied field.     

Study of the solid-liquid-vapour phase equilibria of flexible chain molecules using Wertheim's thermodynamic perturbation theory

The phase diagram of flexible molecules formed by freely-jointed tangent spheres is studied using the first-order thermodynamic perturbation theory of Wertheim for both fluid and solid phases. A mean-field term is added to the free energy of the fluid and solid phase in order to account for attractive dispersion forces. The approach is used to determine the global (solid-liquid-vapour) phase diagrams and triple points of chain molecules of increasing chain length. It is found that the triple point temperature is not affected strongly by the length of the chain, whereas the gas-liquid critical temperature increases dramatically. The asymptotic limits of the phase diagram for infinitely long chains are discussed. The reduced critical temperature of infinitely long chains as given by the mean-field theory is 2/3, and the reduced triple point temperature is 0.048 56, so that an asymptotic value of T _t/T _c = 0.07284 for the ratio of the triple to critical point temperatures is obtained. This indicates that fully-flexible tangent chains present an enormous liquid range. The proposed theory, while being extremely simple, provides a useful insight into the phase behaviour of chain molecules, showing the existence of finite asymptotic limits for the triple and critical point temperatures. However, since n-alkanes present an asymptotic limit of about T _t/T _c, = 0.40, the agreement With experiment is not quantitative. This suggests that fully flexible models may not be appropriate to model the solid phases of real chain molecules.     

Thermodynamic limit of correlation functions in a system of gravitating fermions

We show that the correlation functions in a system of gravitating fermions converge as tempered distributions in the thermodynamic limit, if the system is not at the point of phase-transition. The densities converge to the density of the Thomas-Fermi-theory and are not correlated in the limit.     

Absence of zero-temperature transmission rate of a double-chain tight-binding model for DNA with random sequence of nucleotides in thermodynamic limit

The zero-temperature transmission rate spectrum of a double-chain tight-binding model for real DNA is calculated. It is shown that a band of extended-like states exists only for finite chain length with strong inter-chain coupling. While the whole spectrum tends to zero in thermodynamic limit, regardless of the strength of inter-chain coupling. It is also shown that a more faithful model for real DNA with periodic sugar–phosphate chains in backbone structures can be mapped into the above simple double-chain tight-binding model. Combined with above results, the transmission rate of real DNA with long random sequence of nucleotides is expected to be poor.     

Chiral spin liquid wave function and the Lieb-Schultz-Mattis theorem

We study a chiral spin liquid wave function defined as a Gutzwiller projected BCS state with a complex pairing function. After projection, spontaneous dimerization is found for any odd but finite number of chains, thus satisfying the Lieb-Schultz-Mattis theorem, whereas for an even number of chains there is no dimerization. The two-dimensional thermodynamic limit is consistently reached for a large number of chains since the dimer order parameter vanishes in this limit. This property clearly supports the possibility of a spin liquid ground state in two dimensions with a gap to all physical excitations and with no broken translation symmetry.     

Particle injection into a chain: decoherence versus relaxation for Hermitian and non‐Hermitian dynamics

A model system for the injection of fermionic particles from filled source sites into an empty chain is investigated. The ensuing dynamics for Hermitian as well as for non‐Hermitian time evolution, where the particles cannot return to the bath sites (quantum ratchet), is studied. A non‐homogeneous hybridization between bath and chain sites permits transient currents in the chain. Non‐interacting particles show decoherence in the thermodynamic limit: the average particle number and the average current density in the chain become stationary for long times, whereas the single‐particle density matrix displays large fluctuations around its mean value. Using the numerical time‐dependent density‐matrix renormalization group (t‐DMRG) method it is demonstrated, on the other hand, that sizable density‐density interactions between the particles introduce relaxation which is by orders of magnitudes faster than the decoherence processes.     

Melting and crystallization transitions in organically modified layered silicates studied with differential scanning calorimetry

Melting and crystallization behaviors of the monolayers in the surface modified layered silicates were studied with differential scanning calorimetry. The layered silicates with different layer charge densities were used for the purpose. The ammonium ions forming the monolayer were also varied according to their chemical architecture, chain length and number of chains in the molecular structure. The quality of surface modification of the mineral was quantified using high resolution thermo-gravimetric analysis. At an optimum combination of the cross-sectional area of the ions, chain length as well as chain density, ordered monolayers are formed on the surface which turn to disordered state at a temperature corresponding to the melting of the ordered structure. These transitions are reproducible in nature but may require longer cooling times to regenerate again. The heat flow to the modified mineral (corresponding to melt enthalpy) increased on increasing the chain length as well as chain density, however, the transition temperatures decreased with increasing chain density in the surface modification molecules. The modified minerals were also hygroscopic in nature with first heating cycles showing the evaporation of water, however, with increasing chain length and chain density in the surface modification, the extent of hydrophobicity of the minerals increased. The effect of chemical architecture was not significant as it seemed to be driven by the chain length and capability of the modification molecules to crystallize in general or on the processing conditions used.     

On the thermodynamics of the random one-dimensional Ising chain in a transverse field

For three simple one-dimensional disordered models: (a) the Ising chain with random magnetic moments in a transverse field, (b) the Ising chain with random coupling constants in a transverse field, and (c) the X-Y model with a special type of disorder, the asymptotic equivalence in the thermodynamic limit is proved and some of its consequences are discussed. The spectral density of the finite chain for the model (a) is calculated by Dean's method for several representative cases and the presence of the local modes is indicated. The expressions for the initial susceptibilities for the models (a) and (b) are reviewed and (in two cases) the derivations are simplified.     

Curved polymer and polyelectrolyte brushes beyond the Daoud-Cotton model

We revise the classical Daoud-Cotton (DC) model to describe conformations of polymer and polyelectrolyte chains end-grafted to convex spherical and cylindrical surfaces. In the framework of the DC model, local stretching of chains in the brush does not depend on the degree of polymerization of grafted chains, and the polymer density profile follows a single-exponent power law. This model, however, does not correspond to a minimum in free energy of the curved brush. The nonlocal (NL) approximation exploited in the present paper implies the minimization of the overall free energy of the brush and predicts that the polymer density profile does not follow a single-exponent power law. In the limit of large surface curvature the NL approximation provides the same scaling laws for brush thickness and free energy as the local DC model. Numerical prefactors are however different. Extra extension of chains in the brush interior region leads to larger equilibrium brush thickness and lower free energy per chain. A significant difference between outcomes of the two models is found for brushes formed by ionic polymers, particularly for weakly dissociating (p H-sensitive) polyelectrolytes at low solution salinity.     

Finite-size scaling for correlations of quantum spin chains at criticality

We study the finite-size scaling behavior of two-point correlation functions of translationally invariant many-body systems at criticality. We propose an efficient method for calculating the two-point correlation functions in the thermodynamic limit from numerical data of finite systems. Our method is most effective when applied to a two-dimensional (classical) system which possesses a conformal invariance. By using this method with numerical data obtained from exact diagonalizations and Monte Carlo simulations, we study the spin-spin correlations of the quantum spin-1/2 and-3/2 antifierromagnetic chains. In particular, the logarithmic corrections to power-law decay of the correlation of the spin-1/2 isotropic Heisenberg antiferromagnetic chain are studied thoroughly. We clarify the cause of the discrepancy in previous calculations for the logarithmic corrections. Our result strongly supports the field-theoretic prediction based on the mappings to the Wess-Zumino-Witten nonlinear -model or the sine-Gordon model. We also treat logarithmic corrections and crossover phenomena in the spin-spin correlation of the spin-3/2 isotropic Heisenberg antiferromagnetic chain. Our results are consistent with the Affleck-Haldane prediction that the correlation of the spin-3/2 chain exhibits a crossover to the same asymptotic behavior as in the spin-1/2 chain.     

Structure of polymer solutions at interfaces: a Monte Carlo simulation study

The canonical ensemble Monte Carlo method is applied to study the structure of polymer solutions confined between surfaces. The polymer molecules are modeled as fused-sphere freely rotating chains with fixed bond length and bond angles and the solvent as hard spheres. The simulation results for the configurational and conformational properties of the chains are presented with varying interfacial distances, chain concentrations, and chain lengths. The chains are depleted at the wall at lower density, which, however, becomes less at higher density. With an increase in the interfacial distance, the enhancement/depletion of the chains at the wall becomes more marked. At all interfacial distances and chain lengths, increasing the concentration of the solvent makes the oscillation in the density profile of the chains more pronounced. Conformational properties provide important indications regarding the behaviour of chains as they approach surfaces.     

Thermodynamic limit in a charged plasma in equilibrium

We study the existence conditions for the thermodynamic limit in the chain of BBGKY equations for the equilibrium correlation functions of a charged plasma. It is shown that in order for the thermodynamic limit to exist the charge of the plasma cannot increase faster than the surface area of the plasma. When this condition is satisfied the equilibrium correlation functions of the charged plasma are asymptotically identical to the correlation functions of a neutral plasma in a self-consistent electrostatic field, which depends only on the one-particle correlation functions. For a plasma which is uniform everywhere except in a thin surface region, this field is found in explicit form. For a plasma occupying an infinite half-space, the problem is equivalent to a neutral plasma near a charged wall.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, 11–16, February, 1987.