On the critical behaviour of a surface interacting linear polymer chain

A method based on a real space renormalization group transformation is developed to describe the critical behaviour of a surface interacting linear flexible polymer chain, represented by a self-avoiding walk. It is shown that a lattice model based on a central rule in which the starting point of the walk and the surface are taken to be in the middle of one cell, provides a suitable framework to study both the penetrable and impenetrable surfaces. In contrast to this, a method based on a corner rule in which the starting point of the walk and the surface are fixed to be one corner of a cell cannot describe the behaviour of a chain interacting with an impenetrable surface. The value of crossover exponent found by us for a square lattice are in agreement with those expected to be exact for both the cases.     

On the dynamics of the classical Heisenberg chain

We investigate the space- and time-dependent spin-correlation-function for a linear chain of classical magnetic moments coupled by a purely isotropic exchange interaction. The correlation functions are given in terms of relative variables which at low temperatures turn out to be the normal coordinates appropriate to a system without long-range-order. Analytic expressions for the correlation functions, including correctly the fluctuations responsible for the absence of long-range-order, are given at low temperatures. Well-defined spin wave oscillations exist for wavelengths smaller than the correlation length. In this region our results are consistent with experimental data for the isotropic Heisenberg antiferromagnet TMMC.     

On asymptotic behaviour of four spin correlation functions in a linear chain of spins

Second and fourth moments of the four spin frequency correlation functions S^Z_i(t) S^Z_i+1 (t) S^Z_j+1 〉_k,ω and 〈S^x_i(t) S^z_i+1 (t) S^x_jS^z_j+1 〉_kω have been calculated in the high temperature limit for an arbitrary spin with an isotropic Heisenberg Hamiltonian.     

On the existence of the thermodynamic limit for classical systems with nonspherical potentials

It is shown, that the configurational partition function for a classical system of molecules interacting with nonspherical pair potential is proportionals to the configurational partition function for a system of particles interacting with temperature-dependent spherical k-body potentials (k ?2). Therefore, the thermodynamic limit for nonspherical molecules exists if the effective k-body interaction is stable and tempered. A number of criteria for the nonspherical potential are developed which ensure these properties. In case the nonsphericity is small in a certain sense, stability and temperedness of the angle-averaged nonspherical potential are sufficient to ensure thermodynamic behaviour.     

On the derivation of power-law distributions within classical statistical mechanics far from the thermodynamic limit

We show that within classical statistical mechanics, without taking the thermodynamic limit, the most general Boltzmann factor for the canonical ensemble is a q-exponential function. The only assumption here is that microcanonical distributions have to be separated from the total system energy, which is the prerequisite for any sensible measurement. We derive that all separable distributions are parametrized by a mathematical separation constant Q, which can be related to the non-extensivity q-parameter in Tsallis distributions. We further demonstrate that nature fixes the separation constant Q to 1 for large dimensionality of Gibbs $\Gamma$-phase space. Our results will be relevant for systems with a low-dimensional $\Gamma$-space, for example nanosystems, comprised of a small number of particles, or for systems with a dimensionally collapsed phase space, which might be the case for a large class of complex systems.     

A new criterion for the mixed crystal behaviour in the diatomic linear chain model

Lucovsky, Brodsky, Burstein (LBB) have studied the behaviour of mixed crystals by setting up a criterion for the existence of local mode frequencies in real crystals starting from a diatomic linear chain model. This, while successfully predicting the one and two mode behaviour for some systems fails to predict the mixed mode behaviour. We propose a similar criterion for the existence of gap modes, by demanding that the gap mode predicted by the diatomic linear chain model should lie within the gap of the real three dimensional solid for its existence. It is shown that the gap modes for various systems calculated using this criterion are in reasonable agreement with the experimental values. The infrared behaviour of mixed crystals has to be determined by examining the existence of local as well as gap modes for the two end members of the system. This generalized new criterion successfully predicts the mixed mode behaviour of III–V mixed crystals besides predicting the one and two mode behaviour, observed in infrared absorption of mixed alkali halioes and III-V compounds.     

On the influence of non-analytic terms on the Gell-Mann-Okubo relation for the pseudoscalar mesons

Li and Pagels have demonstrated that one can understand the pseudoscalar mesons coming off the chiral limit in an octet-broken fashion. One might therefore hope to be able to attribute the major part of the error in the Gell-Mann-Okubo relation to the next leading contributions to the masses, which are O(λ²ln λ) where λ is the breaking parameter. In the following the effect that such terms may have is considered in the $\text{(3,}\text{3}\text{) + (}\text{3}\text{,3), (8,8)}\text{and}\text{(6,}\text{6}\text{) + (}\text{6}\text{,6)}$ models of chiral-symmetry breaking. Of these the (8,8) fares the best.     

Phase transition in a linear chain of classical spins with a logarithmic nearest neighbour pair potential

We present the complete calculation of the partition function and correlation functions of a linear array of classical spins coupled by a nearest neighbour logarithmic pair potential. In the case of a ferromagnetic coupling there occurs a phase transition at T_c > 0. The critical exponents of the specific heat C and the magnetic susceptibility χ (in the absence of an external field) are shown to have the non-classical value α = 2 and classical value γ = 1 respectively. The underlying mathematical mechanism of the phase transition is the complete degeneracy of all the eigenvalues of the corresponding integral equation (Kac's mechanism). Below T_c the partition function becomes complex. For antiferromagnetic coupling the free energy is analytic in the whole temperature range and so no phase transition occurs in this case.     

A complementary thermodynamic limit for classical Coulomb matter

The canonical equilibrium measure of classical two-component Coulomb matter with regularized interactions is analyzed in a finite volume. It is shown that, in the mean-field regime, the one-particle density is inhomogeneous on a new characteristic length scale _inh. For a system ofN positive andN negative particles, _inh and the characteristic length scale of correlations _corr (=Debye screening length) are related via _inh=(2N)^1/2 _corr. The major conceptual conclusion that is drawn from this is that one needs two nontrivial complementary thermodynamic limits to define the equilibrium thermodynamics of two-component Coulomb systems. One of them is the standard thermodynamic limit (infinite volume), where one takesN, _corr fixed. Its complementary limit is characterized byN, _inh fixed, and is a finite-volume inhomogeneous mean-field limit. The most prominent new feature in the mean-field thermodynamic limit, which is absent in the standard thermodynamic limit, is an anomalous first-order phase transition where the Coulomb system explodes or implodes, respectively. The phase transition is connected with the existence of a metastable plasma phase far below the ionization temperature.     

Kink solutions of the classical transverse field Ising chain

We investigate stationary and travelling wave solutions of the classical one-dimensional transverse field Ising model. Results are given on the existence, shape and stability of kink solutions and periodic solutions. We review recent analytical results (e.g., the proof of existence of a one-parameter family of stationary kink solutions and the proof of existence of travelling wave kink solutions with nonzero velocity c≠ 0) and extend them by the use of numerical methods. Small oscillations arising in the tails of travelling kink solutions are investigated numerically. In the end, stability analysis puts some light on pinning effects. Received 23 February 2001 and Received in final form 4 October 2001     

The convergence behaviour of expansions in the classical collision theory

In the classical collision theory the scattering angle? depends on the impact parameterb and on the kinetic energyE _r of the relative motion. This angle?(b, E _r ) is expanded for two limiting cases: 1. Expansion in powers of the potentialV(r)/E _r (momentum approximation). 2. Expansion in powers of the impact parameterb (central collision approximation). The radius of convergence of the series depends onb andE _r . It will be given for the following potentialsV(r):

$$A\left( {\frac{a}{r}} \right)^\mu ;Ae^{ - \frac{r}{a}} ;A\frac{a}{r}e^{ - \frac{r}{a}} ;A\left( {\frac{a}{r}} \right)^2 e^{ - \left( {\frac{r}{a}} \right)^2 } .$$     

An exact partition function for the classical Dzialoshinsky-Moriya chain

The exact partition function for the classical anisotropic chain with antisymmetric exchange is evaluated. For certain values of parameters in the hamiltonian, it is seen that the partition function is the same as the classical isotropic chain partition function.     

Asymptotic behaviour of the classical scalar fields and topological charges

The existence and the properties of the limit at spatial infinity are studied for the finite-energy scalar fields with respect to the topological charge introduction. The limit is shown to be constant in time and in almost all spatial directions. The proof of the existence of the limit given by Parenti, Strocchi and Velo is extended to two-dimensional space. A generalized definition of the topological charge is suggested for a -model as an example.     

Correlation inequalities and the thermodynamic limit for classical and quantum continuous systems

We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.Research supported in part by the U.S. National Science Foundation under grant MPS 75-11864A Sloan Foundation Fellow     

Wave function of classical particle in linear potential

We study the problem of classical particle in linear potential using the formalism of Hilbert space and tomographic-probability distribution. We solve the Liouville equation for this problem by finding the density matrix satisfying a von Newmann-like equation in the form of a product of the wave functions. We discuss the relation of the classical solution obtained to quantum mechanics.     

On thermodynamic properties of the chromoplasma

We calculate the contributions of electric and magnetic modes to some thermodynamic functions in SU(2)-gauge theory on the lattice 4×12³. It is shown that the behaviour of the chromoelectric part of the energy E_E can be interpreted within the Ising-type model in agreement with the universality hypothesis. At the same time the behaviour of the magnetic parts of the internal energy and pressure (i.e. E_M and P_M) differs drastically from that of E_E and P_M. The character of the temperature dependence of E_M and P_M exhibited here testifies to the presence of highly nonidentical properties of electric and magnetic excitation modes of chromoplasma and may shed light on the role of unstable modes in gauge theories.