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.     

Superdegenerate point in FCC phase diagram: CVM and Monte Carlo investigations

We investigate the topology of the phase diagram of binary alloys on the fee lattice with first-neighbor antiferromagnetic interactions around the superdegenerate point, where the L1₀ and L1₂ phases meet. We treat the system as a hard-constraint lattice gas, following a procedure previously described by Lebowitzet al. We perform cluster variation method calculations in theT0 limit and Monte Carlo simulations directly atT=0 K on the ground states of the superdegenerate point. We find that: (i) there is no disordered phase in the neighborhood of this point; (ii) a phase L for which two of the four cubic sublattices have the same average occupancy and each of the two others are different appears between L1₀ and L1₂; (iii) the transition L/L1₂ is of first order.     

Infinite-order phase transition in a classical spin system

For an exactly soluble classical spin model with long-range inhomogeneous coupling it is proved that in the absence of external magnetic field the free energy is aC function of the temperature at the critical point.     

The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

Two models for the Freedericksz transition in a fluctuating magnetic field are considered: one is based on a dichotomic and the other on a nonlinear Gaussian noise. Both noises are characterized by a finite correlation time. It is shown that the linear response assumption leading to the best Fokker-Planck approximation in the dichotomic and nonlinear Gaussian cases can be trusted only up to the order ¹ and ⁰, respectively. The role of the corrections to the linear response approximation is discussed and it is shown how to replace the non-Fokker-Planck terms stemming from these corrections with equivalent terms of standard type. This technique is shown to produce perfect agreement with the exact analytical results (dichotomic noise) and to satisfactorily fit the results of analog simulation (nonlinear Gaussian noise).     

Solvable isotherms for a two-component system of charged rods on a line

The Coulomb system consisting of an equal number of positive and negative charged rods confined to a one-dimensional lattice is studied. The grand partition function can be calculated exactly at two values of the coupling constant=q ²/k _B T (q denoting the magnitude of the charges). The exact results lead to the conjecture that in the complex scaled fugacity plane, all the zeros of the grand partition function lie on the negative real axis for<2, on the point=–1 for=2, and on the unit circle for>2. In addition, for>4, we conjecture in general and prove at=4 that the zeros pinch the real axis in the thermodynamic limit, with an essential singularity in the pressure at the reduced density 1/2.     

Spatial fluctuations in reaction-limited aggregation

A general method is used for describing reaction-diffusion systems, namely van Kampen's method of compounding moments, to study the spatial fluctuations in reaction-limited aggregation processes. The general formalism used here and in subsequent publications is developed. Then a particular model is considered that is of special interest, since it describes the occurrence of a phase transition (gelation). The corresponding rate constants for the reaction between two clusters of sizei and sizej areK _ij=ij (i, j=1, 2,). For thediffusion constants D _j of clusters of sizej the following class of models is considered:D _j=D if 1Js andD _j=0 ifj>s. The casess= ands< are studied separately. For the models= the equal-time and the two-time correlation functions are calculated; this modelbreaks down at the gel point. The breakdown is characterized by a divergence of the density fluctuations, and is caused by the large mobility of large clusters. For all models withs< the density fluctuations remain finite att _c, and the equal-time correlation functions in the pre- and in the post-gel stage are calculated. Many explicit and asymptotic results are given. From the exact solution the upper critical dimension in this gelling model isd _c=2.     

Ground state of a spin-phonon system. I. Variational estimates

A study is made of the ground-state energy of a spin-one-half particle in a fieldB and interacting with a phonon bath. The infrared-sensitive case of acoustic phonons with point coupling in three dimensions is characterized by two parameters, a coupling constant andB. Units are used where the high-momentum phonon cutoff is unity. There is a curve (B) separating a symmetry-breaking region with a long-range phonon field from a normal region. Two simple, well-known, approximations are compared. The source theory yields discontinuities in the first derivatives of the energy with respect toB and whenB>e ^–1 and an infinite-order transition whenB<e ^–1, but is trivial in the large- region. The classical theory yields discontinuities in the second derivatives but is trivial in the small- region. An improved variationally fixed ground-state wave function is analyzed. It gives a new (B) curve with an infinite-order transition with continuous energy derivatives whenB<e/(e ²–1/4) and with discontinuous derivatives whenB is larger than this value. It is nontrivial in the entire (B) plane. The crossover to classical behavior occurs near =1/2 forB1. But the wave function does not describe quantum fluctuations in the large- phase. A second way of combining source and classical effects is described. It yields a second-order transition (near =1/2 forB1) everywhere. These theories are special cases of a symmetry-breaking transformation together with a one-mode treatment of quantum fluctuations. The transition is viewed in terms of a single mode with a variable length, coupled dynamically to the spin.     

Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.     

Dynamics of the current-driven Josephson junction

The irreversible macroscopic dynamics of the Josephson junction coupled to external wires acting as a current source is derived rigorously from the underlying microscopic Hamiltonian quantum mechanics. The external systems are treated in the singular coupling limit. The use of this limit is explicitly justified via an interpretation of the singular coupling limit in terms of the relative magnitudes of system, reservoir, and coupling energies. The qualitative behavior of the macroscopic dynamical equations is shown to depend sensitively and crucially on the interaction between the wires and the superconductors and on the size of the wires: the dc Josephson effect only happens when one lets Cooper pairs be driven into the junction by collective (i.e., small) reservoirs.