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.     

Determination of the optimal initial function and the parameters for nonlinear time-delay systems

This paper deals with dynamic systems described by nonlinear differential-difference equations of retarded type. The problem considered is to determine the initial function and certain system parameters which minimize a given cost functional. A computational method is presented and some convergence results are given. Numerical examples of linear and nonlinear systems are also included.     

Pattern formation in diffusion-limited reactions

The conditions for macroscopic segregation ofA andB in a steady-stateA+B 0 reaction are studied in infinite systems. Segregation occurs in one and two dimensions and is marginal ford=3. We note the dependence of these results on the precise experimental conditions assumed in the theory. We also note the difference between these results and our earlier ones for finite systems where the critical dimension isd=2.     

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.     

Small random perturbations of finite- and infinite-dimensional dynamical systems: Unpredictability of exit times

We apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain. We show under general hypotheses that the suitably rescaled exit time converges in the zero-noise limit to an exponential random variable. Then we extend the results to an infinite-dimensional case obtained by adding a small random perturbation to a nonlinear heat equation.     

Viscometry in the study of molecular complexes

Viscometric method has been used to study the interaction between the weak interacting systems such as benzene-CCl₄-cyclohexane, toluene-CCl₄-cyclohexane ando- xylene-CCl₄-cyclohexane. The equilibrium constants and other thermodynamic parameters have been reported and compared with literature data.     

Debye screening for two-dimensional Coulomb systems at high temperatures

The grand canonical ensemble of a two-dimensional Coulomb system with±1 charges is proved to have screening phenomena in its high-temperature region. The Coulomb potential in a finite region is assumed to be (–)^–1, where is the Laplacian with zero boundary conditions on. The hard-core condition is not assumed. The model is set up by separating (–)^–1 into a shortrange part and a long-range part depending on a parameter. The self-energies are subtracted only for the short-range part and therefore a choice of is a choice of subtraction of self-energies. The method of proof is in general the same as that of Brydges-Federbush Debye screening, except that here a modification for the short-range part of the potentials is needed.     

A rigorous control of logarithmic corrections in four-dimensional ϕ4 spin systems

Using Gawedzki and Kupiainen's rigorous block spin transformation method, we study critical phenomena in ⁴ spin systems in four dimensions. In Part I of this work we investigate in detail the renormalization group trajectory of the system not exactly at the critical point.     

Dynamical systems with relaxation time

We define mathematically a class of dynamical systems that exhibit relaxation corresponding to that observed in physical systems, and then show that this class is identical with the class ofK-mixing dynamical systems.     

Consistency of a linear system of inequalities

We show that a consistency check of a linear system of inequalitiesAxb reduces to check whetherQb0 for a certain matrixQ. It is a direct consequence of the Farkas-Minkowski theorem. Thus, when one has to check consistency for different values ofb, one need not run a numerical algorithm for eachb.On leave at the Electronics Research Laboratory of the University of California at Berkeley in a CNRS/NSF Exchange Program.