Ch. Gruber

General properties of polymer systems

We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential.The existence and analyticity properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion.In the special case of Monomer-Dimer systems it is established that no phase transition is possible; moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Padé approximants.Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems.

Sum rules for inhomogeneous Coulomb systems

Using the stationary equilibrium BBGKY hierarchy and some weak spatial decay properties of the correlations, we derive exact sum rules for the equilibrium distribution functions of ionic systems. Our results apply to both homogeneous and nonuniform systems. They show that when there is decay in such systems, then the total excess charge in the vicinity of a given number of fixed ions is zero, and that this excess charge has no dipole nor quadrupole moment. The implications for the static structure factor and for the dielectric tensor are discussed.

Stationary motion of the adiabatic piston

We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures T1 and T2, separated by an adiabatic movable piston whose mass M is much larger than the mass m of the fluid particles. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter e=m/M, and explicitly given up to order e2. In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the fluctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where M=m.

On the equation of state of classical one-component systems with long-range forces

Several definitions of the “pressure” are introduced for one-component systems and shown to be nonequivalent in the presence of a rigid neutralizing background. Relations between these pressures are derived for finite and infinite systems; these relations depend on the asymptotic behavior of the force at infinity, with the Coulomb force at the borderline between different properties. It is argued that only one of those definitions is physically acceptable and its properties are discussed in relation to the asymptotic behavior of the force. It is seen in particular that a knowledge of the state of the infinite system is not sufficient to determine its thermodynamic properties. The results are illustrated by some typical examples.

A sum rule for an inhomogeneous electrolyte

We obtain a new sum rule for the density profile of an electrolyte in contact with a charged flat hard (nonconducting) wall. This, in turn, strongly implies that the decay of the pair correlation near the wall is not faster than (distance)−d−1, where d is the dimension of the system. (AIP)

A new proof of the Stillinger-Lovett complete shielding condition

It is shown that any equilibrium state of classical charged particles with correlation having a spatial decay faster than 1/¦x¦v+2 in dimensionv=2, 3 obeys the Stillinger-Lovett second moment condition. Under the same clustering hypothesis, arbitrary localized external charge distributions are completely shielded.

On the adiabatic properties of a stochastic adiabatic wall: Evolution, stationary non-equilibrium, and equilibrium states

The time evolution of the adiabatic piston problem and the consequences of its stochastic motion are investigated. The model is a one-dimensional piston of mass M separating two ideal fluids made of point particles with mass m⪡M. For infinite systems it is shown that the piston evolves very rapidly toward a stationary non-equilibrium state with non-zero average velocity even if the pressures are equal but the temperatures different on both sides of the piston. For a finite system it is shown that the evolution takes place in two stages: first the system evolves rather rapidly and adiabatically toward a metastable state where the pressures are equal but the temperatures different; then the evolution proceeds extremely slowly toward the equilibrium state where both the pressures and the temperatures are equal. Numerical simulations of the model are presented. The results of the microscopical approach, the thermodynamical equations and the simulations are shown to be qualitatively in good agreement.

Molecule formation and the Farey tree in the one-dimensional Falicov-Kimball model

The ground-state configurations of the one-dimensional Falicov-Kimball model are studied exactly with numerical calculations revealing unexpected effects for small interaction strength. In neutral systems we observe molecular formation, phase separation, and changes in the conducting properties; while in nonneutral systems the phase diagram exhibits Farey tree order (Aubry sequence) and a devils staircase structure. Conjectures are presented for the boundary of the segregated domain and the general structure of the ground states.

Equilibrium properties of classical systems with long-range forces. BBGKY equation, neutrality, screening, and sum rules

We introduce a generalization of the BBGKY equation to define the equilibrium states for systems with long-range forces and study the properties of such states. We show that there are properties typical of short-range forces (shape independence, normal fluctuations, asymptotic behavior of correlation functions) and others which are typical of long-range forces (possible shape dependence, neutrality, sum rules and screening, abnormal fluctuations, boundedness of the internal electric field). If the force decreases at infinity faster than the Coulomb force, the properties will be those typical of short-range forces; on the other hand, if the force decreases at infinity as the Coulomb force or slower, the properties will be those typical of long-range forces.

From the adiabatic piston to macroscopic motion induced by fluctuations

The controversial problem of the evolution of an isolated system with an internal adiabatic wall is investigated with the use of a simple microscopic model and the Boltzmann equation. In the case of two infinite volume one-dimensional ideal fluids separated by a piston whose mass is equal to the mass of the fluid particles we obtain a rigorous explicit stationary non-equilibrium solution of the Boltzmann equation. It is shown that at equal pressures on both sides of the piston, the temperature difference induces a non-zero average velocity, oriented toward the region of higher temperature. It thus turns out that despite the absence of macroscopic forces the asymmetry of fluctuations results in a systematic macroscopic motion. This remarkable effect is analogous to the dynamics of stochastic ratchets, where fluctuations conspire with spatial anisotropy to generate directed motion. However, a different mechanism is involved here. The relevance of the discovered motion to the adiabatic piston problem is discussed.