Philippe A. Martin

Structure of Gibbs states of one-dimensional Coulomb systems

We present a method of computing the Coulomb forces on particles in an infinite configuration of charges in one dimension. The resolution of the apparent nonuniqueness in this problem leads to a structural proof of the translation symmetry breaking in jellium, at all temperatures, and to a related phenomenon of phase nonuniqueness in the two component system. The appropriate generalizations of the DLR and KMS conditions for these states are discussed.

Aggregation dynamics in a self-gravitating one-dimensional gas

Aggregation of mass by perfectly inelastic collisions in a one-dimensional self-gravitating gas is studied. The binary collisions are subject to the laws of mass and momentum conservation. A method to obtain an exact probabilistic description of aggregation is presented. Since the one-dimensional gravitational attraction is confining, all particles will eventually form a single body. The detailed analysis of the probabilityPn(t) of such a complete merging before timet is performed for initial states ofn equidistant identical particles with uncorrelated velocities. It is found that for a macroscopic amount of matter (n→∞), this probability vanishes before a characteristic timet*. In the limit of a continuous initial mass distribution the exact analytic form ofPn(t) is derived. The analysis of collisions leading to the time-variation ofPn(t), reveals that in fact the merging into macroscopic bodies always occurs in the immediate vicinity oft*. Fort>t*, andn large,Pn(t) describes events corresponding to the final aggregation of remaining microscopic fragments.

Dynamics of the Open BCS Model

The dynamics of the strong coupling BCS model, considered as an open system interacting with a thermal bath, is solved rigorously and explicitly in the weak coupling limit and in the infinite-volume limit. The BCS system goes from the normal phase to the ordered phase by bifurcation. Fluctuations around trajectories of intensive observables are Gaussian and Markovian. Thermodynamic phases are global attractors in the physical domain. Structural stability is discussed. The model provides an example of a nonequilibrium statistical mechanical system with phase transition whose irreversible macroscopic dynamics can be calculated exactly from the underlying Hamiltonian quantum mechanics.

On the properties of inhomogeneous charged systems

We give a proof and an extension of equations previously derived by Wertheim and Lovett, Mou and Buff, relating the gradient of the density to an integral of the external force over the pair correlation function; when the system has boundaries it also involves a surface contribution. These equations are derived and used for systems which may contain free charges, dipoles, and a rigid background (jellium). In particular, we derive an equation for the density profile near a plane electrode and we show that the correlation function has to decay no faster than ‖x‖−N(N=space dimension) parallel to the electode.

On the equivalence between KMS-states and equilibrium states for classical systems

It is shown that for any KMS-state of a classical system of non-coincident particles, the distribution functions are absolutely continuous with respect to Lebesgue measure; the equivalence between KMS states and Canonical Gibbs States is then established.

The Casimir effect for the Bose-gas in slabs

We study the Casimir effect for the perfect Bose-gase in the slab geometry for various boundary conditions. We show that the grand canonical potential per unit area at the bulk critical chemical potential

One-dimensional ballistic aggregation: Rigorous long-time estimates

\mu=0

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

has the standard asymptotic form with universal Casimir terms.

Microscopic origin of universality in Casimir forces

Aggregation of mass by perfectly inelastic collisions in a one-dimensional gas of point particles is studied. The dynamics is governed by laws of mass and momentum conservation. The motion between collisions is free. An exact probabilistic description of the state of the aggregating gas is presented. For an initial configuration of equidistant particles on the line with Maxwellian velocity distribution, the following results are obtained in the long-time limit. The probability for finding empty intervals of length growing faster thant2/3 vanishes. The mass spectrum can range from the initial mass up to mass of ordert2/3. Aggregates with masses growing faster thant2/3 cannot occur. Our estimates are in accordance with numerical simulations predictingt−1 decay for the number density of initial masses and a slowert−2/3 decay for the density of aggregates resulting from a large number of collisions (with masses ∼t2/3). Our proofs rely on a link between the considered aggregation dynamics and Brownian motion in the presence of absorbing barriers.

Self-consistent equation for an interacting Bose gas.

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density ρ=ρshort+ρlong into the number density of particles belonging to cycles of finite length (ρshort) and to infinitely long cycles (ρlong) in the thermodynamic limit. For this model we prove that when there is Bose condensation, ρlong is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of ρlong≠ 0 with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas