On a limiting procedure for obtaining physical states for an infinite Fermi system

We propose a limiting procedure for obtaining physical states for an infinite non-relativistic Fermi system. We take the thermodynamic limit of vector states in the Fock representation of the C.A.R. algebra, representing a condensate state of atoms each of which is formed by 4 fermions. In a simplified example considered in detail, the limit state has a simple decomposition into the product of two B.C.S. states. IfB ⁺ is the operator creating the atom from the vacuum |_0F , it is proved that the states obtained by taking the thermodynamic limit of the vector states corresponding to (B ⁺) ⁿ |_0F and respectively, coincide on the gauge-invariant elements of the algebra for a suitable value ofz.Partially supported by C.N.R.     

Long time behaviour of an infinite particle system

The long time behaviour of the semi-infinite Toda lattice is deduced from a set of identities for the squared eigenfunctions of the Toda flow.     

On the definition of a diffusion constant in a nonequilibrium system

A diffusion constant for electrons in a current-carrying semiconductor can be unambiguously defined in nearly uniform systems. For frequency-dependent density gradients it is $$D_{\alpha \beta } (\omega ) \equiv \int\limits_0^\infty {dt e^{i\omega t} \overline {\Delta \upsilon _\alpha (t)\Delta \upsilon _\beta (0),} } $$ where \(\overline {\Delta \upsilon _\alpha (t)\Delta \upsilon _\beta (0)} \) is the velocity correlation function with respect to the steady state in a bias field. This result has been elucidated in the relaxation approximation by different approaches to the diffusion problem. Essential for its derivation is a statistical independence assumption of space and velocities, and in order to get a classical diffusion law of Fick's type certain velocities have to be distributed according to the steady state in a bias field. Diffusion constant and noise temperature are discussed for a few band structures in the relaxation approximation.     

A mechanical definition of the thermodynamic pressure

A dynamical definition of pressure for grand-canonical Gibbs measures in bounded regions Λ is rigorously discussed: It measures the momentum transferred to the walls of the container by the elastically colliding particles. The local pressureP(r, δΛ) so obtained is proportional to the temperature and the local density at the boundaries of Λ. This allows us to obtain a rigorous proof of the virial theorem of Clausius. In this picture the thermodynamic pressureP ^d (Λ) is obtained as the average ofP(r, δΛ) onδΛ. Its relationship with the usual equilibrium pressureP _eq(Λ) = (βsΛ¦)^?1lnZ _Λ (Z _Λ is the grand-canonical partition function) is then discussed. In the particular case in which the regions A are spheres, it is shown that P_d(Λ) converges in average so that, if the limit of P_d(Λ) exists, it equals P_eq, the thermodynamic limit of the equilibrium pressure P_eq(Λ). Finally, convergence ofP _d(Λ) is proven to hold in the particular case of one-dimensional hard cores in the absence of phase transitions.     

H-theorem for an infinite,one-dimensional,hard-point system

We use a well-studied soluble model to define a nonequilibrium entropy. This entropy has all the required properties; in particular, it is not time-reversal invariant, so that its monotonic increase in time also shows up after we perform a velocity inversion experiment.Supported financially by the Belgian Government, Actions de Recherches Concertées, Convention 76/81, II.3.     

On the definition of information

In this note a new system of axioms for the information is given without using the concept of probability. Furthermore, it is shown that the information determines the probability uniquely, except for some degenerate cases.     

Ergodic properties of an infinite one dimensional hard rod system

It is shown that an infinite one dimensional system of hard rods for which the “effective” velocities of the pulses (free velocity plus a drift term due to collisions) are bounded away from some neighborhood of 0 is Bernoulli. This generalizes a result of Sinai who showed that some hard rod systems areK-systems.     

On the nonsymmetric,gravitational collapse of an infinite,dust-filled cylinder

Nonsymmetric terms or angle-dependent terms in the power series expansion of Einstein's equations are considered. It is shown, first, that they do not influence the time evolution of the symmetric terms and, second, that they do not remain bounded as the cylinder collapses. However, the major contribution to the density arising from the symmetric terms makes the nonsymmetric contribution progressively more insignificant as the collapse proceeds.     

An example of an infinite system with zero quantum dynamical entropy

With nary mention of a tree graph, we obtain a cluster expansion bound that includes and vastly generalizes bounds as obtained by extant tree graph inequalities. This includes applications to both two-body and many-body potential situations of the recently obtained new improved tree graph inequalities that have led to the extra 1/N! factors. We work in a formalism coupling a discrete set of boson variables, such as occurs in a lattice system in classical statistical mechanics, or in Euclidean quantum field theory. The estimates of this Letter apply to numerical factors as arising in cluster expansions, due to essentially arbitrary sequences of the basic operations: interpolation of the covariance, interpolation of the interaction, and integration by parts. This includes complicated evolutions, such as the repeated use of interpolation to decouple the same variables several times, to ensure higher connectivity for renormalization purposes, in quantum field theory.This work was supported in part by the National Science Foundation under grant no. PHY-87-01329.     

On the sound field of a shallow spherical shell in an infinite baffle

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Green's function is developed for a shallow spherical shell, which is based upon Reissner's solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Green's function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.     

On the definition of phase transition

It is shown that there exists a phase transition associated with a singularity of the free energy for a model such that for all temperatures the equilibrium state is unique and thus stable with respect to boundary perturbations. It is also shown on this model that there exist phase transitions without symmetry breakdown, which can be related to a phase transition with symmetry breakdown on an equivalent model.     

On the definition of intramolecular motions

The restrictions imposed when defining separate types of internal motions in rigid and nonrigid molecules are considered. These restrictions are based on the symmetry properties of intramolecular dynamics. It is shown that the violation of these restrictions gives rise to physically incorrect consequences.     

On the definition of cavitation intensity

Cavitation intensity has already been used to character the activity or strength of cavitation, and several methods are developed to measure the cavitation intensity. However, the previous definitions of cavitation intensity are often either vague or biased. In this paper, from the point of view of energy, the authors proposed a generalized definition of cavitation intensity, derived an approximate formula to calculate the cavitation intensity and discussed its measure method.     

On the definition of renormalized field products

We study the redefinition of the field products appearing in a Lagrangian and its equations of motion in a Normal Product framework. We propose a method of defining these products, which give the finite Green's functions, in such a way that the canonical derivation of the equations of motion is preserved. This involves the use of the Wilson Expansion in a Dimensionally Regularized form. As an example a ?⁴, ?³, field theory in four dimensions is fully redefined to the 1-loop level.     

On the intrinsic definition of local observables

In a theory of local field algebras satisfying the split property, we introduce an algebra generated by the local energy-momentum operators obtained from the canonical local implementation of translations, and possibly by the local charges operators. We discuss the relations of this algebra to the given algebra of local observables and take some steps toward the characterization of theories where they coincide. In the presence of spontaneously broken symmetries, we present a no-go theorem.     

Quantization of a Hamiltonian system with an infinite number of degrees of freedom

We propose a method of quantization of a discrete Hamiltonian system with an infinite number of degrees of freedom. Our approach is analogous to the usual finite-dimensional quantum mechanics. We construct an infinite-dimensional Schrödinger equation. We show that it is possible to pass from the finite-dimensional quantum mechanics to our construction in the limit when the number of particles tends to infinity. Rigorous mathematical methods are used.     

On the uniqueness of the problem of acoustic diffraction by an infinite plate with local irregularities

The question concerning the uniqueness of the solution to the problem of the acoustic diffraction by an immersed and isolated thin infinite plate with a finite scatterer is studied. It is shown that, to provide the uniqueness of the solution, the conditions at the scatterer must lead to an energy inequality for a source-free field, which determines the absence of the energy-carrying field components at infinity. A formula that generalizes the Sommerfeld formula is obtained and is used to prove the uniqueness of the solution to the problem of diffraction by a plate immersed in an acoustic medium. For the problem of diffraction of a flexural wave by an irregularity of the plate, the uniqueness theorem is proved only for the case of a fixed or hinged edge. When boundary conditions of a general form are imposed on the scatterer in an isolated plate, the uniqueness of the solution is generally lost, which is also corroborated by an example.     

Exchange contribution to the surface energy of an arbitrary fermion system within the infinite barrier model

The surface energy of a many electron system within the infinite barrier model has produced a considerable amount of interest. These studies have provided solutions for the exchange and correlation (within the RPA) surface energies with which approximate treatments of more realistic surfaces have been compared. In this note we present the first generalization, within this model, of the exchange contribution to the surface energy for a general Fermion system of arbitrary interparticle interaction. Such results should be useful for testing approximate methods for a much wider range of Fermion systems.