Expectations and entropy inequalities for finite quantum systems

We prove that the relative entropy is decreasing under a trace-preserving expectation inB(K ¹), and we show the connection between this theorem and the strong subadditivity of the entropy. It is also proved that a linear, positive, trace-preserving map ofB(K) into itself such that 1 decreases the value of any convex trace function.     

Green's functions on finite lattices and their connection to the infinite lattice limit

It is shown that the Green's function on a finite lattice in arbitrary space dimension can be obtained from that of an infinite lattice by means of a translation operator. Explicit examples are given for one- and two-dimensional lattices.     

Velocity correlation functions for finite one-dimensional systems

Certain systems consisting of a one-dimensional gas of a finite number of point particles interacting with a “hard-core” potential are considered.We use the technique developed by Jepsen to calculate exactly the velocity correlation functions for these systems. We discover that after a slow decay for times of the order of the relaxation time, there is a “fast” decay to the equilibrium value on a macroscopic time scale characterized by L/v_TH (L is the length of the container and v_TH the thermal velocity).The dependence of the velocity correlation functions on the initial position of the specified particle is also considered. In particular, the behaviour approaching the boundary of the container is analyzed. These considerations are generalized to systems of higher spatial dimension.     

Few-particle Green's functions for strongly correlated systems on infinite lattices

We show how few-particle Green's functions can be calculated efficiently for models with nearest-neighbor hopping, for infinite lattices in any dimension. As an example, for one-dimensional spinless fermions with both nearest-neighbor and second-nearest-neighbor interactions, we investigate the ground states for up to 5?fermions. This allows us not only to find the stability region of various bound complexes, but also to infer the phase diagram at small but finite concentrations.     

Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points.At a distance r in a ddimensional finite system of size L,the correlation function can be written as the product of|r|~(-(d-2+η))and a finite-size scaling function of the variables r/L and tL~(1/ν),where t=(T-T_c)/T_c,ηis the critical exponent of correlation function,andνis the critical exponent of correlation length.The correlation function only has a sigificant directional dependence when|r|is compariable to L.We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations.We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponentη.     

Simulating infinite systems

A new simulation technique for the dynamics of a disordered network of spins is presented. The method is able to treat the infinite system, avoiding finite size effects. Results for the Sherrington-Kirkpatrick model with asymmetric couplings are given.     

Coherent memory functions for finite systems: Hexagonal photosynthetic unit

Coherent memory functions entering the Generalized Master Equation are presented for a hexagonal model of a photosynthetic unit. Influence of an energy heterogeneity and possible superstructure on an exciton transfer in an antenna system as well as to a reaction center is investigated.One of us (I.B.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency, and Unesco, for hospitality at the International Centre for Theoretical Physics,Trieste.     

On the diffusive nature of entropy flow in infinite systems: Remarks to a paper by Guo-Papanicolau-Varadhan

The hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy (free energy) arguments. The methods of [13] are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise. Following [14, 15] and [3–5] we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended. The flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems [4].This work was supported in part by the Hungarian National Foundation for Scientific Research Grant 1815, NSF Grant DMR 86-12369, and by the Institut des Hautes Etudes Scientifiques     

Isospectral hamiltonian flows in finite and infinite dimensions

A moment map is constructed from the Poisson manifold _A of rank-r perturbations of a fixedN×N matrixA to the dual of the positive part of the formal loop algebra =gl(r)[[, ^–1]]. The Adler-Kostant-Symes theorem is used to give hamiltonians which generate commutative isospectral flows on . The pull-back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in _A. The latter may be identified with flows on finite dimensional coadjoint orbits in and linearized on the Jacobi variety of an invariant spectral curveX _r which, generically, is anr-sheeted Riemann surface. Reductions of _A are derived, corresponding to subalgebras ofgl(r, ) andsl(r, ), determined as the fixed point set of automorphism groupes generated by involutions (i.e., all the classical algebras), as well as reductions to twisted subalgebras of . The theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of PDE's.This research was partially supported by NSF grants MCS-8108814 (A03), DMS-8604189, and DMS-8601995     

Isospectral Hamiltonian flows in finite and infinite dimensions

The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad^*-invariant Poisson submanifolds of the dual of the positive part of the loop algebra is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curveS. These curves are embedded in the total space of a line bundleTP ₁(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles over a suitably desingularized curve may be used to determine both the flow of matricial polynomialsL() and the Hamiltonian flow in the spaceM _N,r×M_N,r in terms of -functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to invariance of the spectral curves and line bundles under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrödinger) equation.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army grant DAA L03-87-K-0110     

Transient elastodynamic response of finite and infinite solid cylinders

The orthogonal eigensolutions for the vibrations of an isotropic finite solid cylinder with a traction-free lateral boundary and rigid-smooth end boundaries are provided. The transient elastodynamic response of this solid cylinder is then constructed using the method of eigenfunction expansion and further extended explicitly and concisely to that of an isotropic infinite solid cylinder. The numerically evaluated analytical solution is shown to compare favorably with that by finite element method (FEM). The effect of external forces on the excitation of each guided wave mode can be quantitatively investigated on the basis of the present solution.     

Logic of infinite quantum systems

Limits of sequences of finite-dimensional (AF)C ^*-algebras, such as the CAR algebra for the ideal Fermi gas, are a standard mathematical tool to describe quantum statistical systems arising as thermodynamic limits of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the thermodynamic potentials, and handle the proliferation of physically inequivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AFC ^*-algebras correspond to countable Boolean algebras, i.e., algebras of propositions in the classical two-valued calculus. We investigate thenoncommutative logic properties of general AFC ^*-algebras, and their corresponding systems. We stress the interplay between Gödel incompleteness and quotient structures in the light of the nature does not have ideals program, stating that there are no quotient structures in physics. We interpret AFC ^*-algebras as algebras of the infinite-valued calculus of Lukasiewicz, i.e., algebras of propositions in Ulam's twenty questions game with lies.     

Effects of finite initial correlation length on scattering structure functions of quenched systems

We examined the effects of a large initial correlation length on the scattering structure function (SSF) when the order parameter is not conserved, using the approach developed by Ohta, Jasnow and Kawasaki. In particular, the SSF exhibits the usual scaling behaviour except that the scaling functions now contain an additional time-dependent parameter. The new scaling functions are computed for various values of the parameter in two and three dimensions.     

Radial position-momentum uncertainties for the infinite circular well and Fisher entropy

We show how the product of the radial position and momentum uncertainties can be obtained analytically for the infinite circular well potential. Some interesting features are found. First, the uncertainty Δr increases with the radius R and the quantum number n, the n-th root of the Bessel function. The variation of the Δr is almost independent of the quantum number n for $n>4$ and it will arrive to a constant for a large n, say $n>4$. Second, we find that the relative dispersion $\mathrm{\Delta }r/〈r〉$ is independent of the radius R. Moreover, the relative dispersion increases with the quantum number n but decreases with the azimuthal quantum number m. Third, the momentum uncertainty Δp decreases with the radius R and increases with the quantum numbers $m>1$ and n. Fourth, the product $\mathrm{\Delta }r\mathrm{\Delta }{p}_r$ of the position-momentum uncertainty relations is independent of the radius R and increases with the quantum numbers m and n. Finally, we present the analytical expression for the Fisher entropy. Notice that the Fisher entropy decreases with the radius R and it increases with the quantum numbers $m>0$ and n. Also, we find that the Cramer–Rao uncertainty relation is satisfied and it increases with the quantum numbers $m>0$ and n, too.     

Geometric partition functions of cellular systems: Explicit calculation of the entropy in two and three dimensions

A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.     

Coherent memory functions for finite molecular systems. A direct method of calculation

A new direct method for calculation of coherent memory functions (MFs) entering the Nakajima-Zwanzig (or GME) equation is presented for rigid molecular chains with natural boundary conditions. The method is illustrated on the chains with three and five molecules and various kinds of imperfections. Obtained results imply a necessity of a more careful interpretation of experimental data (e.g. diffusivity).     

Bogoliubov inequalities for infinite systems

The Bogoliubov inequalities are derived for the infinite volume states describing the thermodynamic limits of physical systems. The only property of the states required is that they satisfy the Kubo-Martin-Schwinger boundary condition.This work was performed under the auspices of the U.S. Atomic Energy Commission.