Stable resonances and signal propagation in a chaotic network of coupled units

We apply the linear response theory developed in \cite{Ruelle} to analyze how a periodic signal of weak amplitude, superimposed upon a chaotic background, is transmitted in a network of non linearly interacting units. We numerically compute the complex susceptibility and show the existence of specific poles (stable resonances) corresponding to the response to perturbations transverse to the attractor. Contrary to the poles of correlation functions they depend on the pair emitting/receiving units. This dynamic differentiation, induced by non linearities, exhibits the different ability that units have to transmit a signal in this network.Comment: 10 pages, 3 figures, to appear in Phys. rev.

Phase transitions with four-spin interactions

Using an extended Lee-Yang theorem and GKS correlation inequalities, we prove, for a class of ferromagnetic multi-spin interactions, that they will have a phase transition(and spontaneous magnetization) if, and only if, the external field $h=0$ (and the temperature is low enough). We also show the absence of phase transitions for some nonferromagnetic interactions. The FKG inequalities are shown to hold for a larger class of multi-spin interactions

Implications of an arithmetical symmetry of the commutant for modular invariants

We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU(3)_k, we classify the modular invariant partition functions when k+3 is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants.Comment: 17 pages, plain TeX, DIAS-STP-92-2

Topics in chaotic dynamics

Various kinematical quantities associated with the statistical properties of dynamical systems are examined: statistics of the motion, dynamical bases and Lyapunov exponents. Markov partitons for chaotic systems, without any attempt at describing ``optimal results''. The Ruelle principle is illustrated via its relation with the theory of gases. An example of an application predicts the results of an experiment along the lines of Evans, Cohen, Morriss' work on viscosity fluctuations. A sequence of mathematically oriented problems discusses the details of the main abstract ergodic theorems guiding to a proof of Oseledec's theorem for the Lyapunov exponents and products of random matricesComment: Plain TeX; compile twice; 30 pages; 140K Keywords: chaos, nonequilibrium ensembles, Markov partitions, Ruelle principle, Lyapunov exponents, random matrices, gaussian thermostats, ergodic theory, billiards, conductivity, gas.

Automorphisms of the affine SU(3) fusion rules

We classify the automorphisms of the (chiral) level-k affine SU(3) fusion rules, for any value of k, by looking for all permutations that commute with the modular matrices S and T. This can be done by using the arithmetic of the cyclotomic extensions where the problem is naturally posed. When k is divisible by 3, the automorphism group (Z_2) is generated by the charge conjugation C. If k is not divisible by 3, the automorphism group (Z_2 x Z_2) is generated by C and the Altsch\"uler--Lacki--Zaugg automorphism. Although the combinatorial analysis can become more involved, the techniques used here for SU(3) can be applied to other algebras.Comment: 21 pages, plain TeX, DIAS-STP-92-4

New Duality Relations for Classical Ground States

We derive new duality relations that link the energy of configurations associated with a class of soft pair potentials to the corresponding energy of the dual (Fourier-transformed) potential. We apply them by showing how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the nature of the ground states of long-ranged potentials and vice versa. They also lead to bounds on the T=0 system energies in density intervals of phase coexistence, the identification of a one-dimensional system that exhibits an infinite number of ``phase transitions," and a conjecture regarding the ground states of purely repulsive monotonic potentials.Comment: 11 pages, 2 figures. Slightly revised version that corrects typos. This article will be appearing in Physical Review Letters in a slightly shortened for

Transfer matrix for spanning trees, webs and colored forests

We use the transfer matrix formalism for dimers proposed by Lieb, and generalize it to address the corresponding problem for arrow configurations (or trees) associated to dimer configurations through Temperley's correspondence. On a cylinder, the arrow configurations can be partitioned into sectors according to the number of non-contractible loops they contain. We show how Lieb's transfer matrix can be adapted in order to disentangle the various sectors and to compute the corresponding partition functions. In order to address the issue of Jordan cells, we introduce a new, extended transfer matrix, which not only keeps track of the positions of the dimers, but also propagates colors along the branches of the associated trees. We argue that this new matrix contains Jordan cells.Comment: 29 pages, 7 figure

On the Thermodynamic Limit in Random Resistors Networks

We study a random resistors network model on a euclidean geometry \bt{Z}^d. We formulate the model in terms of a variational principle and show that, under appropriate boundary conditions, the thermodynamic limit of the dissipation per unit volume is finite almost surely and in the mean. Moreover, we show that for a particular thermodynamic limit the result is also independent of the boundary conditions.Comment: 14 pages, LaTeX IOP journal preprint style file `ioplppt.sty', revised version to appear in Journal of Physics

Note on nonequilibrium stationary states and entropy

In transformations between nonequilibrium stationary states, entropy might be a not well defined concept. It might be analogous to the ``heat content'' in transformations in equilibrium which is not well defined either, if they are not isochoric ({\it i.e.} do involve mechanical work). Hence we conjecture that un a nonequilbrium stationary state the entropy is just a quantity that can be transferred or created, like heat in equilibrium, but has no physical meaning as ``entropy content'' as a property of the system.Comment: 4 page

Correlation Inequalities for Quantum Spin Systems with Quenched Centered Disorder

It is shown that random quantum spin systems with centered disorder satisfy correlation inequalities previously proved (arXiv:cond-mat/0612371) in the classical case. Consequences include monotone approach of pressure and ground state energy to the thermodynamic limit. Signs and bounds on the surface pressures for different boundary conditions are also derived for finite range potentials.Comment: 4 page