Extending the definition of entropy to nonequilibrium steady states

We study the nonequilibrium statistical mechanics of a finite classical system subjected to nongradient forces $\xi$ and maintained at fixed kinetic energy (Hoover-Evans isokinetic thermostat). We assume that the microscopic dynamics is sufficiently chaotic (Gallavotti-Cohen chaotic hypothesis) and that there is a natural nonequilibrium steady state $\rho_\xi$. When $\xi$ is replaced by $\xi+\delta\xi$ one can compute the change $\delta\rho$ of $\rho_\xi$ (linear response) and define an entropy change $\delta S$ based on energy considerations. When $\xi$ is varied around a loop, the total change of $S$ need not vanish: outside of equilibrium the entropy has curvature. But at equilibrium (i.e. if $\xi$ is a gradient) we show that the curvature is zero, and that the entropy $S(\xi+\delta\xi)$ near equilibrium is well defined to second order in $\delta\xi$.Comment: plain TeX, 10 pagesemacs ded

Spectral Characterization of Anomalous Diffusion of a Periodic Piecewise Linear Intermittent Map

For a piecewise linear version of the periodic map with anomalous diffusion, the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the Frobenius-Perron operator are explicitly derived. The evolution of the averages is controlled by real eigenvalues as well as continuous spectra terminating at the unit circle. Appropriate scaling limits are shown to give a normal diffusion if the reduced map is in the stationary regime with normal fluctuations, a L\'evy flight if the reduced map is in the stationary regime with L\'evy-type fluctuations and a transport of ballistic type if the reduced map is in the non-stationary regime.Comment: submitted to Physica D (CHAOTRAN conference proceedings

Characterizing dynamics with covariant Lyapunov vectors

A general method to determine covariant Lyapunov vectors in both discrete- and continuous-time dynamical systems is introduced. This allows to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents.Comment: 4 pages, 3 figures, submitted to Physical Review letter

Comments on the Links between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards

We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface.Comment: 56 pages, 4 eps figures, LaTeX, uses eps

Metastable states and space-time phase transitions in a spin-glass model

We study large deviations of the dynamical activity in the random orthogonal model (ROM). This is a fully connected spin-glass model with one-step replica symmetry breaking behaviour, consistent with the random first-order transition scenario for structural glasses. We show that this model displays dynamical (space-time) phase-transitions between active and inactive phases, as demonstrated by singularities in large deviation functions. We argue that such transitions are generic in systems with long-lived metastable states.Comment: 4.1 pages, 4 fig

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

Jamming probabilities for a vacancy in the dimer model

Following the recent proposal made by Bouttier et al [Phys. Rev. E 76, 041140 (2007)], we study analytically the mobility properties of a single vacancy in the close-packed dimer model on the square lattice. Using the spanning web representation, we find determinantal expressions for various observable quantities. In the limiting case of large lattices, they can be reduced to the calculation of Toeplitz determinants and minors thereof. The probability for the vacancy to be strictly jammed and other diffusion characteristics are computed exactly.Comment: 19 pages, 6 figure