Level Dynamics and Universality of Spectral Fluctuations

The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by establishing a power law for the dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas–Yukawa gas.     

Dynamics and Entropy in the Zhang Model of Self-Organized Criticality

We give a detailed study of dynamical properties of the Zhang model, including evaluation of topological entropy and estimates for the Lyapunov exponents and the dimension of the attractor. In the thermodynamic limit the entropy goes to zero and the Lyapunov spectrum collapses.     

Non-Equilibrium Dynamics of Three-Dimensional Infinite Particle Systems

We show existence and uniqueness for the solutions to the Newton equations relative to a system of infinitely many particles in the space, interacting by means of a positive and short-range potential. The initial conditions are chosen in a set sufficiently large to be the support of any reasonable non-equilibrium state. We extend previous results in one and two dimensions, obtained by Lanford and by Fritz and Dobrushin respectively, many years ago. Received: 7 December 1999 / Accepted: 9 May 2000     

A Class of Integrable and Nonintegrable Mappings and their Dynamics

We analyse a class of mappings which by construction do not belong to the QRT family. We show that some of the members of this class have invariants of high degree. A new linearisable mapping is also identified. A mapping which possesses confined singularities while having nonzero algebraic entropy is presented. Its dynamics are studied in detail and shown to be related intimately to the Fibonacci recurrence.      

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

Squeezing Dynamics for One-Dimensional Infinite Square Well with a Mobile Wall

We show that the dynamics for a particle confined in one-dimensional infinite square well with a mobile boundary can be converted to the case as if the boundary is time-independent at the expense of an appropriate time-dependent Hamiltonian. The Hamiltonian is deduced by the technique of integration within an ordered product of operators.     

Criticality of networks with long-range connections

<正>The formation of giant clusters,namely the percolation phase transition,is one of the most widely studied critical phenomena on networks.The critical behaviors of percolation in oneand two-dimensional lattices have been given in the book[1].For d-dimensional lattices,the critical exponents of percolation change with d until the upper critical dimension du=6,above which they are independent of d and become meanfield like.It is also well known that the critical behaviors     

A Simple Mean Field Model for Social Interactions: Dynamics,Fluctuations, Criticality

We study the dynamics of a spin-flip model with a mean field interaction. The system is non reversible, spacially inhomogeneous, and it is designed to model social interactions. We obtain the limiting behavior of the empirical averages in the limit of infinitely many interacting individuals, and show that phase transition occurs. Then, after having obtained the dynamics of normal fluctuations around this limit, we analyze long time fluctuations for critical values of the parameters. We show that random inhomogeneities produce critical fluctuations at a shorter time scale compared to the homogeneous system.     

Local Critical Perturbations of Unimodal Maps

We introduce a new complete metric in the space of unimodal C ²-maps of the interval, with two maps close if they are close in the C ²-metric and differ only on a small interval containing their critical points. We identify all structurally stable maps in the sense of this metric. They are maps for which either (1) the trajectory of the critical point is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same (minimal) period. We verify the generalized Fatou conjecture for and show that structurally stable maps form an open dense subset of . Partially supported by NSF grant DMS 0456748. Partially supported by NSF grant DMS 0456526.     

The Universality of the Critical Dynamics of the Kinetic Ising Model on the Regular Fractals

The form of the universal scaling law of the critical dynamic exponent, z = D_ƒ + 2/υ, is found on a family of regular fractals by the exact TDRG method. Here, we generate a regular fractal by an anisotropic growing process. Identifying the growing probabilities as the interactions between Ising spins on the fractals, we map the growing probability clouds as a group of the anisotropic Ising Hamiltonians. Applying the RG transformations, we find that the systems of this group of Ising Hamiltonians can be described by two universal static correlation exponents υ₀ = ∞ and υ = 1. So, the growing processes proposed by us capture the essential features in the directed DLA simulations. The studies about their critical dynamic behaviours reveal that unlike the one-dimensional chain the critical dynamics of the kinetic Ising model on the regular fractals is universal. The further discussions show that there is a universal scaling law form of the critical dynamic exponent of the kinetic Ising model, z = D_ƒ + R_max/2υ, on the site models of the regular fractals with R_min = 2. Meanwhile, we discuss Daniel Kandal's correction to the formula of the,critical dynamic exponent in the TDRG method and show that our TDRG calculations are exact.     

Non-equilibrium Dynamics of an Infinite Range XY Model in an External Field

We study the XY model with infinite range interactions in an external magnetic field. The simulations show that in the thermodynamic limit this model does not relax to the thermodynamic equilibrium—instead it becomes trapped in a non-ergodic out-of-equilibrium state. We show how the relaxation towards this non-equilibrium state can be studied using the properties of the collisionless Boltzmann (Vlasov) equation.     

On Hausdorff Dimension of Unimodal Attractors

There exists a universal constant σ<1 such that every attractor of every C⁴ unimodal map with a non-degenerate critical point is an analytic manifold or its Hausdorff dimension is equal to or less than σ.     

Universality of decoherence

We consider environment induced decoherence of quantum superpositions to mixtures in the limit in which that process is much faster than any competing one generated by the Hamiltonian H(sys) of the isolated system. While the golden rule then does not apply we can discard H(sys). By allowing for couplings to different reservoirs, we reveal decoherence as a universal short-time phenomenon independent of the character of the system as well as the bath and of the basis the superimposed states are taken from. We discuss consequences for the classical behavior of the macroworld and quantum measurement: For decoherence of superpositions of macroscopically distinct states H(sys) is always negligible.     

Self-Organized Criticality and Thermodynamic Formalism

We develop a thermodynamic formalism for a dissipative version of the Zhang model of Self-Organized Criticality, where a parameter allows us to tune the local energy dissipation. By constructing a suitable Markov partition we define Gibbs measures (in the sense of Sinai, Ruelle, and Bowen), partition functions, and topological pressure allowing the analysis of probability distributions of avalanches. We discuss the infinite-size limit in this setting. In particular, we show that a Lee–Yang phenomenon occurs in the conservative case. This suggests new connections to classical critical phenomena.     

Criticality in lasers and masers

A critical length for laser self Q-switching and a self oscillation condition for an astrophysical maser are derived by analogy with neutron criticality, with back-scattering of radiation as the coupling process.     

Universality class of interface growth with reflection symmetry

We investigate interface dynamics in 1+1 dimensions, respecting reflection symmetry. In the continuum approach of Kardar, Parisi, and Zhang, the leading nonlinearity is then of the form (h _t)³. On the basis of Monte Carlo simulations for a driven lattice gas, we argue that the nonlinearity is marginally irrelevant. Thus, the universality class is the one of equilibrium interfaces with a purely relaxational bulk dynamics.