Nonlinear Energy Response of Glass Forming Materials

A theory for the nonlinear energy response of a system subjected to a heat bath is developed when the temperature of the heat bath is modulated sinusoidally. The theory is applied to a model glass forming system, where the landscape is assumed to have 20 basins and transition rates between basins obey a power law distribution. It is shown that the statistics of eigenvalues of the transition rate matrix, the glass transition temperature $T_g$, the Vogel-Fulcher temperature $T_0$ and the crossover temperature $T_x$ can be determined from the 1st- and 2nd-order ac specific heats, which are defined as coefficients of the 1st- and 2nd-order energy responses. The imaginary part of the 1st-order ac specific heat has a broad peak corresponding to the distribution of the eigenvalues. When the temperature is decreased below $T_g$, the frequency of the peak decreases and the width increases. Furthermore, the statistics of eigenvalues can be obtained from the frequency dependence of the 1st-order ac specific heat. The 2nd-order ac specific heat shows extrema as a function of the frequency. The extrema diverge at the Vogel-Fulcher temperature $T_0$. The temperature dependence of the extrema changes significantly near $T_g$ and some extrema vanish near $T_x$.Comment: 20 pages, 10 figure

A q-analogue of Catalan Hankel determinants

In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schr\"oder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.Comment: 17 page

Distributed Agreement on Activity Driven Networks

In this paper, we investigate asymptotic properties of a consensus protocol taking place in a class of temporal (i.e., time-varying) networks called the activity driven network. We first show that a standard methodology provides us with an estimate of the convergence rate toward the consensus, in terms of the eigenvalues of a matrix whose computational cost grows exponentially fast in the number of nodes in the network. To overcome this difficulty, we then derive alternative bounds involving the eigenvalues of a matrix that is easy to compute. Our analysis covers the regimes of 1) sparse networks and 2) fast-switching networks. We numerically confirm our theoretical results by numerical simulations