Earthquakes in the length-spectrum Teichm\"uller spaces

Let $X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichm\"uller space $T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $X_0$ such that the ratios of the corresponding simple closed geodesic for the hyperbolic metric on $X_0$ and for the other hyperbolic metric are bounded from the below away from 0 and from the above away from $\infty$ (cf. \cite{ALPS}). This paper studies earthquakes in the length spectrum Teichm\"uller space $T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on earthquake measure $\mu$ such that the corresponding earthquake $E^{\mu}$ describes the hyperbolic metric on $X_0$ which is in the length spectrum Teichm\"uller space. Moreover, we give examples of earthquake paths $t\mapsto E^{t\mu}$, for $t\geq 0$, such that $E^{t\mu}\in T_{ls}(X_0)$ for $0\leq t<t_0$, $E^{t_0\mu}\notin T_{ls}(X_0)$ and $E^{t\mu}\in T_{ls}(X_0)$ for $t>t_0$.Comment: metadata correction, the same version as befor

Heat content and inradius for regions with a Brownian boundary

In this paper we consider $\beta[0; s]$, Brownian motion of time length $s > 0$, in $m$-dimensional Euclidean space $\mathbb R^m$ and on the $m$-dimensional torus $\mathbb T^m$. We compute the expectation of (i) the heat content at time $t$ of $\mathbb R^m\setminus \beta[0; s]$ for fixed $s$ and $m = 2,3$ in the limit $t \downarrow 0$, when $\beta[0; s]$ is kept at temperature 1 for all $t > 0$ and $\mathbb R^m\setminus \beta[0; s]$ has initial temperature 0, and (ii) the inradius of $\mathbb R^m\setminus \beta[0; s]$ for $m = 2,3,\cdots$ in the limit $s \rightarrow \infty$.Comment: 13 page

Invasion Percolation with Temperature and the Nature of SOC in Real Systems

We show that the introduction of thermal noise in Invasion Percolation (IP) brings the system outside the critical point. This result suggests a possible definition of SOC systems as ordinary critical systems where the critical point correspond to set to 0 one of the parameters. We recover both IP and EDEN model, for $T \to 0$, and $T \to \infty$ respectively. For small $T$ we find a dynamical second order transition with correlation length diverging when $T \to 0$.Comment: 4 pages, 2 figure

Efficiency of Energy Conversion in Thermoelectric Nanojunctions

Using first-principles approaches, this study investigated the efficiency of energy conversion in nanojunctions, described by the thermoelectric figure of merit $ZT$. We obtained the qualitative and quantitative descriptions for the dependence of $ZT$ on temperatures and lengths. A characteristic temperature: $T_{0}= \sqrt{\beta/\gamma(l)}$ was observed. When $T\ll T_{0}$, $ZT\propto T^{2}$. When $T\gg T_{0}$, $ZT$ tends to a saturation value. The dependence of $ZT$ on the wire length for the metallic atomic chains is opposite to that for the insulating molecules: for aluminum atomic (conducting) wires, the saturation value of $ZT$ increases as the length increases; while for alkanethiol (insulating) chains, the saturation value of $ZT$ decreases as the length increases. $ZT$ can also be enhanced by choosing low-elasticity bridging materials or creating poor thermal contacts in nanojunctions. The results of this study may be of interest to research attempting to increase the efficiency of energy conversion in nano thermoelectric devices.Comment: 2 figure