The incomplete beta function law for parallel tempering sampling of classical canonical systems

We show that the acceptance probability for swaps in the parallel tempering Monte Carlo method for classical canonical systems is given by a universal function that depends on the average statistical fluctuations of the potential and on the ratio of the temperatures. The law, called the incomplete beta function law, is valid in the limit that the two temperatures involved in swaps are close to one another. An empirical version of the law, which involves the heat capacity of the system, is developed and tested on a Lennard-Jones cluster. We argue that the best initial guess for the distribution of intermediate temperatures for parallel tempering is a geometric progression and we also propose a technique for the computation of optimal temperature schedules. Finally, we demonstrate that the swap efficiency of the parallel tempering method for condensed-phase systems decreases naturally to zero at least as fast as the inverse square root of the dimensionality of the physical system.Comment: 11 pages, 4 figures; minor changes; to appear in J. Chem. Phy

Universal conservation law and modified Noether symmetry in 2d models of gravity with matter

It is well-known that all 2d models of gravity---including theories with nonvanishing torsion and dilaton theories---can be solved exactly, if matter interactions are absent. An absolutely (in space and time) conserved quantity determines the global classification of all (classical) solutions. For the special case of spherically reduced Einstein gravity it coincides with the mass in the Schwarzschild solution. The corresponding Noether symmetry has been derived previously by P. Widerin and one of the authors (W.K.) for a specific 2d model with nonvanishing torsion. In the present paper this is generalized to all covariant 2d theories, including interactions with matter. The related Noether-like symmetry differs from the usual one. The parameters for the symmetry transformation of the geometric part and those of the matterfields are distinct. The total conservation law (a zero-form current) results from a two stage argument which also involves a consistency condition expressed by the conservation of a one-form matter ``current''. The black hole is treated as a special case.Comment: 3

A topological zero-one law and elementary equivalence of finitely generated groups

Let $\mathcal G$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $\sigma$ in the infinitary logic $\mathcal L_{\omega_1, \omega}$, the set of all models of $\sigma$ in $\mathcal S$ is either meager or comeager. In particular, we show that the zero-one law holds for certain natural spaces associated to hyperbolic groups and their generalizations. As an application, we obtain that generic torsion-free lacunary hyperbolic groups are elementarily equivalent; the same claim holds for lacunary hyperbolic groups without non-trivial finite normal subgroups. Our paper has a substantial expository component. We give streamlined proofs of some known results and survey ideas from topology, logic, and geometric group theory relevant to our work. We also discuss some open problems.Comment: Some typos and inaccuracies are corrected, a few minor improvement

Tilt-angle landscapes and temperature dependence of the conductance in biphenyl-dithiol single-molecule junctions

Using a density-functional-based transport method we study the conduction properties of several biphenyl-derived dithiol (BPDDT) molecules wired to gold electrodes. The BPDDT molecules differ in their side groups, which control the degree of conjugation of the pi-electron system. We have analyzed the dependence of the low-bias zero-temperature conductance on the tilt angle phi between the two phenyl ring units, and find that it follows closely a cos^2(phi) law, as expected from an effective pi-orbital coupling model. We show that the tilting of the phenyl rings results in a decrease of the zero-temperature conductance by roughly two orders of magnitude, when going from a planar conformation to a configuration in which the rings are perpendicular. In addition we demonstrate that the side groups, apart from determining phi, have no influence on the conductance. All this is in agreement with the recent experiment by Venkataraman et al. [Nature 442, 904 (2006)]. Finally, we study the temperature dependence of both the conductance and its fluctuations and find qualitative differences between the examined molecules. In this analysis we consider two contributions to the temperature behavior, one coming from the Fermi functions and the other one from a thermal average over different contact configurations. We illustrate that the fluctuations of the conductance due to temperature-induced changes in the geometric structure of the molecule can be reduced by an appropriate design.Comment: 9 pages, 6 figures; submitted to Phys. Rev.

A strong zero-one law for connectivity in one-dimensional geometric random graphs with non-vanishing densities

We consider the geometric random graph where n points are distributed independently on the unit interval [0,1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some transmission range. When F admits a continuous density f which is strictly positive on [0,1], we show that the property of graph connectivity exhibits a strong critical threshold and we identify it. This is achieved by generalizing a limit result on maximal spacings due to Levy for the uniform distribution