Thermodynamic graph-rewriting

We develop a new thermodynamic approach to stochastic graph-rewriting. The ingredients are a finite set of reversible graph-rewriting rules called generating rules, a finite set of connected graphs P called energy patterns and an energy cost function. The idea is that the generators define the qualitative dynamics, by showing which transformations are possible, while the energy patterns and cost function specify the long-term probability $\pi$ of any reachable graph. Given the generators and energy patterns, we construct a finite set of rules which (i) has the same qualitative transition system as the generators; and (ii) when equipped with suitable rates, defines a continuous-time Markov chain of which $\pi$ is the unique fixed point. The construction relies on the use of site graphs and a technique of `growth policy' for quantitative rule refinement which is of independent interest. This division of labour between the qualitative and long-term quantitative aspects of the dynamics leads to intuitive and concise descriptions for realistic models (see the examples in S4 and S5). It also guarantees thermodynamical consistency (AKA detailed balance), otherwise known to be undecidable, which is important for some applications. Finally, it leads to parsimonious parameterizations of models, again an important point in some applications

A Graph Grammar for Modelling RNA Folding

We propose a new approach for modelling the process of RNA folding as a graph transformation guided by the global value of free energy. Since the folding process evolves towards a configuration in which the free energy is minimal, the global behaviour resembles the one of a self-adaptive system. Each RNA configuration is a graph and the evolution of configurations is constrained by precise rules that can be described by a graph grammar.Comment: In Proceedings GaM 2016, arXiv:1612.0105

Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues

We consider the totally asymmetric simple exclusion process (TASEP) on a periodic one-dimensional lattice of L sites. Using Bethe ansatz, we derive parametric formulas for the eigenvalues of its generator in the thermodynamic limit. This allows to study the curve delimiting the edge of the spectrum in the complex plane. A functional integration over the eigenstates leads to an expression for the density of eigenvalues in the bulk of the spectrum. The density vanishes with an exponent 2/5 close to the eigenvalue 0.Comment: 40 page

Combinatorial and topological phase structure of non-perturbative n-dimensional quantum gravity

We provide a non-perturbative geometrical characterization of the partition function of $n$-dimensional quantum gravity based on a coarse classification of riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.Comment: 13 page

Topology Induced Spatial Bose-Einstein Condensation for Bosons on Star-Shaped Optical Networks

New coherent states may be induced by pertinently engineering the topology of a network. As an example, we consider the properties of non-interacting bosons on a star network, which may be realized with a dilute atomic gas in a star-shaped deep optical lattice. The ground state is localized around the star center and it is macroscopically occupied below the Bose-Einstein condensation temperature T_c. We show that T_c depends only on the number of the star arms and on the Josephson energy of the bosonic Josephson junctions and that the non-condensate fraction is simply given by the reduced temperature T/T_c.Comment: 20 Pages, 5 Figure