Monte Carlo Results for Projected Self-Avoiding Polygons: A Two-dimensional Model for Knotted Polymers

We introduce a two-dimensional lattice model for the description of knotted polymer rings. A polymer configuration is modeled by a closed polygon drawn on the square diagonal lattice, with possible crossings describing pairs of strands of polymer passing on top of each other. Each polygon configuration can be viewed as the two- dimensional projection of a particular knot. We study numerically the statistics of large polygons with a fixed knot type, using a generalization of the BFACF algorithm for self-avoiding walks. This new algorithm incorporates both the displacement of crossings and the three types of Reidemeister transformations preserving the knot topology. Its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favor of this ergodicity are given together with a tentative sketch of proof. Assuming this ergodicity, we obtain numerically the following results for the statistics of knotted polygons: In the limit of a low crossing fugacity, we find a localization along the polygon of all the primary factors forming the knot. Increasing the crossing fugacity gives rise to a transition from a self-avoiding walk to a branched polymer behavior.Comment: 36 pages, 30 figures, latex, epsf. to appear in J.Phys.A: Math. Ge

Reversibility Checking for Markov Chains

In this paper, we present reversibility preserving operations on Markov chain transition matrices. Simple row and column operations allow us to create new reversible transition matrices and yield an easy method for checking a Markov chain for reversibility

The Michaelis-Menten-Stueckelberg Theorem

We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the "rarefied gas" limit or the "molecular chaos" hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density ({\em low density of elementary events}). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann $H$-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.Comment: 54 pages, the final version; correction of a misprint in Attachment

Analytical study of tunneling times in flat histogram Monte Carlo

We present a model for the dynamics in energy space of multicanonical simulation methods that lends itself to a rather complete analytic characterization. The dynamics is completely determined by the density of states. In the \pm J 2D spin glass the transitions between the ground state level and the first excited one control the long time dynamics. We are able to calculate the distribution of tunneling times and relate it to the equilibration time of a starting probability distribution. In this model, and possibly in any model in which entering and exiting regions with low density of states are the slowest processes in the simulations, tunneling time can be much larger (by a factor of O(N)) than the equilibration time of the probability distribution. We find that these features also hold for the energy projection of single spin flip dynamics.Comment: 7 pages, 4 figures, published in Europhysics Letters (2005

Noisy Hamiltonian Monte Carlo for doubly-intractable distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician's toolbox as an alternative sampling method in settings when standard Metropolis-Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept-reject step is used to correct the bias. For doubly-intractable distributions -- such as posterior distributions based on Gibbs random fields -- HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept-reject proposals poses difficulty. In this paper, we study the behaviour of HMC when these quantities are replaced by Monte Carlo estimates