The supernova-regulated ISM. I. The multi-phase structure

We simulate the multi-phase interstellar medium randomly heated and stirred by supernovae, with gravity, differential rotation and other parameters of the solar neighbourhood. Here we describe in detail both numerical and physical aspects of the model, including injection of thermal and kinetic energy by SN explosions, radiative cooling, photoelectric heating and various transport processes. With 3D domain extending 1 kpc^2 horizontally and 2 kpc vertically, the model routinely spans gas number densities 10^-5 - 10^2 cm^-3, temperatures 10-10^8 K, local velocities up to 10^3 km s^-1 (with Mach number up to 25). The thermal structure of the modelled ISM is classified by inspection of the joint probability density of the gas number density and temperature. We confirm that most of the complexity can be captured in terms of just three phases, separated by temperature borderlines at about 10^3 K and 5x10^5 K. The probability distribution of gas density within each phase is approximately lognormal. We clarify the connection between the fractional volume of a phase and its various proxies, and derive an exact relation between the fractional volume and the filling factors defined in terms of the volume and probabilistic averages. These results are discussed in both observational and computational contexts. The correlation scale of the random flows is calculated from the velocity autocorrelation function; it is of order 100 pc and tends to grow with distance from the mid-plane. We use two distinct parameterizations of radiative cooling to show that the multi-phase structure of the gas is robust, as it does not depend significantly on this choice.Comment: 28 pages, 22 figures and 8 table

The supernova-regulated ISM. II. The mean magnetic field

The origin and structure of the magnetic fields in the interstellar medium of spiral galaxies is investigated with 3D, non-ideal, compressible MHD simulations, including stratification in the galactic gravity field, differential rotation and radiative cooling. A rectangular domain, 1x1x2 kpc^{3} in size, spans both sides of the galactic mid-plane. Supernova explosions drive transonic turbulence. A seed magnetic field grows exponentially to reach a statistically steady state within 1.6 Gyr. Following Germano (1992) we use volume averaging with a Gaussian kernel to separate magnetic field into a mean field and fluctuations. Such averaging does not satisfy all Reynolds rules, yet allows a formulation of mean-field theory. The mean field thus obtained varies in both space and time. Growth rates differ for the mean-field and fluctuating field and there is clear scale separation between the two elements, whose integral scales are about 0.7 kpc and 0.3 kpc, respectively.Comment: 5 pages, 10 figures, submitted to Monthly Notices Letter

Random Costs in Combinatorial Optimization

The random cost problem is the problem of finding the minimum in an exponentially long list of random numbers. By definition, this problem cannot be solved faster than by exhaustive search. It is shown that a classical NP-hard optimization problem, number partitioning, is essentially equivalent to the random cost problem. This explains the bad performance of heuristic approaches to the number partitioning problem and allows us to calculate the probability distributions of the optimum and sub-optimum costs.Comment: 4 pages, Revtex, 2 figures (eps), submitted to PR

Phase Transition in the Number Partitioning Problem

Number partitioning is an NP-complete problem of combinatorial optimization. A statistical mechanics analysis reveals the existence of a phase transition that separates the easy from the hard to solve instances and that reflects the pseudo-polynomiality of number partitioning. The phase diagram and the value of the typical ground state energy are calculated.Comment: minor changes (references, typos and discussion of results

Symmetry breaking in numeric constraint problems

Symmetry-breaking constraints in the form of inequalities between variables have been proposed for a few kind of solution symmetries in numeric CSPs. We show that, for the variable symmetries among those, the proposed inequalities are but a specific case of a relaxation of the well-known LEX constraints extensively used for discrete CSPs. We discuss the merits of this relaxation and present experimental evidences of its practical interest.Postprint (author’s final draft

Phase transition and landscape statistics of the number partitioning problem

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution, is investigated by examining the energy landscape of this classic optimization problem. This is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure, termed a barrier tree, the leaves and internal nodes of which represent, respectively, the minima and the lowest energy saddles connecting those minima. Here we apply several measures of shape (balance and symmetry) as well as of branch lengths (barrier heights) to the barrier trees that result from the landscape of the NPP, aiming at identifying traces of the easy/hard transition. We find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights. Only the {\it difficulty} measure, given by the maximum value of the ratio between the barrier height and the energy surplus of local minima, succeeded in detecting traces of the phase transition in the tree. In adddition, we show that the barrier trees associated with the NPP are very similar to random trees, contrasting dramatically with trees associated with the $p$ spin-glass and random energy models. We also examine critically a recent conjecture on the equivalence between the NPP and a truncated random energy model

Introduction

Non peer reviewe