Rationalizing Choice Functions by Multiple Rationales

The paper presents a notion of rationalizing choice functions that violate the “Independence of Irrelevant Alternatives” axiom. A collection of linear orderings is said to provide a rationalization by multiple rationales for a choice function if the choice from any choice set can be rationalized by one of the orderings. We characterize a tight upper bound on the minimal number of orderings that is required to rationalize arbitrary choice functions, and calculate the minimal number for several specific choice procedures.

Estimating differential entropy using recursive copula splitting

A method for estimating the Shannon differential entropy of multidimensional random variables using independent samples is described. The method is based on decomposing the distribution into a product of the marginal distributions and the joint dependency, also known as the copula. The entropy of marginals is estimated using one-dimensional methods. The entropy of the copula, which always has a compact support, is estimated recursively by splitting the data along statistically dependent dimensions. Numerical examples demonstrate that the method is accurate for distributions with compact and non-compact supports, which is imperative when the support is not known or of mixed type (in different dimensions). At high dimensions (larger than 20), our method is not only more accurate, but also significantly more efficient than existing approaches

On the Relation between Entropy and Kinetics

Thermodynamic and kinetic properties of materials are usually thought of as unconnected. Relating them is a conceptual challenge with many practical benefits. Here, based on first principles, we derive a rigorous inequality relating entropy and diffusion coefficients. It is universal and applicable to equilibrium as well as non-equilibrium steady states. The relation can be used to obtain useful bounds for the diffusion coefficient (normal or anomalous) from the calculated thermodynamic entropy or, conversely, to estimate the entropy based on measured diffusion coefficients. We demonstrate the validity and usefulness of the relation through several examples. We discuss its broad range of applications, in particular, for non-equilibrium systems.Comment: 22 pages, 6 figure

Detecting and characterizing phase transitions in active matter using entropy

A major challenge in the study of active matter lies in quantitative characterization of phases and transitions between them. We show how the entropy of a collection of active objects can be used to classify regimes and spatial patterns in their collective behavior. Specifically, we estimate the contributions to the total entropy from correlations between the degrees of freedom of position and orientation. This analysis pin-points the flocking transition in the Vicsek model while clarifying the physical mechanism behind the transition. When applied to experiments on swarming Bacillus subtilis with different cell aspect ratios and overall bacterial area fractions, the entropy analysis reveals a rich phase diagram with transitions between qualitatively different swarm statistics. We discuss physical and biological implications of these findings.Comment: 18 pages, 5 figures. arXiv admin note: text overlap with arXiv:2208.0529