Geometrical Ambiguity of Pair Statistics. I. Point Configurations

Point configurations have been widely used as model systems in condensed matter physics, materials science and biology. Statistical descriptors such as the $n$-body distribution function $g_n$ is usually employed to characterize the point configurations, among which the most extensively used is the pair distribution function $g_2$. An intriguing inverse problem of practical importance that has been receiving considerable attention is the degree to which a point configuration can be reconstructed from the pair distribution function of a target configuration. Although it is known that the pair-distance information contained in $g_2$ is in general insufficient to uniquely determine a point configuration, this concept does not seem to be widely appreciated and general claims of uniqueness of the reconstructions using pair information have been made based on numerical studies. In this paper, we introduce the idea of the distance space, called the $\mathbb{D}$ space. The pair distances of a specific point configuration are then represented by a single point in the $\mathbb{D}$ space. We derive the conditions on the pair distances that can be associated with a point configuration, which are equivalent to the realizability conditions of the pair distribution function $g_2$. Moreover, we derive the conditions on the pair distances that can be assembled into distinct configurations. These conditions define a bounded region in the $\mathbb{D}$ space. By explicitly constructing a variety of degenerate point configurations using the $\mathbb{D}$ space, we show that pair information is indeed insufficient to uniquely determine the configuration in general. We also discuss several important problems in statistical physics based on the $\mathbb{D}$ space.Comment: 28 pages, 8 figure

Short periodic orbits theory for partially open quantum maps

We extend the semiclassical theory of short periodic orbits [Phys. Rev. E {\bf 80}, 035202(R) (2009)] to partially open quantum maps. They correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity $R$. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of these kind of systems. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belong to the classical repeller of the open map ($R=0$) are able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of $R$ values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behaviour, providing an explanation in terms of short periodic orbits.Comment: 6 pages, 4 figure

Signatures of the neurocognitive basis of culture wars found in moral psychology data\ud

Moral Foundation Theory (MFT) states that groups of different observers may rely on partially dissimilar sets of moral foundations, thereby reaching different moral valuations on a subset of issues. With the introduction of functional imaging techniques, a wealth of new data on neurocognitive processes has rapidly mounted and it has\ud become increasingly more evident that this type of data should provide an adequate basis for modeling social systems. In particular, it has been shown that there is a spectrum of cognitive styles with respect to the differential handling of novel or corroborating information.\ud Furthermore this spectrum is correlated to political affiliation. Here we use methods of statistical mechanics to characterize the collective behavior of an agent-based model society whose interindividual interactions due to information exchange in the form of opinions, are in qualitative agreement with neurocognitive and psychological data. The main conclusion derived from the model is\ud that the existence of diversity in the cognitive strategies yields different statistics for the sets of moral foundations and that these arise from the cognitive interactions of the agents. Thus a simple interacting agent model, whose interactions are in accord with empirical data about moral dynamics, presents statistical signatures\ud consistent with those that characterize opinions of conservatives and liberals. The higher the difference in the treatment of novel and corroborating information the more agents correlate to liberals.\u

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras \bf{V}, and formulated as a statement about functors Set --> \bf{V}. From this one easily obtains analogs of the results stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega, and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on \Omega. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of 2^{card(\Omega)} copies of itself. That paper also gave an example of a group of cardinality 2^{card(\Omega)} that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently established a large class of such examples. Those results are shown to be instances of a general property of the lattice of solution sets in Sym(\Omega) of sets of equations with constants in Sym(\Omega). Again, similar results -- this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega. Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely to be updated more often than arXiv copy Revised version includes answers to some questions left open in first version, references to results of Wehrung answering some other questions, and some additional new result