In general terms this proposal concerns the behavior of systems with many degrees of freedom. Here one looks for (a) collective behavior of the system as a whole or (b) a statistical description of the behavior that an individual component might exhibit. It is emphasized that (a) and (b) are not disjoint criteria: indeed quite often (but not always) they coincide. The setups that will be used to promote these studies are of four types; all, to a greater or lesser extent, are interrelated and all four have their origins in the description of physical problems. These are (i) [Classical Spin--Systems] Lattice models of spins or particles interacting under certain rules, primarily local, which are known as the laws of classical equilibrium statistical mechanics. (ii) [Quantum Spin--Systems] A similar setup as in (i) but under the rules of quantum equilibrium statistical mechanics. (iii) [Interacting Particle Systems] Models where individual particles or spins occupying lattice positions update their position or state dynamically in time according to stochastic rules that are themselves effected by the existing local configuration. (iv) [Cluster Models/Graphical Representations] Graphical models where, in addition to local rules, configurations are determined, stochastically, according to global considerations or non-local rules. In addition to the standard lattice based large-scale descriptions, some of the work will concern continuum models --as well as continuum limits of lattice models.
A philosophy which became prevalent beginning in the late 1960's is that of universality: Once the basic nature and premise of the constituents and their interactions are determined, the resultant (possible modes of) large?scale, long?time collective behaviors in systems with many degrees of freedom do not depend on the meticulous details of the model. Thus a variety of systems will be investigated where the nature of the interacting constituents is of the simplest type. However, it is anticipated that the full tapestry of collective behavior will be exhibited. Thus, notwithstanding the physical origins of these types of models, it is anticipated, at least by the PI, that these types of models allow for a broad range of applications--even beyond the physical sciences. (Already some of this has come to fruition in the context of a related proposal which concerns the study of criminal behavior and allocation of police resources.) Moreover and perhaps more pertinently: This proposal and its predecessor encompass a broad array of problems with multiple facets providing research opportunities at every level and the successful completion of the program will require the assistance of many collaborators ranging from the undergraduate to the senior research collaborator.
In general terms this research concerned the behavior of systems with many degrees of freedom. From the mathematical perspective the word many is taken to mean either "infinite" or, at best, asymptotically large. Under these circumstances, the detailed behavior of all the constituents is almost a meaningless concept and one looks for (a) collective behavior of the system as a whole or (b) a statistical description of the behavior that an individual component might exhibit. It is emphasized that (a) and (b) are not disjoint criteria. The setups used to promote these studies are of four types; all, to a greater or lesser extent, are interrelated and all four have their origins in the description of physical problems. These are (i) [Classical Spin–Systems] Lattice models of spins or particles interacting under certain rules, primarily local, which are known as the laws of classical equilibrium statistical mechanics. (ii) [Quantum Spin–Systems] A similar setup as in (i) but under the rules of quantum equilibrium statistical mechanics. (iii) [Interacting Particle Systems] Models where individual particles or spins occupying lattice positions update their position or state dynamically in time according to stochastic rules that are themselves effected by the existing local configuration. (iv) [Cluster Models/Graphical Representations] Graphical models where, in addition to local rules, configurations are determined, stochastically, according to global considerations or non–local rules. A philosophy which became prevalent beginning in the late 1960’s is that of universality: Once the basic nature and premise of the constituents and their interactions are determined, the resultant (possible modes of) large–scale, long–time collective behaviors do not depend on the meticulous details of the model. Thus, notwithstanding the physical origins of these types of models, it has been demonstrated, in the context of this award, that these types of models allow for a broad range of applications – even beyond the physical sciences. Hence, while the primary scientific goals lean towards the advancement of the mathematical sciences in the area of probability theory, some of the results turn out to be of direct relevance to the biological and social sciences. Particular examples include applications to the formation of gang territories. In addition, the research has encompassed a broad array of problems with multiple facets providing theoretical (non–numerical based) research opportunities at every level. Successful completion of the program has required the assistance of many collaborators. In the context of this award, the PI has enjoyed collaborations with undergraduate, graduate, postdoctoral, and senior level co–researchers from a variety of cultural backgrounds with both genders adequately represented.
