This grant supports theoretical research on statistical physics. The research is divided into two parts. One part will study the properties of networks. Networks are ubiquitous in our lives and range from the internet to biological and social networks. Network properties can be revealed by applying methods from statistical physics. The second part of the research deals with applying statistical physics methods to more complicated systems such as biological ones.
In particular, this research is focused on structural and dynamical properties of heterogeneous networks and on applying first-passage statistics to elucidate fundamental complex processes.
Intellectual Merit: A prominent feature of heterogeneous networks is that the degree of each node (the number of attached links) has a broad distribution. This feature has profound consequences for physical processes occurring on these networks and on underlying network structure. In the first part of the project, the question of how social balance is achieved in networks that initially contain competing friendly and unfriendly pair-wise interactions will be investigated. The dynamics of consensus formation for the voter model on heterogeneous networks will also be studied, where the underlying broad degree distribution controls the dynamics. On the topic of network structure, the phenomenon of large fluctuations on non-sparse networks will be studied. In addition, the dynamics of specialization will be investigated, where an initially homogeneous network becomes more modular as it grows, a ubiquitous process that underlies both the nature of scientific progress and societal organizations. For all these examples, the master equation approach will be our primary investigative tool. In the second part of this research, classical methods from first-passage statistics will be applied to diverse stochastic processes with biological and fundamental ramifications. One such example is the statistics of cell division, where replicative senescence (essentially cell death) is governed by the length of telomeres contained in each cell. Upon each cell division, the telomere lengths in each daughter cell decreases and the cell dies when telomere length reaches zero. First-passage ideas will be applied to determine the lifetime distribution of a cell population. Another research effort will be devoted to understanding mechanisms for social diversity in competitive populations, where individual agents can gain fitness by competing with less fit agents, or can lose fitness due to inactivity. These interactions give rise to a random walk in fitness space with fitness-dependent hopping probabilities. Basic question about this stochastic process, such as whether a society consisting of such agents is egalitarian or inherently unfair, will be addressed. Finally, the simplest "piston" problem - a finite one-dimensional interval where a massive particle separates two much lighter particles - will be investigated. This idealized system exhibits extremely rich dynamics that may be understood by constructing mappings to equivalent billiard systems and by exploiting first-passage ideas. This approach will be extended to develop insights for real piston systems that contain many particles and also where particles undergo inelastic collisions.
Broader Impacts: The research is ideal for extensive graduate student participation. Many of the problems are relatively easy to conceptualize and important aspects of these projects do not require extensive formal background before a student can become productive. The appropriate investigative methods involve a healthy mix of analytics and numerics, thus providing broad theoretical training for students. The projects outlined will thus contribute substantially to graduate education by providing the basis for the Ph.D. thesis research of several graduate students. Additionally, several international postdoctoral fellows with their own external funding have recently spent 1-2 year periods at Boston University to work on grant-related projects, thus considerably leveraging NSF funds.Finally, it is anticipated that some of the research will be performed in collaboration with colleagues and Los Alamos National Lab (LANL) and at the Santa Fe Institute (SFI). A number of the projects are natural outgrowths of scientific interactions developed by the PI while in residence at LANL during the past year. The projects on opinion dynamics and on social diversity, in particular, mesh well with the current emphasis at SFI and provides a natural segue to collaborative research with the physical and social scientists at this institution.
