Dynamical Mean-Field Theory

The dynamical mean-field theory (DMFT) is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particular, the combination of the DMFT with conventional methods for the calculation of electronic band structures has led to a powerful numerical approach which allows one to explore the properties of correlated materials. In this introductory article we discuss the foundations of the DMFT, derive the underlying self-consistency equations, and present several applications which have provided important insights into the properties of correlated matter.Comment: Chapter in "Theoretical Methods for Strongly Correlated Systems", edited by A. Avella and F. Mancini, Springer (2011), 31 pages, 5 figure

Priorities for synthesis research in ecology and environmental science

Synthesis research in ecology and environmental science improves understanding, advances theory, identifies research priorities, and supports management strategies by linking data, ideas, and tools. Accelerating environmental challenges increases the need to focus synthesis science on the most pressing questions. To leverage input from the broader research community, we convened a virtual workshop with participants from many countries and disciplines to examine how and where synthesis can address key questions and themes in ecology and environmental science in the coming decade. Seven priority research topics emerged: (1) diversity, equity, inclusion, and justice (DEIJ), (2) human and natural systems, (3) actionable and use-inspired science, (4) scale, (5) generality, (6) complexity and resilience, and (7) predictability. Additionally, two issues regarding the general practice of synthesis emerged: the need for increased participant diversity and inclusive research practices; and increased and improved data flow, access, and skill-building. These topics and practices provide a strategic vision for future synthesis in ecology and environmental science

Subsampling scaling

In real-world applications, observations are often constrained to a small fraction of a system. Such spatial subsampling can be caused by the inaccessibility or the sheer size of the system, and cannot be overcome by longer sampling. Spatial subsampling can strongly bias inferences about a system’s aggregated properties. To overcome the bias, we derive analytically a subsampling scaling framework that is applicable to different observables, including distributions of neuronal avalanches, of number of people infected during an epidemic outbreak, and of node degrees. We demonstrate how to infer the correct distributions of the underlying full system, how to apply it to distinguish critical from subcritical systems, and how to disentangle subsampling and finite size effects. Lastly, we apply subsampling scaling to neuronal avalanche models and to recordings from developing neural networks. We show that only mature, but not young networks follow power-law scaling, indicating self-organization to criticality during development.peerReviewe