Active flows and networks

Abstract

Coherent, large scale dynamics in many nonequilibrium physical, biological, or information transport networks are driven by small-scale local energy input. In the first part of this thesis, we introduce and explore two analytically tractable nonlinear models for such active flow networks, drawing motivation from recent microfluidic experiments on bacterial and other microbial suspensions. In contrast to equipartition with thermal driving, we find that active friction selects discrete states with only a limited number of modes excited at distinct fixed amplitudes. When the active transport network is incompressible, these modes are cycles with constant flow; when it is compressible, they are oscillatory. As is common in such network dynamical systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the flow. Spectral graph theory has traditionally prioritized analyzing Laplacians of unweighted networks with specified adjacency properties. For the second part of the thesis, we introduce a complementary framework, providing a mathematically rigorous positively weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of three archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces weak chimera states in Kuramoto-type oscillator networks, tunes or suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network.