We derive the general structure for returning to the steady macroscopic nonequilibrium condition, assuming a first-order relaxation equation obtained as zero-cost flow for the Lagrangian governing the dynamical fluctuations. The main ingredient is local detailed balance from which a canonical form of the time-symmetric fluctuation contribution (aka frenesy) can be obtained. That determines the macroscopic evolution. As a consequence, the linear response around stationary nonequilibrium gets connected with the small dynamical fluctuations, leading to fluctuation-response relations. Those results may be viewed as nonequilibrium extension of the well-known structure where the relaxation to equilibrium is characterized by a (dissipative) gradient flow on top of a Hamiltonian motion.

]]>In the last 10 to 20 years, the study of such eigenvalue optimisation problems has become popular in the specific context of differential operators defined on metric graphs, also known as quantum graphs. Despite being essentially one-dimensional objects, these can display surprisingly rich behaviour, and are often useful as toy models for higher-dimensional problems.

In this talk I will give an informal introduction, first to some of the original ideas behind shape optimisation of domains, then quantum graphs, discussing what kinds of questions one can ask, what techniques are used, and how our understanding of these problems (and what questions are interesting) has developed over the last couple of decades.

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