# Science Focus

At the core of our proposal are three cross-disciplinary themes, which have been identified as unifying aspects of systems operating far from equilibrium:

From the fractal structure of growing trees to the geometric regularity of snowflakes and crystals, pattern formation and spontaneous ordering are classic examples of emergent phenomena, occurring in both equilibrium and non-equilibrium settings. In this proposal, we consider examples as diverse as biological self-assembly1, community formation in social networks2, and zonal flows in plasma and atmospheric turbulence3. Emergence in equilibrium and non-equilibrium quantum systems has also drawn much interest of late. For instance, exotic quasi-particles emerge in topological phases of matter; magnetic monopole excitations govern magnetic relaxation properties in frustrated magnets; and novel phase transitions appear in driven quantum liquids. Similar theoretical methods are used in modelling this wide range of processes, and we aim to facilitate progress by combining expertise from different areas.**Spontaneous development of structure and patterns:**The sudden fracture of materials under strain is of clear industrial importance and has been studied extensively in mathematics, physics and materials science. Similar failure phenomena appear in many other contexts, ranging from earthquakes to the plasma collapses that hinder nuclear fusion research, and the stock-market crashes and epidemics that are predicted by statistical-mechanical analyses of human behaviour. In a very different but related context, sudden changes in the parameters of a strongly correlated quantum system (quantum quench) lead to non-equilibrium phenomena of great interest to quantum information and to the implementation of quantum computing, through the concepts of quantum coherence and entanglement. There are hints of common mathematical structures underlying these diverse phenomena, which we will explore, concentrating particularly on characteristic correlations that can be used to warn of impending systemic failure.**Dynamics of large-scale failure:**A common setting for the appearance of complex emergent behaviour is that of systems with strong external forcing. Examples include turbulence in pipe flow at high Reynolds number, shear-banding in the non-linear rheology of soft-materials4, and non-equilibrium systems like the micro-maser5 and other driven quantum systems. A related issue concerns responses to sudden shocks. These may lead to unexpected behaviour, such as when physical or computational networks respond to breakage or failure of links. Near equilibrium, fluctuation-dissipation theory provides a generic framework for calculating responses; far-from-equilibrium generalisations of such a theory would be extremely valuable, but remain in their infancy.*Responses to strong driving forces and to shocks:*The idea of emergence is central to the study of strongly correlated systems, both quantum and classical. One spectacular example is the quantum Hall effect, where a system of interacting electrons conspires to form excitations with only a fraction of the electron charge. Others such as giant magnetoresistance (the scientific underpinning of hard drives) have demonstrated how the interplay of materials growth, theoretical understanding and technological innovation can impact upon our daily lives. In spite of these successes, the study of strongly correlated systems has a legacy of fundamental models it cannot solve (e.g. the fermionic Hubbard model) and phenomena that it cannot explain (e.g high T*Emergence and non-equilibrium phenomena in condensed matter*:_{c}superconductivity).

Another key issue in strongly correlated physics is modelling and understanding the behaviour of (quantum) systems out of equilibrium. Non-equilibrium systems demonstrate rich new properties and we are still searching for the general overarching principles. Examples range from cold atomic gases to polariton condensates and turbulent superfluids. Cold atomic gases are particularly interesting in that they both provide some of the simplest contexts where to study out of equilibrium quantum phenomena, and they have the ability to simulate Hamiltonians motivated by solid-state phenomenology.

The interplay between the diverse theoretical and experimental approaches involved holdsenormous promise to revolutionise our understanding.

We recognise that these are just initial topics, and the list above is not exhaustive. NetworkPlus activities (meetings and workshops) will be used to formulate a more complete list of topics. These initial topics, we hope, will bring some focus to our discussions of this very diverse scientific problem.