Density Fluctuations and Phase Behaviour in Non-Equilibrium Systems
There is an established theoretical framework within equilibrium statistical physics, where the underlying physical laws are microscopically reversible in time. For such systems, the Boltzmann probability distribution explicitly governs the space of all possible configurations, allowing a thorough understanding of their physical behaviour. In nature, however, thermal equilibrium is more an exception than a rule. Most biological systems are inherently out of equilibrium: the self-propelling nature of organisms such as birds and bacteria, or the existence of an extinction state easily breaks the principle of detailed balance and places the underlying physical system outside of thermal equilibrium. The absence of time reversibility usually prevents calculation of an explicit probability distribution. Nevertheless, there have been significant advances in the development of theories to describe non-equilibrium systems. A major theme is to employ a dynamical field theoretic approach, where coarse-grained density fields evolve via stochastic partial differential equations that encode the spatiotemporal evolution of the system. These field theories can be constructed in two directions: either bottom-up, by coarse-graining the microscopic rules of particle motion, or top-down, by imposing conservation laws and symmetries. This thesis addresses two types of phase behaviour, each described by dynamical field theories developed via one of the two approaches. The first concerns phase transitions into absorbing states, which are configurations that the system can reach but cannot leave. We focus on a class of such transitions that are coupled to particle number conservation and infinitely many absorbing states: the random organisation (RO) universality class. This class connects closely to one encompassing a wide range of models, including conserved sandpiles and growth of elastic interfaces under disordered media. Using renormalisation group analysis in the Doi-Peliti formalism, a bottom-up field-theoretic approach, we characterise analytically the critical behaviour near the transition, highlighting in particular the surprisingly weak density fluctuations at infinite wavelengths, a phenomenon known as hyperuniformity. Importantly, our results elucidate several physical aspects of hyperuniformity, from its emergence via the cancellation of divergent terms, to the significance of conserved diffusional noise. The second line of work presented in this thesis introduces a ternary phase separation model in the presence of non-reciprocal interactions between two of the three phases. These interactions directly violate Newton's third law and cannot be derived from a free energy, making them intrinsic to non-equilibrium systems. We adopt a top-down field-theoretic approach combined with numerical simulations to uncover the rich phenomena in this model, from the contact angles formed at the triple point of phase coexistence to the formation and stability of travelling wave patterns. Analytically, we examine the propagation speed of these travelling waves near equilibrium via perturbation methods. This model paves the way for a generalisation of the equilibrium wetting theory near solid boundaries to active and non-equilibrium systems.
Soft Matter, Non-Equilibrium Statistical Mechanics, Renormalisation Group, Pattern Formation, Field Theories
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