The aim of the project is to better understand the formation of equilibrium-like domains in non-equilibrium systems, their emergence and control. The project will include both analysis of simple systems and the study of natural and technological systems based on real data
Equilibrium is a fundamental concept in statistical physics; it assumes that while the system dynamics is governed by microscopic interactions, some systems eventually reach a state where macroscopic observables remain unchanged. The evolution of such systems is typically driven by the corresponding Hamiltonian energy function and their states converge to the equilibrium Gibbs-Boltzmann distribution. The dynamics of a non-equilibrium system, on the other hand, is not necessarily governed by a process derived from a Hamiltonian and such systems do not converge to an equilibrium state; this characterises many real systems, for instance in the ﬁnancial and biological areas. However, constituents of some of these systems exhibit equilibrium-like behaviour in emerging localised or non-localized domains; notable examples of this behaviour are the emergence of equilibrium-like structures in functional brain networks and neuronal dynamics, and the equilibrium theory of markets. Consequently, such domains may exist under some conditions within many other non-equilibrium systems but are difficult to identify. This project aims to better understand these domains and how best to identify and control them at both theoretical and practical levels.
Equilibrium vs. non-equilibrium:Most systems in statistical physics fall into one of these two categories, they are either equilibrium or non-equilibrium systems. The evolution of both (in discrete time steps) is characterised by a trajectory s(0)→ ∙∙∙∙ →s(t), where s(t) is a microscopic state of the system (microstate) at time t. For a Markovian processes this probability can be decomposed to a chain of transition probabilities from one time step to the next
P[s(0) →∙∙→ s(t)] = W[s(t)|s(t −1)] ×· ··× W[s(1)|s(0)] P(s(0)) (1)
with initial P(s(0)) and transition W[s(t)|s(t−1)] probability distributions. Expectation value of any macroscopic observable represented as a function of the microstates, deﬁning a macrostate, can be computed from the probability distribution (1). Unfortunately, even for highly stylised models of statistical physics this procedure is non-trivial. In equilibrium systems, one assumes that the probability of any microscopic trajectory is invariant under time-reversal; this mostly leads to a property termed detailed balance for the stationary distribution P∞[s] of process (1), where transitions from state s to s’ are balanced by transitions in the opposite direction. For thermodynamic systems, this gives rise to the Gibbs-Boltzmann distribution, P∞[s]~ exp [-E(s)/T] for temperature T and Hamiltonian (or energy) function E(s), which usually follows from the transition probability W[s|s’]. The stationary distributions in systems without detailed balance (when such distributions do exist) are generally much more complicated and are difficult to analyse.
Equilibrium-like domains:While many real systems, for instance in biology, ﬁnance and technology, are inherently not in equilibrium, some of their constituents exhibit equilibrium-like behaviour in emerging localised or non-localised domains. Past studies addressed equilibrium and non-equilibrium systems separately. We have recently illustrated , by examining designed exemplar Ising-spin like models of both densely and sparsely connected topologies, that in a large class of non-equilibrium systems without detailed balance, one can still ﬁnd domains that exhibit equilibrium-likebehaviour. The term equilibrium-like behaviour refers to systems where the average values of macroscopic observables are equal to those of their equilibrium counterparts, assuming that in very large systems only a limited number of observables can be measured. For the simple systems studied, these domains have been shown to emerge either abruptly or gradually, depending on the system parameters, for instance temperature, and disappear, becoming indistinguishable from the remainder of the system for other parameter values. Consequently, such domains may exist, under some conditions, within a non-equilibrium system but may be difficult to identify.
 D. Saad and A. Mozeika, Emergence of Equilibrium-like Domains within Non-equilibrium Ising Spin Systems, Phys. Rev. E 87, 032131 (2013).
The aim of the current project is to explore and develop this novel concept in order to understand and identify the emergence of equilibrium-like domains in ﬁnancial, social or biological systems, examine their dynamics when external conditions change slowly and rigorously prove the existence of this phenomenon in simple models. More speciﬁcally, the project will include the following activities:
(a) Study analytically sparsely connected systems: Recently, we were able to show the emergence of equilibrium-like domains in simple sparse systems of various topologies; however, the dynamics of such models is typically difficult to investigate via the methodology used, that relies on Generating Functional Analysis. We will develop the methods needed to study the dynamics and emergence of equilibrium domains by: (i) examining models that can be studied using variants of the existing method; (ii) exploit recent advances in the dynamical cavity method to analyse sparse systems with such properties. This will provide the theoretical basis needed to better understand the phenomena in sparsely connected systems.
(b) Rigorous analysis of simple systems: While the mathematical methods required for studying the dynamics in disordered systems is generally difficult, relying on generating functional methods that become intractable in many cases, some simple models are mathematically tractable and can be fully investigated rigorously. We will investigate such systems to fully establish the emergence and dynamics of equilibrium domains and their dependence on initial conditions and system parameters.
(c) Real systems: The analysis carried out so far has focussed on simple exemplar models based on physical spin-systems. As part of the project we will identify emerging equilibrium-like domains in real systems and show that they exhibit the behaviour observed in the toy physical systems studied, both qualitatively and quantitatively. We will particularly examine ﬁnancial, biological or social networks.
(d) Identify non-localised domains: Equilibrium-like domains are difficult to identify, especially if the nodes which constitute the domain are non-localised, i.e., not clustered. We will develop methods for identifying such domains and nodes related to them by studying the effect of local perturbations on the different nodes.
(e) Control domain emergence/disappearance: Equilibrium-like domains emerge and disappear depending on the system parameters, such as temperature, noise and topology. We will identify control measures to be utilised for preventing the dissipation of domains. Control measures could take the form of global parameter changes such as temperature, external fields applied to particular sites or topological changes. This will impact on our ability to stabilise such domains and keep them in equilibrium-like states under dynamical and volatile external conditions.
(f) Changing parameters: Many systems that exhibit equilibrium-like behaviour are subject to adiabatically changing parameters and/or interactions, for instance bird-ﬂock dynamics and ﬁnancial markets. We will study how slowly changing external parameters affect the equilibrium-like domains; how they can be controlled and what are the level and speed of these changes that cause the equilibrium-like domains to collapse.
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