Dynamical phase transitions for the activity biased Ising model in a magnetic field

We consider large deviations of the dynamical activity - defined as the total number of configuration changes within a time interval - for mean-field and one-dimensional Ising models, in the presence of a magnetic field. We identify several dynamical phase transitions that appear as singularities in the scaled cumulant generating function of the activity. In particular, we find low-activity ferromagnetic states and a novel high-activity phase, with associated first- and second-order phase transitions. The high-activity phase has a negative susceptibility to the magnetic field. In the mean-field case, we analyse the dynamical phase coexistence that occurs on first-order transition lines, including the optimal-control forces that reproduce the relevant large deviations. In the one-dimensional model, we use exact diagonalisation and cloning methods to perform finite-size scaling of the first-order phase transition at non-zero magnetic field

Giant leaps and long excursions: Fluctuation mechanisms in systems with long-range memory

We analyse large deviations of time-averaged quantities in stochastic processes with long-range memory, where the dynamics at time t depends itself on the value q_t of the time-averaged quantity. First we consider the elephant random walk and a Gaussian variant of this model, identifying two mechanisms for unusual fluctuation behaviour, which differ from the Markovian case. In particular, the memory can lead to large deviation principles with reduced speeds, and to non-analytic rate functions. We then explain how the mechanisms operating in these two models are generic for memory-dependent dynamics and show other examples including a non-Markovian symmetric exclusion process.Comment: longer version (16 pages), with more detailed discussio

Large deviations and optimal control forces for hard particles in one dimension

We analyse large deviations of the dynamical activity in one-dimensional systems of diffusing hard particles. Using an optimal-control representation of the large-deviation problem, we analyse effective interaction forces which can be added to the system, to aid sampling of biased ensembles of trajectories. We find several distinct regimes, as a function of the activity and the system size: we present approximate analytical calculations that characterise the effective interactions in several of these regimes. For high activity the system is hyperuniform and the interactions are long-ranged and repulsive. For low activity, there is a near-equilibrium regime described by macroscopic fluctuation theory, characterised by long-ranged attractive forces. There is also a far-from-equilibrium regime in which one of the interparticle gaps becomes macroscopic and the interactions depend strongly on the size of this gap. We discuss the extent to which transition path sampling of these ensembles is improved by adding suitable control forces

Symmetries and geometrical properties of dynamical fluctuations in molecular dynamics

We describe some general results that constrain the dynamical fluctuations that can occur in non-equilibrium steady states, with a focus on molecular dynamics. That is, we consider Hamiltonian systems, coupled to external heat baths, and driven out of equilibrium by non-conservative forces. We focus on the probabilities of rare events (large deviations). First, we discuss a PT (parity-time) symmetry that appears in ensembles of trajectories where a current is constrained to have a large (non-typical) value. We analyse the heat flow in such ensembles, and compare it with non-equilibrium steady states. Second, we consider pathwise large deviations that are defined by considering many copies of a system. We show how the probability currents in such systems can be decomposed into orthogonal contributions that are related to convergence to equilibrium and to dissipation. We discuss the implications of these results for modelling non-equilibrium steady states

Accelerated relaxation and suppressed dynamic heterogeneity in a kinetically constrained (East) model with swaps

We introduce a kinetically constrained spin model with a local softness parameter, such that spin flips can violate the kinetic constraint with an (annealed) site-dependent rate. We show that adding MC swap moves to this model can dramatically accelerate structural relaxation. We discuss the connection of this observation with the fact that swap moves are also able to accelerate relaxation in structural glasses. We analyse the rates of relaxation in the model. We also show that the extent of dynamical heterogeneity is strongly suppressed by the swap moves.EPSRC funding (to co-author JP Garrahan), see acknowledgement

Ergodicity and large deviations in physical systems with stochastic dynamics

Abstract In ergodic physical systems, time-averaged quantities converge (for large times) to their ensemble-averaged values. Large deviation theory describes rare events where these time averages differ significantly from the corresponding ensemble averages. It allows estimation of the probabilities of these events, and their mechanisms. This theory has been applied to a range of physical systems, where it has yielded new insights into entropy production, current fluctuations, metastability, transport processes, and glassy behaviour. We review some of these developments, identifying general principles. We discuss a selection of dynamical phase transitions, and we highlight some connections between large-deviation theory and optimal control theory. Graphical abstract </jats:sec

Effects of random pinning on the potential energy landscape of a supercooled liquid.

We use energy landscape methods to investigate the response of a supercooled liquid to random pinning. We classify the structural similarity of different energy minima using a measure of overlap. This analysis reveals a correspondence between distinct particle packings (which are characterised via the overlap) and funnels on the energy landscape (which are characterised via disconnectivity graphs). As the number of pinned particles is increased, we find a crossover from glassy behavior at low pinning to a structure-seeking landscape at high pinning, in which all thermally accessible minima are structurally similar. We discuss the consequences of these results for theories of randomly pinned liquids. We also investigate how the energy landscape depends on the fraction of pinned particles, including the degree of frustration and the evolution of distinct packings as the number of pinned particles is reduced.epsr

Unraveling the Large Deviation Statistics of Markovian Open Quantum Systems

We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.This work was supported by EPSRC Grants No. EP/ M014266/1 (J. P. G.) and No. EP/N03404X/1 (F. C. and J. P. G.), and by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. 335266 (ESCQUMA) (F. C.)

Tracer dynamics in one dimensional gases of active or passive particles

Abstract We consider one-dimensional systems comprising either active run-and-tumble particles (RTPs) or passive Brownian random walkers. These particles are either noninteracting or have hardcore exclusions. We study the dynamics of a single tracer particle embedded in such a system—this tracer may be either active or passive, with hardcore exclusion from environmental particles. In an active hardcore environment, both active and passive tracers show long-time subdiffusion: displacements scale as t 1/4 with a density-dependent prefactor that is independent of tracer type, and differs from the corresponding result for passive-in-passive subdiffusion. In an environment of noninteracting active particles, the passive-in-passive results are recovered at low densities for both active and passive tracers, but transient caging effects slow the tracer motion at higher densities, delaying the onset of any t 1/4 regime. For an active tracer in a passive environment, we find more complex outcomes, which depend on details of the dynamical discretization scheme. We interpret these results by studying the density distribution of environmental particles around the tracer. In particular, sticking of environment particles to the tracer cause it to move more slowly in noninteracting than in interacting active environments, while the anomalous behaviour of the active-in-passive cases stems from a ‘snowplough’ effect whereby a large pile of diffusive environmental particles accumulates in front of an RTP tracer during a ballistic run.</jats:p

Duality relations for the ASEP conditioned on a low current

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of currents, parametrized by the driving strength $s_K$, $K \geq 1$, we prove duality relations which arise from the quantum algebra $U_q[\mathfrak{gl}(2)]$ symmetry of the generator of the process with reflecting boundary conditions. Using these duality relations we prove on microscopic level a travelling-wave property of the conditioned process for a family of shock-antishock measures for $N>K$ particles: If the initial measure is a member of this family with $K$ microscopic shocks at positions $(x_1,\dots,x_K)$, then the measure at any time $t>0$ of the process with driving strength $s_K$ is a convex combination of such measures with shocks at positions $(y_1,\dots,y_K)$. which can be expressed in terms of $K$-particle transition probabilities of the conditioned ASEP with driving strength $s_N$.Comment: 26 page