Discreteness Effects in Population Dynamics

We analyse numerically the effects of small population size in the initial transient regime of a simple example population dynamics. These effects play an important role for the numerical determination of large deviation functions of additive observables for stochastic processes. A method commonly used in order to determine such functions is the so-called cloning algorithm which in its non-constant population version essentially reduces to the determination of the growth rate of a population, averaged over many realizations of the dynamics. However, the averaging of populations is highly dependent not only on the number of realizations of the population dynamics, and on the initial population size but also on the cut-off time (or population) considered to stop their numerical evolution. This may result in an over-influence of discreteness effects at initial times, caused by small population size. We overcome these effects by introducing a (realization-dependent) time delay in the evolution of populations, additional to the discarding of the initial transient regime of the population growth where these discreteness effects are strong. We show that the improvement in the estimation of the large deviation function comes precisely from these two main contributions

Finite-size and finite-time effects in large deviation functions near dynamical symmetry breaking transitions

We introduce and study a class of particle hopping models consisting of a single box coupled to a pair of reservoirs. Despite being zero-dimensional, in the limit of large particle number and long observation time, the current and activity large deviation functions of the models can exhibit symmetry-breaking dynamical phase transitions. We characterize exactly the critical properties of these transitions, showing them to be direct analogues of previously studied phase transitions in extended systems. The simplicity of the model allows us to study features of dynamical phase transitions which are not readily accessible for extended systems. In particular, we quantify finite-size and finite-time scaling exponents using both numerical and theoretical arguments. Importantly, we identify an analogue of critical slowing near symmetry breaking transitions and suggest how this can be used in the numerical studies of large deviations. All of our results are also expected to hold for extended systems.Comment: 34 pages, 6 figure

Effect of disorder geometry on the critical force in disordered elastic systems

We address the effect of disorder geometry on the critical force in disordered elastic systems. We focus on the model system of a long-range elastic line driven in a random landscape. In the collective pinning regime, we compute the critical force perturbatively. Not only our expression for the critical force confirms previous results on its scaling with respect to the microscopic disorder parameters, it also provides its precise dependence on the disorder geometry (represented by the disorder two-point correlation function). Our results are successfully compared to the results of numerical simulations for random field and random bond disorders.Comment: 18 pages, 7 figure

Dynamical symmetry breaking and phase transitions in driven diffusive systems

We study the probability distribution of a current flowing through a diffusive system connected to a pair of reservoirs at its two ends. Sufficient conditions for the occurrence of a host of possible phase transitions both in and out of equilibrium are derived. These transitions manifest themselves as singularities in the large deviation function, resulting in enhanced current fluctuations. Microscopic models which implement each of the scenarios are presented, with possible experimental realizations. Depending on the model, the singularity is associated either with a particle-hole symmetry breaking, which leads to a continuous transition, or in the absence of the symmetry with a first-order phase transition. An exact Landau theory which captures the different singular behaviors is derived.Comment: 14 pages, 2 figure

Dynamical phase transitions in the current distribution of driven diffusive channels

We study singularities in the large deviation function of the time-averaged current of diffusive systems connected to two reservoirs. A set of conditions for the occurrence of phase transitions, both first and second order, are obtained by deriving Landau theories. First-order transitions occur in the absence of a particle-hole symmetry, while second-order occur in its presence and are associated with a symmetry breaking. The analysis is done in two distinct statistical ensembles, shedding light on previous results. In addition, we also provide an exact solution of a model exhibiting a second-order symmetry-breaking transition.Comment: 44 pages, 6 figure

Static fluctuations of a thick 1D interface in the 1+1 Directed Polymer formulation

Experimental realizations of a 1D interface always exhibit a finite microscopic width $\xi>0$; its influence is erased by thermal fluctuations at sufficiently high temperatures, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature $T_c(\xi)$. Exploiting the exact mapping between the static 1D interface and a 1+1 Directed Polymer (DP) growing in a continuous space, we study analytically both the free-energy and geometrical fluctuations of a DP, at finite temperature $T$, with a short-range elasticity and submitted to a quenched random-bond Gaussian disorder of finite correlation length $\xi$. We derive the exact `time'-evolution equations of the disorder free-energy $\bar{F}(t,y)$, its derivative $\eta (t,y)$, and their respective two-point correlators $\bar{C}(t,y)$ and $\bar{R}(t,y)$. We compute the exact solution of its linearized evolution $\bar{R}^{lin}(t,y)$, and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder ($\xi=0$), to construct a `toymodel' leading to a simple description of the DP. This model is characterized by Brownian-like free-energy fluctuations, correlated at small $|y|<\xi$, of amplitude $\tilde{D}_{\infty}(T,\xi)$. We present an extended scaling analysis of the roughness predicting $\tilde{D}_{\infty} \sim 1/T$ at high-temperatures and $\tilde{D}_{\infty} \sim 1/T_c(\xi)$ at low-temperatures. We identify the connection between the temperature-induced crossover and the full replica-symmetry breaking in previous Gaussian Variational Method computations. Finally we discuss the consequences of the low-temperature regime for two experimental realizations of KPZ interfaces, namely the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.Comment: 33 pages, 6 figures. The initial preprint arXiv:1209.0567v1 has been split into two parts upon refereeing process. The first part gathers the analytical results and is published (see reference below). It corresponds to the current version of arXiv:1209.0567. The second part gathers the numerical results and corresponds the other arXiv preprint arXiv:1305.236

Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model

We analyze large deviations of the time-averaged activity in the one dimensional Fredrickson-Andersen model, both numerically and analytically. The model exhibits a dynamical phase transition, which appears as a singularity in the large deviation function. We analyze the finite-size scaling of this phase transition numerically, by generalizing an existing cloning algorithm to include a multi-canonical feedback control: this significantly improves the computational efficiency. Motivated by these numerical results, we formulate an effective theory for the model in the vicinity of the phase transition, which accounts quantitatively for the observed behavior. We discuss potential applications of the numerical method and the effective theory in a range of more general contexts.Comment: 20 pages, 10 figure

Finite-time and finite-size scalings in the evaluation of large-deviation functions: Analytical study using a birth-death process

The Giardin\`a-Kurchan-Peliti algorithm is a numerical procedure that uses population dynamics in order to calculate large deviation functions associated to the distribution of time-averaged observables. To study the numerical errors of this algorithm, we explicitly devise a stochastic birth-death process that describes the time evolution of the population probability. From this formulation, we derive that systematic errors of the algorithm decrease proportionally to the inverse of the population size. Based on this observation, we propose a simple interpolation technique for the better estimation of large deviation functions. The approach we present is detailed explicitly in a two-state model.Comment: 13 pages, 1 figure. First part of pair of companion papers, Part II being arXiv:1607.0880

Equilibrium-like fluctuations in some boundary-driven open diffusive systems

There exist some boundary-driven open systems with diffusive dynamics whose particle current fluctuations exhibit universal features that belong to the Edwards-Wilkinson universality class. We achieve this result by establishing a mapping, for the system's fluctuations, to an equivalent open --yet equilibrium-- diffusive system. We discuss the possibility of observing dynamic phase transitions using the particle current as a control parameter