Abstract. We study the early time dynamics of the 2d ferromagnetic Ising model instantaneously quenched from the disordered to the ordered, low temperature, phase. We evolve the system with kinetic Monte Carlo rules that do not conserve the order parameter. We confirm the rapid approach to random critical percolation in a time-scale that diverges with the system size but is much shorter than the equilibration time. We study the scaling properties of the evolution towards critical percolation and we identify an associated growing length, different from the curvature driven one. By working with the model defined on square, triangular and honeycomb microscopic geometries we establish the dependence of this growing length on the lattice coordination. We discuss the interplay with the usual coarsening mechanism and the eventual fall into and escape from metastability.arXiv:1705.06508v3 [cond-mat.stat-mech]

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body interactions drawn from a zero-mean Gaussian probability distribution. In the statistical physics framework, the potential energy is of the so-called p = 2 spherical disordered kind, closely linked to the O(N ) scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable system. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, N → ∞, we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We obtain the global dynamical observables (energy density, self correlation function, linear response function, static susceptibility, etc.) with numerical and analytic methods and we show that, in most cases, they are out of thermal equilibrium. We note, however, that for shallow quenches from the condensed phase the dynamics are close to (though not at) thermal equilibrium à la Gibbs-Boltzmann. Surprisingly enough, in the N → ∞ limit and for a particular relation between parameters the global observables comply Gibbs-Boltzmann equilibrium. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. These are the projections of the vector spin (or particle's position on the sphere) on the eigenvectors of the interaction matrix, the most relevant being those linked to the eigenvalues at the edge of the spectrum. We argue that in a system with infinite size the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N − 1 integrals of motion, notably, their scaling with N , and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of global observables. We elaborate on the role played by these constants of motion in the post-quench dynamics and we briefly discuss the possible description of the asymptotic dynamics in terms of a Generalised Gibbs Ensemble.

We study the bidimensional voter model on a square lattice with numerical simulations. We demonstrate that the evolution takes place in two distinct dynamic regimes; a first approach towards critical site percolation and a further approach towards full consensus. We calculate the time dependence of the two growing lengths, finding that they are both algebraic but with different exponents (apart from possible logarithmic corrections). We analyze the morphology and statistics of clusters of voters with the same opinion. We compare these results to the ones for curvature driven two-dimensional coarsening.

In this contribution we further study the classical disordered p = 2 spherical model with Hamiltonian dynamics, or in integrable systems terms, the Neumann model, in the infinite size limit. We summarise the asymptotic results that some of us presented in a recent publication, and we deepen the analysis of the pre-asymptotic dynamics. We also discuss the possible description of the asymptotic steady state with a Generalised Gibbs Ensemble.

-Binary mixtures prepared in an homogeneous phase and quenched into a two-phase region phaseseparate via a coarsening process whereby domains of the two phases grow in time. With a numerical study of a spin-exchange model we show that this dynamics first take a system with equal density of the two species to a critical percolation state. We prove this claim and we determine the time-dependence of the growing length associated to this process with the scaling analysis of the statistical and morphological properties of the clusters of the two phases.Phase separation is the process whereby a binary mixture of components A and B, initially in a homogeneous phase, demix. This process leads to the coexistence of two phases: one rich in A and the other in B [1][2][3][4][5][6]. The system, initially in an unstable spatially uniform state, progressively coarsens to approach its thermodynamically stable phase-separated state. Such phenomena arise in binary alloys, fluid mixtures, and polymer blends. Recently, the dynamics of phase separation have seen a revival of interest in the context of experimental [7,8] and numerical [9][10][11][12] studies of binary mixtures of Bose gases.The late time dynamics are well understood. In the absence of driving forces, a dynamic scaling regime with statistically self-similar domain morphology sets in. This regime is well-described by an extension of the Lifshitz-Slyozov-Wagner (LSW) theory [13,14], in which the typical domain radius grows as [15] (whereas for scalar non-conserved order parameter dynamics the growing length is also given by a power law but the exponent is z d = 2). Numerical results in favour of this law were published in [15][16][17] for spin-exchange models although the growth-law can be more complex in particle or polymer phase separating systems, see e.g.[18] and references therein. The pre-asymptotic dynamics leading to this regime have not been discussed in detail in the literature. It was noticed in [19] that the low-temperature evolution of a bidimensional 50:50 binary mixture after a quench from infinite temperature shares many points in common with the one generated by Glauber single spin-flip stochastic dynamics satisfying detailed balance [20,21]. On the one hand, an early approach to critical percolation was noticed, although the time needed to reach this state was not studied in detail. On the other hand, a separation of length-scales in the statistics and morphology of finite size cluster areas and domain wall lengths was observed. Linear or planar objects that are smaller than the typical ones,, satisfy dynamic scaling with respect to d (t), while larger objects were found to be very close to the ones of critical percolation.In this Letter we characterise the early stages of the dynamical process. More precisely, we analyse the way in which the system approaches a state with a stable pattern of critical percolating domains. We monitor a number of observables (to be defined in the main part of the text) and we explain how their behaviour constitu...