Large deviations, condensation, and giant response in a statistical system

We study the probability distribution $P$ of the sum of a large number of non-identically distributed random variables $n_m$. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability, is discussed and interpreted in analogy to phase-transitions in Statistical Mechanics. A general expression for $P$ is derived, and its sensitivity to the details of the distribution of a single $n_m$ is worked out. These general results are verified by the analytical and numerical solution of some specific examples.Comment: New improved and extended version (12 pages, 4 figures) with more references and additional discussions. To appear in J. Phys. A: Math. Theo

Coarsening in inhomogeneous systems

This Article is a brief review of coarsening phenomena occurring in systems where quenched features - such as random field, varying coupling constants or lattice vacancies - spoil homogeneity. We discuss the current understanding of the problem in ferromagnetic systems with a non-conserved scalar order parameter by focusing primarily on the form of the growth-law of the ordered domains and on the scaling properties.Comment: 13 pages, 5 figures. This paper is a contribution to the special issue "Coarsening dynamics", Comptes Rendus de Physique, https://sites.google.com/site/ppoliti/crp-special-issu

Fast growth at low temperature in vacancy-mediated phase-separation

We study the phase-separation dynamics of a two-dimensional Ising model where A and B particles can only exchange position with a vacancy. In a wide range of temperatures the kinetics is dominated, during a long preasymptotic regime, by diffusion processes of particles along domain interfaces. The dynamical exponent z associated to this mechanism differs from the one usually expected for Kawasaki dynamics and is shown to assume different values depending on temperature and relative AB concentration. At low temperatures, in particular, domains grow as t^{1/2}, for equal AB volume fractions.Comment: LaTeX, 5 pages, 4 figures, to appear on Phys. Rev.

Condensation of Fluctuations in and out of Equilibrium

Condensation of fluctuations is an interesting phenomenon conceptually distinct from condensation on average. One stricking feature is that, contrary to what happens on average, condensation of fluctuations may occurr even in the absence of interaction. The explanation emerges from the duality between large deviation events in the given system and typical events in a new and appropriately biased system. This surprising phenomenon is investigated in the context of the Gaussian model, chosen as paradigmatical non interacting system, before and after an istantaneous temperature quench. It is shown that the bias induces a mean-field-like effective interaction responsible of the condensation on average. Phase diagrams, covering both the equilibrium and the off-equilibrium regimes, are derived for observables representative of generic behaviors.Comment: 8 pages, 7 figure

The role of initial state and final quench temperature on the aging properties in phase-ordering kinetics

We study numerically the two-dimensional Ising model with non-conserved dynamics quenched from an initial equilibrium state at the temperature $T_i\ge T_c$ to a final temperature $T_f$ below the critical one. By considering processes initiating both from a disordered state at infinite temperature $T_i=\infty$ and from the critical configurations at $T_i=T_c$ and spanning the range of final temperatures $T_f\in [0,T_c[$ we elucidate the role played by $T_i$ and $T_f$ on the aging properties and, in particular, on the behavior of the autocorrelation $C$ and of the integrated response function $\chi$. Our results show that for any choice of $T_f$, while the autocorrelation function exponent $\lambda _C$ takes a markedly different value for $T_i=\infty$ [$\lambda _C(T_i=\infty)\simeq 5/4$] or $T_i=T_c$ [$\lambda _C(T_i=T_c)\simeq 1/8$] the response function exponents are unchanged. Supported by the outcome of the analytical solution of the solvable spherical model we interpret this fact as due to the different contributions provided to autocorrelation and response by the large-scale properties of the system. As changing $T_f$ is considered, although this is expected to play no role in the large-scale/long-time properties of the system, we show important effects on the quantitative behavior of $\chi$. In particular, data for quenches to $T_f=0$ are consistent with a value of the response function exponent $\lambda _\chi=\frac{1}{2}\lambda _C(T_i=\infty)=5/8$ different from the one [$\lambda _\chi \in (0.5-0.56)$] found in a wealth of previous numerical determinations in quenches to finite final temperatures. This is interpreted as due to important pre-asymptotic corrections associated to $T_f>0$.Comment: 25 pages, 15 figures. To appear on Phys. Rev.

Dynamic fluctuations in unfrustrated systems: random walks, scalar fields and the Kosterlitz-Thouless phase

We study analytically the distribution of fluctuations of the quantities whose average yield the usual two-point correlation and linear response functions in three unfrustrated models: the random walk, the $d$ dimensional scalar field and the 2d XY model. In particular we consider the time dependence of ratios between composite operators formed with these fluctuating quantities which generalize the largely studied fluctuation-dissipation ratio, allowing us to discuss the relevance of the effective temperature notion beyond linear order. The behavior of fluctuations in the aforementioned solvable cases is compared to numerical simulations of the 2d clock model with $p=6,12$ states.Comment: 27 pages, 3 figure

Out of equilibrium dynamics of the spiral model

We study the relaxation of the bi-dimensional kinetically constrained spiral model. We show that due to the reversibility of the dynamic rules any unblocked state fully decorrelates in finite times irrespectively of the system being in the unjammed or the jammed phase. In consequence, the evolution of any unblocked configuration occurs in a different sector of phase space from the one that includes the equilibrium blocked equilibrium configurations at criticality and in the jammed phase. We argue that such out of equilibrium dynamics share many points in common with coarsening in the one-dimensional Ising model and we identify the coarsening structures that are, basically, lines of vacancies. We provide evidence for this claim by analyzing the behaviour of several observables including the density of particles and vacancies, the spatial correlation function, the time-dependent persistence and the linear response.Comment: 14 pages 12 figure

Comment on "Scaling of the linear response in simple aging systems without disorder"

We have repeated the simulations of Henkel, Paessens and Pleimling (HPP) [Phys.Rev.E {\bf 69}, 056109 (2004)] for the field-cooled susceptibility $\chi_{FC}(t) - \chi_0 \sim t^{-A}$ in the quench of ferromagnetic systems to and below $T_C$. We show that, contrary to the statement made by HPP, the exponent $A$ coincides with the exponent $a$ of the linear response function $R(t,s) \sim s^{-(1+a)}f_R(t/s)$. We point out what are the assumptions in the argument of HPP that lead them to the conclusion $A<a$.Comment: 4 pages, 4 figure

Heat exchanges in coarsening systems

This paper is a contribution to the understanding of the thermal properties of aging systems where statistically independent degrees of freedom with largely separated timescales are expected to coexist. Focusing on the prototypical case of quenched ferromagnets, where fast and slow modes can be respectively associated to fluctuations in the bulk of the coarsening domains and to their interfaces, we perform a set of numerical experiments specifically designed to compute the heat exchanges between different degrees of freedom. Our studies promote a scenario with fast modes acting as an equilibrium reservoir to which interfaces may release heat through a mechanism that allows fast and slow degrees to maintain their statistical properties independent.Comment: 12 pages, 8 figure

Topological regulation of activation barriers on fractal substrates

We study phase-ordering dynamics of a ferromagnetic system with a scalar order-parameter on fractal graphs. We propose a scaling approach, inspired by renormalization group ideas, where a crossover between distinct dynamical behaviors is induced by the presence of a length $\lambda$ associated to the topological properties of the graph. The transition between the early and the asymptotic stage is observed when the typical size $L(t)$ of the growing ordered domains reaches the crossover length $\lambda$. We consider two classes of inhomogeneous substrates, with different activated processes, where the effects of the free energy barriers can be analytically controlled during the evolution. On finitely ramified graphs the free energy barriers encountered by domains walls grow logarithmically with $L(t)$ while they increase as a power-law on all the other structures. This produces different asymptotic growth laws (power-laws vs logarithmic) and different dependence of the crossover length $\lambda$ on the model parameters. Our theoretical picture agrees very well with extensive numerical simulations.Comment: 13 pages, 4 figure