Linear Glauber model

Stochastic thermodynamics of opinion dynamics models

We show that models of opinion formation and dissemination in a community of individuals can be framed within stochastic thermodynamics from which we can build a nonequilibrium thermodynamics of opinion dynamics. This is accomplished by decomposing the original transition rate that defines an opinion model into two or more transition rates, each representing the contact with heat reservoirs at different temperatures, and postulating an energy function. As the temperatures are distinct, heat fluxes are present even at the stationary state and linked to the production of entropy, the fundamental quantity that characterizes nonequilibrium states. We apply the present framework to a generic-vote model including the majority-vote model in a square lattice and in a cubic lattice. The fluxes and the rate of entropy production are calculated by numerical simulation and by the use of a pair approximation.

Geospatial distributions reflect temperatures of linguistic features

Quantifying the speed of linguistic change is challenging because the historical evolution of languages is sparsely documented. Consequently, traditional methods rely on phylogenetic reconstruction. Here, we propose a model-based approach to the problem through the analysis of language change as a stochastic process combining vertical descent, spatial interactions, and mutations in both dimensions. A notion of linguistic temperature emerges naturally from this analysis as a dimensionless measure of the propensity of a linguistic feature to undergo change. We demonstrate how temperatures of linguistic features can be inferred from their present-day geospatial distributions, without recourse to information about their phylogenies. Thus, the evolutionary dynamics of language, operating across thousands of years, leave a measurable geospatial signature. This signature licenses inferences about the historical evolution of languages even in the absence of longitudinal data.

Grand-canonical solution of semiflexible self-avoiding trails on the Bethe lattice

We consider a model of semiflexible interacting self-avoiding trails (sISATs) on a lattice, where the walks are constrained to visit each lattice edge at most once. Such models have been studied as an alternative to the self-attracting self-avoiding walks (SASAWs) to investigate the collapse transition of polymers, with the attractive interactions being on site as opposed to nearest-neighbor interactions in SASAWs. The grand-canonical version of the sISAT model is solved on a four-coordinated Bethe lattice, and four phases appear: non-polymerized (NP), regular polymerized (P), dense polymerized (DP), and anisotropic nematic (AN), the last one present in the phase diagram only for sufficiently stiff chains. The last two phases are dense, in the sense that all lattice sites are visited once in the AN phase and twice in the DP phase. In general, critical NP-P and DP-P transition surfaces meet with a NP-DP coexistence surface at a line of bicritical points. The region in which the AN phase is stable is limited by a discontinuous critical transition to the P phase, and we study this somewhat unusual transition in some detail. In the limit of rods, where the chains are totally rigid, the P phase is absent and the three coexistence lines (NP-AN, AN-DP, and NP-DP) meet at a triple point, which is the endpoint of the bicritical line.

Percolation and coarsening in the bidimensional voter model

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.

Stochastic approach to equilibrium and nonequilibrium thermodynamics

We develop the stochastic approach to thermodynamics based on stochastic dynamics, which can be discrete (master equation) and continuous (Fokker-Planck equation), and on two assumptions concerning entropy. The first is the definition of entropy itself and the second the definition of entropy production rate, which is non-negative and vanishes in thermodynamic equilibrium. Based on these assumptions, we study interacting systems with many degrees of freedom in equilibrium or out of thermodynamic equilibrium and how the macroscopic laws are derived from the stochastic dynamics. These studies include the quasiequilibrium processes; the convexity of the equilibrium surface; the monotonic time behavior of thermodynamic potentials, including entropy; the bilinear form of the entropy production rate; the Onsager coefficients and reciprocal relations; and the nonequilibrium steady states of chemical reactions.

Aging and fluctuation-dissipation ratio in a nonequilibrium q-state lattice model

A generalized version of the nonequilibrium linear Glauber model with q states in d dimensions is introduced and analyzed. The model is fully symmetric, its dynamics being invariant under all permutations of the q states. Exact expressions for the two-time autocorrelation and response functions on a d-dimensional lattice are obtained. In the stationary regime, the fluctuation-dissipation theorem holds, while in the transient the aging is observed with the fluctuation-dissipation ratio leading to the value predicted for the linear Glauber model.

Fluctuation-dissipation relation for stochastic dynamics without detailed balance

We show that the fluctuation-dissipation relation can be established for a class of stochastic dynamics that lack detailed balance. This class comprises lattice spin models whose time evolution is governed by a master equation with a one-spin-flip transition rate having the up-down symmetry. The relation is obtained by the introduction of a multiplicative perturbation of the transition rate, which reduces to the usual perturbation when detailed balance is fulfilled. As a part of the derivation we set up an equivalent two-spin-flip stochastic dynamics that conserves the magnetization.

Inflow versus outflow zero-temperature dynamics in one dimension

It has been suggested that Glauber (inflow) and Sznajd (outflow) zero-temperature dynamics for the one-dimensional Ising ferromagnet with nearest-neighbor interactions are equivalent. Here we compare the two dynamics from the analytical and computational points of view. We use the method of mapping an Ising spin system onto the dimer RSA model and show that already this simple mapping allows us to see the differences between inflow and outflow zero-temperature dynamics. Then we investigate both dynamics with synchronous, partially synchronous, and random sequential updating using the Monte Carlo technique and compare both dynamics in the number of persistent spins, clusters, mean relaxation time, and relaxation time distribution.

Fluctuation-dissipation theorem and the linear Glauber model

We obtain exact expressions for the two-time autocorrelation and response functions of the -dimensional linear Glauber model. Although this linear model does not obey detailed balance in dimensions d > or = 2, we show that the usual form of the fluctuation-dissipation ratio still holds in the stationary regime. In the transient regime, we show the occurrence of aging, with a special limit of the fluctuation-dissipation ratio, x(infinity) = 1/2, for a quench at the critical point.