Landauer's Principle at Zero Temperature.

Landauer's bound relates changes in the entropy of a system with the inevitable dissipation of heat to the environment. The bound, however, becomes trivial in the limit of zero temperature. Here we show that it is possible to derive a tighter bound which remains nontrivial even as T→0. As in the original case, the only assumption we make is that the environment is in a thermal state. Nothing is said about the state of the system or the kind of system-environment interaction. Our bound is valid for all temperatures and is always tighter than the original one, tending to it in the limit of high temperatures.

Thermodynamic Uncertainty Relations from Exchange Fluctuation Theorems.

Thermodynamic uncertainty relations (TURs) place strict bounds on the fluctuations of thermodynamic quantities in terms of the associated entropy production. In this Letter, we identify the tightest (and saturable) matrix-valued TUR that can be derived from the exchange fluctuation theorems describing the statistics of heat and particle flow between multiple systems of arbitrary dimensions. Our result holds for both quantum and classical systems, undergoing general finite-time nonstationary processes. Moreover, it provides bounds not only for the variances, but also for the correlations between thermodynamic quantities. To demonstrate the relevance of TURs to the design of nanoscale machines, we consider the operation of a 2-qubit swap engine undergoing an Otto cycle and show how our results can be used to place strict bounds on the correlations between heat and work.

Analytical expression for the exit probability of the q-voter model in one dimension.

We present in this paper an approximation that is able to give an analytical expression for the exit probability of the q-voter model in one dimension. This expression gives a better fit for the more recent data about simulations in large networks [A. M. Timpanaro and C. P. C. do Prado, Phys. Rev. E 89, 052808 (2014)] and as such departs from the expression ρ(q)/ρ(q)+(1-ρ)(q) found in papers that investigated small networks only [R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008); P. Przybyla et al., Phys. Rev. E 84, 031117 (2011); F. Slanina et al., Europhys. Lett. 82, 18006 (2008)]. The approximation consists in assuming a large separation on the time scales at which active groups of agents convince inactive ones and the time taken in the competition between active groups. Some interesting findings are that for q=2 we still have ρ(2)/ρ(2)+(1-ρ)(2) as the exit probability and for q>2 we can obtain a lower-order approximation of the form ρ(s)/ρ(s)+(1-ρ)(s) with s varying from q for low values of q to q-1/2 for large values of q. As such, this work can also be seen as a deduction for why the exit probability ρ(q)/ρ(q)+(1-ρ)(q) gives a good fit, without relying on mean-field arguments or on the assumption that only the first step is nondeterministic, as q and q-1/2 will give very similar results when q→∞.

Mean-field approximation for the Sznajd model in complex networks.

This paper studies the Sznajd model for opinion formation in a population connected through a general network. A master equation describing the time evolution of opinions is presented and solved in a mean-field approximation. Although quite simple, this approximation allows us to capture the most important features regarding the steady states of the model. When spontaneous opinion changes are included, a discontinuous transition from consensus to polarization can be found as the rate of spontaneous change is increased. In this case we show that a hybrid mean-field approach including interactions between second nearest neighbors is necessary to estimate correctly the critical point of the transition. The analytical prediction of the critical point is also compared with numerical simulations in a wide variety of networks, in particular Barabási-Albert networks, finding reasonable agreement despite the strong approximations involved. The same hybrid approach that made it possible to deal with second-order neighbors could just as well be adapted to treat other problems such as epidemic spreading or predator-prey systems.

Testing validity of the Kirkwood approximation using an extended Sznajd model.

We revisit the deduction of the exit probability of the one-dimensional Sznajd model through the Kirkwood approximation [F. Slanina et al., Europhys. Lett. 82, 18006 (2008)]. This approximation is peculiar in that, in spite of the agreement with simulation results [F. Slanina et al., Europhys. Lett. 82, 18006 (2008); R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008); A. M. Timpanaro and C. P. C. Prado, Phys. Rev. E 89, 052808 (2014)], the hypothesis about the correlation lengths behind it are inconsistent and fixing these inconsistencies leads to the same results as a simple mean field. We use an extended version of the Sznajd model to test the Kirkwood approximation in a wider context. This model includes the voter, Sznajd, and "United we stand, divided we fall" models [R. A. Holley and T. M. Liggett, Ann. Prob. 3, 643 (1975); K. Sznajd-Weron and J. Sznajd, Int. J. Mod. Phys. C 11, 1157 (2000)] as different parameter combinations, meaning that some analytical results from these models can be used to evaluate the performance of the Kirkwood approximation. We also compare the predicted exit probability with simulation results for networks with 10(3) sites. The results show clearly the regions in parameter space where the approximation gives accurate predictions, as well as where it starts failing, leading to a better understanding of its reliability.

Exit probability of the one-dimensional q-voter model: analytical results and simulations for large networks.

We discuss the exit probability of the one-dimensional q-voter model and present tools to obtain estimates about this probability, both through simulations in large networks (around 10(7) sites) and analytically in the limit where the network is infinitely large. We argue that the result E(ρ) = ρ(q)/ρ(q) + (1-ρ)(q), that was found in three previous works [F. Slanina, K. Sznajd-Weron, and P. Przybyla, Europhys. Lett. 82, 18006 (2008); R. Lambiotte and S. Redner, Europhys. Lett. 82, 18007 (2008), for the case q = 2; and P. Przybyla, K. Sznajd-Weron, and M. Tabiszewski, Phys. Rev. E 84, 031117 (2011), for q > 2] using small networks (around 10(3) sites), is a good approximation, but there are noticeable deviations that appear even for small systems and that do not disappear when the system size is increased (with the notable exception of the case q = 2). We also show that, under some simple and intuitive hypotheses, the exit probability must obey the inequality ρ(q)/ρ(q) + (1-ρ) ≤ E(ρ) ≤ ρ/ρ + (1-ρ)(q) in the infinite size limit. We believe this settles in the negative the suggestion made [S. Galam and A. C. R. Martins, Europhys. Lett. 95, 48005 (2001)] that this result would be a finite size effect, with the exit probability actually being a step function. We also show how the result that the exit probability cannot be a step function can be reconciled with the Galam unified frame, which was also a source of controversy.

Connections between the Sznajd model with general confidence rules and graph theory.

The Sznajd model is a sociophysics model that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favor bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modeled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We state these results and present comparisons between the mean field and simulations in Barabási-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims and some graph theory concepts, together with examples. We also show that there is no qualitative difference in the mean-field results if we require that a group of size q>2, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean field, this would coincide with the q-voter model).

Coexistence of interacting opinions in a generalized Sznajd model.

The Sznajd model is a sociophysics model that mimics the propagation of opinions in a closed society, where the interactions favor groups of agreeing people. It is based in the Ising and Potts ferromagnetic models and, although the original model used only linear chains, it has since been adapted to general networks. This model has a very rich transient, which has been used to model several aspects of elections, but its stationary states are always consensus states. In order to model more complex behaviors, we have, in a recent work, introduced the idea of biases and prejudices to the Sznajd model by generalizing the bounded confidence rule, which is common to many continuous opinion models, to what we called confidence rules. In that work we have found that the mean field version of this model (corresponding to a complete network) allows for stationary states where noninteracting opinions survive, but never for the coexistence of interacting opinions. In the present work, we provide networks that allow for the coexistence of interacting opinions for certain confidence rules. Moreover, we show that the model does not become inactive; that is, the opinions keep changing, even in the stationary regime. This is an important result in the context of understanding how a rule that breeds local conformity is still able to sustain global diversity while avoiding a frozen stationary state. We also provide results that give some insights on how this behavior approaches the mean field behavior as the networks are changed.

Generalized Sznajd model for opinion propagation.

In the last decade the Sznajd model has been successfully employed in modeling some properties and scale features of both proportional and majority elections. We propose a version of the Sznajd model with a generalized bounded confidence rule-a rule that limits the convincing capability of agents and that is essential to allow coexistence of opinions in the stationary state. With an appropriate choice of parameters it can be reduced to previous models. We solved this model both in a mean-field approach (for an arbitrary number of opinions) and numerically in a Barabási-Albert network (for three and four opinions), studying the transient and the possible stationary states. We built the phase portrait for the special cases of three and four opinions, defining the attractors and their basins of attraction. Through this analysis, we were able to understand and explain discrepancies between mean-field and simulation results obtained in previous works for the usual Sznajd model with bounded confidence and three opinions. Both the dynamical system approach and our generalized bounded confidence rule are quite general and we think it can be useful to the understanding of other similar models.