Violation of the zeroth law of thermodynamics in systems with negative specific heat.

We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy, it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear; that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different from the initial temperature.

Systems with negative specific heat in thermal contact: violation of the zeroth law.

Using both numerical simulations and exact expressions for the free energy and microcanonical entropy for a modified Hamiltonian mean-field (HMF) model, we show that when two similar systems with the same intensive parameters but with negative specific heat are weakly coupled, they undergo a process in which the total entropy increases irreversibly. We find that the final equilibrium is such that two phases appear at a temperature (equal in both systems) that is generally different from the initial temperature. We corroborate our results using two different kinds of couplings between the HMF systems. We confirm that our results hold also for the Ising model with long- and short-range interactions, which also has a parameter region with negative specific heat in the microcanonical ensemble. Further, we show that we can couple each system having negative specific heat to a third system that can be used as a thermometer, as long as this thermometer is small enough not to drive the system out of the microcanonical ensemble. Therefore, we show an instance of violation of the zeroth law of thermodynamics.

Phase transitions in systems of self-propelled agents and related network models.

An important characteristic of flocks of birds, schools of fish, and many similar assemblies of self-propelled particles is the emergence of states of collective order in which the particles move in the same direction. When noise is added into the system, the onset of such collective order occurs through a dynamical phase transition controlled by the noise intensity. While originally thought to be continuous, the phase transition has been claimed to be discontinuous on the basis of recently reported numerical evidence. We address this issue by analyzing two representative network models closely related to systems of self-propelled particles. We present analytical as well as numerical results showing that the nature of the phase transition depends crucially on the way in which noise is introduced into the system.

Static pairwise annihilation in complex networks.

We study static annihilation on complex networks, in which pairs of connected particles annihilate at a constant rate during time. Through a mean-field formalism, we compute the temporal evolution of the distribution of surviving sites with an arbitrary number of connections. This general formalism, which is exact for disordered networks, is applied to Kronecker, Erdös-Rényi (i.e., Poisson), and scale-free networks. We compare our theoretical results with extensive numerical simulations obtaining excellent agreement. Although the mean-field approach applies in an exact way neither to ordered lattices nor to small-world networks, it qualitatively describes the annihilation dynamics in such structures. Our results indicate that the higher the connectivity of a given network element, the faster it annihilates. This fact has dramatic consequences in scale-free networks, for which, once the "hubs" have been annihilated, the network disintegrates and only isolated sites are left.

Fluctuation-dissipation theorem for metastable systems.

We show that an appropriately defined fluctuation-dissipation theorem, connecting generalized susceptibilities and time correlation functions, is valid for times shorter than the nucleation time of the metastable state of Markovian systems satisfying detailed balance. This is done by assuming that such systems can be described by a superposition of the ground and first excited states of the master equation. We corroborate our results numerically for the metastable states of a two-dimensional Ising model.

Translocation properties of primitive molecular machines and their relevance to the structure of the genetic code.

We address the question, related with the origin of the genetic code, of why are there three bases per codon in the translation to protein process. As a follow-up to our previous work (Aldana et al., 1998, Martínez-Mekler et al., 1999a,b), we approach this problem by considering the translocation properties of primitive molecular machines, which capture basic features of ribosomal/messenger RNA interactions, while operating under prebiotic conditions. Our model consists of a short one-dimensional chain of charged particles (rRNA antecedent) interacting with a polymer (mRNA antecedent) via electrostatic forces. The chain is subject to external forcing that causes it to move along the polymer which is fixed in a quasi-one-dimensional geometry. Our numerical and analytic studies of statistical properties of random chain/polymer potentials suggest that, under very general conditions, a dynamics is attained in which the chain moves along the polymer in steps of three monomers. By adjusting the model in order to consider present-day genetic sequences, we show that the above property is enhanced for coding regions. Intergenic sequences display a behavior closer to the random situation. We argue that this dynamical property could be one of the underlying causes for the three-base codon structure of the genetic code

Region visited by a spherical Brownian particle in the presence of an absorbing boundary.

We calculate the time dependence of the average volume of a Wiener sausage in the presence of an absorbing boundary in one and three dimensions. In one dimension it is shown that the presence of an absorbing point reduces the time dependence of the average span from being proportional to sqrt[t] in an unbounded space, to being proportional to ln(t) at long times. In three dimensions the average volume increases as sqrt[t] at long times rather than being proportional to t as in free space.

Transport properties of the diluted Lorentz slab.

We study the behavior of a point particle incident on a slab of a randomly diluted triangular array of circular scatterers. Various scattering properties, such as the reflection and transmission probabilities and the scattering time are studied as a function of thickness and dilution. We show that a diffusion model satisfactorily describes the mentioned scattering properties. We also show how some of these quantities can be evaluated exactly and their agreement with numerical experiments. Our results exhibit the dependence of these scattering data on the mean free path. This dependence again shows excellent agreement with the predictions of a Brownian motion model.

Coupled normal heat and matter transport in a simple model system.

We introduce the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system is a Lorentz gas with fixed freely rotating circular scatterers which scatter point particles via perfectly rough collisions. Upon imposing either a temperature gradient and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds. Transport in this system is normal in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are nontrivially coupled, satisfying Onsager's reciprocity relations.

Simple model of a random walk with arbitrarily long memory.

We present a generalization of the persistent random-walk model in which the step at time n depends on the state of the step at time n-T, for arbitrary T. This gives rise to arbitrarily long memory effects, yet by an appropriate transformation the model is tractable by essentially the same techniques applicable to the usual persistent random-walk problem. We apply our results to the specific case of delayed "step" persistence, and analyze its asymptotic statistical properties.

Statistical characterization of random electrostatic potentials

In this work we study statistical properties of random electrostatic potentials generated by one dimensional lattices with random charges. We show that the resulting random potentials are correlated Gaussian processes, satisfying the Lindeberg version of the central limit theorem, if certain restrictions are imposed on the individual potentials generated by the particles on the lattice. Since most of the point-particle electrostatic potentials occurring in nature satisfy the Lindeberg condition, the correlation properties of the random potentials are not arbitrary and must comply with the central limit theorem. Based on this theorem we can obtain explicit expressions for these correlations. We thus are able to give a characterization of a broad class of potentials yielding feasible physical scenarios. We illustrate some consequences of our findings by considering dynamical properties of a test particle interacting with the lattice. We show how the long range correlations generate statistical features in these properties, which are best exhibited when considering different length scales.

Dynamical weight functions for a planar crack

The stress intensity factors are evaluated for a moving planar crack for loadings which vary arbitrarily in time and three dimensions of space. We exploit the adjoint elasticity equation obeyed by the corresponding weight functions, and a new and more universal Wiener-Hopf factorization of the Rayleigh function, this being the central difficulty in such calculations. For the mode II weight function we give further asymptotic results crucial to a subsequent calculation of crack stability with respect to out-of-plane perturbations.

Deterministic transport in ratchets.

We present the deterministic transport properties of driven overdamped particles in a simple piecewise-linear ratchet potential. We consider the effects on the stationary current due to local spatial asymmetry, time asymmetry in the driving force, and we include the possibility of a global spatial asymmetry. We present an extremely simple scheme for evaluating the current that is established on the ratchet within an "adiabatic" approximation, and compare the results with exact numerical integration of the process.