First-principles derivation of density functional formalism for quenched-annealed systems

We derive from first principles (without resorting to the replica trick) a density functional theory for fluids in quenched disordered matrices (QA-DFT). We show that the disorder-averaged free energy of the fluid is a functional of the average density profile of the fluid as well as the pair correlation of the fluid and matrix particles. For practical reasons it is preferable to use another functional: the disorder-averaged free energy plus the fluid-matrix interaction energy, which, for fixed fluid-matrix interaction potential, is a functional only of the average density profile of the fluid. When the matrix is created as a quenched configuration of another fluid, the functional can be regarded as depending on the density profile of the matrix fluid as well. In this situation, the replica-Ornstein-Zernike equations which do not contain the blocking parts of the correlations can be obtained as functional identities in this formalism, provided the second derivative of this functional is interpreted as the connected part of the direct correlation function. The blocking correlations are totally absent from QA-DFT, but nevertheless the thermodynamics can be entirely obtained from the functional. We apply the formalism to obtain the exact functional for an ideal fluid in an arbitrary matrix, and discuss possible approximations for non-ideal fluids.Comment: 19 pages, uses RevTeX

Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics

Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.Comment: Review, 48 pages, 26 figure

Mathematics of evolution

This article is a translation of an article originally published (in Spanish) in Gaceta de la RSME, Vol. 12, no. 4, 2009, pp. 667–686. Published with permission.This work is supported by projects MOSAICO (Ministerio de Educación y Ciencia, Spain) and MODELICO-CM (Comunidad de Madrid, Spain)

Huge Progeny Production during the Transient of a Quasispecies Model of Viral Infection, Reproduction and Mutation

Eigen's quasi-species model describes viruses as ensembles of different mutants of a high fitness "master" genotype. Mutants are assumed to have lower fitness than the master type, yet they coexist with it forming the quasi-species. When the mutation rate is sufficiently high, the master type no longer survives and gets replaced by a wide range of mutant types, thus destroying the quasi-species. It is the so-called "error catastrophe". But natural selection acts on phenotypes, not genotypes, and huge amounts of genotypes yield the same phenotype. An important consequence of this is the appearance of beneficial mutations which increase the fitness of mutants. A model has been recently proposed to describe quasi-species in the presence of beneficial mutations. This model lacks the error catastrophe of Eigen's model and predicts a steady state in which the viral population grows exponentially. Extinction can only occur if the infectivity of the quasi-species is so low that this exponential is negative. In this work I investigate the transient of this model when infection is started from a small amount of low fitness virions. I prove that, beyond an initial regime where viral population decreases (and can go extinct), the growth of the population is super-exponential. Hence this population quickly becomes so huge that selection due to lack of host cells to be infected begins to act before the steady state is reached. This result suggests that viral infection may widespread before the virus has developed its optimal form.This work is part of two research projects: MOSAICO, from Ministerio de Educación y Ciencia (Spain) and MODELICO-CM, from Comunidad Autónoma de Madrid (Spain)

Time Scales in Evolutionary Dynamics

Evolutionary game theory has traditionally assumed that all individuals in a population interact with each other between reproduction events. We show that eliminating this restriction by explicitly considering the time scales of interaction and selection leads to dramatic changes in the outcome of evolution. Examples include the selection of the inefficient strategy in the Harmony and Stag-Hunt games, and the disappearance of the coexistence state in the Snowdrift game. Our results hold for any population size and in the presence of a background of fitness.Comment: Final version with minor changes, accepted for publication in Physical Review Letter

La entropía como creadora de orden

7 págs, 7 figs.-- Número monográfico de la Revista dedicado al Centenario de Ludwig Boltzmann (1844-1906).In spite of the identification between entropy and disorder, there are many phase transitions in which an ordered phase emerges and at the same time entropy increases. In this article it will be shown that this paradox gets resolved by making a literal interpretation of the famous Boltzmann's equation S = k log W. Two examples: freezing of a fluid and demixing of a binary mixture, will illustrate this phenomenon. From them the concept of entropic force or interaction, very useful in polymer or colloid science, will emerge.La investigación del autor está financiada por los proyectos BFM2003-0180 del Ministerio de Educación y Ciencia y UC3M-FI-05-007 de la Universidad Carlos III de Madrid y la Comunidad Autónoma de Madrid, y forma parte del proyecto “Modelización y Simulación de Sistemas No Homogéneos en Materia Condensada”, MOSSNOHO (S-0505/ESP/000299), financiado por la Comunidad Autónoma de Madrid.Publicad

Las matemáticas de la evolución

20 págs, 2 figs.-- Este artículo conmemora los 200 años del nacimiento de Charles Robert Darwin y los 150 desde la publicación de su gran obra, El origen de las especies.En este artículo se proporciona una panorámica de lo que la Matemática ha aportado a la teoría evolutiva. Desde la genética de poblaciones, pasando por los procesos estocásticos y acabando por la teoría de redes complejas, muchos resultados relevantes sobre los mecanismos evolutivos se han obtenido gracias a su descripción matemática. Aún no puede decirse que la Teoría de la Evolución sea una doctrina científica matemáticamente formulada en todos sus detalles, como a Darwin le habría gustado, pero es indudable que cada vez estamos más cerca de ello. Hoy en día podría decirse que los estudios teóricos de los procesos evolutivos son, al menos, tan importantes como los experimentales y que, como Darwin afirmaba, son ellos los que arrojan luz en la oscuridad.El autor agradece a los proyectos MOSAICO y MOSSNOHO-CM el apoyo económico.Publicad

Fundamental-measure density functional for the fluid of aligned hard hexagons: New insights in fundamental measure theory

In this article we obtain a fundamental measure functional for the model of aligned hard hexagons in the plane. Our aim is not just to provide a functional for a new, admittedly academic, model, but to investigate the structure of fundamental measure theory. A model of aligned hard hexagons has similarities with the hard disk model. Both share "lost cases", i.e. admit configurations of three particles in which there is pairwise overlap but not triple overlap. These configurations are known to be problematic for fundamental measure functionals, which are not able to capture their contribution correctly. This failure lies in the inability of these functionals to yield a correct low density limit of the third order direct correlation function. Here we derive the functional by projecting aligned hard cubes on the plane x+y+z=0. The correct dimensional crossover behavior of these functionals permits us to follow this strategy. The functional of aligned hard cubes, however, does not have lost cases, so neither had the resulting functional for aligned hard hexagons. The latter exhibits, in fact, a peculiar structure as compared to the one for hard disks. It depends on a uniparametric family of weighted densities through a new term not appearing in the functional for hard disks. Apart from studying the freezing of this system, we discuss the implications of the functional structure for new developments of fundamental measure theory.Comment: 10 pages, 9 figures, uses RevTeX

Neutral evolution and the acceleration of the molecular clock

Large sets of genotypes give rise to the same phenotype because phenotypic expressionis highly redundant. Accordingly, a population can accept mutations without alteringits phenotype, as long as the genotype mutates into another one on the same set. Bylinking every pair of genotypes that are mutually accessible through mutation, geno-types organize themselves into neutral networks (NN). These networks are known to beheterogeneous and assortative, and these properties affect the evolutionary dynamics ofthe population. By studying the dynamics of populations on NN with arbitrary topol-ogy we analyze the effect of assortativity, of NN (phenotype) fitness, and of networksize. We find that the probability that the population leaves the network is smallerthe longer the time spent on it. This progressive "phenotypic entrapment" entails asystematic increase in the overdispersion of the process with time and an accelerationin the fixation rate of neutral mutations