Publications (52)101.78 Total impact
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ABSTRACT: In this work we study the stochastic process of twospecies coagulation. This process consists in the aggregation dynamics taking place in a ring. Particles and clusters of particles are set in this ring and they can move either clockwise or counterclockwise. They have a probability to aggregate forming larger clusters when they collide with another particle or cluster. We study the stochastic process both analytically and numerically. Analytically, we derive a kinetic theory which approximately describes the process dynamics. One of our strongest assumptions in this respect is the so called wellstirred limit, that allows neglecting the appearance of spatial coordinates in the theory, so this becomes effectively reduced to a zeroth dimensional model. We determine the long time behavior of such a model, making emphasis in one special case in which it displays selfsimilar solutions. In particular these calculations answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time. We compare our analytical results with direct numerical simulations of the stochastic process and both corroborate its predictions and check its limitations. In particular, we numerically confirm the ordering dynamics predicted by the kinetic theory and explore properties of the realizations of the stochastic process which are not accessible to our theoretical approach.04/2014;  [Show abstract] [Hide abstract]
ABSTRACT: Recently, a variational approach has been introduced for the paradigmatic KardarParisiZhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expansion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradientflow counterpart to the KPZ equation. A dynamic renormalization group analysis at oneloop order of this new mesoscopic model yields the KPZ scaling relation alpha+z=2, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a pathintegral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation.Papers in Physics. 01/2014; 5. 
Article: Global existence versus blowup results for a fourth order parabolic PDE involving the Hessian
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ABSTRACT: On considère une équation différentielle qui décrit la croissance épitaxiale d'une couche rugueuse de grains. Il s'agit d'une équation parabolique pour laquelle l'évolution est gouvernée par une compétiton entre le déterminant Hessien de la solution et l'opérateur biharmonique. Ce modèle peut présenter une structure de flux gradient suivant les conditions au bord. On étend d'abord des résultats précédents sur l'existence de solutions stationnaires pour ce modèle avec des conditions de Dirichlet. Pour l'équation d'évolution on démontre l'existence locale de solutions pour tout donné initial et l'existence globale pour des donnés suffisamment démontre. En exploitant les conditions au bord et la structure variationnelle de l'équation, suivant la taille de la donné initial on démontre l'explosion en temps fini et/ou la convergence à une solution stationnaire pour les solutions globales.Journal de Mathématiques Pures et Appliquées. 01/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We present the formal geometric derivation of a nonequilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of nonequilibrium statistical mechanics.European Journal of Applied Mathematics 09/2013; 24(03). · 1.14 Impact Factor 
Article: 10.1016/j.jde.2012.12.012
European Journal of Applied Mathematics 01/2013; · 1.14 Impact Factor 
Article: Author's personal copy
Journal of Differential Equations 01/2013; 254:25152531. · 1.48 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This paper deals with some mathematical models arising in the theory of epitaxial growth of crystal. We focalize the study on a stationary problem which presents some analytical difficulties. We study the existence of solutions. The central model in this work is given by the following fourth order elliptic equation,Δ2u=det(D2u)+λf,x∈Ω⊂R2,conditions on ∂Ω. The framework to study the problem deeply depends on the boundary conditions.Journal of Differential Equations 01/2013; 254:2515–2531. · 1.48 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. Our results depend on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter we prove existence of solutions to this boundary value problem. For large values of the same parameter we prove nonexistence of solutions. We also provide rigorous bounds for the values of this parameter which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions.Mathematical Methods in the Applied Sciences 09/2012; To appear. · 0.78 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The effect of a uniform dilation of space on stochastically driven nonlinear field theories is examined. This theoretical question serves as a model problem for examining the properties of nonlinear field theories embedded in expanding Euclidean FriedmannLema\^{\i}treRobertsonWalker metrics in the context of cosmology, as well as different systems in the disciplines of statistical mechanics and condensed matter physics. Field theories are characterized by the speed at which they propagate correlations within themselves. We show that for linear field theories correlations stop propagating if and only if the speed at which the space dilates is higher than the speed at which correlations propagate. The situation is in general different for nonlinear field theories. In this case correlations might stop propagating even if the velocity at which space dilates is lower than the velocity at which correlations propagate. In particular, these results imply that it is not possible to characterize the dynamics of a nonlinear field theory during homogeneous spatial dilation {\it a priori}. We illustrate our findings with the nonlinear KardarParisiZhang equation.Journal of Physics A Mathematical and Theoretical 08/2012; 46(35). · 1.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: To this day, computer models for stromatolite formation have made substantial use of the KardarParisiZhang (KPZ) equation. Oddly enough, these studies yielded mutually exclusive conclusions about the biotic or abiotic origin of such structures. We show in this paper that, at our current state of knowledge, a purely biotic origin for stromatolites can neither be proved nor disproved by means of a KPZbased model. What can be shown, however, is that whatever their (biotic or abiotic) origin might be, some morphologies found in actual stromatolite structures (e.g. overhangs) cannot be formed as a consequence of a process modelled exclusively in terms of the KPZ equation and acting over sufficiently large times. This suggests the need to search for alternative mathematical approaches to model these structures, some of which are discussed in this paper.Journal of The Royal Society Interface 10/2011; 9(70):105162. · 4.91 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scaledependent fractality, and suggests the advent of a selfsimilar fractal dimension. The centerofmass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the longrange surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilutionfree universality class.Physical Review E 09/2011; 84(3 Pt 1):031131. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Starting from a variational formulation of the KardarParisiZhang (KPZ) equation, we point out some strong constraints and consistency tests, to be fulfilled by realspace discretization schemes. In the light of these findings, the mainstream opinion on the relevance of Galilean invariance and the fluctuationdissipation theorem (peculiar of 1D) is challenged.03/2011;  [Show abstract] [Hide abstract]
ABSTRACT: The stochastic nonlinear partial differential equation known as the KardarParisiZhang (KPZ) equation is a highly successful phenomenological mesoscopic model of surface and interface growth processes. Its suitability for analytical work, its explicit symmetries and its prediction of an exact dynamic scaling relation for a onedimensional substratum led people to adopt it as a 'standard' model in the field during the last quarter of a century. At the same time, several conjectures deserving closer scrutiny were established as dogmas throughout the community. Among these, we find the beliefs that 'genuine' nonequilibrium processes are nonvariational in essence, and that the exactness of the dynamic scaling relation owes its existence to a Galilean symmetry. Additionally, the equivalence among planar and radial interface profiles has been generally assumed in the literature throughout the years. Hereamong other topicswe introduce a variational formulation of the KPZ equation, remark on the importance of consistency in discretization and challenge the mainstream view on the necessity for scaling of both Galilean symmetry and the onedimensional fluctuationdissipation theorem. We also derive the KPZ equation on a growing domain as a first approximation to radial growth, and outline the differences with respect to the classical case that arises in this new situation.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 01/2011; 369(1935):396411. · 2.89 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We obtain a FokkerPlanck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.Physical Review E 07/2010; 82(1 Pt 1):011926. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We explore the selforganization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, such as collective animal motion and opinion spreading dynamics, are also outlined.Physical Review E 07/2010; 82(1 Pt 2):016113. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In order to perform numerical simulations of the KardarParisiZhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a HopfCole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the HopfCole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between realspace and pseudospectral discrete representations. In addition we discuss the relevance of the Galileaninvariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuationdissipation theorem, peculiar of one dimension.Physical Review E 06/2010; 81(6 Pt 2):066706. · 2.31 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The dynamics of one species chemical kinetics is studied. Chemical reactions are modelled by means of continuous time Markov processes whose probability distribution obeys a suitable master equation. A large deviation theory is formally introduced, which allows developing a Hamiltonian dynamical system able to describe the system dynamics. Using this technique we are able to show that the intrinsic fluctuations, originated in the discrete character of the reagents, may sustain oscillations and chaotic trajectories which are impossible when these fluctuations are disregarded. An important point is that oscillations and chaos appear in systems whose meanfield dynamics has too low a dimensionality for showing such a behavior. In this sense these phenomena are purely induced by noise, which does not limit itself to shifting a bifurcation threshold. On the other hand, they are large deviations of a short transient nature which typically only appear after long waiting times. We also discuss the implications of our results in understanding extinction events in population dynamics models expressed by means of stoichiometric relations.SIAM Journal on Applied Dynamical Systems 04/2010; · 1.45 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Scaling and hyperscaling laws provide exact relations among critical exponents describing the behavior of a system at criticality. For nonequilibrium growth models with a conserved drift there exist few of them. One such relation is $\alpha +z=4$, found to be inexact in a renormalization group calculation for several classical models in this field. Herein we focus on the twodimensional case and show that it is possible to construct conserved surface growth equations for which the relation $\alpha +z=4$ is exact in the renormalization group sense. We explain the presence of this scaling law in terms of the existence of geometric principles dominating the dynamics.Journal of Physics A Mathematical and Theoretical 04/2010; · 1.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The asymptotic shape of randomly growing radial clusters is studied. We pose the problem in terms of the dynamics of stochastic partial differential equations. We concentrate on the properties of the realizations of the stochastic growth process and in particular on the interface fluctuations. Our goal is unveiling under which conditions the developing radial cluster asymptotically weakly converges to the concentrically propagating spherically symmetric profile or either to a symmetry breaking shape. We demonstrate that the long range correlations of the surface fluctuations obey a selfaffine scaling and that scale invariance is achieved by means of the introduction of three critical exponents. These are able to characterize the large scale dynamics and to describe those regimes dominated by system size evolution. The connection of these results with mathematical morphogenetic problems is also outlined.Chaos Solitons & Fractals 01/2010; · 1.50 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In order to perform numerical simulations of the KPZ equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a HopfCole transformation applied to a diffusion equation (with \emph{multiplicative} noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on \emph{space} and the HopfCole transformation is \emph{local} both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between realspace and pseudospectral discrete representations. In addition we discuss the relevance of the Galilean invariance violation in these consistent discretization schemes, and the alleged conflict of standard discretization with the fluctuationdissipation theorem, peculiar of 1D. Comment: RevTex, 23pgs, 2 figures, submitted to Phys. Rev. E01/2010;
Publication Stats
257  Citations  
101.78  Total Impact Points  
Top Journals
 Physical Review E (11)
 Physical Review E (4)
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Institutions

2010–2013

Universidad Autónoma de Madrid
 Department of Mathematics
Madrid, Madrid, Spain 
Instituto de Física de Cantabria
Santander, Cantabria, Spain


2008–2009

Spanish National Research Council
 Institute of Fundamental Physics
Madrid, Madrid, Spain


2006–2008

University of Oxford
 Mathematical Institute
Oxford, ENG, United Kingdom


2004–2006

National Distance Education University
 Department of Fundamental Physics
Madrid, Madrid, Spain


2004–2005

University of California, San Diego
 Department of Chemistry and Biochemistry
San Diego, CA, United States
