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S. Franz

Publications and source records attributed to S. Franz.

At least 19 recordsLinked to original sources
arXiv2020

Mixed $hp$ FEM for singularly perturbed fourth order boundary value problems with two small parameters

We consider fourth order singularly perturbed boundary value problems with two small parameters, and the approximation of their solution by the $hp$ version of the Finite Element Method on the {\emph{Spectral Boundary Layer}} mesh from \cite{MXO}. We use a mixed formulation requiring only $C^{0}$ basis functions in two-dimensional smooth domains. Under the assumption of analytic data, we show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through numerical examples, including results using a stronger (balanced) norm.

math.NA↗
arXiv2016

The ALFA Roman Pot Detectors of ATLAS

The ATLAS Roman Pot system is designed to determine the total proton-proton cross-section as well as the luminosity at the Large Hadron Collider (LHC) by measuring elastic proton scattering at very small angles. The system is made of four Roman Pot stations, located in the LHC tunnel in a distance of about 240~m at both sides of the ATLAS interaction point. Each station is equipped with tracking detectors, inserted in Roman Pots which approach the LHC beams vertically. The tracking detectors consist of multi-layer scintillating fibre structures readout by Multi-Anode-Photo-Multipliers.

physics.ins-det↗
arXiv2011

Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions

A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its derivatives are established in the $L_1$ norm. The dependence of these bounds on the small perturbation parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem. The present article is a more detailed version of our recent paper [7].

math.NA↗
arXiv2010

Critical interface: twisting spin glasses at $T_c$

We consider identical copies of spin glasses in finite dimension coupled at the boundaries. This allows to identify the spin glass analogous of twisted boundary conditions in ferromagnetic system and leads to the definition of an interface free-energy that is positively defined and that should scale with a positive power of the system size in the spin glass phase. In this note we study the behavior of the interface at the spin glass critical temperature $T_c$ within mean field theory. We show that the leading scaling of the interface free-energy does not depend on replica symmetry breaking, and can be obtained by simple scaling arguments using a cubic theory for critical spin glasses.

cond-mat.dis-nn↗
arXiv2010

Opportunity and choice in social networks

Our societies are heterogeneous in many dimensions such as census, education, religion, ethnic and cultural composition. The links between individuals - e.g. by friendship, marriage or collaboration - are not evenly distributed, but rather tend to be concentrated within the same group. This phenomenon, called `imbreeding homophily', has been related to either (social) preference for links with own-type individuals (`choice-based' homophily) or to the prevalence of individuals of her same type in the choice set of an individual (`opportunity-based' homophily). Choices determine the network of relations we observe whereas opportunities pertain to the composition of the (unobservable) social network individuals are embedded in and out of which their network of relations is drawn. In this view, we propose a method that, in the presence of multiple data, allows one to distinguish between opportunity and choice based homophily. The main intuition is that, with unbiased opportunities, the effect of choice-based homophily gets weaker and weaker as the size of the minority shrinks, because individuals of the minority rarely meet and have the chance to establish links together. The occurrence of homophily in the limit of very small minorities is therefore an indicator of opportunity bias. We test this idea across the dimensions of race and education on data on US marriages, and across race on friendships in US schools.

physics.soc-ph↗
arXiv2003

Quasi-equilibrium during aging of the 2D Edwards-Anderson model

We test the quasi-equilibrium picture of the aging dynamics -strictly valid in the asymptotic dynamical regime of aging systems- in the pre-asymptotic aging regime of the two dimensional Edwards-Anderson spin glass model. We compare the fluctuation-dissipation characteristic for spin autocorrelation function and response with a corresponding one obtained for a suitably defined new correlation function and its conjugated response. In agreement with the quasi-equilibrium picture we find that after a short transient the two corresponding fluctuation-dissipation ratios (FDR) coincide at equal times. Moreover we show that, as it happens for the usual FDR, the new dynamic FDR at finite time coincides with the static one at finite size.

cond-mat.stat-mech↗
arXiv2001

Exact solutions for diluted spin glasses and optimization problems

We study the low temperature properties of p-spin glass models with finite connectivity and of some optimization problems. Using a one-step functional replica symmetry breaking Ansatz we can solve exactly the saddle-point equations for graphs with uniform connectivity. The resulting ground state energy is in perfect agreement with numerical simulations. For fluctuating connectivity graphs, the same Ansatz can be used in a variational way: For p-spin models (known as p-XOR-SAT in computer science) it provides the exact configurational entropy together with the dynamical and static critical connectivities (for p=3, γ_d=0.818 and γ_s=0.918 resp.), whereas for hard optimization problems like 3-SAT or Bicoloring it provides new upper bounds for their critical thresholds (γ_c^{var}=4.396 and γ_c^{var}=2.149 resp.).

cond-mat.dis-nn↗
arXiv2001

A ferromagnet with a glass transition

We introduce a finite-connectivity ferromagnetic model with a three-spin interaction which has a crystalline (ferromagnetic) phase as well as a glass phase. The model is not frustrated, it has a ferromagnetic equilibrium phase at low temperature which is not reached dynamically in a quench from the high-temperature phase. Instead it shows a glass transition which can be studied in detail by a one step replica-symmetry broken calculation. This spin model exhibits the main properties of the structural glass transition at a solvable mean-field level.

cond-mat.dis-nn↗
arXiv2001

Kob-Andersen model: a non-standard mechanism for the glassy transition

We present new results reflecting the analogies between the Kob-Andersen model and other glassy systems. Studying the stability of the blocked configurations above and below the transition we also give arguments that supports their relevance for the glassy behaviour of the model. However we find, surprisingly, that the organization of the phase space of the system is different from the well known organization of other mean-field spin glasses and structural glasses.

cond-mat.stat-mech↗
arXiv1999

Quasi-equilibrium interpretation of aging dynamics

We develop an interpretation of the off-equilibrium dynamical solution of mean-field glassy models in terms of quasi-equilibrium concepts. We show that the relaxation of the "thermoremanent magnetization" follows a generalized version of the Onsager regression postulate of induced fluctuations. We then find the rationale for the equality between the fluctuation-dissipation ratio and the rate of growth of the configurational entropy close to the asymptotic state, found empirically in mean-field solutions.

cond-mat.dis-nn↗
arXiv1999

The response of glassy systems to random perturbations: A bridge between equilibrium and off-equilibrium

We discuss the response of aging systems with short-range interactions to a class of random perturbations. Although these systems are out of equilibrium, the limit value of the free energy at long times is equal to the equilibrium free energy. By exploiting this fact, we define a new order parameter function, and we relate it to the ratio between response and fluctuation, which is in principle measurable in an aging experiment. For a class of systems possessing stochastic stability, we show that this new order parameter function is intimately related to the static order parameter function, describing the distribution of overlaps between clustering states. The same method is applied to investigate the geometrical organization of pure states. We show that the ultrametric organization in the dynamics implies static ultrametricity, and we relate these properties to static separability, i.e., the property that the measure of the overlap between pure states is essentially unique. Our results, especially relevant for spin glasses, pave the way to an experimental determination of the order parameter function.

cond-mat.dis-nn↗
arXiv1998

Overlap properties and adsorption transition of two Hamiltonian paths

We consider a model of two (fully) compact polymer chains, coupled through an attractive interaction. These compact chains are represented by Hamiltonian paths (HP), and the coupling favors the existence of common bonds between the chains. Using a ($n=0$ component) spin representation for these paths, we show the existence of a phase transition for strong coupling (i.e. at low temperature) towards a ``frozen'' phase where one chain is completely adsorbed onto the other. By performing a Legendre transform, we obtain the probability distribution of overlaps. The fraction of common bonds between two HP, i.e. their overlap $q$, has both lower ($q_m$) and upper ($q_M$) bounds. This means in particuliar that two HP with overlap greater than $q_M$ coincide. These results may be of interest in (bio)polymers and in optimization problems.

cond-mat.stat-mech↗
arXiv1998

Comment on ``Dynamical Heterogeneities in a Supercooled Lennard-Jones Liquid''

In two recent interesting letters evidence was presented for the existence of a growing dynamic correlation length when we approach the glass transition from the liquid phase (a similar divergence is present also in the off-equilibrium dynamics below $T_c$). Here we would like to point out that this phenomenon can be easily predicted using the replica approach.

cond-mat.stat-mech↗
arXiv1997

Basins of attraction of metastable states of the spherical $p$-spin model

We study the basins of attraction of metastable states in the spherical $p$-spin spin glass model, starting the relaxation dynamics at a given distance from a thermalized condition. Weighting the initial condition with the Boltzmann distribution we find a finite size for the basins. On the contrary, a white weighting of the initial condition implies vanishing basins of attraction. We make the corresponding of our results with the ones of a recently constructed effective potential.

cond-mat↗
arXiv1997

Temperature evolution and bifurcations of metastable states in mean-field spin glasses, with connections with structural glasses

The correlations of the free-energy landscape of mean-field spin glasses at different temperatures are investigated, concentrating on models with a first order freezing transition. Using a ``potential function'' we follow the metastable states of the model in temperature, and discuss the possibility of level crossing (which we do not find) and multifurcation (which we find). The dynamics at a given temperature starting from an equilibrium configuration at a different temperature is also discussed. In presence of multifurcation, we find that the equilibrium is never achieved, leading to aging behaviour at slower energy levels than usual aging. The relevance of the observed mechanisms for real structural glasses is discussed, and some numerical simulations of a soft sphere model of glass are presented.

cond-mat.dis-nn↗
arXiv1995

Glassy Mean-Field Dynamics of the Backgammon model

In this paper we present an exact study of the relaxation dynamics of the backgammon model. This is a model of a gas of particles in a discrete space which presents glassy phenomena as a result of {\it entropy barriers} in configuration space. The model is simple enough to allow for a complete analytical treatment of the dynamics in infinite dimensions. We first derive a closed equation describing the evolution of the occupation number probabilities, then we generalize the analysis to the study the autocorrelation function. We also consider possible variants of the model which allow to study the effect of energy barriers.

cond-mat↗