We plan two poster sessions during the conference: on Tuesday and Thursday afternoons
Depending on funds availability, partial financial support may be offered to participants presenting a poster
IF YOU WANT TO PRESENT A POSTER, PLEASE FILL THIS FORM
Deadline: April 1, 2024
How to print your poster in Trieste: you may contact the shop Cartoleria DM, by sending your pdf in A0 format. Two types of printing options are possible: "usual" paper-plastic €15; fabric €25. Please contact the shop at least 24hrs before the pick-up time
TITLES & ABSTRACTS OF POSTER PRESENTATIONS
Pedro Cardoso (University of Bonn)
Hydrodynamics of a d-dimensional long jumps symmetric exclusion with a slow barrier
We obtain the hydrodynamic limit of symmetric long-jumps exclusion in Z^d (for d ≥ 1), where the jump rate is inversely proportional to a power of the jump’s length with exponent γ + 1, where γ ≥ 2. Moreover, movements between Z^d−1 × Z∗− and Z^d−1 × N are slowed down by a factor αn^−β (with α > 0 and β ≥ 1). In the hydrodynamic limit we obtain the heat equation in Rd without boundary conditions or with Neumann boundary conditions, depending on the values of β and γ. The (rather restrictive) condition in previous works (for d = 1) about the initial distribution satisfying an entropy bound with respect to a Bernoulli product measure with constant parameter is weakened or completely dropped.
Bernardo Collufio (GSSI, L'Aquila)
Mathematical and numerical study of first ionization phenomena with coupled Vlasov-type equations in extended phase spaces
We set up a mathematical model for describing first ionization processes of collisionless monoatomic gases inside bounded containers where reflective phenomena occur on the boundaries. We use a kinetic approach, with three unknowns distributions (neutrals, electrons, ions), starting from conservative microscopic dynamics involving a additional energy variable of the phase space. We show that classical macroscopic conservation laws are satisfied by the model and a balance law for total energy is derived. We then provide an existence result in 1D under symmetry and isotropy assumptions. Lastly, some numerical tests are implemented for showing quantitative estimations about the process which could be relevant for practical applications.
Hugo Da Cunha (University of Lyon)
Hydrodynamic limit for a Facilitated Exclusion Process with open boundaries
The one-dimensional Facilitated Exclusion Process (FEP) is a model of interacting particle system whose microscopic dynamics is purely stochastic and which belongs to the class of Kinetically Constrained Lattice Gases. Due to the dynamical constraints, this model displays a phase transition at the critical density 1/2, below which the system is completely frozen. In recent years, the FEP has been extensively studied on the periodic setting, but I will consider it with boundary conditions. We put the system in contact with reservoirs that can inject and remove particles at both ends, regulating the speed of those exchanges thanks to a parameter θ ∈ R. I will describe the hydrodynamic limit of this model in the diffusive timescale, when it starts from its supercritical phase. Namely, the macroscopic density of particles evolves according to a fast diffusion equation with different boundary conditions, such as Dirichlet, Robin or Neumann, depending on the value of the parameter θ. FEP’s behaviour is quite peculiar as, on the contrary to other exclusion models, there is a distinction between two types of particles (active and frozen), and macroscopically the reservoirs act on the density of active particles. This is a joint work with Clément Erignoux and Marielle Simon.
Théophile Dolmaire (University of Bonn)
Inelastic collapse of three particles in dimension $d \geq 2$
The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford’s theorem). In the case of large systems of particles that interact inelastically (sand, snow, interstellar dust), the derivation of the inelastic Boltzmann equation is still open. One major difficulty, already at the microscopic level, comes from the phenomenon of inelastic collapse, when infinitely many collisions take place in finite time. Assuming that the restitution coefficient r is constant, we obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles: either the particles arrange in a nearly-linear chain, or they form a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct explicit initial configurations leading to a nearly-linear collapse in a stable way, such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.
The results are obtained in collaboration with Juan J. L. Velázquez (Universität Bonn).
Francesco Drago (University of L'Aquila)
Formal Power Series and applications in statistical mechanics
Analytic functions give rise to convergent power series. The concept of Formal Power Series (FPS) was invented to work rigorously with non convergent power series. As we are interested in systems with infinitely many degrees of freedom, we consider FPS in infinite dimensional spaces. To that purpose, firstly we need to introduce a basis independent representation of Formal Power Series. Secondly, In order to treat derivatives we will introduce vector valued FPS. In particular we consider FPS which are generating functions of weighted combinatorial species. The latter is particularly useful in applications in statistical mechanics. Our aim is to develop a systematic framework which permits to work rigorously with the graphical expansion of partition and correlation functions. We also want to use this framework to prove that such FPS are actually convergent power series and hence give rise to analytic functions. Indeed, we also present some convergence results we got in this way.
Joint work with Tobias Kuna and Dimitrios Tsagkarogiannis.
Federica Iacovissi (University of L'Aquila)
Large Deviations for rational models and the matrix product ansatz
We introduce the rational model which is a stochastic model for the random generation of words over a given alphabet A. This model defines a probability measure μ_N on A^N which is constructed using a product of matrices associated with each a ∈ A^N . By enlarging the state space, we show a natural approach to compute the large deviations rate functional for the measure μ_N . In the finite case, we discuss specific examples according to the properties of the matrices.
Furthermore, we show that, by selecting particular alphabets and matrices, we can obtain the matrix product ansatz representation of the boundary-driven TASEP, and we discuss the approach for deriving a new large deviations rate functional for the process.
Josué Knorst (Unicamp & ENSTA Paris)
Systems of particles with singular interaction under multiplicative common noise
We consider a system of particles interacting via a singular kernel (Keller-Segel, for instance) and subject to common noise in addition to individual noises. The common noise is multiplicative, i.e., has the form of $\sigma_t(x)B_t$. The limiting equation for such a system is an SPDE, for which we prove well-posedness via classical arguments (Krylov’s Lp -theory for SPDEs). We prove the propagation of chaos by classical compacticity arguments and provide quantitative rates in the case of spatial invariant noise.
Chun Yin Lam (University of Münster)
Variational convergence of exchange-driven particle system in thermodynamic limit
We consider the thermodynamic limit of mean-field stochastic particle systems on a complete graph. The evolution of occupation number at each vertex is driven by particle exchange with its rate depending on the population of the starting vertex and the destination vertex, including zero-range and misanthrope processes. We show that under a detailed balance condition and suitable growth assumptions on the exchange rate, the evolution equation of the law of the particle density can be seen as a generalised gradient flow equation related to the large deviation rate functional. We show the variational convergence of the gradient structures based on the energy dissipation principle, which coincides with the large deviation rate function of the finite system. The convergence of the system in this variational sense is established based on compactness of the density and flux and Γ-lower-semicontinuity of the energy dissipation functional along solutions to the continuity equation. The driving free energy Γ-converges in the thermodynamic limit, after taking possible condensation phenomena into account.
Anna Macaluso (University of Parma)
Multi-temperature hydrodynamic limits derived from a mixed Boltzmann-BGK model
We present a mixed Boltzmann–BGK model for inert mixtures of monatomic gases that combines the positive features of both Boltzmann and BGK formulations. In particular, it preserves the collision detail of the Boltzmann model in the dominant process between intra-species molecules, and it takes advantage of the computational manageability of the BGK descriptions in the remaining collisional phenomenon. In the regime dominated by intra-species collisions, the multi-velocity and multi-temperature hydrodynamic equations are derived with a classical Chapman-Enskog expansion technique up to Navier-Stokes order. The resulting equations constitute a system of balance laws for the main macroscopic fields and are coupled by proper source terms taking into account the inter-species interactions. At the Navier-Stokes level, Newton and Fourier laws are recovered, and the viscosity and heat conductivity coefficients are explicitly computed in terms of the microscopic parameters.
We consider a family of exclusion processes defined on the discrete interval with weak boundary interaction that allows the creation and annihilation of particles on a neighborhood of radius L of the boundary under very general rates. We prove that the hydrodynamic equation is the heat equation with non-linear Robin boundary conditions. We present a particular choice of boundary rates for which we have multiple stationary solutions but for which it still holds the uniqueness of the solution of its hydrodynamic equation. We also prove the associated dynamical large deviations principle. Joint work with Beatriz Salvador and Claudio Landim.
Giorgio Martalò (University of Parma)
A BGK reactive model for mixtures of monatomic gases. Consistency and asymptotic trend to equilibrium
We propose a kinetic BGK-type model for a mixture of four monatomic gases, undergoing a bimolecular and reversible chemical reaction. The elastic and reactive interactions are described separately by distinct relaxation terms and the mechanical operator is the sum of binary BGK contributions, one for each pair of interacting species. In this way, the model separately incorporates the effects of mechanical processes and chemical reactions and retains the effects of inter-species interactions which are proper for the mixture. The dependence of Maxwellian attractors on the main macroscopic fields is explicitly expressed by assuming that the exchange rates for momentum and energy of mechanical and chemical operators coincide with the ones of the corresponding Boltzmann terms. Under suitable hypotheses, the entropy dissipation is proven by means of an H-functional prescribing the relaxation of the distribution functions to the equilibrium. Assuming isotropic distribution functions, we perform numerical simulations for the macroscopic fields to investigate how the dynamics push the mixture towards thermalization and chemical equilibrium. Simulations show that, when initial temperatures are far from equilibrium, the relaxation occurs at a later stage and the classical H-Boltzmann functional is not monotone during the initial transient.
Brune Massoulié (University of Paris Dauphine)
Transience time and mixing time for the facilitated exclusion process and the SSEP with traps
The facilitated exclusion process (FEP) is a particle system, where particles evolve on a discrete lattice, making random jumps while obeying local constraints. Because of these constraints, the process will almost surely become blocked (frozen), or reach an absorbing set of configurations after a finite time. This process can thus be seen as a toy model for the liquid-solid transition. In our work, we study the timescale after which these events occur, by introducing a new representation of this process. We therefore study a nonreversible variant of the simple symmetric exclusion process (SSEP), where particles can be destroyed by traps. This allows to make precise estimates on the transience time of both the FEP and the SSEP with traps, we also establish cutoff for the mixing time of the latter, and this opens the way for estimating precisely the mixing time of the FEP.
Dominik Nowak (University of Basel)
The Lorentz Gas in the Weak Coupling Regime: Derivation of the Linear Landau- Vlasov Equation
We study the dynamics of a test particle in a system of N randomly distributed stationary spherical obstacles (scatterers) in dimensions d ≥ 2. We assume that the test particle’s motion is influenced by two contributing factors. One contribution comes from collisions with scatterers, whose interaction potential is modelled by εαU(r/ε), where α ∈ (0,1/2] and U is radially symmetric and strictly decreasing. The second factor is a long range force field of mean-field type generated by the collection of all scatterers in the system. Although such systems are well understood when considering either collisions or an external force field, studying these two driving forces simultaneously leads to a combination of both local and non-local effects, which introduces new technical difficulties. In the weak coupling regime, we prove that for α ∈ (0, (d − 1)/8) the test particle’s probability density converges to the solution of the linear Landau-Vlasov equation as ε → 0.
Emanuele Pasqui (University of Padova)
Extremes and entropic repulsion for the Gaussian Free Field with bond disorder
Randomly fluctuating interfaces naturally arise in the context of coexistence of two homogeneous phases and are studied in a variety of statistical mechanics models. The Gaussian process on the integer lattice with zero mean and covariances given by the Green function of the simple random walk on the lattice is known as the (Lattice) Gaussian Free Field. It has been determined by [1] that in dimension $d>=3$ the probability of the event that all the spins of the field are positive in a box of volume $N^d$ decays exponentially at speed $N^{d-2} \log N$. In our work we focus on the Gaussian Free Field with bond disorder, in which the underlying graph is the integer lattice with weights sampled according to a stationary and ergodic distribution. We aim at revisiting the above results in this random environment, computing explicitly the corresponding constants. We also discuss the phenomenon of entropic repulsion, that is the equality between the asymptotic behaviour of the field under the conditioning on the event of positive spins on a box of volume $N^d$, and the law of a stationary Gaussian field with mean shifted by a height of order $\sqrt{log N}$.
References
[1] Erwin Bolthausen, Jean-Dominique Deuschel, and Ofer Zeitouni, Entropic repulsion of the lattice free field, Communications in Mathematical Physics 170 (1995), no. 2, 417-443.
Giacomo Passuello (University of Padova)
Mixing cutoff for simple random walks on the Chung-Lu digraph
In this paper, we are interested in the mixing behaviour of simple random walks on inhomogeneous directed graphs. We focus our study on the Chung-Lu digraph, which is an inhomogeneous network that generalizes the Erdős–Rényi digraph. In particular, under the Chung-Lu model, edges are included in the graph independently and according to given Bernoulli laws, so that the average degrees are fixed. To guarantee the existence of a unique reversible measure, which is implied by the strong connectivity of the graph, we assume that the average degree grows logarithmically in the size n of the graph. In this weakly dense regime, we prove that the total variation distance to equilibrium displays a cutoff behaviour at the entropic time of order logn/loglogn. Moreover, we prove that on a precise window, the cutoff profile converges to the Gaussian tail function. This is qualitatively similar to what was proved for the directed configuration model, where degrees are deterministically fixed. In terms of statistical ensembles, our analysis provides an extension of these cutoff results from a hard to a soft-constrained model.
Beatriz Salvador (IST Lisbon)
From duality to correlations
We propose a redefinition, based on stochastic duality, of the usual k-points correlation function for models for which a certain type of duality function is available. As an example, we give the symmetric simple partial exclusion and inclusion processes, the independent random walkers, the Brownian Energy Process, the Harmonic model, and the multispecies exclusion, all of the above considered in a one-dimensional lattice with open boundary. For such models, we show that the time evolution of the redefined k-points correlation function is described by a closed partial differential equation that can be written in terms of the generator of a k-dimensional random walk, whose jump rates are model-dependent. As a consequence, we deduce an asymptotic independence which many models share, such as the ones mentioned above. The case k = 2 is joint work with Patricia Gonçalves and the generalization for k ≥ 3 is ongoing work.
Carel Wagenaar (TU Delft)
A dynamic asymmetric exclusion process
The asymmetric exclusion process ASEP(q,N), which allows N particles per site, can be generalized by adding a dynamic parameter. This process, dynamic ASEP, has rates consisting of the ASEP(q,N)-rates multiplied by a factor containing a height function. The latter makes the rates of dynamic ASEP interpolate between ASEP(q,N) and ASEP(1/q,N). Moreover, dynamic ASEP has many interesting (orthogonal) dualities which generalize many other Markov dualities. The duality functions on top of this hierarchy are Askey-Wilson polynomials: a family of explicit orthogonal polynomials that generalize many other orthogonal polynomials. Just as is the case with ASEP(q,N), dynamic ASEP has an intimate connection with the quantum algebra Uq(sl(2)).