Browsing by Author "Theil, Florian"
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- ItemDistribution of Cracks in a Chain of Atoms at Low Temperature(Cham (ZG) : Springer International Publishing AG, 2021) Jansen, Sabine; König, Wolfgang; Schmidt, Bernd; Theil, FlorianWe consider a one-dimensional classical many-body system with interaction potential of Lennard–Jones type in the thermodynamic limit at low temperature 1/β∈(0,∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of Nexp(−βesurf/2) with esurf>0 a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.
- ItemDistribution of cracks in a chain of atoms at low temperature(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Jansen, Sabine; König, Wolfgang; Schmidt, Bernd; Theil, FlorianWe consider a one-dimensional classical many-body system with interaction potential of Lennard--Jones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(-β e surf /2) with e surf > 0 a surface energy.
- ItemEmergence of Structures in Particle Systems: Mechanics, Analysis and Computation(Zürich : EMS Publ. House, 2018) Schmidt, Bernd; Stefanelli, Ulisse; Theil, FlorianThe meeting focused on the last advances in particle systems. The talks covered a broad range of topics, ranging from questions in crystallization and atomistic systems to mesoscopic models of defects to machine learning approaches and computational aspects.
- ItemInterplay of Analysis and Probability in Applied Mathematics(Zürich : EMS Publ. House, 2015) König, Wolfgang; Theil, Florian; Zimmer, JohannesThis workshop brought together analysts and probabilists working on problems at some of the many interfaces of these two fields. Most of the problems discussed during the meeting have their origin in physics or chemistry. The workshop was grouped around the four themes (1) condensation in random structures, (2) disordered systems, (3) discrete-to-continuum transitions, (4) atomistic and molecular systems.
- ItemInterplay of Analysis and Probability in Applied Mathematics(Zürich : EMS Publ. House, 2018) Dirr, Nicolas; König, Wolfgang; Theil, FlorianThis workshop continued to foster the collaboration between researchers working in analysis and probability, respectively. Some core areas, in which this happens with high success, belong to the objectives of this meeting: stochastic homogenization of various quantities in random media and random operators, metastability in several particle models with stochastic input that are triggered by physics reasonings, emergence of macroscopic effects in large random structures like graphs or permutations. A main feature present was the exploration of the benefit of a high-level combination of methods from both fields: analysis and probability.
- ItemSurface Energy and Boundary Layers for a Chain of Atoms at Low Temperature(Berlin ; Heidelberg : Springer, 2021) Jansen, Sabine; König, Wolfgang; Schmidt, Bernd; Theil, FlorianWe analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard–Jones type. The pressure (stress) is assumed to be small but positive and bounded away from zero, while the temperature β- 1 goes to zero. Our main results are: (1) As β→ ∞ at fixed positive pressure p> 0 , the Gibbs measures μβ and νβ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals E¯ bulk and E¯ surf. The minimizer of the surface functional corresponds to zero temperature boundary layers; (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of E¯ surf; (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts; (4) Bounds on the decay of correlations are provided, some of them uniform in β. © 2020, The Author(s).
- ItemSurface energy and boundary layers for a chain of atoms at low temperature(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Jansen, Sabine; König, Wolfgang; Schmidt, Bernd; Theil, FlorianWe analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As _x000C_ the temperature goes to zero and at fixed positive pressure, the Gibbs measures _x0016__x000C_for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in _x000C_the inverse temperature.