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- ItemThermal effects in gravitational Hartree systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Aki, Gonca L.; Dolbeault, Jean; Sparber, ChristofWe consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb-Newton interaction. For a given mass $M>0$, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T^*>0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature T_c in (0, T^*) above which mixed states appear.
- ItemThe sharp interface limit of the Van der Waals-Cahn-Hilliard phase model for fixed and time dependent domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Dreyer, Wolfgang; Kraus, ChristianeWe study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory. For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition ...
- ItemLarge deviations for the capacity in dynamic spatial relay networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hirsch, Christian; Jahnel, BenediktWe derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.
- ItemSpace-time large deviations in capacity-constrained relay networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hirsch, Christian; Jahnel, Benedikt; Patterson, RobertWe consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.