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Type: Text × Subject: phase transition × WGL Institute: WIAS ×

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Now showing 1 - 10 of 21
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    A large-deviations principle for all the cluster sizes of a sparse Erdős-Rényi graph
    (New York, NY [u.a.] : Wiley, 2021) Andreis, Luisa; König, Wolfgang; Patterson, Robert I. A.
    [For Abstract, see PDF]
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    On unwanted nucleation phenomena at the wall of a VGF chamber
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Dreyer, Wolfgang; Duderstadt, Frank; Eichler, Stefan; Naldzhieva, Margarita
    This is preliminary study on a phenomenon that happens during crystal growth of GaAs in a vertical gradient freeze (VGF) device. Here unwanted polycrystals nucleate at the chamber wall and move into the interior of the crystal. This happens within an undercooled region in the vicinity of the triple point, where the liquid-solid interface meets the chamber wall. The size and shape of that region is modelled by the Gibbs-Thomson law, which will be rederived in this paper. Hereafter we identify the crucial parameter, whose proper adjustment may minimize the undercooled region.
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    On the clustering property of the random intersection graphs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Yao, Xin; Chen, Jinwen; Zhang, Changshui; Li, Yanda
    A random intersection graph mtlmcalG_V,W,p is induced from a random bipartite graph mtlmcalG^*_V,W,p with vertices classes mtlV, mtlW and the edges incident between mtlv in V and mtlw in W with probability mtlp. Two vertices in mtlV are considered to be connected with each other if both of them connect with some common vertices in mtlW. The clustering properties of the random intersection graph are investigated completely in this article. Suppose that the vertices number be mtlN = mabsV and mtlM=mabsW and mtlM = N^alpha, p=N^-beta, where mtlalpha > 0,, beta > 0, we derive the exact expressions of the clustering coefficient mtlC_v of vertex mtlv in mtlmcalG_V,W,p. The results show that if mtlalpha < 2beta and mtlalpha neq beta, mtlC_v decreases with the increasing of the graph size; if mtlalpha = beta or mtlalpha geq 2beta, the graph has the constant clustering coefficients, in addition, if mtlalpha > 2beta, the graph connecChangshui Zhangts almost completely. Therefore, we illustrate the phase transition for the clustering property in the random intersection graphs and give the condition that mtlriG being high clustering graph.
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    Interface conditions for limits of the Navier-Stokes-Korteweg model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hermsdörfer, Katharina; Kraus, Christiane; Kröner, Dietmar
    In this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situ
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    Continuum percolation in a nonstabilizing environment
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Jhawar, Sanjoy Kumar; Vu, Anh Duc
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].
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    A large-deviations approach to gelation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Andreis, Luisa; König, Wolfgang; Patterson, Robert
    A @large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t = 1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdos-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdos-Rényi graphs are connected.
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    Sliding modes for a phase-field system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Barbu, Viorel; Colli, Pierluigi; Gilardi, Gianni; Marinoschi, Gabriela; Rocca, Elisabetta
    In the present contribution the sliding mode control (SMC) problem for a phasefield model of Caginalp type is considered. First we prove the well-posedness and some regularity results for the phase-field type state systems modified by the statefeedback control laws. Then, we show that the chosen SMC laws force the system to reach within finite time the sliding manifold (that we chose in order that one of the physical variables or a combination of them remains constant in time). We study three different types of feedback control laws: the first one appears in the internal energy balance and forces a linear combination of the temperature and the phase to reach a given (space dependent) value, while the second and third ones are added in the phase relation and lead the phase onto a prescribed target phi*. While the control law is non-local in space for the first two problems, it is local in the third one, i.e., its value at any point and any time just depends on the value of the state.
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    Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert; Rocca, Elisabetta; Schimperna, Giulio
    In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.
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    Bilinear coagulation equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heydecker, Daniel; Patterson, Robert I.A.
    We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time tg ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.
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    A compressible mixture model with phase transition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christiane
    We introduce a new thermodynamically consistent diffuse interface model of AllenCahn/NavierStokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a YoungLaplace and a Stefan type law.