We present Chapman-Enskog and Hilbert expansions applied to the O(v/c) Boltzmann equation for the radiative transfer of neutrinos in core-collapse supernovae. Based on the Legendre expansion of the scattering kernel for the collision integral truncated after the second term, we derive the diffusion limit for the Boltzmann equation by truncation of Chapman-Enskog or Hilbert expansions with reaction and collision scaling. We also give asymptotically sharp results obtained by the use of an additional time scaling. The diffusion limit determines the diffusion source in the isotropic diffusion source approximation (IDSA) of Boltzmann's equation [M. Liebendörfer, S.which the free streaming limit and the reaction limit serve as limiters. Here, we derive the reaction limit as well as the free streaming limit by truncation of Chapman-Enskog or Hilbert expansions using reaction and collision scaling as well as time scaling, respectively. Finally, we explain why limiters are a good choice for the definition of the source term in the IDSA.

In this paper, we discuss the possible generalizations of the social influence with recurrent mobility (SIRM) model [Phys. Rev. Lett. 112, 158701 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.158701]. Although the SIRM model worked approximately satisfying when U.S. election was modeled, it has its limits: it has been developed only for two-party systems and can lead to unphysical behavior when one of the parties has extreme vote share close to 0 or 1. We propose here generalizations to the SIRM model by its extension for multiparty systems that are mathematically well-posed in case of extreme vote shares, too, by handling the noise term in a different way. In addition, we show that our method opens alternative applications for the study of elections by using an alternative calibration procedure and makes it possible to analyze the influence of the "free will" (creating a new party) and other local effects for different commuting network topologies.

Abstract. In this paper we give an introduction to the Boltzmann equation for neutrino transport used in core collapse supernova models as well as a detailed mathematical description of the Isotropic Diffusion Source Approximation (IDSA) established in [6]. Furthermore, we present a numerical treatment of a reduced Boltzmann model problem based on time splitting and finite volumes and revise the discretization of the IDSA in [6] for this problem. Discretization error studies carried out on the reduced Boltzmann model problem and on the IDSA show that the errors are of order one in both cases. By a numerical example, a detailed comparison of the reduced model and the IDSA is carried out and interpreted. For this example the IDSA modeling error with respect to the reduced Boltzmann model is numerically determined and localized.

This is the published version of a paper published in .Citation for the original published paper (version of record):Michaud, J. (2017) Continuous time limits of the utterance selection model. (2006)]. This is motivated by the fact that the Fokker-Planck continuous time limit derived in the original version of the USM is only valid for a small range of parameters. We investigate the consequences of relaxing these constraints on parameters. Using the normal approximation of the multinomial approximation, we derive a continuous time limit of the USM in the form of a weak-noise stochastic differential equation. We argue that this weak noise, not captured by the Kramers-Moyal expansion, cannot be neglected. We then propose a coarse-graining procedure, which takes the form of a stochastic version of the heterogeneous mean field approximation. This approximation groups the behavior of nodes of the same degree, reducing the complexity of the problem. With the help of this approximation, we study in detail two simple families of networks: the regular networks and the star-shaped networks. The analysis reveals and quantifies a finite-size effect of the dynamics. If we increase the size of the network by keeping all the other parameters constant, we transition from a state where conventions emerge to a state where no convention emerges. Furthermore, we show that the degree of a node acts as a time scale. For heterogeneous networks such as star-shaped networks, the time scale difference can become very large, leading to a noisier behavior of highly connected nodes. Phys

A puzzling fact about linguistic norms is that they are mainly stable, but the conventional variant sometimes changes. These transitions seem to be mostly S-shaped and, therefore, directed. Previous models have suggested possible mechanisms to explain these directed changes, mainly based on a bias favoring the innovative variant. What is still debated is the origin of such a bias. In this paper, we propose a refined taxonomy of mechanisms of language change and identify a family of mechanisms explaining self-actuated language changes. We exemplify this type of mechanism with the preference-based selection mechanism that relies on agents having dynamic preferences for different variants of the linguistic norm. The key point is that if these preferences align through social interactions, then new changes can be actuated even in the absence of external triggers. We present results of a multi-agent model and demonstrate that the model produces trajectories that are typical of language change.