RESUMEN
SignificanceThe analysis of complex systems with many degrees of freedom generally involves the definition of low-dimensional collective variables more amenable to physical understanding. Their dynamics can be modeled by generalized Langevin equations, whose coefficients have to be estimated from simulations of the initial high-dimensional system. These equations feature a memory kernel describing the mutual influence of the low-dimensional variables and their environment. We introduce and implement an approach where the generalized Langevin equation is designed to maximize the statistical likelihood of the observed data. This provides an efficient way to generate reduced models to study dynamical properties of complex processes such as chemical reactions in solution, conformational changes in biomolecules, or phase transitions in condensed matter systems.
Asunto(s)
Simulación de Dinámica Molecular , Funciones de VerosimilitudRESUMEN
The characterization of intermittency in turbulence has its roots in the refined similarity hypotheses of Kolmogorov, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in an attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, we investigate the Gaussian multiplicative chaos formalism, which requires the construction of a log-correlated stochastic process X_{t}. The fractional Gaussian noise of Hurst parameter H=0 is of great interest because it leads to a log correlation for the logarithm of the process. Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a stochastic model: X_{t}=∫_{0}^{∞}Y_{t}^{x}k(x)dx, where Y_{t}^{x} is an Ornstein-Uhlenbeck process with speed of mean reversion x and k is a kernel. A regularization of k(x) is required to ensure stationarity, finite variance, and logarithmic autocorrelation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models. To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature X_{t}^{N}=∑_{i=1}^{N}ω_{i}Y_{t}^{x_{i}}, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.
RESUMEN
We study a generalization of the Wright-Fisher model in which some individuals adopt a behavior that is harmful to others without any direct advantage for themselves. This model is motivated by studies of spiteful behavior in nature, including several species of parasitoid hymenoptera in which sperm-depleted males continue to mate despite not being fertile. We first study a single reproductive season, then use it as a building block for a generalized Wright-Fisher model. In the large population limit, for male-skewed sex ratios, we rigorously derive the convergence of the renormalized process to a diffusion with a frequency-dependent selection and genetic drift. This allows a quantitative comparison of the indirect selective advantage with the direct one classically considered in the Wright-Fisher model. From the mathematical point of view, each season is modeled by a mix between samplings with and without replacement, and analyzed by a sort of "reverse numerical analysis", viewing a key recurrence relation as a discretization scheme for a PDE. The diffusion approximation is then obtained by classical methods.