Harmonic oscillator Brownian motion: Langevin approach revisited

The original strategy applied by Langevin to Brownian movement problem is used to solve the case of a free particle under a harmonic potential. Such straightforward strategy consists in separating the noise termin the Langevin equation in order to solve a deterministic equation associated with the Mean Square Displacement (MSD). In this work, to achieve our goal we first calculate the variance for the stochastic harmonic oscillator and then the MSD appears immediately. We study the problem in the damped and lightly damped cases and show that, for times greater than the relaxation time, Langevin's original strategy is quite consistent with the exact theoretical solutions reported by Chandrasekhar and Lemons, these latter obtained using the statistical properties of a Gaussian white noise. Our results for the MSDs are compared with the exact theoretical solutions as well as with the numerical simulation.

Deterministic Models

This chapter talks about deterministic models as expressions of ecological hypotheses. These models are based on a deterministic equation or equations making predictions that can be compared with observations. The nature of the model ensures that there is no uncertainty in such predictions. Hence, the chapter first discusses the different ways that deterministic models have been used in ecology; identifying three main ones: theoretical, empirical, and simulation. It then outlines a few deterministic models widely used in ecology. Rather than attempting a comprehensive treatment of the subject, the chapter instead introduces a set of core functions that are widely used to represent ecological process across all subdisciplines.

Deterministic and Wind/Wave Modeling: A Comprehensive Approach to Deterministic and Probabilistic Descriptions of Ocean Waves

Deterministic Modeling of ocean surface rogue waves is often done with highly complex spectral codes for the nonlinear Schrödinger equation and its higher order versions, the Zakharov equation or the full Euler equations in two-space and one-time dimensions. Wind/Wave Modeling is normally conducted with a kinetic equation derived from a deterministic equation: the nonlinear four wave interactions are normally computed with the Discrete Interaction Approximation (DIA) algorithm, the Webb-Resio-Tracy (WRT) algorithm or the full Boltzmann integral. I give an overview of these methods and show how a fully self-consistent approach can simultaneously yield all of these methods while computing a multidimensional Fourier series that contains rogue wave packets as “coherent structures” or “nonlinear Fourier components” in the theory. The methods also lead to hyperfast codes in which deterministic evolution is millions of times faster than traditional spectral codes on a large multicore computer. This method could lead the way to an ideal future in which there are single codes that can simultaneously compute the deterministic and probabilistic evolution of surface waves.

Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density G λ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time s i (r) onwards. The value of s i (r) can be obtained for each r and i as the unique root of a deterministic equation.

Maximizing the Expected Duration of Owning a Relatively Best Object in a Poisson Process with Rankable Observations

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density Gλ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time si(r) onwards. The value of si(r) can be obtained for each r and i as the unique root of a deterministic equation.