Effective dynamics using conditional expectations

The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable $\xi(x)$, where $x$ describes the configuration of the system in a high-dimensional space $\R^n$, and $\xi$ is a smooth function with value in $\R$ (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on $x \in \R^n$, the equilibrium properties on $\xi(x)$ are completely determined by the free energy. On the other hand, the question of the effective dynamics on $\xi(x)$ is much more difficult to address. Starting from an overdamped Langevin equation on $x \in \R^n$, we propose an effective dynamics for $\xi(x) \in \R$ using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics

Asymptotic analysis for the generalized langevin equation

Various qualitative properties of solutions to the generalized Langevin equation (GLE) in a periodic or a confining potential are studied in this paper. We consider a class of quasi-Markovian GLEs, similar to the model that was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem (invariance principle), short time asymptotics and the white noise limit are studied. Our proofs are based on a careful analysis of a hypoelliptic operator which is the generator of an auxiliary Markov process. Systematic use of the recently developed theory of hypocoercivity \cite{Vil04HPI} is made.Comment: 27 pages, no figures. Submitted to Nonlinearity

Dimension reduction for systems with slow relaxation

We develop reduced, stochastic models for high dimensional, dissipative dynamical systems that relax very slowly to equilibrium and can encode long term memory. We present a variety of empirical and first principles approaches for model reduction, and build a mathematical framework for analyzing the reduced models. We introduce the notions of universal and asymptotic filters to characterize `optimal' model reductions for sloppy linear models. We illustrate our methods by applying them to the practically important problem of modeling evaporation in oil spills.Comment: 48 Pages, 13 figures. Paper dedicated to the memory of Leo Kadanof

Ergodic properties of quasi-Markovian generalized Langevin equations with configuration dependent noise and non-conservative force

We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted $L^{\infty}$ spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force

Influence of age, social patterns and nasopharyngeal carriage on antibodies to three conserved pneumococcal surface proteins (PhtD, PcpA and PrtA) in healthy young children

The acquisition of specific antibodies is paramount to protect children against pneumococcal diseases, and a better understanding of how age, ethnicity and/or Streptococcus pneumoniae (Spn) nasopharyngeal carriage influence the acquisition of antibodies to pneumococcal surface proteins (PSP) is important for the development of novel serodiagnostic and immunisation strategies. IgG antibody titres against three conserved PSP (PhtD, PcpA and PrtA) in the sera of 451 healthy children aged 1 to 24months from Israel [Jewish (50.1%) and Bedouin (49.9%)] were measured by enzyme-linked immunosorbent assay (ELISA), while nasopharyngeal swabs from these children were assessed for the presence of Spn. Globally, anti-PhtD and anti-PrtA geometric mean concentrations (GMC; EU/ml) were high at <2.5months of age [PhtD: 35.3, 95% confidence interval (CI) 30.6-40.6; PrtA: 71.2, 95 % CI 60-84.5], was lower at 5-7months of age (PhtD: 10, 95 % CI 8-12.4; PrtA: 17.9, 95 % CI 14.4-22.1) and only increased after 11months of age. In contrast, an increase in anti-PcpA was observed at 5-7months of age. Anti-PcpA and anti-PrtA, but not anti-PhtD, were significantly higher in Bedouin children (PcpA: 361.6 vs. 226.3, p = 0.02; PrtA: 67.2 vs. 29.5, p < 0.001) in whom Spn nasopharyngeal carriage was identified earlier (60% vs. 38% of carriers <6months of age, p = 0.002). Spn carriage was associated with significantly higher anti-PSP concentrations in carriers than in non-carriers (p < 0.001 for each PSP). Thus, age, ethnicity and, essentially, nasopharyngeal carriage exert distinct cumulative influences on infant responses to PSP. These specific characteristics are worthwhile to include in the evaluation of pneumococcal seroresponses and the development of new PSP-based vaccine

The diffusion of a new service: Combining service consideration and brand choice

We propose an individual-level model of a two-stage service diffusion process. In the first stage, customers decide whether to "consider" joining the service. This (Consideration) stage is modeled by a hazard model. Customers who decide to consider the service move on to the Choice stage, wherein they choose among the service alternatives and an outside No Choice option. This stage is modeled by a conditional Multinomial Logit model. The service provider does not observe the transition in the first stage of potential customers who have yet to choose a brand. Such potential customers may have started to consider joining the service, yet chose the outside alternative in each period thereafter. One of the main contributions of the model is its ability to distinguish between these two non-adopter types. We estimated the model using data on the adoption process of newly introduced service plans offered by a commercial bank. We employed the hierarchical Bayes Monte Carlo Markov Chain procedure to estimate individual as well as population parameters. The empirical results indicate that the model outperforms competing models in breadth of analysis, model fit, and prediction accuracy

A weak characterization of slow variables in stochastic dynamical systems

We present a novel characterization of slow variables for continuous Markov processes that provably preserve the slow timescales. These slow variables are known as reaction coordinates in molecular dynamical applications, where they play a key role in system analysis and coarse graining. The defining characteristics of these slow variables is that they parametrize a so-called transition manifold, a low-dimensional manifold in a certain density function space that emerges with progressive equilibration of the system's fast variables. The existence of said manifold was previously predicted for certain classes of metastable and slow-fast systems. However, in the original work, the existence of the manifold hinges on the pointwise convergence of the system's transition density functions towards it. We show in this work that a convergence in average with respect to the system's stationary measure is sufficient to yield reaction coordinates with the same key qualities. This allows one to accurately predict the timescale preservation in systems where the old theory is not applicable or would give overly pessimistic results. Moreover, the new characterization is still constructive, in that it allows for the algorithmic identification of a good slow variable. The improved characterization, the error prediction and the variable construction are demonstrated by a small metastable system

An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. As in standard projection operator methods, a set of resolved variables is selected to capture the slow, macroscopic behavior of the system, and the family of quasi-equilibrium probability densities on phase space corresponding to these resolved variables is employed as a statistical model. The macroscopic dynamics of the mean resolved variables is determined by optimizing over paths of these probability densities. Specifically, a cost function is introduced that quantifies the lack-of-fit of such paths to the underlying microscopic dynamics; it is an ensemble-averaged, squared-norm of the residual that results from submitting a path of trial densities to the Liouville equation. The evolution of the macrostate is estimated by minimizing the time integral of the cost function. The value function for this optimization satisfies the associated Hamilton-Jacobi equation, and it determines the optimal relation between the statistical parameters and the irreversible fluxes of the resolved variables, thereby closing the reduced dynamics. The resulting equations for the macroscopic variables have the generic form of governing equations for nonequilibrium thermodynamics, and they furnish a rational extension of the classical equations of linear irreversible thermodynamics beyond the near-equilibrium regime. In particular, the value function is a thermodynamic potential that extends the classical dissipation function and supplies the nonlinear relation between thermodynamics forces and fluxes

Multiscale integration schemes for jump-diffusion systems

We study a two-time-scale system of jump-diffusion stochastic differential equations. We analyze a class of multiscale integration methods for these systems, which, in the spirit of [1], consist of a hybridization between a standard solver for the slow components and short runs for the fast dynamics, which are used to estimate the effect that the fast components have on the slow ones. We obtain explicit bounds for the discrepancy between the results of the multiscale integration method and the slow components of the original system

Learning to mentalize: a mediational approach for caregivers and therapists

Mentalization-based therapies (MBTs) are rigorous, theoretically grounded, and evidence-based interventions. However, dissemination of this psychodynamically informed intervention lags behind that of more skills-based therapies because of a lack of concrete operationalization of its key components. In this proof-of-concept paper, we describe how the learning (mediational) components of an educational intervention, the Mediational Intervention for Sensitizing Caregivers (MISC), can operationalize key components of MBTs in behaviorally anchored ways. We suggest that the process of the recovery of mentalizing can be operationalized through five learning components: focusing, affecting, expanding, rewarding, and regulating. In operationalizing the process of rebuilding mentalizing using these observable, behaviorally anchored concepts focusing on creating epistemic trust, we hope to increase the accessibility of MBTs to a wider audience