Effective transport properties for flashing ratchets using homogenization theory

We study effective transport properties of Brownian motor models of molecular motors. The effective drift and diffusivity can be calculated by solving cell problems, given explicitly by homogenization theory. We briefly describe how this approach is equivalent to theWang-Peskin-Elston (WPE) [3] numerical algorithm for calculating effective transport properties of flashing ratchets. For an on-off flashing ratchet we examine the optimization of the Peclet number as a function of the free parameters of the system. We also present a numerical method for solving the cell equations for a flashing ratchet with Gaussian multiplicative noise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Entropy Production Rate of the Coupled Diffusion Process

Coupled diffusion processes model the systems of molecular motors, in which chemical free energy is converted to mechanical energy. Entropy production rate is a crucial term in calculating the efficiency of the motors. In this paper, we calculate the entropy production rates of coupled diffusion processes.     

Stokes efficiency of molecular motors with inertia

A molecular motor utilizes chemical free energy to generate a unidirectional motion in a viscous media. The stochastic motion of a motor is governed by a Langevin equation coupled to the chemical occupancy state. The change of chemical occupancy state is governed by a discrete Markov process. The Stokes efficiency was introduced to measure how “efficiently” the motor uses chemical free energy to drive through the surrounding fluid. For the overdamping case where the effect of inertia is ignored, it was proved that the Stokes efficiency is bounded by 100% [H. Wang, G. Oster, The Stokes efficiency for molecular motors and its applications, Europhysics Letters 57 (2002) 134–140]. Here we present a proof for the general case.     

A homogenization approach to flashing ratchets

The paper continues the program of the authors to develop a mathematical framework to understand and characterize the notion of “asymmetric” potentials, which has been introduced to explain how molecular motors work, considering flashing ratchets, i.e., molecules diffusing in a potential with periodic switches. The mathematical model is a Fokker–Planck equation with a space–time periodic potential and diffusion of order of magnitude compatible with the period of the potential. After performing a homogenization analysis of the problem the “asymmetric” potentials are characterized by the property that the solution, which models the molecule density, concentrates on one end of the domain. Finally explicit examples are presented exhibiting that the concentration phenomena (motor effect) takes place are presented. The proof uses techniques from the theory of viscosity solutions for the Hamilton–Jacobi equation which, in the homogenization limit, defines the effective hamiltonian.     

Numerical solution of a PDE model for a ratchet-cap pricing with BGM interest rate dynamics

In this paper we present a new numerical method to price an interest rate derivative. The financial product consists of a particular ratchet cap contract which contains a set of ratchet caplets. For this purpose, we first pose the PDE pricing model for each ratchet caplet by means of Feynman-Kac theorem. The underlying interest rates are the forward LIBOR rates, the dynamics of which are assumed to follow the recently introduced BGM (LMM) market model. For the set of PDEs associated to the ratchet caplets pricing problems, we propose a second order Crank-Nicolson characteristics time discretization scheme combined with a finite element discretization in the interest rate variables. In order to illustrate the performance of the numerical methods, we present an academic test and a real example of a particular ratchet cap pricing. In the second case, a comparison between the results obtained by Monte Carlo simulation and the proposed method is presented.     

Individual and collective behavior of vibrating motors interacting through a resonant plate

We report on experiments of many small motors—cell phone vibrators—glued to and interacting through a resonant elastic plate. We find that the motors tend to avoid frequencies that are just higher than the resonances of a plate, preferring instead frequencies just below those resonances. As a result, motors interacting through a resonant plate exhibit hysteresis in their frequency versus driving voltage. We also find that the stability of a single motor near a resonance is different from the stability of a group of motors near a resonance. When the driving voltage is constant and the transient behavior of the system has passed, we find that the average frequency of all the motors is constant.     

A multiscale molecular switch model

A first-principles mathematical simulation of molecular switching of the isomerization reaction in a naphthalocyanine molecule on the basis of a current-induced tunnel microscope is presented. The Gibbs free energy surface and the reaction pathway are analyzed in terms of metadynamics using the numerical quantum mechanical Car-Parrinello molecular dynamics (CPMD) code. The calculations were performed on an IBM Blue Gene/P supercomputer at Moscow State University’s Faculty of Computational Mathematics and Cybernetics. The height of the energy barrier that must be overcome to achieve the isomerization reaction is found along an introduced coordination variable. A multiscale model of Ehrenfest molecular dynamics states is proposed on the basis of an excited electron. The model uses the coordination direction of the reaction on the free energy surface obtained in calculations for a complete molecule.     

A tensor model for liquid crystals on a spherical surface

Rod-like molecules confined on a spherical surface can organize themselves into nematic liquid crystal phases. This can give rise to novel textures displayed on the surface, which has been observed in experiments. An important theoretical question is how to find and predict these textures. Mathematically, a stable configuration of the nematic fluid corresponds to a local minimum in the free energy landscape. By applying Taylor expansion and Bingham approximation to a general molecular model, we obtain a closed-form tensor model, which gives a free energy form that is different from the classic Landau-de Gennes model. Based on the tensor model, we implement an efficient numerical algorithm to locate the local minimum of the free energy. Our model successfully predicts the splay, tennis-ball and rectangle textures. Among them, the tennis-ball configuration has the lowest free energy.     

Intermittency and deterministic diffusion in chaotic ratchets

We consider the problem of deterministic transport of particles in an asymmetric periodic ratchet potential of the rocking type. When the inertial term is taken into account, the dynamics can be chaotic and modify the transport properties. We calculate the bifurcation diagram as a function of the amplitude of forcing and analyze in detail the crisis bifurcation that leads to current reversals. Near this bifurcation we obtain intermittency and anomalous deterministic diffusion.     

Analytic solution for ratchet guaranteed minimum death benefit options under a variety of mortality laws

We derive a number of analytic results for GMDB ratchet options. Closed form solutions are found for De Moivre’s Law, Constant Force of Mortality, Constant Force of Mortality with an endowment age and constant force of mortality with a cutoff age. We find an infinite series solution for a general mortality laws and we derive the conditions under which this series terminates. We sum this series for at-the-money options under the realistic Makeham’s Law of Mortality.     

Droplet solutions in the diblock copolymer problem with skewed monomer composition

A droplet solution characterizes the lamellar phase of a diblock copolymer when the two composing monomers maintain a skewed ratio. We study the threshold case where the free energy of a droplet solution is comparable to the free energy of the constant solution. Using a Lyapunov-Schmidt reduction approach, adapted to calculus of variations, we prove the existence of a free energy local minimizer with a given number of droplets. Also determined are the free energy, the droplet location, and the droplet size. Supported in part by a Direct Grant from CUHK and an earmarked Grant of RGC of Hong Kong.     

Branching Random Walks in Space–Time Random Environment: Survival Probability,Global and Local Growth Rates

We study the survival probability and the growth rate for branching random walks in random environment (BRWRE). The particles perform simple symmetric random walks on the d-dimensional integer lattice, while at each time unit, they split into independent copies according to time–space i.i.d. offspring distributions. The BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity called the free energy is well studied. We discuss the survival probability (both global and local) for BRWRE and give a criterion for its positivity in terms of the free energy of the associated DPRE. We also show that the global growth rate for the number of particles in BRWRE is given by the free energy of the associated DPRE, though the local growth rate is given by the directional free energy.     

动态多任务双边道德风险契约研究

针对管理活动的动态性与多任务性的特点,将解聘补偿与解聘倾向引入动态多任务契约设计中,构建了基于解聘补偿的动态多任务双边道德风险契约。通过数理推导分析的方法给出了最优契约设计,声誉效应和棘轮效应的度量,探讨了解聘倾向对于契约设计的影响。结果表明解聘倾向的引入对于委托人的道德风险约束是有效的,但是对于代理人的道德风险约束则取决于声誉效应与棘轮效应的大小。在第2期契约中,解聘倾向对固定支付的影响取决于代理人保留收入与解聘补偿的差额。而第1期的契约设计要受到解聘补偿,声誉效应与棘轮效应三者的综合影响。任务关联性对契约设计影响以及相应的实证分析是未来的研究方向。     

Soliton-Stripe Patterns in Charged Langmuir Monolayers

We consider a charged Langmuir monolayer problem where electrostatic interaction forces undulations in the molecular concentration of the monolayer. Using the -convergence theory in singular perturbative variational calculus, we prove the existence of soliton-stripe lamellar patterns as one-dimensional local minimizers of the free energy, which are characterized by sharp domain walls delineating fully segregated dense liquid and dilute gas regions of the monolayer.     

Low-Temperature Localisation for Continuous Spin-Systems

We study the free energy of continuous spin-systems on Z ^d , in the framework of Laplace integrals and transfer operators. Under a weak coupling condition, we show that the free energy in the low-temperature limit is determined, up to an exponentially small error, by the restriction to a neighbourhood of global minima of the energy. We have results for some single- and double-well problems.     

Energy estimates for reaction-diffusion processes of charged species

We consider electro-reaction-diffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate. The same properties are obtained for a fully implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electro-reaction-diffusion system which is dissipative (the free energy decays monotonously). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The mean-field approximation in quantum electrodynamics: The no-photon case

We study the mean-field approximation of quantum electrodynamics (QED) by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal ordering or choice of bare electron/positron subspaces. Neglecting photons, we properly define this Hamiltonian in a finite box [−L/2; L/2)³, with periodic boundary conditions and an ultraviolet cutoff λ. We then study the limit of the ground state (i.e., the vacuum) energy and of the minimizers as L goes to infinity, in the Hartree-Fock approximation. In the case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as L goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy. © 2006 Wiley Periodicals, Inc.     

Quenched point-to-point free energy for random walks in random potentials

We consider a random walk in a random potential on a square lattice of arbitrary dimension. The potential is a function of an ergodic environment and steps of the walk. The potential is subject to a moment assumption whose strictness is tied to the mixing of the environment, the best case being the i.i.d. environment. We prove that the infinite volume quenched point-to-point free energy exists and has a variational formula in terms of entropy. We establish regularity properties of the point-to-point free energy, and link it to the infinite volume point-to-line free energy and quenched large deviations of the walk. One corollary is a quenched large deviation principle for random walk in an ergodic random environment, with a continuous rate function.     

Lambert function and a new non-extensive form of entropy

We propose a new way of defining entropy of a system, whichgives a general form that is non-extensive like Tsallis entropy,but is linearly dependent on component entropies, like Renyientropy, which is extensive. This entropy has a conceptuallynovel but simple origin and is mathematically easy to defineby a very simple expression involving a derivative. It leadsto a probability distribution function involving the Lambertfunction resulting from optimizing the entropy, which has hithertonever appeared in this context, and is somewhat more complexthan the Shannon or Boltzmann form, but is nevertheless mathematicallyquite tractable. We have compared it numerically with the Tsallisand Shannon entropies. We have also considered constraints onthe energy spectra imposed by the properties of the Lambertfunction, which are absent in the Shannon form. It may turnout to be a more appropriate candidate in a physical situationwhere the probability distribution does not suit any of thepreviously defined forms, especially when the probability densityfunction sought is expected to be stiffer than that resultingfrom maximizing the other entropies. We consider the problemof defining free energy and other thermodynamic functions whenthe entropy is given as a general function of the probabilitydistribution, including that for non-extensive forms. We thenfind that the free energy, which is central to the determinationof all other quantities of interest in a thermodynamic context,can be obtained uniquely, at least numerically, even when itis the root of a transcendental equation. In particular, weexamine the cases of the Tsallis form and the new form proposedby us. We compare the free energy, the internal energy and thespecific heat of a simple system of two energy states for eachof these forms and find significant departures for some quantities,while some others are less sensitive to the parametrization.