Calculations of He+H2+ (v=l,j=l) reaction system are carded out on a new potential energy surface (PES) at different collision energies with the quasi-classical trajectory (QCT) method. The results of the reaction pro...Calculations of He+H2+ (v=l,j=l) reaction system are carded out on a new potential energy surface (PES) at different collision energies with the quasi-classical trajectory (QCT) method. The results of the reaction probability and the cross section curves show an obvious oscillatory structure attributed to the resonances caused by the potential well of the Hell2+ complex. The three angular distributions P(θr), P(φr) and P(θr,(φr) as well as four polarization dependent differential cross sections (PDDCSs) are calculated, respectively. Results indicate that the collision energy has great influence on both the vector correlation of k-k', k-j', k-k'-j' and the PDDCSs of the title reaction. The rotational polarization of product Hell+ presents diverse characteristics at dif- ferent collision energies. The rotational angular momentum vectorsj' of product are both aligned and oriented. Furthermore, the product Hell+ tends to scatter forward correspondingly as the collision energy increases.展开更多
This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stoc...This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.展开更多
A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original s...A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It6 stochastic differential equations. The It6 formula is then carried out to obtain the It6 stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged It6 stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.展开更多
Numerical prediction of turbulent mixing can be divided into two subproblems: to predict the geometrical extent of a mixing region and to predict the mixing properties on an atomic or molecular scale, within the mixin...Numerical prediction of turbulent mixing can be divided into two subproblems: to predict the geometrical extent of a mixing region and to predict the mixing properties on an atomic or molecular scale, within the mixing region. The former goal suffices for some purposes, while important problems of chemical reactions(e.g. flames) and nuclear reactions depend critically on the second goal in addition to the first one. Here we review recent progress in establishing a conceptual reformulation of convergence, and we illustrate these concepts with a review of recent numerical studies addressing turbulence and mixing in the high Reynolds number limit. We review significant progress on the first goal, regarding the mixing region, and initial progress on the second goal, regarding atomic level mixing properties. New results concerning non-uniqueness of the infinite Reynolds number solutions and other consequences of a renormalization group point of view, to be published in detail elsewhere, are summarized here.The notion of stochastic convergence(of probability measures and probability distribution functions) replaces traditional pointwise convergence. The primary benefit of this idea is its increased stability relative to the statistical "noise" which characterizes turbulent flow. Our results also show that this modification of convergence, with sufficient mesh refinement, may not be needed. However, in practice, mesh refinement is seldom sufficient and the stochastic convergence concepts have a role.Related to this circle of ideas is the observation that turbulent mixing, in the limit of high Reynolds number, appears to be non-unique. Not only have multiple solutions been observed(and published) for identical problems, but simple physics based arguments and more refined arguments based on the renormalization group come to the same conclusion.Because of the non-uniqueness inherent in numerical models of high Reynolds number turbulence and mixing, we also include here numerical examples of validation. The algorithm we use here has two essential components. We depend on Front Tracking to allow accurate resolution of flows with sharp interfaces or steep gradients(concentration or thermal), as are common in turbulent mixing problems. The higher order and enhanced algorithms for interface tracking, both those already developed, and those proposed here, allow a high resolution and uniquely accurate description of sample mixing problems. Additionally, we depend on the use of dynamic subgrid scale models to set otherwise missing values for turbulent transport coefficients, a step that breaks the non-uniqueness.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos. 10504017 and 10874104)the Key Project of Chinese Ministry of Education (Grant No. 206093)
文摘Calculations of He+H2+ (v=l,j=l) reaction system are carded out on a new potential energy surface (PES) at different collision energies with the quasi-classical trajectory (QCT) method. The results of the reaction probability and the cross section curves show an obvious oscillatory structure attributed to the resonances caused by the potential well of the Hell2+ complex. The three angular distributions P(θr), P(φr) and P(θr,(φr) as well as four polarization dependent differential cross sections (PDDCSs) are calculated, respectively. Results indicate that the collision energy has great influence on both the vector correlation of k-k', k-j', k-k'-j' and the PDDCSs of the title reaction. The rotational polarization of product Hell+ presents diverse characteristics at dif- ferent collision energies. The rotational angular momentum vectorsj' of product are both aligned and oriented. Furthermore, the product Hell+ tends to scatter forward correspondingly as the collision energy increases.
基金Project supported by the National Natural Science Foundation of China (No.10325101, No.101310310)the Science Foundation of the Ministry of Education of China (No. 20030246004).
文摘This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11172233,10932009,61171155Natural Science Foundation of Shannxi Province under Grant No.2012JM8010the Doctorate Foundation of Northwestern Polytechnical University under Grant No.CX201215
文摘A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It6 stochastic differential equations. The It6 formula is then carried out to obtain the It6 stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged It6 stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.
基金supported in part by the Nuclear Energy University Program of the Department of Energy,project NEUP-09-349,Battelle Energy Alliance LLC 00088495(subaward with DOE as prime sponsor),Leland Stanford Junior University 2175022040367A(subaward with DOE asprime sponsor),Army Research Office W911NF0910306This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory,which is supported by the Office of Science of the U.S.Department of Energy under contract DE-AC02-06CH11357.Stony Brook University Preprint number SUNYSB-AMS-12-04
文摘Numerical prediction of turbulent mixing can be divided into two subproblems: to predict the geometrical extent of a mixing region and to predict the mixing properties on an atomic or molecular scale, within the mixing region. The former goal suffices for some purposes, while important problems of chemical reactions(e.g. flames) and nuclear reactions depend critically on the second goal in addition to the first one. Here we review recent progress in establishing a conceptual reformulation of convergence, and we illustrate these concepts with a review of recent numerical studies addressing turbulence and mixing in the high Reynolds number limit. We review significant progress on the first goal, regarding the mixing region, and initial progress on the second goal, regarding atomic level mixing properties. New results concerning non-uniqueness of the infinite Reynolds number solutions and other consequences of a renormalization group point of view, to be published in detail elsewhere, are summarized here.The notion of stochastic convergence(of probability measures and probability distribution functions) replaces traditional pointwise convergence. The primary benefit of this idea is its increased stability relative to the statistical "noise" which characterizes turbulent flow. Our results also show that this modification of convergence, with sufficient mesh refinement, may not be needed. However, in practice, mesh refinement is seldom sufficient and the stochastic convergence concepts have a role.Related to this circle of ideas is the observation that turbulent mixing, in the limit of high Reynolds number, appears to be non-unique. Not only have multiple solutions been observed(and published) for identical problems, but simple physics based arguments and more refined arguments based on the renormalization group come to the same conclusion.Because of the non-uniqueness inherent in numerical models of high Reynolds number turbulence and mixing, we also include here numerical examples of validation. The algorithm we use here has two essential components. We depend on Front Tracking to allow accurate resolution of flows with sharp interfaces or steep gradients(concentration or thermal), as are common in turbulent mixing problems. The higher order and enhanced algorithms for interface tracking, both those already developed, and those proposed here, allow a high resolution and uniquely accurate description of sample mixing problems. Additionally, we depend on the use of dynamic subgrid scale models to set otherwise missing values for turbulent transport coefficients, a step that breaks the non-uniqueness.
