The linear Boltzmann equation describes the macroscopic transport of a gas of non-interacting point particles in low-density matter. It has wide-ranging applications, including neutron transport, radiative transfer, semiconductors, and ocean wave scattering. Recent research shows that the equation fails in highly correlated media, where the distribution of free path lengths is non-exponential. We investigate this phenomenon in the case of polycrystals whose typical grain size is comparable with the mean free path length. Our principal result is a new generalized linear Boltzmann equation that captures the long-range memory effects in this setting. A key feature is that the distribution of free path lengths has an exponential decay rate, as opposed to a power-law distribution observed in a single crystal.
{"title":"Generalized Linear Boltzmann Equations for Particle Transport in Polycrystals","authors":"J. Marklof, Andreas Strombergsson","doi":"10.1093/AMRX/ABV004","DOIUrl":"https://doi.org/10.1093/AMRX/ABV004","url":null,"abstract":"The linear Boltzmann equation describes the macroscopic transport of a gas of non-interacting point particles in low-density matter. It has wide-ranging applications, including neutron transport, radiative transfer, semiconductors, and ocean wave scattering. Recent research shows that the equation fails in highly correlated media, where the distribution of free path lengths is non-exponential. We investigate this phenomenon in the case of polycrystals whose typical grain size is comparable with the mean free path length. Our principal result is a new generalized linear Boltzmann equation that captures the long-range memory effects in this setting. A key feature is that the distribution of free path lengths has an exponential decay rate, as opposed to a power-law distribution observed in a single crystal.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"46 1","pages":"274-295"},"PeriodicalIF":0.0,"publicationDate":"2015-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79744587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A numerical integrator is proposed for solving the multiconfiguration time-dependent Hartree (MCTDH) equations of motion, which are widely used in computations of molecular quantum dynamics. In contrast to existing integrators, the proposed algorithm does not require inverses of illconditioned density matrices and obviates the need for their regularization, allowing for large stepsizes also in the case of near-singular density matrices. The nonlinear MCTDH equations are split into a chain of linear differential equations that can all be efficiently solved by Lanczos approximations to the action of Hermitian-matrix exponentials, alternating with orthogonal matrix decompositions. The integrator is an extension to the Tucker tensor format of recently proposed projector-splitting integrators for the dynamical low-rank approximation by matrices and tensor trains (or matrix product states). The integrator is time-reversible and preserves both the norm and the total energy.
{"title":"Time Integration in the Multiconfiguration Time-Dependent Hartree Method of Molecular Quantum Dynamics","authors":"C. Lubich","doi":"10.1093/AMRX/ABV006","DOIUrl":"https://doi.org/10.1093/AMRX/ABV006","url":null,"abstract":"A numerical integrator is proposed for solving the multiconfiguration time-dependent Hartree (MCTDH) equations of motion, which are widely used in computations of molecular quantum dynamics. In contrast to existing integrators, the proposed algorithm does not require inverses of illconditioned density matrices and obviates the need for their regularization, allowing for large stepsizes also in the case of near-singular density matrices. The nonlinear MCTDH equations are split into a chain of linear differential equations that can all be efficiently solved by Lanczos approximations to the action of Hermitian-matrix exponentials, alternating with orthogonal matrix decompositions. The integrator is an extension to the Tucker tensor format of recently proposed projector-splitting integrators for the dynamical low-rank approximation by matrices and tensor trains (or matrix product states). The integrator is time-reversible and preserves both the norm and the total energy.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"1 1","pages":"311-328"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83850839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work existence results on nonlinear first order as well as doubly nonlinear second order evolution equations involving the fractional p-Laplacian are presented. The proofs do not exploit any monotonicity assumption but rely on a compactness argument in combination with regularity of the Galerkin scheme and the nonlocal character of the operator.
{"title":"On the Evolutionary Fractional p-Laplacian","authors":"Dimitri Puhst","doi":"10.1093/AMRX/ABV003","DOIUrl":"https://doi.org/10.1093/AMRX/ABV003","url":null,"abstract":"In this work existence results on nonlinear first order as well as doubly nonlinear second order evolution equations involving the fractional p-Laplacian are presented. The proofs do not exploit any monotonicity assumption but rely on a compactness argument in combination with regularity of the Galerkin scheme and the nonlocal character of the operator.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"72 1","pages":"253-273"},"PeriodicalIF":0.0,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86256817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Analytical Approach for the Growth Rate of the Variance of the Deformation Related to an Elasto-Plastic Oscillator Excited by a White Noise","authors":"A. Bensoussan, C. Feau, L. Mertz, S. Yam","doi":"10.1093/AMRX/ABU008","DOIUrl":"https://doi.org/10.1093/AMRX/ABU008","url":null,"abstract":"","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"19 1","pages":"99-128"},"PeriodicalIF":0.0,"publicationDate":"2014-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90157847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a sequence of continuations which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.
{"title":"Existence of Travelling Waves for a Reaction–Diffusion System with a Line of Fast Diffusion","authors":"Laurent Dietrich","doi":"10.1093/AMRX/ABV002","DOIUrl":"https://doi.org/10.1093/AMRX/ABV002","url":null,"abstract":"We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a sequence of continuations which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"1 1","pages":"204-252"},"PeriodicalIF":0.0,"publicationDate":"2014-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88826272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
queue, with 0 < ν < 1. We let (N1, N2) be the numbers of customers in the two parallel queues, and let p(m, n) = Prob[N1 = m, N2 = n] be the joint queue length distribution in the steady state. The two arrival rates are λ1 and λ2, the exponential server works at rate μ, and ρ1 = λ1/μ, ρ2 = λ2/μ. If N1 > N2 (N1 < N2), the server works on the first (second) queue, but if N1 = N2, the server works at rate μν on the first queue and rate μ(1 − ν)
{"title":"On the Nonsymmetric Longer Queue Model II: Marginal and Conditional Probabilities","authors":"C. Knessl, Haishen Yao","doi":"10.1093/AMRX/ABU002","DOIUrl":"https://doi.org/10.1093/AMRX/ABU002","url":null,"abstract":"queue, with 0 < ν < 1. We let (N1, N2) be the numbers of customers in the two parallel queues, and let p(m, n) = Prob[N1 = m, N2 = n] be the joint queue length distribution in the steady state. The two arrival rates are λ1 and λ2, the exponential server works at rate μ, and ρ1 = λ1/μ, ρ2 = λ2/μ. If N1 > N2 (N1 < N2), the server works on the first (second) queue, but if N1 = N2, the server works at rate μν on the first queue and rate μ(1 − ν)","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"22 1","pages":"1-47"},"PeriodicalIF":0.0,"publicationDate":"2014-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74560911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The parallel replica dynamics, originally developed by A.F. Voter, eciently simulates very long trajectories of metastable Langevin dynamics. We present an analogous algorithm for discrete time Markov processes. Such Markov processes naturally arise, for example, from the time discretization of a continuous time stochastic dynamics. Appealing to properties of quasistationary distributions, we show that our algorithm reproduces exactly (in some limiting regime) the law of the original trajectory, coarsened over the metastable states.
{"title":"The Parallel Replica Method for Simulating Long Trajectories of Markov Chains","authors":"D. Aristoff, T. Lelièvre, G. Simpson","doi":"10.1093/amrx/abu005","DOIUrl":"https://doi.org/10.1093/amrx/abu005","url":null,"abstract":"The parallel replica dynamics, originally developed by A.F. Voter, eciently simulates very long trajectories of metastable Langevin dynamics. We present an analogous algorithm for discrete time Markov processes. Such Markov processes naturally arise, for example, from the time discretization of a continuous time stochastic dynamics. Appealing to properties of quasistationary distributions, we show that our algorithm reproduces exactly (in some limiting regime) the law of the original trajectory, coarsened over the metastable states.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"44 1","pages":"332-352"},"PeriodicalIF":0.0,"publicationDate":"2014-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83721132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the global well-posedness and we study the linear response for a system of two coupled equations composed of a Dirac equation for an infinite rank operator and a nonlinear wave or Klein-Gordon equation.
{"title":"Global Well-Posedness for a Nonlinear Wave Equation Coupled to the Dirac Sea","authors":"Julien Sabin","doi":"10.1093/AMRX/ABU004","DOIUrl":"https://doi.org/10.1093/AMRX/ABU004","url":null,"abstract":"We prove the global well-posedness and we study the linear response for a system of two coupled equations composed of a Dirac equation for an infinite rank operator and a nonlinear wave or Klein-Gordon equation.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"24 19 1","pages":"312-331"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88694966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the stochastic optimization of the temporal shape of a laser pulse for material ablation. The temperature of the material subject to the laser pulse is modeled by a Stefan-like heat transfer equation. From the surface temperature is then determined the ablation depth of the studied material, which is the quantity to optimize for a given laser pulse fluence. Several numerical tests are presented for laser ablation of silicon, which is the material of interest here.
{"title":"Optimization of the Temporal Shape of Laser Pulses for Ablation","authors":"Pascal Turbis, E. Lorin, A. Cournoyer","doi":"10.1093/AMRX/ABU001","DOIUrl":"https://doi.org/10.1093/AMRX/ABU001","url":null,"abstract":"This paper is devoted to the stochastic optimization of the temporal shape of a laser pulse for material ablation. The temperature of the material subject to the laser pulse is modeled by a Stefan-like heat transfer equation. From the surface temperature is then determined the ablation depth of the studied material, which is the quantity to optimize for a given laser pulse fluence. Several numerical tests are presented for laser ablation of silicon, which is the material of interest here.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"76 1","pages":"244-274"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86641430","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By using ergodic theory of subadditive processes and variational convergence, we study the macroscopic behavior of a thin 3D composite made up of high-conductivity fibers that are randomly distributed according to a stochastic point process in a bounded open set of ℝ3. The thickness of the body, the conductivity and the size of the cross sections of the fibers depend on a small parameter e. The variational limit functional energy obtained when e tends to 0 is deterministic and depends on two variables: one is the solution of a variational problem posed in a 2D bounded open set and describes the behavior of the medium; the other captures the limit behavior of suitably rescaled solutions in the fibers when the thickness and the size section become increasingly thin and the conductivity of the fibers becomes increasingly large.
{"title":"Two-Dimensional Deterministic Model of a Thin Body with Randomly Distributed High-Conductivity Fibers","authors":"G. Michaille, A. Nait-Ali, S. Pagano","doi":"10.1093/AMRX/ABT007","DOIUrl":"https://doi.org/10.1093/AMRX/ABT007","url":null,"abstract":"By using ergodic theory of subadditive processes and variational convergence, we study the macroscopic behavior of a thin 3D composite made up of high-conductivity fibers that are randomly distributed according to a stochastic point process in a bounded open set of ℝ3. The thickness of the body, the conductivity and the size of the cross sections of the fibers depend on a small parameter e. The variational limit functional energy obtained when e tends to 0 is deterministic and depends on two variables: one is the solution of a variational problem posed in a 2D bounded open set and describes the behavior of the medium; the other captures the limit behavior of suitably rescaled solutions in the fibers when the thickness and the size section become increasingly thin and the conductivity of the fibers becomes increasingly large.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"7 1","pages":"122-156"},"PeriodicalIF":0.0,"publicationDate":"2013-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78168855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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