@Article{CSIAM-AM-1-155, author = {Qiang and Du and qd2125@columbia.edu and 6585 and Department of Applied Physics and Applied Mathematics, Columbia University, New York 10027, USA. and Qiang Du and Xiantao and Li and xli@math.psu.edu and 13141 and Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA. and Xiantao Li and Liming and Yuan and lzy104@psu.edu and 7424 and Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania, 16802, USA. and Liming Yuan}, title = {Analysis of Coarse-Grained Lattice Models and Connections to Nonlocal Interactions}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {1}, pages = {155--185}, abstract = {

We study coarse-grained models of some linear static lattice models with interactions up to second nearest neighbors. It will be demonstrated how nonlocal interactions, as described by a nonlocal kernel function, arise from a coarse-graining procedure. Some important properties of the nonlocal kernels will be established such as its decay rate and positivity. We also study the scaling behavior of the kernel functions as the level of coarse-graining changes. In addition, we suggest closure approximations of the nonlocal interactions that can be expressed in local PDE forms by introducing auxiliary variables.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0009}, url = {http://global-sci.org/intro/article_detail/csiam-am/16798.html} }