Local order metrics for many-particle systems across length scales

Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math>-dimensional Euclidean space <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msup><b:mi mathvariant="double-struck">R</b:mi><b:mi>d</b:mi></b:msup></b:math> across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>n</d:mi></d:math>-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the local number variance <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msubsup><e:mi>σ</e:mi><e:mi>N</e:mi><e:mn>2</e:mn></e:msubsup><e:mrow><e:mo>(</e:mo><e:mi>R</e:mi><e:mo>)</e:mo></e:mrow></e:mrow></e:math> associated with a spherical sampling window of radius <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>R</f:mi></f:math> (which encodes pair correlations) and an integral measure derived from it <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:msub><g:mi mathvariant="normal">Σ</g:mi><g:mi>N</g:mi></g:msub><g:mrow><g:mo>(</g:mo><g:msub><g:mi>R</g:mi><g:mi>i</g:mi></g:msub><g:mo>,</g:mo><g:msub><g:mi>R</g:mi><g:mi>j</g:mi></g:msub><g:mo>)</g:mo></g:mrow></g:mrow></g:math> that depends on two specified radial distances <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi>R</i:mi><i:mi>i</i:mi></i:msub></i:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msub><j:mi>R</j:mi><j:mi>j</j:mi></j:msub></j:math>. Across the first three space dimensions (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>d</k:mi><k:mo>=</k:mo><k:mn>1</k:mn><k:mo>,</k:mo><k:mn>2</k:mn><k:mo>,</k:mo><k:mn>3</k:mn></k:mrow></k:math>), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mi>R</l:mi></l:math>. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>R</m:mi></m:math>. These local order metrics could also aid in the inverse design of structures with prescribed length-scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments of the number of points within a spherical window of radius <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mrow><n:mi>R</n:mi></n:mrow></n:math> [S. Torquato , ] to devise even more sensitive order metrics.