Particlevaries

Particlevaries is a theoretical construct used in models of particle systems to quantify local variation among particles within a region. In this framework, particlevaries denotes a spatially dependent field that records how much a chosen particle property deviates from its local mean.

Mathematically, let p_i(t) be a property of particle i at time t, and let R(x) be a neighborhood around position x containing N_R(x) particles. The local particlevaries is defined as Pvar(x,t) = (1/N_R) sum_{i in R} [p_i(t) - p̄_R(t)]^2, where p̄_R(t) = (1/N_R) sum_{i in R} p_i(t) is the local mean. Equivalently, Pvar can be described as a field by evaluating this statistic across space using a sliding window, kernel, or Voronoi cell partitioning.

Interpretation: high Pvar indicates strong heterogeneity in the chosen property within the neighborhood, while low Pvar implies uniformity. The concept can be applied to any particle property, such as velocity, kinetic energy, charge, or mass distribution.

Computation and estimation: In simulations, Pvar is computed by binning space into cells and calculating the variance of p within each cell. In experimental data, local estimates rely on finite sampling or smoothing techniques to infer the dispersion of p in space.

Applications and context: Particlevaries is used to study granular materials, colloids, plasmas, and multiphase or heterogeneous flows where local heterogeneity affects transport, reaction rates, or mechanical response. It complements global measures by highlighting spatial dispersion rather than just averages.

History and terminology: The term particlevaries is not a universally standardized term in mainstream physics; in this article it is presented as a conceptual construct for describing local dispersion in particle systems. See also variance, heterogeneity, and local order parameters.