Coarse-graining

Coarse-graining replaces fine-scale degrees of freedom with averaged or effective variables, producing a description of a system at larger scales.

Renormalization group: A framework for studying how a system’s description changes as its scale of observation changes. It formalizes repeated coarse-graining and tracks which features persist across scales.
Degrees of freedom: Independent variables needed to specify the state of a physical or mathematical system. These are the fine-scale quantities that coarse-graining combines, averages, or eliminates.
Molecular dynamics: A computational method that simulates the motion of atoms or molecules over time. Coarse-grained molecular models reduce atomic detail to simulate larger systems or longer times.
Microscopic model: A model that represents a system through its individual particles or other fine-scale constituents. It retains detail that a coarse-grained model deliberately replaces or omits.
Block-spin transformation: A lattice procedure that combines spins in spatial blocks into effective variables. It gives a concrete example of replacing many microscopic variables with one block variable.
Statistical mechanics: The framework that connects macroscopic properties of matter to the statistical behavior of microscopic states. Coarse-graining often bridges microscopic states and macroscopic thermodynamic descriptions.
Lattice Boltzmann method: A computational fluid-dynamics method that evolves discrete particle-distribution functions on a lattice. Its distribution functions provide a mesoscopic description between molecular motion and fluid equations.
Agent-based model: A simulation in which individual agents follow rules and generate system-level behavior through their interactions. It preserves distinct entities where coarse-grained descriptions may use population averages.
Marginalization: The operation of summing or integrating over variables that are not retained in a probability distribution. Discarded degrees of freedom are often removed mathematically by marginalizing over them.
Ensemble average: An average over the possible states of a system, weighted by their probabilities. It is one common way to turn microscopic variation into a representative quantity.

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