Entropy: (1) The former trouble with particle-tracking simulation, and (2) A measure of computational information penalty

Abstract

Traditional random-walk particle-tracking (PT) models of advection and dispersion do not track entropy, because particle masses remain constant. However, newer mass-transfer particle tracking (MTPT) models have the ability to do so because masses of all compounds may change along trajectories. Additionally, the probability mass functions (PMF) of these MTPT models may be compared to continuous solutions with probability density functions, when a consistent definition of entropy (or similarly, the dilution index) is constructed. This definition reveals that every discretized numerical model incurs a computational entropy. Similar to Akaike’s (1974, 1992) entropic penalty for larger numbers of adjustable parameters, the computational complexity of a model (e.g., number of nodes or particles) adds to the entropy and, as such, must be penalized. Application of a new computational information criterion reveals that increased accuracy is not always justified relative to increased computational complexity. The MTPT method can use a particle-collision based kernel or an adaptive kernel derived from smoothed-particle hydrodynamics (SPH). The latter is more representative of a locally well-mixed system (i.e., one in which the dispersion tensor equally represents mixing and solute spreading), while the former better represents the separate processes of mixing versus spreading. We use computational means to demonstrate the fitness of each of these methods for simulating 1-D advective-dispersive transport with uniform coefficients.