Statistical error in particle simulations of low Mach number flows

Abstract

Abstract-We present predictions for the statistical error due to finite sampling in the presence of thermal fluctuations in molecular simulation algorithms. Expressions for the fluid velocity, density and temperature are derived using equilibrium statistical mechanics. The results show that the number of samples needed to adequately resolve the flowfield scales as the inverse square of the Mach number. The theoretical results are verified for a dilute gas using direct Monte Carlo simulations. The agreement between theory and simulation verifies that the use of equilibrium theory is justified.

Heat transfer in micro devices packaged in partial vacuum

“…In particular, the NSF equations do not take into account the thermal stress, i.e., σ T ij ∝ − µ p ∂q i ∂x j , which appears in the stress balance equation (4). For the lower portion of the vertical surface (0 ≤ y 0.6), q y is greater than σ xy , which, according to (11), induces a downward motion in the fluid and thus the primary eddies. However, close to the top plate shear stress dominates over heat flux which produces an upward motion in gas, hence the formation of the secondary vortices.…”

“…Direct numerical solution of the Boltzmann equation gives an accurate microscopic description of gas flows at all Knudsen numbers, but requires detailed information on microscopic phase space, and thus typically huge computational times [11]. The direct simulation Monte Carlo (DSMC) method, proposed by Bird [12], is another commonly used numerical method for simulation of high Knudsen number gas flows.…”

Comparison of Kinetic Theory and Hydrodynamics for Poiseuille Flow

“…Being a stochastic algorithm, DSMC measurements have statistical variation but this variation is that of spontaneous fluctuations and so confidence intervals (error bars) may be evaluated using statistical mechanics. (17) Although the geometry of the problem is simple, some care must be taken to formulate and implement boundary conditions that are equivalent in the NS and DSMC computations. The boundary conditions for the Navier-Stokes equations are: at the entrance (x=−L x /2),…”

The impact of subcontinuum gas conduction on topography measurement sensitivity using heated atomic force microscope cantilevers

“…This can be further reduced by increasing the number of particles per cell and/or the number of samples as the statistical error is inversely proportional to the square root of the number of particles per cell and the number of samples. 24 …”