Math 2.0

Fluid mechanics Definition

Direct simulation Monte Carlo

Builds on Monte Carlo methodBoltzmann equationKnudsen number

Definition

Consider a dilute gas in a region Ω⊆ℝ3\Omega \subseteq \R^3, divided into cells. Binary collisions are modelled by a cross section σ(∣g∣,⋅)\sigma(\abs{g}, \cdot) depending on the relative speed ∣g∣\abs{g}. Each of NN simulated particles has a position in Ω\Omega and a velocity in ℝ3\R^3, and represents WW real molecules. Direct simulation Monte Carlo advances the particles by a time step Δt\Delta t by repeating the following steps.

  1. Move: each particle moves freely, x↦x+vΔtx \mapsto x + v \Delta t, with the boundary conditions applied at the walls.
  2. Collide: in each cell, random pairs of particles in the cell are chosen. Each pair collides with probability proportional to ∣g∣\abs{g} times its total cross section. The constant is fixed by WW, Δt\Delta t and the cell volume.
  3. Update: a colliding pair gets new velocities sampled from the collision model, conserving momentum and kinetic energy.
  4. Sample: density, velocity and temperature in each cell are estimated by averages over its particles.

Remarks

The splitting into free motion and local collisions mirrors the two terms of the Boltzmann equation. Wagner proved in 1992 that Bird's scheme converges to a solution of the Boltzmann equation. The limit is taken as the number of particles tends to infinity and the cells and the time step shrink. The cells should be smaller than the mean free path, and Δt\Delta t smaller than the mean collision time, so the method is costly at small Knudsen numbers.

Example: for hard spheres of diameter dd, the total cross section is πd2\pi d^2. The post-collision relative velocity has length ∣g∣\abs{g} and a direction uniform on the sphere, and the centre of mass velocity is unchanged.

Non-example: molecular dynamics integrates Newton's equations, with deterministic collisions between particles at contact. It is not DSMC, because its collisions do not come from random pair selection within cells.

Reading path

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  1. Boltzmann equation concept
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  4. Direct simulation Monte Carlo definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo
About this page
TagECVNFXD1A2
URIm2://ECVNFXD1A2
Kinddefinition
WritingW0 stub
MSC76P05, 65C05, 82C40
DefinesDirect simulation Monte Carlo
UsesMonte Carlo method, Boltzmann equation, Knudsen number
Changed3 November 2026

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