Particle transport B: Monte Carlo methods versus moment equations

This is the second of three articles introducing the basic concepts of the Monte Carlo method for radiation transport calculations. In this article, we’ll consider an alternative to the Monte Carlo treatment of the two-dimensional random walk: the derivation and solution of a transport equation. Here, we define an appropriate quantity averaged over a random distribution of particles and seek a differential equation that describes how the quantity varies. For this calculation, the quantity is the average density of particles n(x,y,t) in the plane with units of number/m^2. The quantity is plotted in Fig. 1 (this figure also appeared in the preceding article). To make a direct comparison with the Monte Carlo results, we must carefully set model constraints:

  • Although the density may vary in space, the distribution of particle velocities is the same at all points. Particles all have constant speed v0 and there is an isotropic distribution of direction vectors.
  • There is a uniform-random background density of scattering objects.
  • Equation 8 of the previous article gives the probability distribution of a (the distance particles travel between collisions) in terms of the mean-free-path ?.

Figure 1. Particle density as a function of radius (distance from the source), with mean-free-path equal to 1.0 and 100 collisions. The solid line is the solution of the two-dimensional diffusion equation and the points are the results of the Monte Carlo solution.


We want to find how the density changes as particles perform their random walk. Changes occur if, on the average, there is a flow of particles (a flux) from one region of space to another. If the density n is uniform, the same number of particles flow in one direction as the other, so the average flux is zero. Therefore, we expect that fluxes depend on gradients of the particle density. We can find the dependence using the construction of Fig. 2. Assume that the particle density varies in x near a point x0. Using a coordinate system with origin at x0, the first order density variation is given by Eq. 9. The goal is to find an expression for the number of particles per second passing through the line element ?y. To carry out derivation, we assume the following two conditions:


  • The material is homogeneous. Equivalently, ? has the same value everywhere.
  • Over scale length ?, relative changes in n are small.




Construction to relate flux to density gradient

Figure 2. Construction to relate flux to density gradient.

Using polar coordinates shown centered on the line element, consider an element in the plane of area (r ??)(?r)$. We want to find how many particles per second originating from this region pass through ?y. We can write the quantity as the product Jx ?y, where Jx is the linear flow density in units of particles/m-s. On the average, every particle in the calculation volume has the same average number of collisions per second, given by Eq. 10. The rate of scattering events in the area element equals ? times the number of particles in the area (Eq. 11). The fraction of scattered particles aimed at the segment is given by Eq. 12.

Finally, the probability that a particle scattered out of the area element reaches the line element was given in the previous article as exp(-r/?). Combining this expression with Eqs. 10 and 11, we can determine the current density from all elements surrounding the line segment. Taking the density variation in the form of Eq. 13 leads to the expression of Eq. 14. The integral of the first term in brackets equals zero, so that only the term proportional to the density gradient contributes. Carrying out the integrals, the linear current density is given by Eq. 15. The planar diffusion coefficient (with units m^2/s) is given by Eq. 16. Generalizing to possible variations in both x and y, can write Eq. 15 in the form of Eq. 17.  This relationship between the vector current density and the gradient of density is called Fick’s first law. Equation 18 lists Fick’s second law, a statement of conservation of particles. In the equation, the quantity ?•J is the divergence of flux from a point and S is the source of particles at that point (particles/m^2-s). Equation 18 is the diffusion equation for particles in a plane. It states that the density at a point changes in time if there is a divergence of flux or a source or sink.

We are now ready to compare the predictions of the model with the Monte Carlo results of the previous section. Equation 19 gives is solution to the diffusion equation for particles emission from the origin of the plane. The quantity r equals ?(x^2 + y^2). We can verify Eq. 19 by direct substitution by using the cylindrical form of the divergence and gradient operators and taking D as uniform in space. In order to make a comparison with the Monte Carlo calculation, we pick a time value t0 = Nc ?/v0 and evaluate A based on the condition of Eq. 20. The resulting expression for the density at time t0 is given by Eq. 21. The prediction of Eq. 21 s plotted as the solid line in Fig. 1. The results from the two methods show close absolute agreement.

Finally, we can determine the theoretical 1/e radius of the particle cloud from Eq. 21 to yield Eq. 22. In a random walk, the particle spread increases as the square root of the number of transits between collisions. For Nc = 100, the value is re/? ? 14.1.


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