Hermite filtering and form factors
Introduction
In several domains of physical sciences Monte Carlo simulations turn out to be very often the last or the sole resort whenever the behaviour of a system made by a large number of objects must be studied. In some cases––accelerator/plasma physics and molecular dynamics are just a few relevant examples––one needs to model the dynamics of a huge number (≈1010–1011) of mutually interacting particles under the action of external fields [2]. In such contexts, direct integration of trajectories/evaluation of fields is both uninteresting––since only statistical properties need to be accounted for––and infeasible, even for diluted systems where self-fields effects (mutual interactions among the particles) can be neglected. On the other hand, because of practical computer limitations, simulations are usually bounded to a few thousand or a few millions “macroparticles”, each mimicking a fairly large number (≈104–106) of real objects. As a consequence, non-physically large fields may develop when macroparticles pass close to each other, either posing stability concerns and introducing unphysical collisional contributions into the model.
Unfortunately, numerical artifacts exhibit a logarithmic scaling with the number of macroparticles and can probably be suppressed only by resorting to “real” simulations. Since such an approach is merely not viable, one must devise suitable techniques to suppress the numerical noise and achieve fields regularization. In some cases the numerical method itself reduces the noise. For example, whenever the fields are to be computed on the vertices of a regular mesh, the size of the cell defines a cutoff in the frequency domain, even though the eventuality of a macroparticle passing very close to a mesh point must be considered and accounted for (usually by weighted summation of contributions to the fields from the neighbour vertices or through other sharing algorithms).
In this paper, the classical approach of giving to the elementary object both a size and a shape (in either coordinate and/or momentum space) is deepened and thoroughly described. In fact the features of the function used to dress the macroparticle largely determines the filtering properties needed to keep numerical artifacts under control. In many cases, the fulfilment of some requirements relevant to the underlying phenomena being described must be taken into account. An illuminating example is the electron dynamics of a relativistic charged beam, where form factors lending themselves to a covariant evaluation of self-fields are clearly preferable (see [5]).
The outline of this contribution is the following: in Section 2 the concept of form factor is discussed with respect to the classical problem of representing a generic, time-varying distribution as a superposition of functions. Some requirements are derived on the filtering properties in the frequency domain that make a systematic approach possible. In Section 3 it is shown as these remarks apply to the particular case of the form factor most widely used for its filtering properties: the gaussian function. In Section 5 an interesting example is presented illustrating the capabilities of the gaussian-like form factors in reproducing the relevant features of a distribution yet preserving unmodified the desired properties as low-pass filters. Lack of space prevents describing the extension to a number of dimensions higher than one. The last section is devoted to final remarks and acknowledgments.
Section snippets
Distributions and form factors
Consider a (macroscopically) continuous “charge” distribution and n “elementary charges” q1,q2,…,qn at x1,x2,…,xn each characterized by a form factor ξ(x−xi) (i=1,2,…,n) so that the approximated charge distribution:is reasonably accurate (in a sense to be made more precise). Assuming thatit follows that for the total “charge” to be conserved the form factor must be normalized:Consider now the trivial identity:
Hermite filtering
Consider a complete set of orthogonal polynomials [1, Orthogonal Polynomials] Hn(x) with respect to a given weight function W(x)(>0):and a sth degree polynomial Qs(x) such that:In order to derive Qs(x) explicitly one can exploit first the completeness of orthogonal set to express either Qs(x) and the generic power xl as a linear combination of Hn(x):and then the orthogonality
Gaussian-like form factors
Let us consider the functionas a particular instance of the form factors discussed above. Fig. 1, in which are shown the plots of χs(x) for s=8 (solid line) and s=4 (dashed line), suggests that for higher values of s the function narrows, lending itself to be used to describe the features of a distribution at a finer scale according to the remarks discussed in Section 2. In the limit s→∞ the form factor tends to the Dirac δ:The (4.2) can be proved by
An application
Consider the distribution:where g(x,μ,σ) denotes the normalized gauss functionIt can be easily shown that the mean value of distribution (5.1) is given by:while the r.m.s. deviation reads:Consider now the following discrete approximations of (5.1):andwhere
Final remarks
A class of functions is introduced featuring interesting properties with respect to Fourier transform. These functions exhibit scaling characteristics that ease the task of designing particle form factors with the right filtering properties necessary for smoothing unphysical effects (high frequency noise and other numerical artifacts) in Monte Carlo simulations relevant, e.g., for computational accelerator physics and molecular dynamics. Other area of applications could be image and motion
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