Spectra

The function ` spectra()` calculates several spectra for
the output fields. The function begins by performing a Fourier
transform on the field and derivative grids, thus generating
lattices of Fourier values and . At each gridpoint in
Fourier space the function then calculates
,
, , and . The formulas for the
occupation number and the energy spectrum are given
below.

The ` spectra()` function doesn't output these values for each
grid point, but rather groups them into bins. The size of the bins is
determined by the lattice spacing . This means for example that in
three dimensions the bin () includes only the point
while the next bin (
) includes the
neighboring points. In one dimension there should be one point per
bin. The four quantities listed above are each averaged over each bin.

The output of this function is in several files. The first,
`spectratimes_ext`, simply lists the times at which
spectra were recorded. There is also one file per output field
that contains the spectra. These files are called
`spectrafield_number_ext`. They contain seven
columns listing, for each bin, , the number of points in the
bin, ,
,
, ,
and . (The quantities , , and are
defined below). Note that the four spectra (the last four columns)
are calculated at each point on the lattice and then averaged for
the bin, whereas and are given for the bottom of
the bin.

At each time for which spectra are recorded the
` spectra()` function will write one row for each bin into
each of the spectra files. When examining the output it's easy to
tell where one time ends and another begins because the
frequencies (first column) will monotonically increase at each
time and then reset to zero when a new spectrum begins.

The number of points in each bin is given to make it possible to take
sums of different quantities over the lattice. For example Parseval's
theorem for a discrete Fourier transform states

(5.4) |

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This documentation was generated on 2008-01-21