LATTICEEASY is primarily designed for calculations of processes
occurring in three dimensional space. When lower numbers of dimensions
are used the modes are designed to approximately reproduce the
behavior of a one or two dimensional slice through a three dimensional
space. Thus in our discussion of the number and energy spectra we will
consider the three dimensional case. The definitions of and
used for one and two dimensional runs will be given in section 5.4.
The occupation number is defined to be an adiabatic
invariant of the field evolution, whose integral
corresponds in the large amplitude limit to classical number
density. The energy density
gives the spectrum of energy
in different Fourier modes. In order to give a sensible definition
of these quantities in an expanding universe it is necessary first
to switch to conformal coordinates (defined below) in which the
field equations take the form of an undamped oscillator. When the
frequency
of this oscillator is changing adiabatically
then a nearly constant occupation number
can be calculated.
So in the following sections we convert the field equation to
conformal coordinates, define the quantities
and
,
and then convert back to physical coordinates in order to
apply the program rescalings and thus derive formulas for
and
in program units.
Before doing any of that, however, we start by discussing the properties of Fourier transforms of classical fields. It turns out that in order to define sensible intrinsic quantities like occupation number the Fourier transforms need to be rescaled in order to take account of the finite size and spacing of the lattice. These rescalings are derived below. Some of this discussion duplicates parts of section 6.3.2 but is presented here for completeness.
The vectors and
are shown without vector
notation except where needed for clarity. An ordinary
refers
to the potential while
is used to mean volume. Dots
indicate differentiation with respect to
while primes denote
differentiation with respect to rescaled time, either
(conformal time) or
(program time). The usage should be
clear from context. Angle brackets indicate spatial averages over
the lattice box.