freestream
Free streaming and Landau matching for boostinvariant hydrodynamic initial conditions.
freestream
is a Python implementation of preequilibrium free streaming for heavyion collisions, as described in
 J. Liu, C. Shen, U. Heinz, "Preequilibrium evolution effects on heavyion collision observables", PRC 91 064906 (2015), arXiv:1504.02160 [nuclth].
 W. Broniowski, W. Florkowski, M. Chojnacki, A. Kisiel, "Freestreaming approximation in early dynamics of relativistic heavyion collisions", PRC 80 034902 (2009), arXiv:0812.3393 [nuclth].
Installation
Simply run
pip install freestream
The only requirements are numpy (1.8.0 or later) and scipy (0.14.0 or later).
Usage
freestream
has an objectoriented interface through the FreeStreamer
class, which takes three parameters:
freestream.FreeStreamer(initial, grid_max, time)
where

initial
is a square array containing the initial state, 
grid_max
is the x and y maximum of the grid in fm, i.e. half the grid width (see following example), 
time
is the time to free stream in fm/c.
The initial
array must contain a twodimensional (boostinvariant) initial condition discretized onto a uniform square grid.
It is then interpreted as a density profile of noninteracting massless partons at time τ = 0+.
The grid_max
parameter sets the outermost edge of the grid, not the midpoint of the outer grid cell, e.g.
 A 200 × 200 grid with a max of 10.0 fm has cell edges at 10.00, 9.90, ..., +10.00 and cell midpoints at 9.95, 9.85, ..., +9.95.
 A 201 × 201 grid with a max of 10.05 fm has cell edges at 10.05, 9.95, ..., +10.05 and cell midpoints at 10.00, 9.90, ..., +10.00.
This is the same definition as the trento gridmax
parameter.
It is very important that the grid max is set correctly to avoid superluminal propagation.
Suppose initial
is an n × n initial condition array with a grid max of 10.0 fm and we want to free stream for 1.0 fm.
We first create a FreeStreamer
object:
import freestream
fs = freestream.FreeStreamer(initial, 10.0, 1.0)
We can now extract the various quantities needed to initialize hydro from fs
.
Energymomentum tensor T^{μν}
Tuv = fs.Tuv()
Tuv
is an n × n × 3 × 3 array containing the full tensor at each grid point.
If we only want a certain component of the tensor, we can pass indices to the function:
T00 = fs.Tuv(0, 0)
T00
is an n × n array containing T^{00} at each grid point.
This is purely for syntactic convenience: fs.Tuv(0, 0)
is equivalent to fs.Tuv()[:, :, 0, 0]
.
Energy density e and flow velocity u^{μ}
e = fs.energy_density() # n x n
u = fs.flow_velocity() # n x n x 3
We can also extract the individual components of flow velocity:
u1 = fs.flow_velocity(1) # n x n
Again, this is equivalent to fs.flow_velocity()[:, :, 1]
.
Shear tensor π^{μν} and bulk pressure Π
The shear pressure tensor π^{μν} works just like T^{μν}:
pi = fs.shear_tensor() # n x n x 3 x 3
pi01 = fs.shear_tensor(0, 1) # n x n
The bulk viscous pressure Π depends on the equation of state P(e). By default, the ideal EoS P(e) = e/3 is used:
bulk = fs.bulk_pressure()
The bulk pressure is in fact zero with the ideal EoS, but there will be small nonzero values due to numerical precision.
To use another EoS, pass a callable object to bulk_pressure()
:
bulk = fs.bulk_pressure(eos)
For example, suppose we have a table of pressure and energy density we want to interpolate.
We can use scipy.interpolate
to construct a spline and pass it to bulk_pressure()
:
import scipy.interpolate as interp
eos_spline = interp.InterpolatedUnivariateSpline(energy_density, pressure)
bulk = fs.bulk_pressure(eos_spline)
Other notes
The code should run in a few seconds, depending on the grid size. Computation time is proportional to the number of grid cells (i.e. n^{2}).
Ensure that the grid is large enough to accommodate radial expansion. The code does not check for overflow.
FreeStreamer
returns references to its internal arrays, so do not modify them in place—make copies!
Testing and internals
FreeStreamer
uses a twodimensional cubic spline (scipy.interpolate.RectBivariateSpline) to construct a continuous initial condition profile from a discrete grid.
This is very precise provided the grid spacing is small enough.
The spline sometimes goes very slightly negative around sharp boundaries; FreeStreamer
coerces these negative values to zero.
The script test.py
contains unit tests and generates visualizations for qualitative inspection.
To run the tests, install nose and run:
nosetests v test.py
There are two unit tests:
 Comparison against an analytic solution for a symmetric Gaussian initial state (computed in Mathematica).
 Comparison against a randomlygenerated initial condition without interpolation.
These tests occasionally fail since there is a random component and the tolerance is somewhat stringent (every grid point must agree within 0.1%). When a test fails, it will print out a list of ratios (observed/expected). Typically the failures occur at the outermost grid cell where the system is very dilute, and even there it will only miss by ~0.2%.
To generate visualizations, execute test.py
as a script with two arguments, the test case to visualize and a PDF output file.
There are three test cases:

gaussian1
, a narrow symmetric Gaussian centered at the origin. 
gaussian2
, a wider asymmetric Gaussian offset from the origin. 
random
, a randomlygenerated initial condition (this is not in any way realistic, it's only for visualization).
For example:
python test.py gaussian1 freestream.pdf
will run the gaussian1
test case and save results in freestream.pdf
.
The PDF contains visualizations of the initial state and everything that FreeStreamer
computes.
In each visualization, red colors indicate positive values, blue means negative, and the maximum absolute value of the array is annotated in the upper left.
Animations
The included script animate.py
generates animations (like the one at the top of this page) from initial conditions saved in HDF5 format (e.g. trento events).
It requires python3 with matplotlib and h5py, and of course freestream
must be installed.
To animate a trento event, first generate some events in HDF5 format then run the script:
trento Pb Pb 10 o events.hdf ./animate.py events.hdf event_0 freestream.mp4
The first argument is the HDF5 filename, the second is the dataset to animate, and the last is the animation filename.
Run ./animate.py help
for more information including options for the animation duration, framerate, colormap, etc.