freestream 1.0.1

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 object-oriented 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 two-dimensional (boost-invariant) initial condition discretized onto a uniform square grid. It is then interpreted as a density profile of non-interacting 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 --grid-max 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.

Tuv = fs.Tuv()

T00 = fs.Tuv(0, 0)

e = fs.energy_density()  # n x n
u = fs.flow_velocity()  # n x n x 3

u1 = fs.flow_velocity(1)  # n x n

pi = fs.shear_tensor()  # n x n x 3 x 3
pi01 = fs.shear_tensor(0, 1)  # n x n

bulk = fs.bulk_pressure()

bulk = fs.bulk_pressure(eos)

import scipy.interpolate as interp

eos_spline = interp.InterpolatedUnivariateSpline(energy_density, pressure)
bulk = fs.bulk_pressure(eos_spline)

Testing and internals

FreeStreamer uses a two-dimensional 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 randomly-generated 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 randomly-generated 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.