Free streaming for heavy-ion collision initial conditions.

## Project description

*Free streaming and Landau matching for boost-invariant hydrodynamic initial conditions.*

freestream is a Python implementation of pre-equilibrium free streaming for heavy-ion collisions, as described in

J. Liu, C. Shen, U. Heinz, “Pre-equilibrium evolution effects on heavy-ion collision observables”, PRC 91 064906 (2015), arXiv:1504.02160 [nucl-th].

W. Broniowski, W. Florkowski, M. Chojnacki, A. Kisiel, “Free-streaming approximation in early dynamics of relativistic heavy-ion collisions”, PRC 80 034902 (2009), arXiv:0812.3393 [nucl-th].

## Installation

Simply run

pip install freestream

The only requirements are numpy (1.8.0 or later) and scipy.

## 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 (this is the same definition as trento).
For example:

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.

**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.

### Energy-momentum tensor *T*^{μν}

`Tuv = fs.Tuv()`

Tuv is now 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 now 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 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.

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