Fluidization conditions

Currently there are two implemented conditions for selecting hadrons from a collision as input for a hydrodynamic evolution, controlled by Type. Namely, they are Constant_Tau, which relies on the hadron's hyperbolic time, and Dynamic, where the condition is that the energy density around the hadron exceeds a defined threshold. In both cases, particles that obey the fluidization condition are removed from the evolution and written to the Initial_Conditions output, which must be included in the config.

Constant tau

The hyperbolic time is taken from the Proper_Time field in the Initial_Conditions subsection when configuring the output. If this information is not provided, the default value corresponds to the passing time of the two nuclei, where all primary interactions are expected to have occured:

\[ \tau_0 = (r_\mathrm{p} \ + \ r_\mathrm{t}) \ \left(\left(\frac{\sqrt{s_\mathrm{NN}}} {2 \ m_\mathrm{N}}\right)^2 - 1\right)^{-1/2} \]

Therein, \( r_\mathrm{p} \) and \( r_\mathrm{t} \) denote the radii of the projectile and target nucleus, respectively, \( \sqrt{s_\mathrm{NN}}\) is the collision energy per nucleon and \( m_\mathrm{N} \) the nucleon mass. Note though that, if the passing time is smaller than 0.5 fm, the default proper time of the hypersurface is taken to be \(\tau = 0.5\ \mathrm{fm}\) as a minimum bound to ensure the proper time is large enough to also extract reasonable initial conditions at RHIC/LHC energies. If desired, this lowest possible value can also be specified in the configuration file with the Lower_Bound field. This is best applied to higher beam energies, where the majority of the system is expected to behave as a fluid starting with a Bjorken picture.

Dynamic with energy density

Hydrodynamics is in general applicable for systems in or close to equilibrium. A hadron gas will always be driven towards equilibration, but this will be faster if the temperature or density is higher. This can be effectively captured by conditioning the fluid-like behavior to the local energy density: if it is higher than a given value, then this region can be considered a fluid. By default, the threshold energy density is set to 0.5 GeV/fm³, but this can be controlled with the Energy_Density_Threshold. This procedure is based on Akamatsu:2018olk [1], where particles that suffered elastic collisions are not fluidizable, but here they are included by default. If desired, this can be changed with the Fluidizable_Processes key.

The threshold condition is evaluated at every time step in a lattice centered at the origin that starts with a fixed length of 40 fm in each direction (for zero Minimum_Time), but grows linearly every 5 fm after the first 20 fm until Maximum_Time, such that even particles at the speed of light are always contained in the lattice. The number of cells is fixed, meaning that each cell increases in size.

Type — string, required

Type of initial conditions provided. Possible values are:

"Constant_Tau" → a hypersurface of constant \(\tau\) is used.
"Dynamic" → regions with sufficient energy density become fluid cells, with its particles written to the IC output.

The parameters for each are described below. If a key that does not match the type is present in the configuration file, SMASH will throw.

Parameters for fluidization at constant tau

Lower_Bound — double, optional, default = 0.5

Lower bound in fm for the IC proper time if Proper_Time is not provided. It is only used if the constant tau initial condition is active.

Proper_Time — double, optional, default = \(f(t_{np})\)

Proper time in fm at which hypersurface is created. Its default value depends on the nuclei passing time \(t_{np}\) as follows,

\[ f(t_{np})=\begin{cases} \mathrm{\texttt{Lower_Bound}} & t_{np} \le \mathrm{\texttt{Lower_Bound}}\\ t_{np} & t_{np} > \mathrm{\texttt{Lower_Bound}} \end{cases}\;. \]

It is only used if the constant tau initial condition is active.

pT_Cut — double, optional, default = No cut is done

If set, employ a transverse momentum cut for particles contributing to the initial conditions for hydrodynamics. A positive value in GeV is expected. Only particles characterized by \(0<p_T<\mathrm{\texttt{pT_Cut}}\) are printed to the output file. A value of 0 corresponds to no cut. It is only used if the constant tau initial condition is active.

Rapidity_Cut — double, optional, default = No cut is done

If set, employ a rapidity cut for particles contributing to the initial conditions for hydrodynamics. A positive value is expected and the cut is employed symmetrically around 0. Only particles characterized by \(|\mathrm{\texttt{Rapidity_Cut}}|<y\) are printed to the output file. A value of 0 corresponds to no cut. It is only used if the constant tau initial condition is active.

Parameters for dynamic fluidization

Energy_Density_Threshold — double, optional, default = 0.5

Set the minimum energy density in GeV/fm³ for a particle to be considered fluid. It is only used if the dynamic initial condition is active.

Minimum_Time — double, optional, default = 0

Set the minimum time in fm for a particle to be considered fluid. If larger than 10 fm, the initial lattice size also increases. It is only used if the dynamic initial condition is active.

Maximum_Time — double, optional, default = 100

Set the maximum time in fm for a particle to be considered fluid. For efficiency in production runs, it is recommended to set to a lower value. It is only used if the dynamic initial condition is active.

Fluidization_Cells — int, optional, default = 80

Fixed number of cells in each direction to select fluidizing particles. Ideally the cell should be small enough for a meaningful interpolation,

Fluidizable_Processes — list of strings, optional, default = "All"

Determines which process types can have outgoing particles as fluidizable. Possible values are:

"All"
"Elastic": Elastic \(2\to2\)
"Decay": All \(1\to N\) processes
"Inelastic": All \(N\to1\) processes
"SoftString"
"HardString"

The argument for allowing string processes to produce fluidizable hadrons, even though they break detailed balance, is that the system is expanding, so the fragmentation products are driven towards equilibration when the medium becomes large enough, which happens if the fluidization happens after their formation time.

Formation_Time_Fraction — double, optional, default = 1.0

Fraction of the formation time after which a particle can fluidize. It is is non-negative, and can assume values above 1. Setting it to 0 corresponds to ignoring formation time. This is only relevant if string fragmentation can produce fluidizable particles.

Extracting initial conditions for hydrodynamic evolution

The following example configures the initial conditions for hydrodynamics for a Au+Au collision at \(\sqrt{s_{NN}}=200\ \mathrm{GeV}\) at midrapidity ( \(-1<y<1\)). In addition, the extended OSCAR2013 and ASCII outputs are enabled.

Output:
    Initial_Conditions:
        Format: ["ASCII","Oscar2013"]
        Extended: True
Modi:
    Collider:
        Projectile:
            Particles: {2212: 79, 2112: 118} #Gold197
        Target:
            Particles: {2212: 79, 2112: 118} #Gold197
        Sqrtsnn: 200
        Initial_Conditions:
            Type: "Constant_Tau"
            Rapidity_Cut: 1