Virtual Heat Bath Method

The DSLAM model is a microcanonical simulation of a particle and Ąeld ensemble. All equations of motion, mean Ąeld interactions, elastic particle collisions and particle production and annihilation processes conserve energy and momentum. To simulate a thermal box which can change its temperature, the simulation has to be extended to allow canonical simulations, thus allowing a thermal exchange of energy while conserving the particle number in this canonical process. In total however, the particle number can change due to internal annihilation and pair-production processes. A canonical process can be implemented with a heat reservoir.

The most simple ansatz would be the implementation of reservoir walls. Whenever particles reach the wall, they are replaced by thermal particles from the reservoir. This ansatz has some major drawbacks. The rate of energy exchange is directly proportional to the particle density and the number of particles touching the wall. Additionally, in a coupled medium artiĄcial correlations will be generated. This can be visualized by a temperature change of the system. The system will cool down near the walls and will build up a temperature gradient, see Ągure8.1. Conversely, if the medium has physical correlation through interactions, these correlations get suddenly washed out near the reservoir walls because of the newly generated, uncorrelated particles.

A workaround is a heat bath, which is implemented by virtual particles. This kind of heat bath can be described as a second, inĄnitely large particle reservoir which exists in addition to the physical particles. These two particle species can interact via microscopic interactions. However, the second particle species is not directly simulated but thermal particles are sampled according

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Figure 8.1: Visualization of a small system embedded in a thermal heat bath, which is realized by reservoir walls which emit thermal particles back into the system when a physical particle leaves the system through the walls. This type of heat bath creates inhomogeneities and artiĄcial correlations by building up a temperature gradient for a system with a strongly coupled medium.

to an equilibrium Boltzmann distribution function on demand if energy should be exchanged with the heat bath, therefore the name virtual heat bath.

For every particle in a cell, a potential interaction partner is sampled from an equilibrium-distribution function f(p, T)_bath. The temperature of the distribution function is a parameter of the heat bath. Employing a microscopic collision kernel, the interaction probability can be calculated. If an interaction happens, the particle and the virtual heat bath particle interact and exchange energy and momentum. The amount of exchange energy can be positive or negative and depends on the collision kinematics. After some time the distribution function of the system will align with the thermal distribution function.

In case of the DSLAM model, the thermal heat bath distribution function was chosen to be the Boltzmann distribution, which is the thermal equilibrium distribution for the quarks if the system can equilibrate via elastic interactions

f(p, T)_bath = exp⁽−E T

)

. (8.24)

Chapter 8 Numerical Implementation-Details of the DSLAM Model

Figure 8.2: Visualization of the virtual heat bath method. Physical particles (yellow) can interact with virtual particles of the heat bath (blue). Virtual particles are not propagated but are according to a thermal distribution function for every possible interaction partner. Interaction probabilities are calculated using microscopic cross sections. In case of an interaction, particles and virtual particles exchange energy and momentum. Within the time evolution the distribution function of the physical particles will converge to the equilibrium distribution of the heat bath.

The total particle density of the heat bath particles scales with the third power of the temperature, increasing the efective reaction rate with the heath bath

∫ dE f(p, T)_bath =n_bath . (8.25)

For every particle in a cell, a virtual partner particle is sampled from (8.24) and for every pair the interaction probability is calculated. For an isotropic cross sectionσ_bath is analogous to the one of elastic scatterings (8.18) and reads

P_bath= s·σ_bath(T) E1Ebath

∆t

∆³x (8.26)

whereE1 is the energy of the physical particle,E_bath the energy of the thermal particle andsas the Mandelstam variable.

With a constant cross sectionσ_bath(T) =σ_bath the interaction rate between the heat bath and the physical particles scales with the particle number and the density of the heat bath n_bath, which itself depends on the bath temperature.

In case the temperature of the heat bath should change within a simulation but the interaction rates should be kept constant, the cross section should be chosen temperature dependent as well.

For

σ_bath(T) = σˆ_bath

2T³ (8.27)

the interaction rate stays with the heat bath and the particles independent of the temperature.

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Figure 8.2 shows a visualization of the principle of the virtual heat bath. The method of a virtual-particle bath mimics a canonical heat bath. By sampling virtual particles from the heat bath, the exchanged amount energy between particles and ŞrealŤ particles has a stochastic character. This method has several advantages. Spatial anisotropies are not generated, any energy exchange between particles and the heat bath is distributed evenly across all particles.

Additionally, a Ąne grained control over the heat bath is given by the two parameters temperature T and the heat bath cross sectionσ_bath. The ratio of the system- and heat bath temperature measures the hardness of the interactions

r_bath = T_bath

T_medium (8.28)

and gives the average energy exchange between the two reservoirs. The cross section σbath

determines the time in which the system equilibrates because it is proportional to the reaction rate between the particles and the heat bath while. The heat bath method is used in the DSLAM model to control the temperature of a thermal box. It is both used to keep the system at a constant temperature and to change the system temperature, for example to drive the system from a hot phase through the chiral-phase transition to a cold phase. All energy which is released in the phase transition can be absorbed by the heat bath. Calculations with the virtual heat bath method are used in Section 3.2.1in which a particle ensemble thermalizes with the heat bath, see Figure 3.4and 3.5.