7.5 Additional properties of the Ąeld-particle method
7.5.2 Field-Field Interaction
The particle-Ąeld method was motivated and introduced with the scope of interactions between a Ąeld and an ensemble of particles. The implementation for the damping process σ →qq¯ can be used to extend this method for interactions between diferent Ąelds. In case of the linearσ-model, the interaction between the σ-Ąeld and the pionic Ąeld π are given by the interaction potential
Lσ,π = λ2
2 σ2π2 . (7.109)
The general equation of motion describe this interaction in the mean-Ąeld approach. Additional, interactions in terms of discrete interaction processes like
σ ↔ππ (7.110)
can be described and implemented with the particle-Ąeld method to implement additional processes which generate non-deterministic Ąeld distributions and additional thermal modes on the Ąelds. The implementation would be the same as described in Section 7.4.4. Particle distributions fσ andfπ are derived at a possible interaction point from the local ĄeldŠs properties.
These distribution function are used to calculate microscopic interaction probabilities, realizations of these interactions are calculated using Monte-Carlo techniques. For every interaction, the energy and momentum exchange is calculated and is fed back to the Ąelds using the described methods. An interaction is then described by a discrete energy and momentum transfer from one Ąeld to another Ąeld while the total energy and momentum stays constant.
7.5.3 Influence of Test-Particles and the Interaction Volume
The test-particle ansatz was used in this work to solve the Vlasov- and Boltzmann-equation. An analytic distribution function is discretized by a Ąnite number of test-particles which represent a statistical ensemble of the original distribution function. Additionally, by rescaling the particle number with an artiĄcial multiplierNtest, which linearly scales the number of particles. The total energy of the system is independent of the test-particles, which rescales the energy per particle with this factor. Every particle is propagated with its physical mass, energy and momentumm, E and P, however energy and momentum are rescaled within collisions and energy-momentum exchanges:
Eˆi= Ei
Ntest (7.111)
A higher Ntest leads to a lower exchanged energy for two particles in a collision. The same holds for particle creation and annihilation. For physical particles, the needed energy for pair production ¯qq is at least
min (Eqq¯ ) = 2mq (7.112)
125
With test particles, this energy threshold is rescaled:
min (Eqq¯ ) = 2mq
Ntest (7.113)
In the particle-Ąeld method, this test-particle scaling has a notable impact. Both the total energy and momentum of every particle is scaled with Ntest, see (7.111). This changes the absolute values for the energy and momentum transferred from and to the Ąeld. The ratio of the energy to momentum stays constant
Eˆi Pˆi = Ei
Pi , (7.114)
which leads to a relative scaling of the Gaussian parametrization, but its shape and movement is not altered.
However, the minimal pair-production threshold has an impact on the system dynamics. With (7.113) being the smallest energy the wave equation can dissipate, Ntest has a direct impact as a cut of energy dissipation. A small number of the test-particle multiplier could lead to a loss of damping of a Ąeld if the Ćuctuations and energy density on the Ąeld is too weak to create a particle pair. On the other hand, a very large number in the test-particle multiplier repatriates the system back to a continuous damping, like in the Langevin-equation.
The second important parameter for the particle production threshold is the interaction volume which is given by the width of the Gaussian parametrizationσx. A larger width leads to a larger interaction volume and therefore includes a larger volume of the Ąeld in the interaction. The larger this volume the more energy can be dissipated from the Ąeld. The maximum amount of energy which can be dissipated in a single interaction is given by
max (∆E) =∫
V dx ϵ(x)∏
i
exp [
−(x−xint)2 2σi2
]
. (7.115)
With the simple approximation of a constant energy density ϵ(x) =ϵand for a spherical Gaussian σi=σ, equation (7.115) becomes
max (∆E) = 4πϵ∫
V dr r2 exp (
−r2 2σ
)
= (2π)3/2ϵ σ3 .
(7.116)
The maximal possible energy which can be dissipated by an interaction scales linearly with the volume. By comparing (7.113) with (7.116) the threshold energy densityϵmin, at which the Ąeld has enough energy to create particle-pairs, can be derived as
ϵmin= 1
√2 π3/2 mq σ3Ntest
. (7.117)
Chapter 7 Particle-Wave Interaction Method A higher test-particle number and a larger interaction volume decreases the lowest energy density at which the Ąeld starts to be damped and dissipated. This behavior is not a numerical artifact but consistent with the test-particle ansatz and the discrete damping. The more massive and energetic the particle pairs become, the more energy is needed to create them. This leads to a more ŞsteplikeŤ damping, as it was shown in the example of the discretely damped harmonic oscillator.
7.5.4 No-Momentum-Approximation
The particle-Ąeld method allows to transfer arbitrary amounts of energy and momentum to and from Ąelds, described by the four parameters ∆E and ∆P. The numerical implementation leads to a system of four coupled, non-linear equations which have to be solved consistently. As an approximation to reduce the numerical complexity, the energy-momentum transfer can be approximated to have zero momentum exchange
∆E = ∆E ∆P= 0 . (7.118)
The full problem for the energy-momentum equations (7.24) and (7.25) reduces to the single scalar and non-linear equation for the energy (7.24). This reduces the computational costs dramatically because a completely diferent set of solvers can be used, the number of equations to solve reduce and some additional numerical optimizations can be used. A numerical optimization, which can be employed with this approximation is discussed in the Section 8.7.
This zero-momentum approximation is applicable, in cases where a net-momentum transfer is not important for the system dynamics or the dynamics is described by a large superposition with a zero net-transfer of momentum. Typical examples are the simulation of a Langevin-like equation, in which momentum transfers are not considered or a thermal medium in rest.
However, this approximation still allows full controllable and discrete interactions with energy conservation.
7.5.5 Numerical Errors
Interactions between particles and Ąelds employ a relative large number of numerical operations on the Ąelds. In this subsection a rough estimation of the numerical errors is given.
Modern computer systems represent Ćoating point numbers by using the IEEE Standard for Floating-Point Arithmetic (IEEE 754). Employing double-precision 64-bit precision, all numbers are represented in the form
x=m·2e (7.119)
127
with a 11 bit precision for the exponent e and a 52 bit precision for the mantissa m. While a double-precision number is deĄned over a very large range (from 2.23·10−308 to 1.8·10308), the precision within numerical operations is much smaller than implied by the precision of the exponent. For typical arithmetic operations like ±and ×, the expected error per operation due round of efects is in the order of magnitude
ϵ= 2−53≈5.7·10−16. (7.120)
An interaction between a particle pair and the scalar Ąeld afects a grid region ofN = 403= 64·103 grid points, leading to a numerical error of about 6·10−12. This numerical error is still four orders of magnitude smaller than the error given by the non-linear solver, which Ąnds its solution with a precision of about ≈10−8. The presented test-calculations simulated roughly 107 collisions, corresponding to 20 times the number of particles. For a rough estimation, we assume the numerical errors in each simulation run to have a random total and maximal error of
ϵmax= 10−8·√
404·107 ≈10−3 (7.121)
Numerical calculations have shown a relative precision of about 10−5, see for example Figure 7.14.
Note that beside the errors given through round-of efects, the partial diferential equation solver can generate numerical noise when employing strong gradients, difering from the exact solution of the equations of motion. These mathematical errors can exceed the numerical errors in non-equilibrium scenarios. The used solving scheme in DSLAM is the Leap-Frog solver, whose properties are explained in Section 8.3.2. This solving scheme conserves energy over a period of several time steps and has a numerical solving precision of ∆t2. A more detailed discussion on numerical errors can be found in [130,131].
Chapter 8
Numerical Implementation-Details of the DSLAM Model
Essentially, all models are wrong, but some are useful.
George Edward Pelham Box
This chapter describes the numerical realization and implementation of the DSLAM model (Dynamical Simulation of a LinearSigmA Model). The main focus does not lie on the physical properties of the underlying linearσ-model but on the numerical methods and algorithms which have been employed as well the used programming techniques and software packages.
This chapter is organized as follows: Section8.1 explains the basic architecture and used software libraries. Numerical details on the single components of the simulation are described in Section 8.2.
8.1 Software Architecture and Programming Techniques
The DSLAM model was written with help of diferent programming languages and employs several external libraries.
Python was chosen as the primary developing language https://www.python.org/ because of its clean syntax and very fast developing cycles. It is an interpreted language, which is in general not as performant as a compiled language like C, but employing the correct techniques, it can be used for eicient numerical calculations. Many recent projects within high performance computing (HPC) are realized using Python [132,133]
129