7.3 Particle-Field Interaction
7.3.2 Parameterization of interactions
Without further constraints, equation (7.24) and (7.25) have either no solutions or inĄnitely many. The problem is the fact that (7.24) and (7.25) reduce equations with many degrees of freedom to four scalar values,E andP. A classical Ąeld on a numerical grid has as many degrees of freedom as it has grid points, in general ∼Nd, in the continuous case the number of degrees are continuous and non-countable. Even if the system rests in a box and has a high-frequency cutof, the general statement is
NDOF(ϕ,ϕ˙)≫4 (7.26)
The Ąrst approach to this problem was an algorithm, which assumes an initial random perturbation and evolves dynamically by progressively minimize the energy and momentum diference in the equations with the constrain to minimize the needed strength of the disturbance, as well. For Ćat Ąelds or Ąelds with weak waves on it, the algorithm showed nice Şinteraction bumpsŤ. In general 97
however, it was not numerically stable, had no guaranteed convergence and had problems with strongly asymmetric Ąelds. Additionally, this solution was not very elegant as it left no control over the interaction space and lacked of a physical motivation.
To tackle these problems, the degrees of freedom of the numerical problem have to be reduced to a controllable system. The trick is to introduce an interaction parameterization which has a smooth spatially proĄle, is spatial limited and has exact four parameters, which can be mapped to momentum and energy properties.
In Appendix A diferent possible parameterization are discussed. To summarize this appendix chapter, the numerical parametrization forδϕ(x, tk) has to be as smooth as possible. A natural choice would be the use of a single point like excitation on the grid. This is inefective for two reasons: point like excitations are unsteady points on a numerical grid and lead to numerical artifacts like the Gibbs-phenomena, artiĄcial exciting high frequencies and violating energy conservation.
A useful and robust parametrization is a three dimensional, moving Gaussian wave packet, δϕ(x,v) =A0
3
∏
i
exp [
−(xi−vi˜t)2 2σi2
]\
\
\
\
\˜
t→0
. (7.27)
The variablesvdeĄne the direction and velocity of the Gaussian wave packet andA0 the strength of the interaction. The three position arguments xi are Ąxed by the interaction position. The time-parameter ˜tis needed to deĄne and calculate the derivatives for the energy and momentum in (7.13) and (7.14),
∂tδϕt→0= −A0
3
∏
i
[(cvixi σi2
) exp
((xi)2 2σi2
)]
(7.28)
∂xiδϕt→0= −A0xi σi2
3
∏
i
[ exp
((xi)2 2σi2
)]
(7.29)
The three widths of the Gaussianσi are free parameters, and can be Ąxed to a single spherical radius by σx=σy =σz =σ. It determines the interaction volume and should be chosen to Ąt the system scale. It has an impact on the minimal scale of possible modes in the system, as we will see in Section7.4.3.
With this parametrization, a wave packet is added to the Ąeld after an interaction. Figure 7.3 shows an example of a Ąeld kick, which is parameterized with this Gaussian parametrization.
The Ąeld gets a small ŞbumpŤ at the interaction point.
To ĄndA0 andvin the parameterization (7.27), which solve (7.24) and (7.25) for a given ∆E and ∆P, the four coupled and non-linear equations have to be solved with a numerical equation solver. ∆E and ∆Pare given by the physical interaction, the derivation of these quantities will
Chapter 7 Particle-Wave Interaction Method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
0 0.2 0.4 0.6 0.8 1
Scalar field [a.u.]
spatial position [a.u.]
Φ(ti) Φ(ti)+δΦδΦ
Figure 7.3: Visualization of the interaction principle, as described in (7.27). The energy of the initial Ąeldϕ(t) is changed by a parameterized Ąeld variationδϕ. The resulting Ąeldϕ(t) +δϕ is increased by a given energy ∆E and momentum ∆P. The traveling direction of the Gaussian is depending on the momentum.
be explained in various examples in the next chapters. A simple visualization of this principle is given in Figure 7.3, which shows the local modiĄcation of the Ąeld by a Gaussian.
An important implementation issue for ĄndingA0 andv is the energy dependence of the Ąeld and which part of the energy-contributions are dominating. The general deĄnition of the energy (7.13) is again
E =∫
V
d3x ϵ(x) =∫
V
d3x [1
2ϕ˙2+1 2
(∇⃗ϕ)2+U(ϕ)⎢, (7.30) Depending on the system dynamics, the dominant part of the equation can difer. In case of a damped harmonic oscillator, the potential energy U(ϕ) is as import as the kinetic energy of the systemŠs energy; therefore it is favorable to change the potential energy in case of an interaction.
In case of the linear σ-model, the change of the potential energy is quite weak in comparison to chagne of the kinetic parts of the Ąeld for a small disturbance. If the σ Ąeld is equilibrated and shows thermal noise, most of the energy is stored in the kinetic parts of the energy:
1
2( ˙ϕ)2+1
2(∇ϕ)2≫U(ϕ) . (7.31)
In such a case a more appropriate interaction-parametrization would change mainly the kinetic energy of the system and not the potential energy. This becomes even more important if the 99
interaction should also remove energy from the system. If the parametrization would change the potential energy, adding of energy would always be possible. In contrast, removing energy would be impossible in most of the time, if the Ąeld holds most of its energy as kinetic energy, not potential energy.
A possible Gaussian parametrization in three dimensions, which changes mainly the kinetic energy of the system could have exactly the same form as (7.27) but keeps ϕ untouched and adds all its energy to ˙ϕ. This has a very nice side efect: adding of energy does not generates an immediate Gaussian wave packet, which appears Şout of the blueŤ on the Ąeld, but rather creates a kick which evolves the interaction bump with time. In this case, the wave equation stays continuous in space and additionally in time.
Removing of energy works according to the same principle, just the other way around. Energy is taken from the Ąeld by reducing the kinetic part of the Ąeld ˙ϕby damping it with the Gaussian ansatz.
Finally, the parameterization (7.27) should be seen as a classical approximation which is valid for small velocities. The parameterization is not Lorentz invariant, resulting in spatial extent which does not depend on the velocity. This has an interesting efect, in case of a 3D system, the maximal momentum to energy ratio which can be generated with (7.27) is max{PE}= 1/2. The parameterization can be extended with a Lorentz boost, for example along the x-direction for v= (vx,0,0)
δϕ(x, t) =A0exp (
−γ2(x−vxt)2 2σ2
)
×exp (
−y2+z2 2σ2
) (7.32)
The Lorentz-boost leads to a disc shaped deformation of the initially spherical Gaussian. With the boost, the momentum to energy ratio of (7.32) has the correct relativistic limit
v→1lim
⎭P E
}
= 1. (7.33)
Atv= 0.3 both solutions difer by a factor of about 18%, for small velocities they are nearly the same, and (7.27) can be used as a safe approximation.
Calculations for the momentum to energy ratio of the two parameterizations (7.27) and (7.32) are given in AppendixA.
Chapter 7 Particle-Wave Interaction Method