In the last section the momentum over energy ratio for a spherical Gaussian in three dimensions was calculated. This calculation neglected any relativistic efects, resulting in a upper momentum to energy ratio of 1/2. Furthermore momentum and energy of any relativistic object would grow to inĄnity for β→1, which is not the case for the non-relativistic calculation in the last section.
To address this issue, we will boost the Gaussian along the velocity direction. For simplicity we assume the velocity to be in line with our coordination-system in x-direction
v=
⎛ ˆ ˆ
∐ v 0 0
⎞ ˆ ˆ ˆ
. (A.35)
The non-relativistic form of the spherically symmetric 3D Gaussian becomes ϕ(x, t) =A0exp
(−(x−vt)2 2σx2
) exp
(−y2−z2 2σyz2
)
. (A.36)
Both the space and time are boosted in x-direction:
x′ =γx , (A.37)
t′ =γt , (A.38)
1/γ=√1−β , (A.39)
β= v2
c2 =v2 . (A.40)
Boosting (A.36) leads to
ϕ(x, t) =A0exp
(−γ2(x−vt)2 2σ2
) exp
(−y2−z2 2σ2
)
. (A.41)
173
Performing the same calculations for the energy and momentum as in the previous section, we arrive at the energy to momentum ratio with
P
E = 2v
3−v2 = −2√ β
3−β . (A.42)
We have performed a Lorentz boost on the Gaussian, still equation (A.42) looks very similar to (A.31) with the diference of the minus sign. However, for v → 1 we now get the correct relativistic limit
max(P E
)v→1= 1 . (A.43)
Figure A.5 shows the ratio P/E in dependence to the relativistic velocity. Additionally, the
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Momentum to Energy Ratio
Wave Packet Velocity vx
Figure A.5: Ratio of the momentum to energy transfer of a 3D spherical Gaussian with a Lorentz-boost.
energy and momentum of the Gaussian are now boosted and can reach arbitrary large values for v→1. In equation (A.41)x and twere boosted. This boost, which introduced an additionalγ2 in the equation could be absorbed in a boosted ˜σx:
˜ σx= σ
γ (A.44)
This boost in x-direction can be reinterpreted in a smaller Gaussian width in x-direction, compressing the spherical Gaussian to a thin disk for high velocities v → 1. In a numerical simulation the Lorentz-boost has to be employed in all three dimensions, extending (A.23) with
Appendix A Numerical Properties of Different Interaction Parameterizations
the general Lorentz-transformation ϕ(x, t) =A0
3
∏
i
exp
⋃
⨄− 1 2σ2i
( 4
∑
µ
Λµνxν )2⋂
⎦ . (A.45)
However, for small velocities or small momentum to energy ratios, the non-boosted version can be used as an approximation. For P/E <0.3 the error is about 18.6%. Additionally, one has to be careful with the numerics for high velocities. For v →1 the Gaussian becomes a slim disk which can lead to discretization artifacts if the efective ˜σ becomes of the order of magnitude of the grid size.
175
Appendix B
Numerical Sampling
Maybe life is random, but I doubt it.
Steven Tyler
Numerical Sampling Methods
In physics many systems are deĄned and described by distribution functions. Prominent distribu-tion funcdistribu-tions are the Gaussian distribudistribu-tion, the exponential distribudistribu-tion as well as Boltzmann, Fermi- or Bose-distributions.
Distribution functions are handy in statistical physics and many models deĄne evolution equations for distribution functions. The Boltzmann function is a typical example for such a theory. Other models are the Fokker-Planck equation [153], the master equation or the Black-Scholes equation for stock-option pricing [154] and many more. For a given distribution function, one can derive time-evolution equations within these theories.
In general, distribution functions are continuous objects with a statistical interpretation. Most of the time, they are deĄned as probability density functions f(x) with the following properties:
Boundedness : ∫ dx f(x) =c , (B.1)
with c= 1 in case of a probability-density function or with a physical value forc, for example the particle number in case of the Boltzmann distribution.
For a probability distribution function, the expectation value for an observableA(x) is deĄned as
⟨A⟩=∫ A(x)f(x)dx . (B.2)
177
This comes handy for analytic calculations, for numerical computations a discrete realization of a distribution function is needed. The process of generating a Ąnite number of elements with properties which follow a given distribution function is called sampling
f(x)−−−−−→Sampling 1 N
N
∑
i=1
δ(x−xi), (B.3)
which reasembles the distribution function again in the continuous limit:
N→∞lim 1 N
N
∑
i
δ(x−xi) =f(x) (B.4)
There is no single method for sampling, in fact it is an own discipline in numerical mathematics and various methods exist, depending on the properties of the functions to sample. All methods, which have been used in this thesis is listed in the following. The common point of all functions are the need for good random numbers.
Random-Number Generation
Sampling is directly related to random-number generation. In theory, a distribution function could be sampled with a deterministic pattern over the sampling range. However, for Monte-Carlo simulations one is interested in exploring the physical phase space of a model with probability methods. Therefore it is of interest to generate slightly diferent versions of a calculation, even though the initial conditions are the same. In general, the initial conditions are given by a distribution function and because of the too high numerical complexity of propagating the exact solution, the initial conditions are sampled and the subset of Monte-Carlo realizations are evaluated. Multiple runs with diferent sampling realizations lead to diferent results in the end which can be used for statistical analysis.
In all cases, Monte-Carlo calculations make heavy use of random numbers. For simulations, the generated random numbers should have a high statistical quality with a long period and for practical use should be fast in generation. In contrast to cryptographic applications, Monte Carlo simulations do not need an unpredictable stream of numbers.
A good choice for a general-purpose generator is the patent free Mersenne-Twister MT19937 [140]. It is based on a linear recurrence matrix on a Ąnite Ąeld and has a period of 2219937−1.
The chosen implementation implementation is the SSE2 optimized dSFMT implementation [155]
which generated Ćoating-point numbers in the interval [0,1).
Appendix B Numerical Sampling
B.1 Inversion Method
The inversion method is a straight forward, numerically stable and fast method for the exact sampling of a distribution function. Its drawback is its limitation to distribution functions which have an invertible cumulative distribution function.
The cumulative distribution function for a random variable withX ∈R withF(x) =P(X≤x) can be calculated from the probability distribution function f(x) via an integration
FX(x) =∫ x
−∞
f(τ) dτ . (B.5)
Per deĄnition,FX(x)∈[0,1].
For any random number ξ, the generated numberXf follows the distributionf if
Xf :=F−1(ξ) ¶ξ∈[0,1]♢ , (B.6)
with ξ as an uniformly distributed random number. The following shows the sampling of an exponential function with the probability distribution function
f(x) = exp (−γx) forx∈R+ , (B.7)
and the normed cumulative distribution function F(x) =∫ x
0 dx′ f(x′) = 1
γ [1−exp (−γx)] . (B.8)
By inverting and substituting ξ= 1−ξ′ the following generating function is deĄned x=F−1(ξ) =−ln(ξ)
γ ¶ξ∈(0,1]♢ . (B.9)
A sample of multiplex will follow an exponential distribution.