## New PDF release: Relativistic Kinetic Theory

By Groot S.R., van Leeuwen W.A., van Weert Ch.G.

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The mass of the proton and are given by p + γCMB → by mp c 2 ≈ 940 MeV and m ≈ 1230 MeV. 58) and evaluate this energy numerically. Hint: Use the following assumptions: (i) The most favorable kinematic situation is when the directions of the incident proton and photon are opposite, (ii) assume that the energy of the incident photon is ECMB = kB TCMB with T = 3 K. 7kB TCMB , while the minimum “effective mass” needed for pion production is actually Mmin = mp + mπ . What is the minimum proton energy needed for pion production under these assumptions?

The expansion coefﬁcients an , bn can be obtained by multiplying both sides of Eqn. 17) by cos(mπx/L), sin(mπx/L) respectively and integrating over one cycle. 19) The an (bn ) thus measure the “overlap” of the desired waveform with one of the basic cosine or sine solutions. 1. 21) nπ where n = 1, 2, . . and so forth. In this case, the odd sin(Nπ x/L) terms cannot contribute to an even function, so that the bn vanish for symmetry reasons. 3. Fourier series approximations, FN (x), to a square wave pulse for N = 1, 3, 5, 21.

For k1 ≈ k2 ≈ k the resulting waveform is a plane wave, cos(kx), modulated by the “beat envelope”, 2A cos( kx), where k ≡ (k1 − k2 )/2. This factor gives complete destructive interference when x = (2n + 1)π/2 k; this implies that successive interference (“beat”) minima will be separated by x = π/ k or x k ≈ π ≥ O(1). This is our ﬁrst example of a quite general feature, namely that • The degree of localization of a wave packet in space ( x) making use of interference effects is inversely correlated with the spread in available k values ( k).

### Relativistic Kinetic Theory by Groot S.R., van Leeuwen W.A., van Weert Ch.G.

by Ronald

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