The chemical potential arises in the first law of thermodynamics as
$$dE = TdS -pdV + \mu dN$$
and hence it represents the energy added to the system when one changes the particles in the system at constant entropy and volume. I don't see why the chemical potential has to vanish in equilibrium...
In cosmological perturbation theory one can make an argument that very early on the relation between dark-matter perturbations and photon-temperature perturbations satisfies
$$\delta = 3\Theta_0 + \text{constant}.$$
If ##\text{constant} = 0## we call these primordial perturbations adiabatic and...
In Dodelson's "Introduction to Modern Cosmology" at p. 61 he introduces a non- equilibrium number density
$$n_i = g_i e^{\mu_i/T} \int \frac{d^3p}{(2\pi)^3} e^{-E_i/T}$$
and an equilibrium number density
$$n_i^{(0)} = g_i \int \frac{d^3p}{(2\pi)^3} e^{-E_i/T},$$
from which it follows that the...
In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that...
When dealing with cosmological perturbations, there are a lot of different notions that are thrown around in the literature like statistical homogeneity and isotropy. However, these terms are often not motivated and clearly defined.
Could anyone recommend any good references where these notions...
Thanks for the reply! From what you've now said, how would one go on to argue (fairly rigorously) that the ##a_{lm}##-s for a given ##l## must be drawn from the same probability distributions?
In Dodelson's "Modern Cosmology" on p.241 he states that the ##a_{lm}##-s -- for a given ##l##-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of ##m##. Dodelson does not explain...
Thanks a lot for your help vanhees71! Might I ask you one(three) last questions?
My motivation for reading up on the Boltzmann equation is related to the study of cosmology.
When computing first order versions of the Boltzmann equations for the CMB temperature fluctuations, one assumes a form...
Why does it make sense to use Lagrange multipliers here? Lagrange multipliers implies that we are extremizing under the energy-momentum constraint, but how do we know that the extremization of this function corresponds to $$\psi+\psi_2-\psi_1'-\psi_2'=0$$? Does it make sense that this function...
Thanks! I'll certainly check them out.
So in the light of local entropy always increasing, is the statement from Dodelson above a reference to equilibrium in the sense that the entropy has reached its maximum? Is this the definition of kinetic equilibrium? How does this tie in with "scattering...
Ah, I actually already had a look at them!
Since you are here, might I take the opportunity to ask you to elaborate on how you obtained equation 1.4.10?
In section 3.1 of Dodelson's "Modern Cosmology", after introducing the Boltzmann equation, in the second paragraph of page 60 the author states that:
"The first, most important realization is that scattering processes typically enforce kinetic equilibrium.
That is, scattering takes place so...
In section 3.1 of Dodelson's "Modern Cosmology", after introducing the Boltzmann equation, in the second paragraph of page 60 the author states that:
"The first, most important realization is that scattering processes typically enforce kinetic equilibrium.
That is, scattering takes place so...
How would I go ahead and proving that the geodesic will in fact not wander out of U? Furthermore, if that is proven, my argument only shows that we can get to map any other poing in ##U## back to ##p##. How would I now extend this to any two points in this region -- not just ##p## and any other...
I agree with you that what the paper states is that one can find two neighbourhoods ##U,V## where ##p\in U \subset V## such that for any ##q \in U\ \text{Im}(\gamma)## there exists a unique future pointing geodesic, as well as a unique past pointing geodesic -- that stays within ##V## --...