All Questions
Tagged with cosmology statistical-mechanics
77
questions
0
votes
1
answer
67
views
Why can we use Bose-Einstein statistics in this expression for number density
In a system with $N$ particles in some volume $V$ in contact with a reservoir of temperature $T$, we find that
$$\bar{n_i}=\frac{g_i}{e^{\frac{{\epsilon}_i -\mu}{kT}} \pm 1}$$
depending on whether the ...
- 129
4
votes
2
answers
139
views
Why can photon be treated like gas?
In Cosmology, especially when studying Cosmic Dynamic, sometime we will treat photons as gas to calculate its pressure, but according to my understanding, photon and gas are nothing alike. Why can ...
- 445
5
votes
1
answer
81
views
Net particle number density for relativistic particles at finite chemical potential (tricky integral)
Question: How does one show that the chemical potential of relativistic fermions is negligible at high energies? In particular, I would like to show that the difference between the particle density $n$...
- 1,198
2
votes
5
answers
1k
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A few questions about Boltzmann brains [closed]
So I've been thinking about Boltzmann brains and there are some questions which I wasn't able to answer online:
Given that Boltzmann brains only last for a split second, is experience (the notion or ...
- 185
2
votes
1
answer
195
views
Determining free Fermi/Bose gas chemical potentials, given temperature, number density, and baryon density
I'm attempting to recreate some plots of individual particle chemical potentials from S. Reddy et. al.'s paper on neutrino interactions in hot and dense matter (of the same title). The specifics aren'...
- 799
0
votes
0
answers
32
views
Poincaré recurrence in a closed universe? [duplicate]
Is it possible that the Poincaré recurrence applies to a closed universe (with a finite spacetime)? If it is, would this mean that a closed universe could eventually reach the same state as its ...
- 2,472
8
votes
2
answers
3k
views
Do Boltzmann brain thought experiments suggest literally anything can form randomly?
Do Boltzmann brain thought experiments suggest literally anything can form randomly?
What are the limitations to what random fluctuations can form? Literally any physical, material object?
Lastly, I ...
user353810
0
votes
1
answer
64
views
Integral on the solid angle of a function of the direction [closed]
I need to compute the integral
$$\int \frac{d^3q}{(2\pi)^3} \frac{q}{E}n^in^jn^k \frac{\partial g}{\partial x^i}$$
where $n^i$, $n^j$ and $n^k$ are the unit vectors and $g$ is a function of the ...
- 308
1
vote
1
answer
94
views
The content of early universe and max limit on the number of particles
When the size of the observable universe was small as a proton what was the physical content of it?
Was there a limit of particles in such a tiny volume with enormous high density and energy?
Can we ...
- 489
5
votes
3
answers
926
views
Why does the CMB have a spectrum like a black-body radiation?
Equilibrium distributions of particles (Maxwell, Boltzmann, Saha) are achieved by the particle collisions. On the other hand, photons do not interact with each other. From the introductory course in ...
- 51
1
vote
0
answers
36
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Why are two assumptions for the approximation from the BE, FD to MB distribution and $T$ invariance in the Boltzmann equation reasonable?
As the title says, I can not understand whether the assumptions
are reasonable.
If an interaction $1+2\leftrightarrow 3+4$ is taken into account, the number variance in time is proportional to
$$|\...
4
votes
1
answer
133
views
Question about deriving the amount of pressure due to an element of momentum space in cosmology (Baumann Cosmology Book Eq. 3.10)
In Daniel Baumann's cosmology book Eq. (3.10) and cosmology lecture notes Eq. (3.2.18) he states that the pressure in the early universe can be defined as
$$P=\frac g{(2\pi)^3}\int d^3 p\ f(p)\times\...
- 394
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votes
0
answers
40
views
Is the 2nd law of thermodynamics more of a likelihood than a law? [duplicate]
Given enough time, even extremely unlikely events are certain to happen. This includes spontaneous reduction in disorder.
As an example, given enough time, all the particles of a gas in a container ...
- 1,421
1
vote
0
answers
41
views
Internal degrees of freedom of a Weyl spinor
In statistical mechanics, we have formulas like
$$
n
=
\frac{g}{2\pi^2}
\int_m^\infty
\frac{E(E^2 - m^2)^{1/2}}{e^{(E - \mu)/T} \pm 1}\ dE
.
$$
where $g$ is the internal degrees of freedom. What is ...
- 574
1
vote
2
answers
266
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What is the theoretical value of this phase space invariant?
So I wanted know how to theoretically calculate this phase space invariant (equation $3.31a$ )$R$ in our universe (FLRW metric) during the cosmological nucleosysthesis:
$$R = \int_{p} \frac{\mathcal{...
- 1,389
2
votes
1
answer
169
views
What does the arrow of time and entropy say about the universe and repetition?
This question What is the relationship between how time is viewed in thermodynamics and how time is viewed in general relativity? is close to what I was wondering, but it didn't get into repetition ...
- 123
3
votes
1
answer
1k
views
Conservation of entropy in cosmology
I've been trying to follow the procedure that some books give in order to prove that the entropy of the universe is conserved (S is constant).
It usually goes like this:
Consider the second law of ...
1
vote
0
answers
37
views
Axions approaching thermal distribution
I am reading Sikivie's paper on Axion Cosmology. I have the equation:
$$\frac{\mathrm{d}}{\mathrm{dt}}[R^3(n_a^{th}-n_a^{eq})] = -\Gamma R^3(n_a^{th}-n_a^{eq}) \tag{1}$$
where $R(t)$ is the ...
- 1,222
4
votes
0
answers
97
views
What kind of matter's energy density scales as the inverse of the scale factor
We know that radiation energy density scales as $a^{-4}$ with EoS parameter ($w=\frac{1}{3}$), matter as $a^{-3}$ with ($w=0$), curvature as $a^{-2}$ with ($w=-\frac{1}{3}$).
Then which kind of matter ...
- 682
0
votes
1
answer
71
views
Manipulating the Boltzmann equation for baryons
The Boltzmann equation for baryons is
$$m_p\frac{\partial (n_b u_b^j)}{\partial t} + 4Hm_pn_bu_b^j + \frac{m_pn_b}{a} \frac{\partial \Psi}{\partial x^j} = F_{e\gamma}^j(\vec{x},t)\ \ \ \ \ \ \ \ eqn.(...
- 1,222
1
vote
0
answers
322
views
Fermi-Dirac vs. Maxwell-Boltzmann distribution in the early universe plasma
From my studies, I remember that the quantum effects relative to the bosonic or fermonic nature of the particles play a role only in the conditions of degenerate gas: when the plasma is very dense and ...
15
votes
4
answers
3k
views
If dark matter can't lose kinetic energy, then why is it not traveling at relativistic speeds?
I have read this question:
The only way you can do this is to remove kinetic energy from the system.
With normal matter this is done through electromagnetic interactions, which turn the kinetic ...
- 29k
1
vote
0
answers
192
views
Energy density of particle species in thermal equilibrium
I am reading the book Kolb and Turner "The Early Universe". In the thermodynamics section they mention that the total energy density of different species in equilibrium is
$$\rho=T^4\sum_{\...
user255856
1
vote
1
answer
53
views
Number of interaction of early universe
I am reading An introduction to particle dark matter by Stefano Profumo. I am struggling on a derivation.
As given from the text, in the early universe $\frac{\dot{T}}{T}=-\frac{\dot{a}}{a}=-H$, ...
- 567
7
votes
5
answers
556
views
Can the "Boltzmann brain" concept inform discussions of cosmology?
The Boltzmann brain was originally discussed as a sort of thought-experiment or aid
to reflection on what might possibly happen in the universe. Its first discussion was
in the context of thermal ...
- 59.6k
2
votes
2
answers
151
views
Negative Temperature and bounded Energy of the Universe
I have just learned about the pretty cool idea about negative temperature (although it is "hotter" then positive temperature...) of a two level system. The condition for having negative ...
- 2,794
10
votes
10
answers
11k
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If we were able to prove that the universe is infinite, wouldn't that statistically prove that there is no other forms of life?
I want to begin my explanation using abstract mathematical explanation to repetition possibility by taking independent samples $X_n$ from some continuous probability distribution: https://math....
- 368
5
votes
2
answers
797
views
Can we really apply the second law to the entire universe?
I do not doubt the second law in general, just if it rigorously applied to the entire universe. Here's why I ask this
2nd law - restricted to isolated systems: "The second law may be formulated ...
- 601
60
votes
9
answers
8k
views
How do different definitions of entropy connect with each other?
In many places over the Internet, I have tried to understand entropy.
Many definitions are presented, among which I can formulate three (please correct me if any definition is wrong):
Entropy = ...
- 1,137
2
votes
0
answers
65
views
Bose-Einstein statistics in de Sitter spacetime
The metric in de Sitter spacetime with $(+,-,-,-)$ as metric signature is, in cartesian coordinates:
\begin{equation}
ds^2=dt^2-a^2(t)(dx^2+dy^2+dz^2).
\end{equation}
Where $a(t)=e^{Ht}$. How do I ...
- 2,613
2
votes
2
answers
286
views
Why can't primordial fluctuations be super-"horizon" without inflation?
I am trying to understand why, in cosmology, it is said that the presence of fluctuations at scales above the Hubble distance would not be expected in the absence of inflation or something like it.
We ...
- 59.6k
5
votes
1
answer
98
views
Decoupled Temperature for photons: why is it 0.25 $\rm eV$ rather than 13.6 $\rm eV$?
When calculating the decoupled temperature of photons using Saha' equation for the following process:
\begin{equation}
e^- p\longleftrightarrow H\gamma
\end{equation}
we find that $T_{dec}=3000$ K$=0....
- 699
3
votes
2
answers
557
views
On Boltzmann brain formation
As a popular thought conundrum it is said that if our universe can randomly fluctuate into a self-conscious formation (called Boltzmann brain), then given enough time it will do so and hence the ...
- 79
4
votes
1
answer
280
views
What happens here when the Second Law of Thermodynamics doesn't hold true?
In the 1920s, theoretical physicists, most notably Albert Einstein, considered the possibility of a cyclic model for the universe as an (everlasting) alternative to the model of an expanding universe. ...
- 1,764
2
votes
0
answers
87
views
How is the entropy of the universe defined?
My understanding is that to define the entropy of a system what you have to do is as follows:
Define the boundaries of your system.
Define a set of "microstates" of the system.
Define a partition of ...
- 225
0
votes
1
answer
218
views
Chemical equilibrium for a $2-2$ scattering and Boltzmann equation in cosmology and astroparticle physics
Definition of chemical equilibrium According to the definition of chemical equilibrium in Wikipedia, it is a situation where the rate of the forward reaction is same as the rate of backward reaction ...
- 26.8k
0
votes
1
answer
303
views
Energy density, pressure and temperature for massive neutrinos in cosmology
I want to be able to numerically compute the mean energy density and pressure for a massive neutrino species in cosmology, at any given scale factor $a$. These are given in terms of the distribution ...
- 121
6
votes
1
answer
1k
views
Derivation of relativistic pressure
As you can find in many cosmology textbooks, the relativistic pressure in quantum statistical mechanics can be witten as below:
$$p=g \int \frac{d^3P}{(2 \pi \hbar)^3} \frac{c^2 |\mathbf{P}|^2}{3E(\...
- 83
-1
votes
2
answers
153
views
How much energy is required by a planet like Earth to support and maintain life as we know it?
How much energy is required by a planet like Earth to support and maintain life as we know it and could this energy be provided by a star the size of the moon, located where the moon is without ...
-1
votes
3
answers
93
views
If the moon produced energy like the Sun would it radiate the same energy and light on Earth as the Sun? [closed]
I would like to know if the moon where to shine like the sun would the Earth get the same energy and light as it is getting now from the Sun?
2
votes
0
answers
158
views
How could the universe's expansion "remove the mean gravitational potential"?
I'm citing W. C. Saslaw's The Distribution of the Galaxies: Gravitational Clustering in Cosmology, chapter 25, where he adresses (what seems to be a Newtonian approximation of) the thermodynamic ...
- 2,463
9
votes
1
answer
446
views
How to understand that correlators measure physical correlation?
Background In physics, we always come across $n$-point correlators (e.g. 2-point correlators). For instance, in phase transitions, one is interested in finding the correlation function between order ...
- 26.8k
2
votes
1
answer
91
views
No Temperature in an Expanding Universe?
The statistical definition of temperature as $\bigg(\dfrac{\partial \ln \Omega}{\partial E}\bigg)^{-1}$ inherently assumes the existence of a well-defined energy. To my understanding, a well-defined ...
user87745
3
votes
1
answer
312
views
How to compute Coulomb scattering rate(in early universe)?
I want to compare Coulomb scattering rate with Hubble's expansion rate.
To compute Coulomb scattering rate, I know that
$$ \Gamma = n_p\sigma v_{rel} $$
Here, $\sigma = \frac{e^{4}}{(4\pi\...
- 624
1
vote
0
answers
32
views
What might be the equillibrium Distribution Function of Dark Matter?
Can we assign a thermal distribution function to Dark Matter(not necessarily known ones)?
Is it possible that they won't even follow thermal distribution function and maybe a function of something ...
- 624
4
votes
1
answer
353
views
Pressure in Scott Dodelson's book Modern Cosmology
I am completely at a loss in deriving the equation 2.62 given in Scott Dodelson's Modern Cosmology book. That is the following expression for the pressure of Fermi-Dirac and Bose-Einstein ...
- 568
2
votes
1
answer
349
views
Does ΛCDM imply infinitely many Boltzmann brains?
In this paper, Sean Carroll basically argues that the ΛCDM/'flat lambda' cosmological model has to be wrong since it implies that we're probably Boltzmann brains (and therefore shouldn't trust our ...
- 77
-3
votes
1
answer
225
views
Must the origin of the universe be non-physical? [duplicate]
Since the 2nd law of thermodynamics says the universe's entropy is constantly increasing, then its initial entropy must have been smaller than it is today.
However, no known physical law can reduce ...
- 173
1
vote
0
answers
73
views
Statistical physics in cosmolgy
In https://ned.ipac.caltech.edu/level5/Sept03/Trodden/Trodden4_2.html and also in Mo et al. (Galaxy Formation and Evolution) the number density, energy density and pressure of the different particle ...
- 317
1
vote
2
answers
180
views
Kolb and Turner's expresion for number density Eqn. (3.46) and pressure in Eqn. (3.48)
Having defined the phase space distribution function $f(\textbf{r},\textbf{p},t)$ in $\mu-$space, one can express the information that there are $N$ particles in the volume $V$ through the condition \...
- 26.8k