What is Boltzmann equation: Definition and 40 Discussions
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.
The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.
The equation arises not by analyzing the individual positions and momenta of each particle in the fluid but rather by considering a probability distribution for the position and momentum of a typical particle—that is, the probability that the particle occupies a given very small region of space (mathematically the volume element
d
3
r
{\displaystyle \mathrm {d} ^{3}{\bf {r}}}
) centered at the position
r
{\displaystyle {\bf {r}}}
, and has momentum nearly equal to a given momentum vector
p
{\displaystyle {\bf {p}}}
(thus occupying a very small region of momentum space
d
3
p
{\displaystyle \mathrm {d} ^{3}{\bf {p}}}
), at an instant of time.
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). See also convection–diffusion equation.
The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising.
Boltzmann entropy definition is given by: $$ S = k_B lnW $$ where ##W## is the weight of the configuration which has the maximum number of microstates.
This equation is used everywhere in statistical thermodynamics and I saw it in the derivation of Gibbs entropy. However, I can't find the...
I've been trying for a very long time to show that the following integral:
$$ I_D=2{\displaystyle \int} \, {\displaystyle \prod_{i=1}^3} d \Pi_i \, (2\pi
)^4\delta^4(p_H-p_L-p_R) |{\cal M}({e_L}^c e_R \leftrightarrow h^*)|^2
f_{L}^0f_{R}^0(1+f_{H}^0). $$
can be reduced to one dimension:
$$
I_D...
I'm trying to follow Scott Dodelson's Modern Cosmology. Specifically Chapter 3. Coverage of the subject of baryogenesis appears to be missing from Dodleson's book, so I'm trying to reconstruct things on my own.
This represents the formula:$$n_p[T]=g_p\space (\frac{k_b\space m_p\space...
Let's look at the Boltzmann equation
$$ \frac {p_{i}} {p_{j}} = e^{\frac{E_{j}-E_{i}} {kT}},$$
and take infinitely high temperature, the RHS becomes 1. I interpreted that this means every energy level is occupied by equal number of electrons. But if T is high enough, wouldn't the hydrogen atom...
Hello guys, i am currently studying for my msc thesis in theoretical physics and i need to find the derivation of relativistic Boltzmann equation, any suggestions ? Because i ve searched for papers/books for it and couldn't find anything. Any ideas?
Thx in advance
Hi,
I am working on a Boltzmann equation question and I know that the solution I am looking for is that:
log(nij/nji)=log(gij/gji)-Eij(eV)(5040/T)
The only thing I don't understand is why log(e-Eij/kt) = θ = 5040/T
From what I have read in textbooks, it is just a given, but I really want to...
Dear community,
I am studying some equations related to the acceleration of cosmic rays(CRs) in magnetized plasma and I have seen a couple the equations I am not able to understand.
First, I see that it is used as time-dependent Boltzmann equation for the CRs
∂ƒ/∂t + (vx + u)∂ƒ/∂x -...
Hello! I read today, in the context of DM, about the Boltzmann equation: $$L[f]=C[f]$$ where ##L[f]## is the Liouville operator (basically ##\frac{df}{dt}##), with ##f(x,v,t)## being the phase-space distribution of the system and ##C[f]## being the collision operator. I am a bit confused about...
I have been studying the statistical mechanics' viewpoint of fluid dynamics by considering the derivation of Navier-Stokes' equations from the Boltzmann equation involving the whole Chapman-Enskog expansion. It is clear that through this process, it is possible to account for the dependence of...
I’ve been reading up about Boltzmann transport equations, and the concept of molecular chaos has come up, in which one assumes the velocities of particles are assumed to be uncorrelated. I’m a bit confused about the concept though. In what sense do the velocities become correlated in the first...
From wikipedia,
The general equation is $$\frac{df}{dt} = (\frac{∂f}{∂t})_{force}+(\frac{∂f}{∂t})_{diff}+(\frac{∂f}{∂t})_{coll}$$
where the "force" term external force, the "diff" term represents the diffusion of particles, and "coll" is the collision term.
So shouldn't be df/dt=0 when it is...
Hi, as you can see at the end of the picture/attached file collision integral is approximated to a discrete sum. Could you express how this approximation is derived?
I've seen the derivation where:
## \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec {v} \cdot \vec {\nabla} f + \vec {a} \cdot \vec \nabla_{\vec{v}} f ##
Although I was told this should more generally be written as:
## \frac {df}{dt} = \frac {\partial f}{\partial t} + \vec {\nabla}...
Homework Statement
You are performing an experiment to validate the Stefan Boltzmann equation. What irradiance would you measure at a temperature of 109C? The emissivity of your thermal heat source is 0.81 and your thermopile measures 0 W/m2 at 27 C when directed towards a blackbody. Submit...
Homework Statement
I must calculate chemical potential using the Boltzmann equation in relaxation time approximation $$f=f^0-\tau v_z^2 \partial f^0/\partial z,$$ where ##f^0## is given as
$$f^0 = 2(\frac{m}{2\pi\hbar})^3 \frac{1}{\exp{\beta(z)(\frac{mv^2}{2}-\mu(z))}+1}$$
I have to consider...
Hi everyone,
I plan to do a simulation of a Boltzmann equation with experimentally known scattering between two particles. Initially I intend to incorporate the scattering into the collision integral and use Lattice Boltzmann Equation (LBE) afterwards. But I only see LGBK (DnQb) which requires...
Homework Statement
Consider the motion of charged particles, in one dimension only, in
the presence of an electric potential V ( x). Show, by direct substitution,
that a function of the form
f=f(1/mv^2 + qV)
is a solution of the Boltzmann equation under steady-state conditions.
Homework...
There was an equation I saw before and I think it pertains to Boltzmann and thermodynamics. I think it describes the entropy of a system. From what I can remember, it involves the symbol omega to denote micro states, k for a constant, and a logarithm somewhere. Anyways, hopefully some one knows...
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...
Hi guys, I recently started reading/working through Scott Dodelson's Modern Cosmology in preparation for a Masters course I'm taking next year and one of the exercises has stumped me and (arghhh!) its not one of the solved ones in the back!
It is in Chapter 4 (The Boltzmann equations) and is...
Homework Statement
Starting from the kinetic equation for the distribution function F*(t, r, v) of some
labelled particle admixture in a gas, derive the self-diffusion equation
∂n*/∂t = D∇2n*
for the number density n*(t,r) = ∫d3vF*(t,r,v) of the labelled particles. Derive also the expression...
I’m searching for an introduction to relativistic Boltzmann equation. (Sorry, I know this is a question about learning material)
I’ve read an excellent script from David Tong about non-relativistic kinetic theory (from Liouville to Navier-Stokes using Boltzmann equation (see link below)...
I'm currently reading about the Boltzmann equation, used for the early Universe.
The equation I end up with, after some simplifications is the following:
\begin{equation}
a^{-3}\frac{d}{dt}\left(n_1a^3\right) = n_1^{(0)}n_2^{(0)}\langle\sigma v\rangle\left[\frac{n_3 n_4}{n_3^{(0)}n_4^{(0)}} -...
Hi there. I have a couple of questions regarding the derivation of the Boltzmann equation in Dodelson for photons when given scalar overdensity perturbations to the FRW metric.
To start with, let ##\Theta(\vec{x})## denote the temperature perturbations to the Bose-Einstein distribution of the...
I am little bit confused about derivation for Boltzmann equation for electron
look at this link http://relativity.livingreviews.org/Articles/lrr-2008-10/articlesu25.html
which is final boltazmann equation ?
Below is part of derivation of the Boltzmann equation in an electric and magnetic field.
I don't understand how to arrive at the bottom equation though. It is known that the dependence of the original distribution function is the given. My idea is to use chain rule but I don't see how to get a...
Homework Statement
Solve the Boltzmann equation for a homogeneous plasma with not external forces present when the collision term is
\frac{\partial f(v,t)}{\partial t} = -\nu (f(v,t) - f_0(v)),
where \nu and f_0 are constants.
Homework Equations
Boltzmann equation
\frac{\partial...
Homework Statement
Hi.
I have a course where I am supposed to show how to get to the Boltzmann equation from a nonequilibrium distrubution function.
At the moment I'm kinda lost to how this is done, so hopefully a hint or two would be of some help :)Homework Equations
The nonequilibrium...
Why I can't calculate the same answer as the solution below? I use the value as what the below solution showed.
Q:
Where Nj is the number of atoms in excited state, No is the number of atoms in the
ground state, Pj and Po are constants determined by the number of states having equal
energy...
In the Boltzmann equation,
{\bf{L}}\left[ f \right] = {\bf{C}}\left[ f \right], the right which is the collision term and in general it is{\bf{C}}[f]=\sum\limits_{p,p_1} {|Amplitiude|^2}{f(p)}. and explicitly, the collision term for decaying process is
\begin{eqnarray}
{\bf{C}}\left[ f...
Hi
I am trying to make the following equation to get Boltzmann equation which I write below.
f(\mathbf{x}+\frac{\mathbf{p}}{m}dt,\mathbf{p} + \mathbf{F}dt,t+dt) \,d\mathbf{x}\,d\mathbf{p}
- f(\mathbf{x},\mathbf{p},t)d\mathbf{x}\,d\mathbf{p} =
\left. \frac{\partial...
I've never heard of this before. Maybe I should learn more physics:
http://www.upenn.edu/pennnews/news/university-pennsylvania-mathematicians-solve-140-year-old-boltzmann-equation-gaseous-behaviors
http://www.pnas.org/content/107/13/5744.abstract?sid=08ced618-f372-4c0d-a80c-64786d92c3ff
I am studying kinetic theory. The Boltzmann equation can be derived from the liouville equation, I want to know how to deduce it , which books explore the question.
I am reading a book( the title is kinetic theory, the author is R.L.Liboff), in this book the method proposed by Prigogine is used...
Homework Statement
If we assume entropy is a function of the multiplicity, \Omega, (S=k*f(\Omega)) show that that function f(\Omega) is ln(\Omega).Homework Equations
The Attempt at a Solution
\Omega can be written as N!/ni!. By using stirling's approximation, this becomes \Omega=...
Homework Statement
Consider a pure hydrogen hydrogen gas ata tempeature of 10080 K. What is the ratio of the populations of the ground state(n=1) to the first excited state(n=2). Note that the energy difference is 1.634e-18 joules between these two states. At what temperature would both...