Frequently asked questions and Boltzmann equation: Difference between pages

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==Registration and content creation==
{{stub}}


===How do I register as a user?===
The Boltzmann equation is a nonlinear evolution equation first put forward by Ludwig Boltzmann to describe the configuration of particles in a gas, but only statistically. However, this equation and related equations are used in other physical situations, such as in optics. The corresponding linear inverse problem is also used in tomography <ref name="Bal2009"/>
In order to be able to create or edit the pages, you must be a registered user. We welcome all users to register. Unfortunately we have not yet figured out how to set a public registration page and at the same time avoid spammers. So right now the only way to register is by sending an email to one of the administrators: [[User:Nestor| Nestor Guillen]], [[User:RayAYag |Ray Yang]], [[User:Russell|Russell Schwab]] or [[User:Luis|Luis Silvestre]].


===How do I edit a page?===
In reality, the Boltzmann equation is not a single equation but a family of equations, where one obtains different equations depending on the nature of the interaction between particles (see below). Although there has been a lot of progress in the analysis of the Cauchy problem under many circumstances, the broad understanding of the equation and the dynamics of its solutions remains largely incomplete. For an overview of the mathematical issues revolving around this equation see for instance <ref name="Vil2002"/>. A basic reference is also <ref name="CerIllPul1994"/>.
There is an edit button on the top that lets you edit each page. You only see it if you are a registered user.


===How do I create a new page?===
== The classical Boltzmann equation ==  
Every time a non existent page is referenced, the link appears red. If you click the red link, you will edit the new page. You need to be a registered user to perform this action.


===Where can I learn how to use a wiki?===
As explained originally by Boltzmann in the probabilistic description of a gas, we assume that the probability that a particle in a gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by some function
Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.


The [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Help:Editing_FAQ dispersive wiki] has a good FAQ section which may be worth reading first.
\begin{equation*}
\int_A f(x,v,t)dxdy.
\end{equation*}


===How do I write a bibliographical reference?===
Then, under certain (natural) physical assumptions, Boltzmann derived an evolution equation for $f(x,v,t)$. In particular,  if one imposes $f$ at time $t=0$ then $f$ should  solve the Cauchy problem
You can use the website http://math.uchicago.edu/~luis/bib.html to generate the references from a BibTeX entry.


The website http://zeteo.info/ used to provide this service as well but seems to be down.
\begin{equation}\label{eqn: Cauchy problem}\tag{1}
\left \{ \begin{array}{rll}
\partial_t f + v \cdot \nabla_x f  & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+,\\
f  & = f_0 & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \{ 0 \}.
\end{array}\right.
\end{equation}


Look at the pages that are already created as an example of how to make the list of bibliography.
where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by


===Are there any rules on what we can write?===
\begin{equation*}
Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_*.
\end{equation*}


These are the guidelines:
here $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write


# Do not write anything offensive or derogative.
\begin{align*}
# Avoid using words like ''outstanding'', ''remarkable'', ''groundbreaking'' or ''tour de force'' when describing a result.
v' & = v-(v-v_*,e)e\\
# If you think that an article is a triviality or is wrong, it is better not to include it in the citations.
v'_* & = v_*+(v-v_*,e)e
\end{align*}


Making a contribution to the wiki is fairly simple and it can take an arbitrarily small amount of time. Most of the articles are currently not perfect. You can add a paragraph here and there if you have little time. Or you can add a new article which just states a result and hope that someone will pick up the rest.
and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.


When writing an article, also keep the following priorities in mind.
== Collision Invariants ==
# It has to be '''easy to read'''. This is the top priority.
# It should be clear what is proved and what is not. But see comment below.
# Avoid too much technicalities. If the assumptions of a general result are too complicated, it is ok to just list the major examples.
# Give references to the papers where theorems are proved.
# Explain the ideas of the proofs when appropriate.
# If a result is a nonlocal version of a classical theorem, mention it.


===Would a user registration ever be revoked?===
The Cauchy problem \ref{eqn: Cauchy problem} enjoys several conservation laws, which in the Boltzmann literature are known as collision invariants. Take $\phi(v)$ to be any of the following functions


Most likely no. Although it could happen if it is used to spam or if one of the guidelines above is grossly disobeyed.
\begin{equation*}
\phi(v) = 1, \;\;v,\;\; \tfrac{|v|^2}{2}
\end{equation*}
\begin{equation*}
\text{(the first and  third ones are real valued functions, the second one is vector valued)}
\end{equation*}


===Ok. I registered and want to contribute. What can I do?===
and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have


Right now, in [[Mwiki:Current events|Current Events]] there is a '''to do''' list. Click on the links and edit the pages. The red links denote that there is a page needed that was not even started.
\begin{equation*}
\frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0
\end{equation*}


The organization of the wiki is not fully established. We may need some extra index pages or categories.
according to what $\phi$ we pick this equation corresponds to  conservation of mass, conservation momentum or conservation of energy.


There are several pages already. But the wiki is still in a very premature state. Most pages need some more work. The idea of having a wiki is that no version of a page will ever be a final version. However, right now they make that very apparent.
== The Landau Equation ==


if you don't know how to start, you can use the pages that are already written as a sample.
A closely related evolution equation is the [[Landau equation]]. For Coulumb interactions, the corresponding collision kernel $B$ always diverges, instead in this case, one uses an equation (which is an asymptotic limit of Boltzmann equation) first derived by Landau,


\begin{equation*}
f_t + x\cdot \nabla_y f = Q_{L}(f,f)
\end{equation*}


== About the philosophy of the site ==
where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as


=== Who wrote all this? ===
\begin{equation*}
The users of the wiki. Several people.
Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2
\end{equation*}


=== Whom do the pages belong to? ===
where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.  
Nobody.


=== What if I disagree with something that the wiki says? ===
You are free to edit its content. If you do not, we will be offended with you.


=== Why did you create a wiki? ===
Note that when $f$ is independent of $x$ the above equation becomes second-order parabolic equation where the coefficients depend non-locally on $f$, in particular, one has an apriori estimate for all higher derivatives of $f$ in terms of its $L^\infty$ and $L^1$ norms (via a bootstrapping argument).
If this wiki project works well, it may become a massive reference which is always up to date. It can potentially be better than a book. For that we need several people involved and willing to edit the articles.


=== Isn't every wiki doomed to fail? ===
== References ==
There are other scientific wiki projects online which can serve as examples: the [http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page dispersive wiki], the [http://www.wikiwaves.org/index.php/Main_Page water wave wiki] and also two quantum physics wikis called [http://qwiki.stanford.edu/index.php/Main_Page Qwiki] and [http://www.quantiki.org/wiki/Main_Page Quantiki].
{{reflist|refs=


The success of a wiki page depends on the contributions made by the users. The current experience with scientific wikis shows a questionable level of success. On the other hand, there are non-scientific wikis which are tremendously successful, for example: the [http://wikitravel.org/ travel guide wiki], the [http://harrypotter.wikia.com/wiki/Main_Page Harry Potter wiki], the [http://recipes.wikia.com/wiki/Recipes_Wiki recipes wiki], the [http://starwars.wikia.com/wiki/Main_Page Star Wars wiki], the [http://www.miwiki.net/ Monkey island wiki], the [http://baseball.wikia.com/ Baseball wiki] or the [http://www.mariowiki.com/ Super Mario wiki], among many others.
<ref name="Bal2009">{{Citation | last1=Bal | first1=G. | title=Inverse transport theory and applications | publisher=IOP Publishing | year=2009 | journal=Inverse Problems | volume=25 | issue=5 | pages=053001}}</ref>


It would be interesting to understand this distinction of success between the non-scientific wikis and the scientific ones.
<ref name="Vil2002">{{Citation | last1=Villani | first1=C. | title=A review of mathematical topics in collisional kinetic theory | publisher=[[Elsevier]] | year=2002 | journal=Handbook of mathematical fluid dynamics | volume=1 | pages=71–74}}</ref>


=== Why should I spend time writing on this wiki? ===
<ref name="CerIllPul1994">{{Citation | last1=Cercignani | first1=Carlo | last2=Illner | first2=R. | last3=Pulvirenti | first3=M. | title=The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106) | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1994}}</ref>
If you are a mathematician who has done some research in the area, you definitely want people to know about your results. If you write an easy to read reference in this wiki, that would help more people know about your work and how it is related with other results in the area. Just be careful not to overplay the importance of your own results (or you may be banned from editing again). The appropriate thing to do is to write about all the related results by other people as much as you write about yours. Also remember to follow the rules above in the [[#Are there any rules on what we can write?|writing guidelines]].
}}
 
If you are a student learning the subject, writing in this wiki may help you understand the topics better (especially if someone comes after you to correct you). Moreover, if you are learning the subject, you probably appreciate the existence of this wiki more than others and are willing to contribute back.
 
There is an interesting video about open science here [http://www.youtube.com/watch?v=DnWocYKqvhw]

Revision as of 20:11, 18 October 2013

This article is a stub. You can help this nonlocal wiki by expanding it.

The Boltzmann equation is a nonlinear evolution equation first put forward by Ludwig Boltzmann to describe the configuration of particles in a gas, but only statistically. However, this equation and related equations are used in other physical situations, such as in optics. The corresponding linear inverse problem is also used in tomography [1]

In reality, the Boltzmann equation is not a single equation but a family of equations, where one obtains different equations depending on the nature of the interaction between particles (see below). Although there has been a lot of progress in the analysis of the Cauchy problem under many circumstances, the broad understanding of the equation and the dynamics of its solutions remains largely incomplete. For an overview of the mathematical issues revolving around this equation see for instance [2]. A basic reference is also [3].

The classical Boltzmann equation

As explained originally by Boltzmann in the probabilistic description of a gas, we assume that the probability that a particle in a gas lies in some region $A$ of phase space $\mathbb{R}^d\times \mathbb{R}^d$ at time $t$ is given by some function

\begin{equation*} \int_A f(x,v,t)dxdy. \end{equation*}

Then, under certain (natural) physical assumptions, Boltzmann derived an evolution equation for $f(x,v,t)$. In particular, if one imposes $f$ at time $t=0$ then $f$ should solve the Cauchy problem

\begin{equation}\label{eqn: Cauchy problem}\tag{1} \left \{ \begin{array}{rll} \partial_t f + v \cdot \nabla_x f & = Q(f,f) & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \mathbb{R}_+,\\ f & = f_0 & \text{ in } \mathbb{R}^d \times \mathbb{R}^d \times \{ 0 \}. \end{array}\right. \end{equation}

where $Q(f,f)$ is the Boltzmann collision operator, a non-local operator given by

\begin{equation*} Q(f,f)(v) = \int_{\mathbb{R}^d}\int_{\mathbb{S}^{d-1}} B(v-v_*,e) (f(v')f(v'_*)-f(v)f(v_*) d\sigma(e) dv_*. \end{equation*}

here $d\sigma$ denotes the Hausdorff measure on $\mathbb{S}^{d-1}$, and given $v,v_* \in \mathbb{R}^d$ and $e \in \mathbb{S}^{d-1}$ we write

\begin{align*} v' & = v-(v-v_*,e)e\\ v'_* & = v_*+(v-v_*,e)e \end{align*}

and $B$, which is known as the Boltzmann collision kernel, measures the strength of collisions in different directions.

Collision Invariants

The Cauchy problem \ref{eqn: Cauchy problem} enjoys several conservation laws, which in the Boltzmann literature are known as collision invariants. Take $\phi(v)$ to be any of the following functions

\begin{equation*} \phi(v) = 1, \;\;v,\;\; \tfrac{|v|^2}{2} \end{equation*} \begin{equation*} \text{(the first and third ones are real valued functions, the second one is vector valued)} \end{equation*}

and let $f(x,v,t)$ be any classical solution to \ref{eqn: Cauchy problem}, then we have

\begin{equation*} \frac{d}{dt}\int_{\mathbb{R}^d\times \mathbb{R}^d} f(x,v,t) \phi(v)\;dx\;dv = 0 \end{equation*}

according to what $\phi$ we pick this equation corresponds to conservation of mass, conservation momentum or conservation of energy.

The Landau Equation

A closely related evolution equation is the Landau equation. For Coulumb interactions, the corresponding collision kernel $B$ always diverges, instead in this case, one uses an equation (which is an asymptotic limit of Boltzmann equation) first derived by Landau,

\begin{equation*} f_t + x\cdot \nabla_y f = Q_{L}(f,f) \end{equation*}

where now $Q_{L}(f,f)$ denotes the Landau collision operator, which can be written as

\begin{equation*} Q_{L}(f,f) = \text{Tr}(A[f]D^2f)+f^2 \end{equation*}

where $A[f]$ is the matrix valued operator given by convolution with the matrix kernel $K(y)= (8\pi|y|)^{-1}\left ( I -\hat y\otimes \hat y)\right )$, $\hat y = y/|y|$.


Note that when $f$ is independent of $x$ the above equation becomes second-order parabolic equation where the coefficients depend non-locally on $f$, in particular, one has an apriori estimate for all higher derivatives of $f$ in terms of its $L^\infty$ and $L^1$ norms (via a bootstrapping argument).

References

  1. Bal, G. (2009), "Inverse transport theory and applications", Inverse Problems (IOP Publishing) 25 (5): 053001 
  2. Villani, C. (2002), "A review of mathematical topics in collisional kinetic theory", Handbook of mathematical fluid dynamics (Elsevier) 1: 71–74 
  3. Cercignani, Carlo; Illner, R.; Pulvirenti, M. (1994), The Mathematical Theory of Dilute Gases (Applied Mathematical Sciences vol 106), Berlin, New York: Springer-Verlag