# Velocity Distribution

Hello All,

Why in LAMMPS the velocities are created based on a Uniform or Gaussian distribution but not the Maxwell-Boltzmann ?

Thank you.

Best regards,

Leila

Easy to generate, I guess.

Yi

And what is the difference between Maxwell distribution for velocities and Gaussian, actually?

Oleg

12.02.2015, 19:09, "Chitsazi, Leila (MU-Student)" <[email protected]>:

By Maxwell-Boltzmann distribution I mean the Maxwell-Boltzmann speed distribution (Maxwell distribution) not the Maxwell-Boltzmann velocity vector distribution.

Best,
Leila

Yi,

Thank you. But I’m not sure if that’s the only reason. Is it ?

Best,

Leila

It is the same. If you have gaussian distribution for velocity components (as it is with velocity command), you have the Maxwell speed distribution.

Oleg

12.02.2015, 19:42, "Chitsazi, Leila (MU-Student)" <[email protected]>:

Here a link to Oleg’s theorem for memory refreshment.
https://casper.berkeley.edu/astrobaki/index.php/Maxwellian_velocity_distribution

Carlos

I know that Maxwell speed distribution can be formulated from Maxwell-boltzmann velocity vector distribution.

Leila

Thank you. I read it before

That was a question that we were talking in our group meeting yesterday. One can be formulated from the other one. So does it matter which one to use to create the distribution of velocities ?

Best,

Leila

My first guess would be no because MB is the result of integrating over the gaussians thus, when you integrate you always loose information. Is like trying to know the dimensions of a body by only knowing its volume. As a general rule you can’t retrieve info after integration has been performed unless extra knowledge is incorporated.
Carlos

PS: Pure mathematicians may feel tempted to comment further…

Carlos,

Thank you for your reply. So based on what Yi also said it’s easier to use Maxwell-Boltzmann velocity distribution. I read in LAMMPS that “for Gaussian distribution velocities are created in processor-independent way and gives the same initial state independent the no. of processors”. Do you think using MB velocity distribution versus the other one has anything to do with what is mentioned above.

Best,

Leila

When you use the velocity command, you’re allocating components of the velocity, not the speed. The only way you can do that is by drawing velocities randomly from the velocity distribution - knowing the distribution of speeds won’t help you.

Not being a pure matematician…

In MD code you need the velocity distribution anyway (each velocity component for each atom is set independently). One could - technically - reconstruct random velocities from speeds by picking a value from MB speed distribution and then combining this absolute value of the velocity vector with two uniformly distributed polar coordinates and finding projections of the resultant vector on x, y, z axes (we know that angles are distributed uniformly - the “extra knowledge”). But there is no reason to start from speed distribution (backwards) and do all the excessive work.

Oleg

12.02.2015, 20:17, “Carlos Campana” <[email protected]…24…>:

Oleg, Carlos and Yi,

Best,

Leila

Thanks.

Best,

Leila

OK but my answer was trying to be more general, beyond MD, Lammps etc. In general the knowledge of “only” certain higher-level probability distribution cannot give you back a lower-level distribution from which the former one was obtained. I suppose your solution works but yes, you have added the extra knowledge I was talking about. I took Leila’s Q more from a philosophical point of view.
Carlos

there is one thing in this discussion that i feel is missing: for all
but a few extreme cases, it doesn't really matter much what kind of
distribution is initially assigned. the system is not (yet) in
equilibrium and while the equilibrium is approached, the distribution
will naturally adjust to what it is supposed to be, irrelevant of what
was initially assigned. especially, if you factor in that different
atoms may be further away or closer to potential energy that would be
consistent with an equilibrium configuration.

axel.

Totally agree

Best,
Oleg

Carlos Campana <[email protected]…24…> 12 февраля 2015 г. 21:04:46 написал:

Dear Axel,
Can you explain a little bit about those few extreme cases ?
Thank you.

Best,
Leila

Leila, you probably don’t want this to turn into a weird discussion on strange(pathological) periodic solutions to n-body equations (or numerics for that matter).

If you are interested in why what Axel says is typically true, you might read a few articles/books on Henri Poincare, Alexander Lyapunov, Kolmogorov, Sinai, etc. It is interesting stuff but not really appropriate or productive for the mailing list I feel.