The Boltzmann distribution is well known.  However, my question is
this:  what is the time based differential equation of the two
population states as they approach the Boltzamm distribution?

Say my initial population states are N1 and N2.  We know what the
final population states have to be.  What is dN1/dt and dN2/dt in
mathematical terms?

labboy

Evolution of distribution (let's call it f rather then n (which looks
like an integer)) is given by

SEARCH TERM: Boltzman Equation

which is given e.g. here as equation (1)

http://www-ncce.ceg.uiuc.edu/tutorials/bte_dd/html/node1.html

The field is call kinetic theory, transport theory or irreversible
statistical mechanics
http://astron.berkeley.edu/~jrg/ay202/node26.html


The relation of this field of applications to mechanics contains some still
unresolved questions, sometime called 'conundrum of irreversibility', which
refers to difficulty to reconcile time symetry of dynamics with
irreversible behaviour of real materials, which exhibits 'arrow of
time'

http://dmoz.org/Society/Philosophy/Philosophy_of_Science/Physics/Time_and_Timelessness/

This is a very good question for a person who his wasting his or her
time in the lab. Is touches on  unresolved problems in the foundation
of statistical mechanics.

Hedgie

---

Request for Answer Clarification by labboy-ga on 08 Feb 2006 13:21 PST

My question did not relate to fluid flow, rather to the relative
population of different energy levels.  I believe you are confusing
the "Boltzman Equation" with "Boltzman distribution".

Please reference: http://en.wikipedia.org/wiki/Population_inversion

Hence, if I have two energy levels with population N1 and N2, which
does NOT satisfy the Boltzman distribution, then what is the time
evolving population of each state?

Please clarify

---

Clarification of Answer by hedgie-ga on 08 Feb 2006 23:00 PST

Dear labboy,

I am most certainly not confusing
the "Boltzman Equation" with the "Boltzman distribution".
even though both are named after the same Ludwig Boltzman
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Boltzmann.html

I gave you the answer in terms of the classical kinetic theory of simple gases
http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution
since you did not mention that your are particularly interested in the
energy distributions of the internal degrees of freedom

 Classical energy distribution are applicable to the internal states 
 of the gas molecules while quantum distributions may be required in
some applications.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/disfcn.html

The equilibration of internal degrees of a dilute gas would still depend
on intermolecular collision, and whole process is still described by
Boltzman equation, just the  mathematics gets  more complex, e,g,

Non-Equilibrium Solutions of a Hierarchy of Discrete Velocity Models
in the Transport Theory of Dilute Gases Including Internal Degrees of
Freedom

https://online.tu-graz.ac.at/tug_online/fdb_detail.ansicht?cvfanr=F10163&cvorgnr=37&sprache=2

Quantum kinetic Boltzmann equation taking into account the ...
A derivation of the quantum Boltzmann equation is given for identical particles
with internal degrees of freedom. It is shown that the off-diagonal (with ...
http://link.aip.org/link/?JTPHES/84/457/1


... The Non-Equilibrium Equations and the Relaxation of the Internal Degrees of
Freedom ... Application of the Boltzmann Equation to a Jet of Monoatomic Gas ...
http://www.worldscibooks.com/engineering/p387.htm

Due to mathematical complexity, solutions are usually numerical.

SEARCH TERMS: Boltzmann equation, internal degrees of freedom

For flow of fluid one usually uses Navier-Stokes eq.
http://scienceworld.wolfram.com/physics/Navier-StokesEquations.html

which, while it can be derived from the Boltzman equation, is more simple.

 In some applications, one may need to differentiate between processes
on two scales:
 
 1) Evolution to thermodynamical equilibrium, which (for a given
molecule) involves many collision with other molecule.  That process
is described by Boltzman Eq.,  (in which the collision integral may 
include internal degrees of freedom, and be quite complex),

2) isolated molecule, in free flight, between the collisons, may
undergo internal transitions, and emit photon. This  quantum
mechanical process,
is governed by a different equation. There is a constant probability
of spontaneous transitions, so eq Ndot = alpha * N given in your
reference.

 The combined process, is core core of the semi-classical model which
Einstein developed when deriving Black Body radiation:
http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_6/advanced/t6_1_5.html
 Search Terms: Einstein coefficients

 There is no general solutions for complex and nonequilibrium cases. 
 (Einstein assumed gas of linear oscilators in equilibrium).

Search terms: laser pumping dynamics, noneqilibrium plasma
 will bring you studies of specific applications and illustrations  such as
 http://www.iap.uni-bonn.de/lehre/ss01_laserphysics/LaserRatesApplet.html

Hedgie