Multi-Species Saha Equation

On 25 Jun, 21:42, John Schutkeker
wrote:
Thomas Smid wrote groups.com:

I don't know how you would obtain a 14th order polynomial here. Pair
reactions shouldn't result in anything higher than quadratic (at least
not with the methods I know), and that shouldn''t depend on the number
of constituents in the first place. Or am I getting something wrong
here?

It's because you've got 7 coupled sets of pair reactions, each generating a
quadratic, and they all have to be solve simultaneously for the components.
I finally see how to set up the equation, although I haven't written it
down yet. Basically, I have to take your dot/outer product expressions
above and feed them into the matrix method for solving n equations in n
unknowns.

That will yield the master equation for the polynomial, which could
hopefully then be fed into the new equations for solving polynomials of
arbitrary order. I really do need to start writing this stuff down, before
these insights slip my mind. Aren't there several different matrix methods
for solving coupled sets of equations of identical form?

It seems you are trying to substitute the equations into each other,
which isn't really a method I have ever considered for solving an
extended system of equations. I have so far always used successfully
the iterative method for solving any system of equations, whether
linear or not. The point is that most other methods (e.g. the usual
matrix inversion) can only be used for linear systems, but not like
here for non-linear equations. The iterative method doesn't bother
about whether your equations are linear in the unknowns or not, it
will (according to my experience) converge in any case if the system
of equations is fully determined (which should be the case if you are
dealing with a realistic equation and you haven't made any mistake in
the corresponding formulae). Also, it works still well in cases where
the matrix would be close to singular (which would for instance lead
to problems with matrix inversion). With the iterative method it just
would take more iterations then.

By the way, I have added now the separately posted equilibrium
equation (and its explanation and solution) to my program
documentation under http://www.plasmaphysics.org.uk/programs/iondens.htm
..

Also, a slight correction to what I said earlier: the primary
production rate q(i) can of course be zero for some constituents. As
long as it is different from zero for at least one constituent, one
still has a primary ionization source. The ionization of the other
constituents would then exclusively be produced by secondary charge
exchange reactions.

Thomas

Thomas Smid wrote in
ups.com:

On 25 Jun, 21:42, John Schutkeker
wrote:
Thomas Smid wrote
groups.com:

I don't know how you would obtain a 14th order polynomial here.
Pair reactions shouldn't result in anything higher than quadratic
(at least not with the methods I know), and that shouldn''t depend
on the number of constituents in the first place. Or am I getting
something wrong here?

It's because you've got 7 coupled sets of pair reactions, each
generating a quadratic, and they all have to be solve simultaneously
for the components. I finally see how to set up the equation,
although I haven't written it down yet. Basically, I have to take
your dot/outer product expressions above and feed them into the
matrix method for solving n equations in n unknowns.

That will yield the master equation for the polynomial, which could
hopefully then be fed into the new equations for solving polynomials
of arbitrary order. I really do need to start writing this stuff
down, before these insights slip my mind. Aren't there several
different matrix methods for solving coupled sets of equations of
identical form?

It seems you are trying to substitute the equations into each other,
which isn't really a method I have ever considered for solving an
extended system of equations. I have so far always used successfully
the iterative method for solving any system of equations, whether
linear or not. The point is that most other methods (e.g. the usual
matrix inversion) can only be used for linear systems, but not like
here for non-linear equations. The iterative method doesn't bother
about whether your equations are linear in the unknowns or not, it
will (according to my experience) converge in any case if the system
of equations is fully determined (which should be the case if you are
dealing with a realistic equation and you haven't made any mistake in
the corresponding formulae).

Going numerical is an irreversible process, and trading equations for
humongous tables generates entropy faster than Maxwell's Demon. Once
you've traded equations for tables, you can never get the equations
back, meaning you can never again do algebra on them, to use them in
another application. After arithmetic, formal logic and Occam's Razor,
algebra is the most powerful tool known to science, meaning that you've
thrown the baby out with the bathwater.

Fraidy cats like you make more work for nonlinear scientists than we can
ever possibly handle, because when you see a nonlinear equation, you
immediately assume that it can't be solved, so you dive for cover and
hide behind the computer. Sibnce about '86, nonlinear science has
become bigger than Galactus, and the purpose of the computer is not to
replace solutions to nonlinear equations, but to guide them. I should
thank you for making work for me, because I'll never be out of a job,
but I'm suffering from nonlinear information overload. I've got so many
good project ideas that I can't give them all away.

Also, it works still well in cases where
the matrix would be close to singular (which would for instance lead
to problems with matrix inversion). With the iterative method it just
would take more iterations then.

If your matrix is singular, you're at a critical point, and are trying
to solve an eigenvalue problem. That requires different solution
methods than matrix inversion, but they're still matrix methods.

Bruce Scott TOK ] wrote in
:

john Schutkeker wrote:

Bruce Scott TOK ] wrote in
:

John Schutkeker wrote:

point of curiousity would be about what conditions in a plasma
chemical reactor vessle could cause such a divergence. Or rather,
not what [...]

In a tokamak, the usual reason is transport (transport time scale
for the local layer shorter than tau_e times M_i/m_e)

What's the "local layer"?

You usually see T_i T_e in the edge regions of a tokamak.

At a point, measure |grad log T_e|^{-1} and define it as L_perp
(assuming if, as usual, T_e is the profile with the steepest
gradient). Your local layer is a layer with thickness L_perp centered
on your point. If you want to be pedantic about nonlocality, then
measure T_e at the last closed flux surface and then move in until
you've reached one e-folding. The distance is then defined as L_perp.
The transport time of the local layer is given by L_perp divided by
the transport diffusivity. This is usually about twice tau_e M_i/m_e
for usual cases (excepting Alcator C-Mod, which has an unusually high
edge density).

The energy confinement time of the edge layer is usually a bit shorter
than the resistive diffusion time of the same layer, which is why the
MHD equilibrium you have there is resistive.

For more detail see B Scott, Phys Plasmas 10 (2003) 963

Hey Bruce, do you know if it has been proven that turbulent transport
definitely gives the Alcator scaling for energy confinement time, ie.
tau_E ~ n?

Thomas Smid wrote:
n/N = sqrt(F*Q/alpha/N).
So the degree of ionization does here for instance not directly depend
on the local temperature T at all.

Except that the recombination coefficient is temperature-dependent
(proportional to T^-1.5 classically but I think less steeply after
quantum corrections).

The general picture is that in equilibrium, for each species, the rate
of ionizations (from all sources) has to equal the number of
recombinations.

Steve Willner wrote:
The general picture is that in equilibrium, for each species, the rate
of ionizations (from all sources) has to equal the number of
recombinations.

Sorry... clicked 'send' too soon. That last should be "...equal the
_rate_ of recombinations."