Johnson Nyquist Noise

Andor wrote:
> Hello Clay,
>
> thanks for your public domain work (including the other papers on your
> site - I especially liked the oscillator paper).

You are welcome and thanks. I hope people find them useful.


>
> I have a query with the Nyquist noise paper. You develop your formular
> in differential form, for example
>
> P(f) = h f / ( exp(h f / k T) - 1) df,
>
> keeping the integral operator df. Further down you integrate and get
>
> P_bar = integral_0^b P(f) df.
>
> which, when substituting the term above becomes
>
> P_bar = integral_0^b h f / ( exp(h f / k T) - 1) df^2,
>
> which is probably not what you want (note the df^2 term). Can't you
> just delete the differential operator in your derivation and set for
> example
>
> P(f) = h f / ( exp(h f / k T) - 1)?
>
> Regards,
> Andor


Hello Andor,

Thanks for pointing out my sloppiness. The thing about the df term is
wanting to show during the density of states transformation that we are
going from the summation to an integration, so I really don't want to
drop the df. So I fixed the last set of equations so as to not have a
df squared term. I think this is okay since we modified something
inside of a summation to become something inside of an integration.
This pretty much follows the standard development in statistical
physics texts for handling the summation.

Thanks,
Clay

Clay wrote:
> Andor wrote:
> > Hello Clay,
>
> > thanks for your public domain work (including the other papers on your
> > site - I especially liked the oscillator paper).You are welcome and thanks.
> I hope people find them useful.

I certainly did.

> Thanks for pointing out my sloppiness. The thing about the df term is
> wanting to show during the density of states transformation that we are
> going from the summation to an integration, so I really don't want to
> drop the df. So I fixed the last set of equations so as to not have a
> df squared term. I think this is okay since we modified something
> inside of a summation to become something inside of an integration.

Except that the notation of summation does not require a "df". I would
prefer that P(f) be a proper power density function, which one can
integrate over f and use the standard notation

P_bar = integral_0^b P(f) df.

Also, when you plot P(f) the "df" operator magically disappears, and
P(f) becomes a standard function. Strictly speaking, you are plotting

P(f)  / df = h f / ( exp(h f / k T) - 1)

and not

P(f) = h f / ( exp(h f / k T) - 1)T df,

except that P(f) / df has a different meaning altogether (never mind
the factor k T).

> This pretty much follows the standard development in statistical
> physics texts for handling the summation.

You must admit that something like

integral_0^b P(f)

looks awful. I never did like statistical physics (except quantum
information theory of course :-).

Regards,
Andor

Andor wrote:
>
> You must admit that something like
>
> integral_0^b P(f)
>
> looks awful. I never did like statistical physics (except quantum
> information theory of course :-).
>
> Regards,
> Andor

Hello Andor,

Yes I admit, it looks odd to say the least. I'll see if I can work out
a "cleaner" way of doing this. Stat. Phys. is a slippery subject to say
the least. I have a book[1] that takes a nonstandard approach, and the
author wants to refute much of the standard paradigm about how Stat.
Phys. is approached. When I get time I'm going to revisit his ideas and
see if I can sort them out.

Thanks,
Clay

[1] Lavenda, Bernard, "Statistical Physics, a Probabilistic Approach"
John Wiley, 1991