J.-I. IZPURA

461

served noise in TE. This becomes apparent when one

“reads” the circuit of Figure 1 with Thermodynamics in

mind: the suspicious dependence of the noise density

4

i

Sf kTR

A2/Hz on the resistance R* it is going to

drive, is the required one to have a mean square voltage

noise

2

vt kTCV2 independent of R* for the

common case of circuits where

12π

cQ

RC f

.

Since fQ > 6 THz at room T, any stray capacitance added

to the Cd = τd/R of a resistor [4] is enough to make true

this cQ

f

*

4kTR

condition where the noise density

v (V2/Hz) is pro-

portional to R* whereas the circuit bandwidth is inversely

proportional to R*. This means that R* has nothing to do

with the integral of

2

Sf R

*

4

v

Sf kTR

vfrom f 0 to f , which

is:

Sf

2

vt kTCV2, thus indicating that the root mean

square (rms) voltage noise does not depend on R*, but on

C. Hence, Fluctuations of energy in C are the mechanism

that generates the Nyquist noise

i

Sf4kT R

A2/Hz

of a resistor, thus giving its advancing character to this

A-model for electrical noise contending that the circuit

element that generates the Johnson noise in resistors is

their capacitance. The complete view in time domain of

these Fluctuations of electrical energy followed by Dis-

sipations of the energy unbalance set by each Fluctuation

is given in [9].

Therefore, the FDT relates the mean square fluctuating

force (voltage) to the friction factor (conductance) in a

noise system provided fluctuations of energy can take

place in it. The non-null mass M (or inertia) of the large

particle sets this Degree of Freedom in [3] that in our

A-model is due to C. This is why we found hard to apply

the FDT to the pure resistance of [1], whose C = 0 (e.g. a

large particle of M = 0 in [3]), poses some troubles be-

cause the collision of a small particle with a “big one” of

null mass wouldn’t be a collision (e.g. no transfer of ki-

netic energy nor momentum would take place). This

non-sense situation for M = 0 shows the non-sense pro-

posal of [1] with C = 0, where only instantaneous fluc-

tuations of energy (e.g. δtc = 0) could store some energy

in such “device”, fluctuations that do not exist [7]. Note

that the assumption of this sort of “energy-conserving”

fluctuations in [3] (δtc = 0) led to the aforesaid paradox.

To conclude we will say that words like: totally elastic,

energy conserving, totally dissipative, pure resistance,

pure capacitance and so on must be used with care be-

cause they can exclude other phenomena that are essen-

tial to understand the problem at hand. This is so for

magnitudes like electrical power that depending on the

existence in time and on the change with time of the

electrical voltage has two orthogonal terms, none of

which describes completely the noisy system. Appendix

I shows active and reactive power as two orthogonal

terms of instantaneous power linked with Dissipation and

Fluctuation of electrical energy in resistors and capaci-

tors. It is worth reading where the electrical energy fluc-

tuating thermally was stored in the compound device

Nyquist used to apply Thermodynamics [7]. It was stored

in the Susceptances of a lossless Transmission Line,

whence it may be seen that two opposed susceptances

(capacitive and inductive) cancelling mutually at fre-

quency f0, don’t cancel their ability to store electrical

energy at this f0. Considering the need for this ability to

store energy in noisy systems as it appeared in the Clas-

sical works of Nyquist and Callen & Welton, we will

recover the Physical sense in this Fluctuation-Dissipation

field where electrical noise is perhaps its best known

exponent. And to end this paper showing the agreement

between Quantum Physics and Classical Thermodynam-

ics in the noise field we will say that this kind of agree-

ment also appears in other fields of Physics [10].

5. Acknowledgements

We wish to thank Prof. E. Iborra, head of the GMME, for

encouraging chats about the meaning of “well known” in

research. We also thank Prof. H. Solar for his cordial

welcome to give a talk on these ideas at CEIT, in the

Universidad de Navarra.

6. References

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