Journal of Modern Physics, 2011, 2, 457-462
doi:10.4236/jmp.2011.26055 Published Online June 2011 (http://www.SciRP.org/journal/jmp)
Revisiting the Classics to Recover the Physical Sense in
Electrical Noise*
Jose-Ignacio Izpura
Group of Microsystems and Electronic Materials (GMME-CEMDATIC),
Universidad Politécnica de Madrid (UPM), Madrid, Spain
Received March 1, 2011; revised April 17, 2011; accepted May 3, 2011
Abstract
This paper shows a physically cogent model for electrical noise in resistors that has been obtained from
Thermodynamical reasons. This new model derived from the works of Johnson and Nyquist also agrees with
the Quantum model for noisy systems handled by Callen and Welton in 1951, thus unifying these two
Physical viewpoints. This new model is a Complex or 2-D noise model based on an Admittance that consid-
ers both Fluctuation and Dissipation of electrical energy to excel the Real or 1-D model in use that only con-
siders Dissipation. By the two orthogonal currents linked with a common voltage noise by an Admittance
function, the new model is shown in frequency domain. Its use in time domain allows to see the pitfall be-
hind a paradox of Statistical Mechanics about systems considered as energy-conserving and deterministic on
the microscale that are dissipative and unpredictable on the macroscale and also shows how to use properly
the Fluctuation-Dissipation Theorem.
Keywords: Quantum-Compliant, Noise Model, Fluctuation-Dissipation Theorem, Cause-Effect,
Action-Reaction
1. Introduction
Looking for the state of the art on shot noise in devices
we found a publication [1] whose authors contend that
electrical charge being piled-up in a pure resistance R
devoid of any capacitance C, generates shot noise. Since
a pure resistance is not a noisy device accordingly to the
well known work of Callen & Welton [2] nor to the clas-
sical Fluctuation-Dissipation Theorem (FDT) derived
from [2] that we find hard to apply to a pure resistance R,
we decided to consider in some detail the academic
viewpoint on the FDT given in a recent paper [3]. Here,
the FDT was nicely explained by a mechanical example
whose analogy with our Admittance-based model (A-
model) for electrical noise, led us to write this paper.
This A-model that agreeing with [2] would be a “Quan-
tum-compliant” model for electrical noise, excels the
model in use today for resistors and has explained re-
cently 1/f excess noise in Solid-State devices [4] and
flicker noise in vacuum ones [5] as simple consequences
of thermal noise in these devices.
These striking phrases of [3]: “one of the great para-
doxes of statistical mechanics is how a system can be
energy conserving and deterministic on the (molecular)
microscale, and yet dissipative and unpredictable/random
on the macroscale. The paradox is resolved by recogniz-
ing that both the dissipation and the random fluctuations
are associated with incomplete information about the
state of the system.” led us to use our A-model to put
some physical sense in this noise-fluctuation field where
some laws of Physics are infringed inadvertently. The
A-model we found by using Thermodynamics to study
noise in resistors [4] is implicit in the results of Johnson
[6] and Nyquist [7]. The circuit it uses will be dealt with
in Section 2. In Section 3 we will consider the example
of [3] under our A-model to show the pitfall leading to
the aforesaid paradox. Section 4 will show how the FDT
should be understood and applied in today’s research on
electrical noise.
2. A Physically Cogent Model for Electrical
Noise in Resistors and Capacitors
*Work supported by the Spanish CICYT under the MAT2010-18933
p
rojec
t
, by the Comunidad Autónoma de Madrid through its IV-PRICIT
Program, and by the European Regional Development Fund (FEDER). The origin of the FDT is the Quantum Mechanical treat-

458 J.-I. IZPURA
ment of a noisy system done by Callen and Welton [2] in
1951 where the need for a Complex Impedance (or Ad-
mittance) function to describe such a noisy system is so
apparent (see Equations 2.12 to 2.15, 3.6, 4.6 and 4.7 in
[2]) that a paper like [1] transgressing this important re-
sult indicates that the Physical sense of Classical works
like [6,7,2] has been lost. The Admittance required by
Thermodynamics in our A-model that agrees with the
one required in [2] is the circuit shown in Figure 1. Al-
though it reminds a circuit used today for noise in resis-
tors, it has a capacitance C (see Section VI of [4]) and it
is more general because R* includes both ohmic and
non-ohmic (e.g. differential) resistances that define the
Conductance
 
1GR



 or real part of the Ad-
mitance
 


Yj GjB

 

vt
 we will use to handle
the noise voltage as the time-varying signal link-
ing noise current through the Conductance


G

with
noise current through the Susceptance

B

at each
frequency

2πf  .
Capacitance C is required by Thermodynamics in or-
der to have the energy store (Degree of Freedom) to have
energy liable to fluctuate thermally in the device, linked
with the fluctuating voltage observed as Johnson noise in
resistors or kT/C noise in capacitors. This unavoidable C
comes from the non null dielectric permittivity of any
material (vacuum included) existing between two ohmic
contacts or terminals put at some distance in our actual
world [4,5]. The need to include in R* both ohmic and
non-ohmic resistances also is dictated by Thermody-
namics in order to keep the thermal fluctuation of kT/2
Joules in the Degree of Freedom that C represents. This
meaning used in [8] to present the kT/C noise as the
noise coming from the Thermal fluctuation of kT/2
Joules in the Degree of Freedom that C represents, al-
lows writing:
 
22
11
22
kT
kTCv tv tC  (1)
thus showing the essential role of C to generate noise in
the circuit of Figure 1 and the secondary role of R* as a
spectrum shaper to make (1) true [4,5]. This summarizes
the advancing character of our A-model where the John-
son noise of resistors comes from their capacitance, a
feature suggested in 1951 [2], but ignored in 2004 [1].
Considering any voltage noise in time as syn-
thesized from sinusoidal terms (Fourier synthesis)
each noise current through the conductance G =
1/R linked with will be a sinusoidal current
in-phase with n v given by:

vt

n
vt

P
it

n
vt

t



Pt n
it (see
Figure 2). The noise current linked with
Gv

Q
it


n
vt
will be a sinusoidal noise current in-quadrature with
given by: Qn it . For this positive B
due to C, a current will have +90˚ phase advance

n
vt


Q
it

vtjB
C
22
4or kT AC
R
Hz s





Vt
A
B
R

Figure 1. Complex Admittance used in the A-Model for the
electrical noise of resistors and capacitors.
Figure 2. Electrical circuit that representing Johnson noise
in resistors or kT/C noise in capacitors, allows studying
other noises obeying the Fluctuation-Dissipation Theorem.
respect to its sinusoidal . Therefore, the time inte-
gral of displacement currents like in C will have
the same phase of

n
vt

Q
it


n
vt.
This coincidence of the aforesaid phases suggests, as
our AM considers, that is the Cause that inte-
grated in time by C, creates the Effect that syn-
thesizes the Johnson noise observed in resistors or the
kT/C noise observed in capacitors. This can be seen viv-
idly in time domain, but this falls out of this paper and
will be published elsewhere [9].

Q
it

n
vt
To deal with electrical noise due to Fluctuations
(changes in time) of the content of electrical energy in
two-terminal devices as resistors and capacitors, we will
use the instantaneous power
 


in
ptv tit, where






PQ
iti tji t is the current through the device.
Note that this same function (or better said, its average
value


i
pt ) was used in Equation 2.13 of [2]. To
work in time domain, the Nyquist spectral density 4kT/R
A2/Hz derived from [7] (4kT/R* A
2/Hz in our model to
keep the kT/C noise of C [4,5]) is replaced by its time
counterpart: a random current iNy(t) with zero mean
shown in Figure 2, whose Kirchoff’s law on top we will
write as:

 
n
nNy
vt
CGvti
t
 
t
(2)
Equation (2) states that any band-limited


Ny
it will
create a Displacement Current in C and a Conduction
Current in R*. To create a pure Displacement current, a
δ-like current


Ny
it of infinite bandwidth (BW  )
or null duration (δtc  0) is needed. The null time
elapsed in this case (δtc  0) allows writing the time
integral of (2) as:
built
q
vC

 (3)
where the weight q of the δ-like current


Ny
it (thus a

J.-I. IZPURA
459
charge) creates instantaneously a voltage step built vqC


in C, or produces a Fluctuation of q2/(2C) Joules in the
energy of C provided it was discharged. Thus, a pure
Fluctuation of energy is not possible in resistors because
their is band-limited as Nyquist showed by an
upper limit for their 4kT/R A
2/Hz density due to Quan-
tum Mechanical reasons [7].

Ny
it
Although very pure Fluctuations of energy result in
this circuit for displacement currents with short δtc, they
always have a non null Dissipation due to the conduction
current during the non null δtc elapsed. To say it bluntly:
an electron leaving one plate with a kinetic energy of
q2/(2C) Joules exactly in a discharged C is unable to cre-
ate built v

t
qC
V in C because part of its q2/(2C) J is
dissipated during its non null transit time δtc. Thus, no
paradox exists if we find this system dissipative in the
macroscale with a huge amount of such transits giving
rise to noticeable dissipation of energy. Assuming, how-
ever, that these passages of single electrons between
plates are energy-conserving events because the term
n has no time to dissipate a noticeable amount of
energy during each short transit time δtc, we would find a
paradox due to this wrong assumption that we would
have done for a parallel-plate capacitor in vacuum with
plates at temperature T before knowing that it is false due
to the R* created by thermoionic emission [5]. This ele-
mental charge noise of one electron in C is the basis of
[9] and the reason for the charge noise power units (C2/s)
that appears in Figure 1 as equivalent (in time domain)
to the well known A2/Hz units in frequency one.
Gv
3. The Fluctuation-Dissipation Theorem and
the A-Model for Electrical Noise
Let us consider the Fluctuation-Dissipation Theorem
explained by the motion of a large particle of mass M
through a sea of small particles having random motions
[3]. These small particles exert, through collisions, a
random force

F
t

on the large one whose velocity at
time t relative to the mean motion of the small ones is
. The mean drag force on the large particle is pro-
portional to

vt

vt

t
through the friction factor  and the
fluctuating random component is proportional, with a
coefficient of proportionality
 , to the so called statistical
white noise . Copying Equation (1) of [3], we
have:
v
Mv
t



 (4)
Following [3], “The aim of the fluctuation-dissipation
theorem is to relate
 to the statistics of .” Using
a white noise process for the mean square fluctu-
ating force

t  

t
2
0
F is obtained in Equation (10) of [3],
which is:


2
22
0
2
c
kT
Ft
t

   (5)
where δtc is the contact time during a collision. Below
this Equation it is written: “This is the required statement
of the fluctuation-dissipation theorem, relating as it does
the mean square fluctuating force to the friction factor.”
[3].
Let us rewrite (2) in this form:
 
()
Ny
vt
CGvtitGvt
t



 
 (6)
where the noise current has been replaced by the
product of a white noise

Ny
it


t

and the constant 2kT


given by Equation (9) of [3] to have a complete, formal
analogy between (4) and (6). From this, the electrical
counterpart of (5) is:
  
2
22 2
414
2
Ny
c
Q
c
kT
itt t
kT kT
f
t
RR




 


(7)
where


2
Ny
it has been approximated by the product
of the flat density


4
i
Sf kTR

 A
2/Hz by its band-
width fQ (Hz) derived from [7] as explained below.
Taking the equivalent bandwidth fQ for


i
Sf from
kT ≈ hfQ (e.g. fQ ≈ kT/h where h is the Plank constant) (7)
suggests that the interaction time behind the Nyquist
noise of density


4
i
Sf kTR

 A
2/Hz is: δtc ≈ 0.08
ps at room T. This


2
Ny
it value that is not difficult to
calculate (e.g.


2
10
Ny
it  10 A
2 or 10 Arms at room
T for R* = 1 kΩ), is hard to be observed, however, being
its time integral as v(t) (Effect) all we will observe due to
the unavoidable C of any two-terminal device used to
sense


2
Ny
it in our actual world. Leaving aside this
reflection about the Measurement of this


2
Ny
it noise,
let’s go to the analogy between (2) and (4). Note the
correspondences (voltage-velocity) and (current-force)
we have because the (voltage  current) product is the
instantaneous power


i
pt
Ny
i
in the circuit due to the
random noise generator (the time counterpart of
4kT/R* A2/Hz) whereas the (velocity  force) product
represents the instantaneous power acting on the
mass M due to the sea of small particles hitting it along
one spatial direction.

t

i
pt
Therefore, results concerning Fluctuation and Dissipa-
tion of electrical energy that our A-model links with
voltage noise of resistors (Johnson noise) and capacitors
(kT/C noise), can be applied to the velocity noise of the
large particle of mass M. The first one we will use is the

460 J.-I. IZPURA
impossibility of energy-conserving collisions whose col-
lision time is not null (δtc > 0) to keep finite the in tera c-
tion power. Due to this, the system of [3] can’t be taken
as “energy conserving and deterministic on the micro-
scale” as it was done in the premise of the paradox at
hand. The dissipative action of the
 v term of (4) never
vanishes for δtc > 0 (except for
 = 0, which is not the
case). Thus, the paradox comes from its false premise.
We found a similar paradox or hypothetical device in [5]
trying to build a pure capacitance C in Thermal Equilib-
rium (TE) at some temperature T because thermoionic
emission is enough to create a differential resistance R*
shunting C to form an energy relaxing cell bearing the
Admittance that Quantum Mechanics requires for a noisy
system [2].
Therefore, the proposal of a noisy resistor offering a
pure Resistance R in TE at some temperature T [1] does
not make physical sense because it is a hypothetical de-
vice that infringes Thermodynamics and Quantum Phys-
ics. Without Susceptance, this device is unable to store
electrical energy liable to fluctuate or it lacks the eingen-
states of electrical energy required in [2] to allow a
Quantum treatment of noise. Even more: the hypothetical
device of an R without C infringes Special Relativity too
because the null dielectric permittivity of the material
required to have C = 0 between two neighbour terminals
would allow the speed of the light to be boundless in it
[5].
Once solved the paradox recognizing that it comes
from a wrong premise that would require an infinite in-
teraction power, let us say that since Fluctuation is linked
with a current in quadrature respect to


n
vt and Dis-
sipation is linked with a current in phase, the phase be-
tween observed Effect and its Cause is what defines if
energy mostly Fluctuates (as electrical energy in C or
kinetic energy of M) or if it is being Dissipated (by a
current
 
Pn
it vtR

 in the device of Figure 2 or
by the drag force


vt  in [3]). The study of the exam-
ple handled in [3] could benefit from these ideas by us-
ing the circuit of Figure 2 as the mechanical impedance

2π

Z
jf or transfer function velocity/force at each
frequency of the large particle in the viscous sea of small
ones. Thus, let us consider the large particle moving back
and forth due to a sinusoidal force

2π

F
jf as one of
the Fourier components of the random force


t  

in
(4). At low frequencies, the term of (4) proportional to
the time derivative of the velocity is small,
the Cause

2πvj f

2

π
F
jf and the Effect tend to
be sinusoids in-phase and thus, the energy brought by the
force F as time passes is mostly dissipated by the friction
represented by the drag force . The inertial stor-
age of energy by low-f components of F acting on M is
small, thus being Dissipation-like forces because Cause
(F) and observed Effect (v) tend to be in-phase.

2πvj f


vt 
At high-f, however, where

2π
C
f
fM  , the
transfer of energy to and from the mass M accelerated
sinusoidally dominates over the energy spent against the
drag force


vt  . This tends to put the sinusoidal force


2π
F
jf with a phase advance of +90˚ respect to the
sinusoidal velocity


2πvj f. Cause (Force) and its Effect
(velocity) tend to be in quadrature and thus, high-fre-
quency components of forces acting on M will be Fluc-
tuation-like forces. Collisions where a small particle
moves fast respect to the large mass M it hits during δtc >
0 will produce a brief force packet whose Fourier com-
ponents mostly will be of this type. This allows to con-
sider these collisions as Fluctuation-like or very elastic
ones, but not as energy-conserving collisions because dis-
sipation never vanishes due to their collision time δtc > 0
needed to have a finite interaction power.
Besides the above, the velocity distribution of the
small particles in [3] leads to find some of them with
very low velocities relative to . A small particle and
the large one approaching at a few nanometres per year
not only suggest a larger collision time or an enhance-
ment of the role of the time integral of the real part of

vt


i
pt respect to the role of its imaginary part, but also
they suggest another reason avoiding elastic collisions.
At this low relative velocity, the large and the small par-
ticles can join in the collision by internal forces as gravi-
tational ones, thus having an inelastic collision of very
long δtc where the conservation of momentum leads to
lose energy as it is known from textbooks. This effect
due to internal forces overseen at first sight would
change the mass M and the simplicity of the problem, of
course, but illustrates the weakness of the premise about
energy-conserving collisions between masses. It also
illustrates vividly the drag action of the large mass M on
the small particles at low relative velocity in a way that
the nearly zero-phase relation between sinusoidal Cause
(F) and Effect (v) does not suggest so clearly.
4. On the Application of the
Fluctuation-Dissipation Theorem
in Noise Physics
The proper application of the FDT in Physics requires
knowing its physical meaning. Equations (5) and (7)
show that the FDT relates the mean square fluctuating
force to the friction factor or the mean square fluctuating
current to the Conductance G = 1/R* with no reference to
the Degree of Freedom allowing Fluctuations of energy
(mechanical or electrical). The mass (inertia) or the ca-
pacitance are irrelevant provided they exist in the system
(e.g. M > 0, C > 0) because once the Degree of Freedom
exists, its kT/2 thermal fluctuation will define the ob-

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 kTCV2 independent of R* for the
common case of circuits where

12π
cQ
f
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
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 kTCV2, 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
Sf4kT 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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462
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Appendix I
  

 
2
2
sin
2
sin cos
E
E
C
UtA t
U
A
Ct
t

t


 


(9) In the circuit of Figure 2, for a sinusoidal


n
vt the
current is sinusoidal whereas the current

P
it


Q
it
is a cosine current advanced +90˚ respect to


n
vt.
Therefore, the instantaneous power delivered by
the generator in Figure 2 has an Active power
always positive

i
pt

Ny
it



n P
pt vti

it

t
iP

vt
and a Reactive
power iQnQ entering and leaving peri-
odically the Susceptance of the device (e.g. oscillating at
2f that is the way a Susceptance stores electrical energy
at a given frequency f). If, and only if, this storage exists
in the device, the stored energy can Fluctuate in it. Let us
show how Active and Reactive power are, respectively,
Dissipation of electrical energy in R without storage and
Fluctuation of electrical energy in C without dissipation.

pt
From (8) and (9) we conclude that the instantaneous
power entering C is used to vary (fluctuate) the electrical
energy stored in C. No part of is dissipated, thus
linking Fluctuations of electrical energy in Resistors or
Capacitors with currents in quadrature with

iQ
pt


n
vt
whereas Dissipations of electrical energy in these de-
vices will come from currents in-phase with


n
vt. From
the current




Pn
it vtR

 that produces in
the resistance R*, the instantaneous power

n
vt


t
iP
p en-
tering R* is:
 


2
sin sin
1sin2
iP
A
pt Att
R
At
R






(10)
Let’s consider



sin
n
vt At



n
vt
in Figure 2, where
the angular frequency ω is 2π times the natural frequency
f: ω = 2πf. From this existing in C and from the
current it produces in it:



Qn
it

Cvtt , the
instantaneous power

iQ
pt entering C is: Contrarily to (9) whose mean value is null because
energy entering C is released subsequently, energy al-
ways enters R* because (10) is a positive power of mean
value


22
avg . All the instantaneous power en-
tering R is thus dissipated and no part of it is stored.
 



 
2
sin cos
sin cos
iQ
pt AtACt
A
Ct t





 (8)
PAR
On the other hand, the fluctuating electrical energy
being stored in C by will be:

n
vt