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The origin of the Johnson noise of resistors is reviewed by a new model fitting in the Fluctuation-Dissipation framework and compared with the velocity noise in Brownian motion. This new model handling both fluctuations as well as dissipations of electrical energy in the Complex Admittance of any resistor excels current model based on the dissipation in their conductance. From the two orthogonal currents associated to a sinusoidal voltage in an electrical admittance, the new model that also considers the discreteness of the electrical charge shows a Cause-Effect dynamics for electrical noise. After a brief look at systems considered as energy-conserving and deterministic on the microscale that are dissipative and unpredictable on the macroscale, the arrow of time is discussed from the noise viewpoint.

This paper improves a recent work [

Each passage of a single electron between plates of C, which means a sudden fluctuation ∆E of electric field between the terminals of a resistor, was called a Thermal Action (TA) [

For

Because the most likely situation when a TA occurs is a discharged C (the expected value of Johnson noise is null) we will use Equation (1) as a “useful mean value” for the energy involved in TAs. The case with a small charge in C due to preceding TAs falls out of the scope of this preliminary paper. Each sudden TA setting a voltage step of

This ordainment (TA first) → (DR next) can exist in a Complex Admittance but not in the real Resistance that uses current noise model for resistors. Considering TAs (reactive currents) and DRs (conduction currents) under the same voltage noise we have two orthogonal powers. The first one is reactive power P_{C} entering the resistor “by” C and the second is active power P_{R} leaving it “by” R (see [_{R} will be the mean square voltage noise ^{2}/Hz (Nyquist noise) or the charge noise of power

Using Equation (1) as explained, an average rate _{C} that enters this device on average will be _{C}_{ }must equate in TE the P_{R} leaving the resistor due to its R, we have [

Thus, a resistor in TE would collect an average power P_{C}_{ }by its C acting as a receiving antenna for TAs and would release an equal amount (on average) of electrical power converted into heat by its R. Note this picture for resistance R as a random set of chances to convert electrical energy into another form: usually phonons, but it could be photons as well (think of a radiation resistance). This discrete nature of R, or better said: of the electrical noise itself, allows a direct explanation for the phase noise of electronic oscillators (their line broadening for example) as due to the

The sum in power of TA’s done in Equation (2) (e.g.

If a TA is a small fluctuation of electric field between the terminals of the 2TD, thus a time-varying electric field, then it becomes a fluctuation of electromagnetic field between two points of the resistor (its terminals) at distance d in space. As Einstein showed in 1905, an electromagnetic signal departing midway two points separated in space by a distance d defines simultaneity at these two points. Added to this, the relativity principle he extended to all physical measurements means that noise ones cannot detect absolute motion in space. This means that each TA will be born “midway” the two terminals of the resistor. Otherwise the sense and the speed

of the absolute motion of the resistor would affect the electrical noise measured between terminals. Thus, the arrival of a TA in the terminals as an electrical voltage will define simultaneity for two observers measuring electrical voltage at these terminals. This way, each TA becomes an instantaneous event

This means that the noise pulses of _{C}. This

The complex impedance used in [

Considering a voltage noise

To study the voltage noise ^{2}/Hz by a random current

Equation (3) states that any band-limited

where the weight q (thus a charge) of the δ-like current

Equations (3) and (4) were obtained in [^{2}/Hz density due to Quantum Mechanical reasons [

For

This solves the paradox in Brownian motion but the null

To resume our reasoning, let us recall the formal analogy between Johnson and Brownian noises that use to be studied by the Langevin equation. As we have written below Equation (3), a δ-like current

Equation (5) formally equal to Equation (3) suggests that the role of inertial mass M in Brownian motion would be similar to the role of electrical capacitance C in electrical noise. Nevertheless, taking C = 0 to study electrical noise in resistors would be rule whereas taking a large particle of mass M = 0 to study its Brownian motion would be the exception (to our knowledge). Leaving aside this “curious situation” let us resume the reasoning of [

where

where we had to approximate the mean square ^{2}/Hz by an upper frequency

Given the null duration

Our firm belief that the device used to measure influences the obtained result led us to consider that when two terminals are placed at distance d, capacitance C is born and a new degree of freedom becomes available for the aforesaid electron confined in

Assuming that for each TA taking place in the resistor an energy

where

Therefore, the product of the resistance _{2TD} sensed this voltage, the active power it would convert into heat on average (e.g. the mean power P_{R} dissipated by R_{2TD}) would be

leaving as heat this hypothetical device should not surpass the maximum reactive power

Equating P_{Q} to the power P_{R} dissipated by R_{2TD} in these conditions and using Equation (8) for

If a _{2TD} would be a resistance that multiplied by C gives an estimation of the minimum time interval between TA’s for an electron able to give a TA in the device. This _{2TD}. This periodic voltage wave only is an unlike situation because due to the randomness of the TAs, there would be an added “jitter” in each

It is worth noting that with these limits (^{2}/Hz decaying as _{str}, the stray capacitance added by the wires connecting the resistor to the front-end stage of the noise voltmeter or spectrum analyser.

Taking ^{2}/Hz decaying as

Observed on the screen of an oscilloscope Johnson noise looks continuous as time passes and independent of the sense time flows. We would say that this random waveform would not show an arrow of time, thus being similar to the undamped oscillation of [

Arrived at this point, let us consider these sentences about vacuum in [^{2}/Hz up to its cut-off frequency given by

This 2TD also would be a capacitor at temperature T that would show a mean square noise voltage of ^{2}. No matter we consider Johnson or

Let us note here that TAs are fluctuations of energy in electromagnetic form, thus a form “not tied” to the time flow required to observe dissipation of electrical energy: recall the way a pure fluctuation of electrical energy was obtained in Equation (4) by freezing the time flow around an instant. From this idea we could say that TAs causing noise are not in the “world” (axis) where their effects requiring a time flow to be observed and measured are (see

The time-asymmetry viewed as an arrow of time associated to dissipation [

“Dissipation is born” also means that we start to have conduction currents in-phase with any sinusoidal voltage synthesizing Johnson noise in the resistor. This is Ohm’s Law leading to a striking “arrow of energy conversion” in sinusoidal regime. We mean that the product of a sinusoidal voltage by a sinusoidal current due to the existence of such voltage (thus proportional to it or in-phase with it) always leads to dissipation of electrical energy as time passes no matter the sense of this passage. This is so because the mean value of such product (mean active power) always is a non-negative value. This electrical energy per unit time must leave the 2TD where these magnitudes are found in phase because for voltage and current in-phase, this energy loses its electrical form and it no longer can remain in the electrical admittance of the 2TD. This means leakage of electrical energy in the circuit as time passes, hence the name “dissipation” for this leakage, usually in the form of heat.

Quite the contrary, the product of a sinusoidal voltage by a sinusoidal current that is in-quadrature with it, only gives a sinusoidal fluctuation of electrical energy in the 2TD where these sinusoidal magnitudes are found. This is a lossless fluctuation at twice the frequency of the voltage or current terms. In this case the electrical energy keeps its electrical form as it fluctuates with time. This in-quadrature condition appears when the current is proportional to the change with time of the voltage (e.g. to

Thus the quotation we would add to this sentence: “the arrow of time” should also exist in the microscopic world” [

Our firm belief that electrical noise in resistors should reveal the discrete nature of electrical signals led us to propose a new model on its origin. This model that is discrete in time treats electrons as electromagnetic waves (displacement currents) across two-terminal devices like resistors. In some circumstances, this model could excel or complement other models based on point charges wandering and colliding within the solid matter.

This new model on the origin of electrical noise considers that the two-terminal device required for its measurement becomes a system that offers degrees of freedom for the energy form that gives the measurable effect. For Johnson noise this form is electrical energy, hence the key role played by the electrical capacitance C between terminals of resistors. This C would define the packets of electromagnetic energy to be converted into an electrical form located in the resistor by its complex admittance. Once the energy is located in this way, the consequences associated to energy/mass located in a volume of space would start to be observed.

One of these consequences is the appearance of dissipation of the electrical energy stored in C by its inner electric field and the appearance of the arrow of time related with this process. Since our noise model shows dissipation for each microscopic event building Johnson noise in resistors but this macroscopic noise would not show such a clear arrow of time when viewed by an oscilloscope, this new noise model we have used could help to better understand the arrow of time at microscopic level in other noisy processes.

This work was partially supported by the European Comission through the 7^{th} Framework Program, by the RAPTADIAG project HEALTH-304814 and by the Ministerio de Economía y Competitividad del Gobierno de España through the MAT2010-18933 project.