# Observable and Unobservable Mechanical Motion

## Abstract

**:**

## 1. Introduction

## 2. Thermodynamic Approach to Classical Mechanical Motion

**r**) for the force acting on this particle at the location $\mathit{r}$ as it moves along its trajectory inside the system.

## 3. Mechanical Motion in the Quantum Domain

## 4. Least Action Principles and Observational Overhead

## 5. Dissipative Overhead and Irreversibility

_{2}molecule is assumed to be coming from the left (1), moving with the mean thermal velocity ${v}_{th}=\sqrt{5{k}_{B}T/M}$, hitting an ${N}_{2}{}^{+}$ ion in the center (2), which is initially at rest, and accelerating it during the time ${\tau}_{int}={d}_{N2}/{v}_{th}$ from $v=0$ to $v={v}_{th}$, thus moving the ion by one mean-free distance to the right and bringing the neutral molecule to rest at the former position of the ${N}_{2}{}^{+}$ ion (3).

- (i)
- the fractional loss of mechanical energy, $\delta {E}_{rad}\left({\tau}_{int}\right)={E}_{rad}\left({\tau}_{int}\right)/{E}_{th}$, in each ion–molecule interaction,
- (ii)
- the number of collisions ${n}_{diss}=1/\delta {E}_{rad}$ required to transfer the mean thermal energy of the moving ion toward its environment where it becomes dissipated,
- (iii)
- the total time, ${\tau}_{diss}={n}_{diss}{\tau}_{coll}$, required for this energy transfer,
- (iv)
- the length of the diffusion path, ${L}_{diss}=\sqrt{\left(\frac{1}{6}\frac{{\lambda}^{2}}{{\tau}_{coll}}\right){\tau}_{diss}}$, covered in the time ${\tau}_{diss}$.

_{2}molecules and, concomitantly, much smaller emission probabilities during each gas–kinetic interaction. With this caveat in mind, we assume that the bottom line of results in Table 1 reasonably approximates the real level of irreversibility in the gas–kinetic process.

## 6. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Photon Detection

**Figure A1.**(

**a**) Signal photon (red vertical arrow) entering a gap between two metal plates. Photons absorbed inside the negatively biased emitter electrode release signal photoelectrons (blue horizontal arrows). Emitter and collector electrodes maintained at the ambient temperature ${T}_{D}$ produce blackbody radiation (red cloud), producing noise electrons. Signal and noise currents together produce externally observable detector currents ${I}_{D}\left(t\right)$; (

**b**) processes of photoelectron excitation, signal generation, and energy dissipation as seen in a band diagram.

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**Figure 1.**(

**a**) Particle dropping to ground without any observations being made; (

**b**) particle dropping to ground with sunlight being scattered into the eye of an observer; (

**c**) electron moving up and down a center-fed antenna producing pulses of radiation which can be detected at a distance; (

**d**) emission of a photon from an atom with the photon carrying away signatures (${E}_{ph}$, $\u0127$) of the electronic transition to some remote point of detection.

**Figure 2.**(

**a**) Circular motion of a charged particle in a conservative central force field; (

**b**) same process as in (

**a**) but allowing for radiation emission from accelerated charges; (

**c**) same process as in (

**b**) with the emitted radiation being absorbed in a macroscopic heat reservoir of temperature ${T}_{E}$, thereby producing an amount of entropy $dS=\frac{d{E}_{rad}}{{T}_{E}}$ and an increase $d\left(mi\right)=\frac{1}{\mathrm{ln}\left(2\right)}\frac{d{E}_{rad}}{{k}_{B}{T}_{E}}$ in missing information concerning the internal state of motion inside the reservoir walls; (

**d**) absorption of radiation in a radiation detector producing a piece of macroscopically observable information that is accessible to outside observers and providing an information gain $d{i}_{D}<d\left(mi\right)$.

**Figure 3.**(

**a**) Radiation-less motion in an excited Bohr atom; (

**b**) same process as in (

**a**) but with finite radiative lifetime and emission energies following the Rydberg formula; (

**c**) same process as in (

**b**) with the emitted radiation being absorbed in a macroscopic heat reservoir of temperature ${T}_{E}$ (${T}_{E}\mathsf{\Delta}{S}_{mn}$ = $\mathsf{\Delta}{E}_{mn}$); (

**d**) absorption of emitted radiation in a radiation detector operated at a temperature ${T}_{E}$.

**Figure 4.**(

**a**) Uncertainty in localization of a photon travelling out from the emitting atom in the center; (

**b**) resulting loss in locational information as a function of time after photon emission.

**Figure 5.**(

**a**) Photon detector turning potential information ${i}_{pot}$ into missing information $m{i}_{D}$ and transiently producing macroscopically observable events with informational value ${i}_{D}<{i}_{pot}$; (

**b**) photon cascade produced by de-excitation of a highly excited H-atom, de-exciting from a quantum state with $m=100$ toward its ground state with $n=1$ with a step size of $\mathsf{\Delta}n=1$; (

**c**) potential information carried with the emitted photons and measured relative to the detector temperature of ${T}_{E}=300\text{}\mathrm{K}$. The graded blue background indicates the impact of thermal detector noise, with the limit of ${i}_{pot}=1\text{}bit$ denoting a conventional signal-to-noise ratio of $SN=1$.

**Figure 6.**Possible trajectories of a particle moving from points A to B in the absence of any external force fields: full line, physically realized motion; dashed lines, alternative routes connecting A and B but prohibited by the principle of least action.

**Figure 7.**Molecule–ion interaction inside a gas of temperature $T$ leading to an exchange of motional energy between the molecule and ion and causing the emission of a pulse of electromagnetic radiation upon interaction (2).

**Table 1.**Relative loss of motional energy $\delta {E}_{rad}\left({\tau}_{int}\right)$ due to radiation damping during gas–kinetic collisions, number of collisions $({n}_{diss})$ required for a complete transfer of the mean thermal ion energy into radiation, total time required for a complete energy transfer (${\tau}_{diss}$), and length of ionic diffusion path expected to be covered during time ${\tau}_{diss}$ (${L}_{diss}$).

Interaction | $\mathit{\delta}{\mathit{E}}_{\mathit{r}\mathit{a}\mathit{d}}\left({\mathit{\tau}}_{\mathit{i}\mathit{n}\mathit{t}}\right)$ | ${\mathit{n}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{s}}$ | ${\mathit{\tau}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{s}}\text{}\left(\mathbf{s}\right)$ | ${\mathit{L}}_{\mathit{d}\mathit{i}\mathit{s}\mathit{s}}\text{}\left(\mathbf{cm}\right)$ |
---|---|---|---|---|

Gas–ion | $6.90\times {10}^{-16}$ | $1.45\times {10}^{15}$ | $1.95\times {10}^{6}$ | $1.08\times {10}^{3}$ |

Gas–electron | $7.97\times {10}^{-9}$ | $1.26\times {10}^{8}$ | $0.17$ | $0.32$ |

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**MDPI and ACS Style**

Müller, J.G. Observable and Unobservable Mechanical Motion. *Entropy* **2020**, *22*, 737.
https://doi.org/10.3390/e22070737

**AMA Style**

Müller JG. Observable and Unobservable Mechanical Motion. *Entropy*. 2020; 22(7):737.
https://doi.org/10.3390/e22070737

**Chicago/Turabian Style**

Müller, J. Gerhard. 2020. "Observable and Unobservable Mechanical Motion" *Entropy* 22, no. 7: 737.
https://doi.org/10.3390/e22070737