# Information Contained in Molecular Motion

## Abstract

**:**

## 1. Introduction

## 2. Entropy of an Ideal Gas

## 3. Gas-kinetic Interactions

## 4. Supervised and Unsupervised Measurement Processes

## 5. Gases with Internal Degrees of Freedom

## 6. Conclusions and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Wave-packet widths $w\left(t\right)$ relative to the average mean-free path ${d}_{av}$ as a function of time (

**a**) and reduced time $\tau =t/{\tau}_{coll}\left({E}_{kin}\right)$ (

**b**) after a gas-kinetic collision has occurred. The individual curves denote solutions for increasing molecular kinetic energies in units of ${E}_{th}=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{k}_{B}T$. Molecular parameters correspond to normal air (N

_{2}, O

_{2}).

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**Figure 1.**Sketch of the molecular motion inside an ideal gas enclosed in a volume $V$ and maintained at a temperature $T$.

**Figure 2.**Variation of ${i}_{loc}\left({E}_{kin},t\right)$ (Equation A12) during successive gas-kinetic collisions as a function of reduced time ${\tau}_{red}=t/{\tau}_{coll}$, where ${\tau}_{coll}={\tau}_{coll}\left({E}_{kin}\right)$ represents the energy-dependent collision time at which follow-on collisions are likely to take place. The individual curves denote solutions for increasing molecular kinetic energies in units of ${E}_{th}$. Molecular parameters correspond to normal air (N

_{2}, O

_{2}).

**Figure 3.**Sketch of molecular motion inside single-molecule gases designed to allow for supervised (

**a**) and un-supervised (

**b**) observations. Quantum-mechanical dispersion of the molecular wavefunctions is indicated by circles of increasing size. Forward and backward travels have been vertically displaced for ease of presentation.

**Figure 4.**Molecular entropies as a function of the reduced inverse temperature $x=\epsilon /{k}_{B}T$ as plotted in information units: (blue) rotations and vibrations following Bose-Einstein and (red) electronic excitations following Fermi-Dirac statistics. The dashed black line denotes the low-temperature approximation to both kinds of entropies, i.e., to excitation energies $\epsilon \gg {k}_{B}T$.

**Table 1.**Molecular entropies relating to molecular rotations, vibrations and electronic excitations as derived from the simplified models of molecular rotators, vibrators and electronic two-level systems [13].

Model | Molecular Entropy |
---|---|

Molecular rotator Molecular vibrator | ${s}_{BE}\left(\omega ,T\right)={k}_{B}\left[\frac{\left(\frac{\hslash \omega}{{k}_{B}T}\right)exp\left(-\frac{\hslash \omega}{{k}_{B}T}\right)}{1-exp\left(-\frac{\hslash \omega}{{k}_{B}T}\right)}-ln\left[1-exp\left(-\frac{\hslash \omega}{{k}_{B}T}\right)\right]\right]$ |

Electronic two-level system | ${s}_{FD}\left({E}_{el},T\right)={k}_{B}\left[\frac{\left(\frac{{E}_{el}}{{k}_{B}T}\right)exp\left(-\frac{{E}_{el}}{{k}_{B}T}\right)}{1+exp\left(-\frac{{E}_{el}}{{k}_{B}T}\right)}+ln\left[1+exp\left(-\frac{{E}_{el}}{{k}_{B}T}\right)\right]\right]$ |

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Müller, J.G. Information Contained in Molecular Motion. *Entropy* **2019**, *21*, 1052.
https://doi.org/10.3390/e21111052

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Müller JG. Information Contained in Molecular Motion. *Entropy*. 2019; 21(11):1052.
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Müller, J Gerhard. 2019. "Information Contained in Molecular Motion" *Entropy* 21, no. 11: 1052.
https://doi.org/10.3390/e21111052