# Landauer’s Principle a Consequence of Bit Flows, Given Stirling’s Approximation

## Abstract

**:**

## 1. Introduction

## 2. Algorithmic Information Theory

#### 2.1. Uncertainty Due to Phase Space Graining

#### 2.2. The Conservation of Bits and Reversibility

#### 2.3. The Source of the Apparent Entropy Increase in an Isolated System

## 3. Landauer’s Principle and Energy per Bit

- While Landauer’s argument is about erasing a discrete entity “the bit” that specifies the actual state of a system, the argument, which is usually expressed in terms of the change to the Shannon or Boltzmann entropies is not clear cut, as a bit is a property of the collection of allowable microstates. For example, Landauer [1] discusses erasure from the Boltzmann/Shannon perspective where a computation shifts the set of allowable input states to the set of allowable output states, by mapping 8-bits on to 4-bits. This dissipates $1.18{k}_{B}T$ Joules. However, in contrast, the algorithmic perspective sees only an entropy change of one bit, as 3 bits (representing 8) drops by an integer to 2 bits (representing 4), clearly indicating that the minimum energy change for one bit is ${k}_{B}ln2T=0.6931{k}_{B}T$ Joules.
- Chandrasekhar [11], assuming the equipartition of energy, has shown that in an isolated system, the energy to drive a Brownian particle from a stored energy state produced by a gravitational field to the most probable set of states, corresponds to a thermodynamic entropy change of ${k}_{B}$ nats or ${k}_{B}ln2$ bits per particle. (Here “nats‘’ stands for natural units of information.) This is Landauer’s principle if one assumes the bundle of energy carrying a nat behaves like a Brownian particle. This can be envisaged as the entropic force dissipating the stored energy and, in so doing, increasing the kinetic energy of the system.
- Again, the Johnson-Nyquist noise in a communication channel, represented by a resistance capacitance network, given the equipartition principle, is ${k}_{B}T$ Joules per mode of oscillation. This corresponds to ${N}_{0}$, the noise power per bandwidth. If the signal energy per bit of data is ${E}_{b}$, Shannon has shown that the optimum capacity for a noisy communication channel at temperature T is ${E}_{B}/{N}_{0}\ge {k}_{B}ln2T$ as more energy is needed to identify a bit as the temperature rises. This can be related to Landauer’s principle by determining the energy cost of sending a signal of one bit carried by a pressure pulse using a gas tube as the communication channel. This would require at least ${k}_{B}ln2T$ Joules, otherwise the pulse could not be identified above the thermal background noise.

## 4. The Validity of Landauer’s Principle

Number of States | ${\mathbf{log}}_{2}{\mathit{n}}_{\mathit{e}}^{\left(\mathit{k}\right)}$ $\mathit{H}\left({\mathit{e}}_{\mathit{i}}^{\left(\mathit{k}\right)}\right)$ | ${\mathbf{log}}_{2}{\mathit{n}}_{\mathit{p}}^{\left(\mathit{k}\right)}$ $\mathit{H}\left({\mathit{p}}_{\mathit{i}}^{\left(\mathit{k}\right)}\right)$ | $\mathit{H}({\mathit{e}}_{\mathit{i}}^{\left(\mathit{k}\right)})+\mathit{H}({\mathit{p}}_{\mathit{i}}^{\left(\mathit{k}\right)})$ | Instruction Bits | Temp $\times {10}^{-6}$ | |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 105.551 | |

1 | 1 | 4.585 | 4.585 | 9.170 | 96.381 | 2.66 |

2 | 576 | 8.229 | 8.229 | 16.458 | 89.093 | 3.35 |

3 | 90,000 | 11.344 | 11.344 | 22.689 | 82.862 | 3.91 |

*** | *** | *** | ** | |||

12 | $5.42\times {10}^{17}$ | 29.276 | 29.636 | 58.912 | 46.639 | 7.89 |

13 | $3.85\times {10}^{18}$ | 30.633 | 31.106 | 61.738 | 43.812 | 8.30 |

14 | $2.45\times {10}^{19}$ | 31.903 | 32.508 | 64.41 | 41.140 | 8.69 |

15 | $1.42\times {10}^{20}$ | 33.093 | 33.849 | 66.942 | 38.609 | 9.09 |

*** | *** | *** | ** | |||

43 | $3.19\times {10}^{30}$ | 43.057 | 58.274 | 101.331 | 4.220 | 19.72 |

44 | $3.79\times {10}^{30}$ | 42.700 | 58.881 | 101.581 | 3.970 | 20.09 |

45 | $4.32\times {10}^{30}$ | 42.293 | 59.476 | 101.779 | 3.781 | 20.47 |

46 | $4.73\times {10}^{30}$ | 41.837 | 60.061 | 101.898 | 3.653 | 20.84 |

47 | $4.95\times {10}^{30}$ | 41.329 | 60.636 | 101.964 | 3.586 | 21.21 |

48 | $4.96\times {10}^{30}$ | 40.768 | 61.201 | 101.969 | 3.582 | 21.58 |

49 | $4.76\times {10}^{30}$ | 40.154 | 61.756 | 101.910 | 3.641 | 21.96 |

50 | $4.37\times {10}^{30}$ | 39.485 | 62.302 | 101.787 | 3.764 | 22.33 |

51 | $3.21\times {10}^{30}$ | 38.759 | 62.839 | 101.598 | 3.953 | 22.70 |

52 | $3.21\times {10}^{30}$ | 37.974 | 63.367 | 101.342 | 4.209 | 23.07 |

53 | $2.56\times {10}^{30}$ | 37.129 | 63.887 | 101.017 | 4.534 | 23.44 |

54 | $1.95\times {10}^{30}$ | 36.222 | 64.399 | 100.621 | 4.930 | 23.81 |

*** | *** | *** | ** | |||

67 | $1.54\times {10}^{26}$ | 16.576 | 70.417 | 86.993 | 16.724 | 28.63 |

68 | $3.75\times {10}^{25}$ | 14.097 | 70.837 | 84.934 | 18.558 | 29.00 |

69 | $7.31\times {10}^{24}$ | 11.344 | 71.252 | 82.596 | 22.955 | 29.37 |

70 | $1.12\times {10}^{24}$ | 8.229 | 71.662 | 79.891 | 25.660 | 29.74 |

71 | $1.19\times {10}^{23}$ | 4.585 | 72.067 | 76.652 | 28.899 | 30.11 |

72 | $6.53\times {10}^{21}$ | 0 | 72.467 | 72.467 | 33.084 | 30.48 |

Total | $5.94\times {10}^{31}$ |

#### 4.1. The Archetypical Model to Illustrate the Algorithmic Perspective

#### 4.1.1. The Contributions to the Algorithmic Entropy of the Model System

#### 4.1.2. Derivation of Landauer’s Principle

## 5. The Landauer Principle at the Quantum Limit

- $I({S}^{\prime}:{R}^{\prime})$ is the mutual information between the system and the reservoir. It is this term that irreversibly increases the entropy. From the algorithmic entropy perspective discussed here, $I({S}^{\prime}:{R}^{\prime})$ corresponds to the algorithmic mutual information embodied in the bits in the residual program or history as in Equation (3). The algorithmic approach interprets the loss of mutual information as adding an extra $H\left({g}_{t}\right)$ bits to the environment. It is necessary to track these bits to maintain reversibility as when the structures carrying these bits disintegrate, the bits dissipate as heat, leading to an increase in the algorithmic entropy of the reservoir.
- $D\left({\rho}_{R}^{\prime}\right|\left|{\rho}_{R}\right)$ is the relative entropy before and after the erasure, given ${\rho}^{\prime}$ and $\rho $. $D\left({\rho}_{R}^{\prime}\right|\left|{\rho}_{R}\right)$ corresponds to the free energy increase of the reservoir during the erasure process. As this term is included where capability exists to do work, it will be zero with an appropriate definition of heat as argued by Bera et al. [15]. These authors do not consider energy that transfers into a reservoir to be heat when work can still be extracted from the energy. From the perspective here, these bits are potential entropy only, and do not align with the realized entropy in the momentum degrees of freedom.

## 6. The Heat Capacity

#### 6.1. The Algorithmic Equivalent of the Einstein Heat Capacity

#### 6.2. Background to the Debye Approach to the Heat Capacity

- The wavelength $\lambda $ of the fundamental mode of vibration of a crystal lattice is given by $\lambda =c/\nu $, where $\nu $ is the vibrational frequency and c the wave velocity. At low temperatures, the wave velocity is assumed to be constant for different modes of vibration. If the wave number ${q}_{m}$ of the m-th mode, is the reciprocal of the wavelength, it follows that, with the assumed constant velocity, the frequency of the m-th mode is $m\nu ={\nu}_{m}$,
- The quantum of energy needed to excite the m-th mode is ${\widehat{\u03f5}}_{m}=h{\nu}_{m}$. Where there are $n\left({q}_{m}\right)$ phonons in a lattice vibration, the vibration at frequency ${\nu}_{m}$ carries energy of $(n\left({q}_{m}\right)+1/2)h{\nu}_{m}$. Here h is Plank’s constant. Ignoring the zero-point energy, the energy of the vibration depends on the number of phonons, i.e., ${\u03f5}_{m}=n\left({q}_{m}\right)h{\nu}_{m}=n\left({q}_{m}\right){\widehat{\u03f5}}_{m}$. The total number of phonons in the mode is $n\left({q}_{m}\right)={\u03f5}_{m}/{\widehat{\u03f5}}_{m},$ where ${\u03f5}_{m}$ is the energy of the mode.
- In a crystal, there are many possible modes as ${q}_{m}$ is a vector with components along the x, y and z axes. In which case, the wavelength of the vibration is the reciprocal of the size of the way vector denoted by $|{q}_{m}|$. However, the actual amplitude of particular mode of vibration depends on the number of phonons denoted by $n\left({q}_{m}\right)$ in the mode. Overall, the total number of modes in the crystalline solid must equal the number of atoms or lattice points.
- The total energy U of the complete vibrational system is given by summing over all the modes.$$U=\sum _{m}{\widehat{\u03f5}}_{m}n\left({q}_{m}\right).$$

#### 6.3. Evaluating $n\left({q}_{m}\right)$ the Number of Modes with Energy ${\u03f5}_{m}$

## 7. The Consistency of the Model System

## 8. Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**Showing how programme, stored energy and momentum bits interchange as the microstate of the isolated system moves away from an ordered configuration.

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Devine, S.
Landauer’s Principle a Consequence of Bit Flows, Given Stirling’s Approximation. *Entropy* **2021**, *23*, 1288.
https://doi.org/10.3390/e23101288

**AMA Style**

Devine S.
Landauer’s Principle a Consequence of Bit Flows, Given Stirling’s Approximation. *Entropy*. 2021; 23(10):1288.
https://doi.org/10.3390/e23101288

**Chicago/Turabian Style**

Devine, Sean.
2021. "Landauer’s Principle a Consequence of Bit Flows, Given Stirling’s Approximation" *Entropy* 23, no. 10: 1288.
https://doi.org/10.3390/e23101288