# Entropy, information and Ludwig Boltzmann

## Gerhard Fasol CEO, Eurotechnology Japan KK, Board Director, GMO Cloud KK. former faculty Cambridge University and past Fellow, Trinity College Cambridge

### Ludwig Boltzmann 20 February 1844 – 5 September 1906

We use Ludwig Boltzmann’s results every day. Here are some examples:

- The definition of the units of temperature, Kelvin, Celsius, are directly linked to Boltzmann’s constant
- The Stefan-Boltzmann radiation law tells us that the total energy emitted by a black body per unit surface area is proportional to the 4th power of the temperature, and allows us to measure temperatures at a distance. For example, the temperature of the surface of the sun can be measured using the Stefan-Boltzmann radiation law
- Boltzmann’s formula S = k log W links the macroscopic Entropy with the probability (W = Wahrscheinlichkeit) of a macrostate
- Boltzmann’s transport equations are used for many purposes, to simulate carrier transport in semiconductor devices, and to design airplanes, turbine blades and cars
- Ludwig Boltzmann’s philosophy of nature contributes to our understanding of nature and our world

Ludwig Boltzmann was proposed several times for the Nobel Prize: 1903, 1905 and three times in 1906, the year he took his life in Duino, Italy.

Ludwig Boltzmann achieved his Matura, Austria’s high-school examination required to enter University education at the age of 19 in 1863.

In 1865, at the age of 21, he published his first research paper entitled “Über die Bewegung der Elektrizität in krummen Flächen” (electricity in curved surfaces). It was the dawn of our electrical age, Maxwell created his Maxwell’s equations in 1861-1862, and on 15 February 1883, 20 years later, Tokyo Dentsu KK received the license to start its electricity business in Tokyo.

Among Ludwig Boltzmann’s teachers were Josef Loschmidt and Jozef Stefan.

Josef Loschmidt proposed structures for 300 chemical compounds including benzene, he determined the number of gas molecules in a given volume and the Loschmidt constant is named after him.

Jozef Stefan created the Stefan-Boltzmann Law with Ludwig Boltzmann, and used it to determine the temperature of the surface of the sun.

Ludwig Boltzmann traveled extensively, was in correspondence and discussions and scientific exchange with most major scientists of the time. He also moved professionally:

- University of Vienna
- 1867-1869 Privat-Dozent
- 1869-1873 Full Professor of Mathematical Physics in Graz
- 1873-1876 Full Professor of Mathematics in Vienna
- 1876-1890 Full Professor at University of Graz, Head of the Institute of Physics
- 1887-1888 Rektor (President) of the University of Graz
- 1890-1894 Professor University of München
- 1894-1900 Professor University of Vienna
- 1900-1902 Professor of Theoretical Physics University of Leipzig
- 1902- Professor University of Vienna

Ludwig Boltzmann supported and worked with women:

One of Ludwig Boltzmann’s students was Lise Meitner (November 1878 – 27 October 1968). Lise Meitner was part of Otto Han’s team that discovered nuclear fission, Otto Hahn was awarded the Nobel Prize. Lise Meitner was the second woman to earn a PhD degree in Physics at the University of Vienna. The Element 109, Meitnerium is named about Lise Meitner.

The first President of Osaka University (1931-1934), Nagaoka Kantaro (1865 – 1950) was Ludwig Boltzmann’s student in München around 1892-1893.

## The unit of temperature, Celsius or Kelvin, is directly linked to Boltzmann’s constant k

Currently the unit of Temperature Kelvin is defined as follows:

One Kelvin is defined such that the temperature of the triple point of water is exactly 273.16 Kelvin.

For this definition to be reproducible, the water needs to be defined: its defined as VSNOW = Vienna Standard Mean Ocean Water.

While this definition may have been best at the time it was set, clearly its not sufficient for today.

When the SI system of physical units will be redefined next year, the definition of the unit of temperature, Kelvin will be:

Kelvin is defined such, that the numerical value of the Boltzmann constant k is equal to exactly 1.380650 x 10^-23 JK^-1.

Thus the unit of temperature Kelvin is directly linked to Boltzmann’s constant.

For more details, see: Boltzmann constant and the new SI system of units

## What is Entropy?

Entropy measures information, entropy is the measure of information.

Macro-states, determined for example by the macroscopic quantities pressure (p), Volume (V), or Temperature (T), or number of particles (N), contain a very large number of micro-states.

Boltzmann’s Entropy S = k logarithm of the phase volume(= the probability) of a macro-state in terms of the possible micro-states.

## Different faces of Entropy

Entropy has many faces

- thermodynamic entropy, is a macroscopic state parameter of a system in equilibrium, like temperature, pressure, volume. However, can we measure entropy directly?
- microscopic, statistical entropy
- Boltzmann Entropy: S = k log W
- Gibbs entropy

- information theory
- Shannon’s entropy

## Shannon’s entropy

Shannon: “I thought of calling it “information”. But the word was overly used, so I decided to call it “uncertainty”. When I discussed it with John von Neumann, he had a better idea:

- in the first place your uncertainty has been used in statistical mechanics (ie by Boltzmann) under that name, so it already has a name
- in the second place, and more importantly, no one knows what entropy really is, so in a debate you will always have the advantage

## What can we learn from Ludwig Boltzmann?

- Empower young people, recognize and support talent early
- LB published first scientific work at age 21
- Full Professor at 25
- Head of Department at 32
- President of University at 43

- Talent is not linear – talent is exponential
- Move around the world. Connect. Interact.
- Empower women (LB promoted many women)
- Don’t accept authority for authority’s sake
- Science/physics issues need to be treated with the methods of physics/science
- No dogmas
- Support entrepreneurs (LB supported airplane developers before airplanes existed)

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