A random walk…

[This was material for a very speculative broad-audience talk I gave some years ago; artwork by Francesco Vedovato]

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Brownian motion was observed by british botanist Robert Brown (1827) and used by Einstein (1905) in thermodynamics. The mean position is zero, the mean distance from origin grows with the square root of time (position isn’t distance!). Good for describing diffusion, like when you drip some drops of black ink in a glass of water and see it spreading over time. Diffusion is an irreversible process.


Brownian motion was also used by Bachelier (1900) for stocks (a version called Geometric Brownian motion hat grows on average due to inflation – Leopardi’s “magnifiche sorti e progressive”). In both cases there is a lack of information about details, modelled through noise. We don’t know the behavior of all agents on a stock market and why they do what they do. For pollen grains: lack of knowledge of detailed interactions with underlying gas molecules.


Forget about pollen grains and think about your room, which is intrinsically an open system: With tiny draughts from the windows, cosmic rays penetrating through walls, cockroaches from the tub, people stepping inside… Detailed interaction with the environment is impossible to attain. The room belongs to an environment (the house) which belongs to an environment (the city) which belongs to a country … to the Universe. What does the Universe belong to? We come back to this later…


Back to the room: if entropy is a measure of disorder, what is the entropy of my room? My mother thinks my room is very messy. It looks messy to her mental order. I think it’s not because I know exactly where things are! My mother lacks knowledge about my room state, so her measure of entropy is different from mine. Entropy is a measure of ignorance. But ignorance with respect to what? Is entropy subjective?


We could acquire information about every single object within the room, then every single atom, then describe all nucelar and subnuclear interactions, then all gravitational fields, then all quantum gravitational interactions with a theory we don’t know yet… Up to what? Strings? Loops? Can we define ignorance with respect to an ultimate “atom” of reality? Well, most people who don’t work with strings think they are a dead theory (while most people working on it thinks it is still alive). Let’s concentrate on loops.


At the heart of Quantum Gravity lies the Wheeler-De Witt equation: the Hamiltonian constraint contains the laws of physics, the wave function of the Universe describes the state of the system (one of many possible states). All of that vanishes! That’s the difference with the Schroedinger equation, where time appears.


Each portion of the universe is a clock for each other portion of the Universe (including us). The whole Universe is not a physical system! Because we can’t do physics in there. But each portion of the Universe is necessarily open. Dissipation is necessary to define time, measurement and physics. Notice that clocks have always been defined either by referring to an outer environment, or by going into the innermost depths.


… of thermodynamics:

1 – You can’t win (you can’t obtain more energy than you poured in)
2 – You can’t tie (actually dissipation occurs making energy less usable)
3 – You can’t quit the game (you cannot perfectly isolate a system)

(there’s a zero-th law but let’s leave it aside).


How entropy as ignorance increases:

– if you are given the first snapshot, you can tell right away that the system has not yet relaxed. You can tell that it is some very early time in the evolution of the system.

– if you look at the second, you can still say that it must be sooner than some time

– if you look at the last, you can’t say anything: it might be a snapshot from any moment.

Entropy growth and loss of information determine the direction of time. They lead towards thermal equilibrium.


OK for where time goes. But how is it measured? By cycles. Irreversible cycles cannot go to equilibrium. Constant production of entropy is necessary to have structures. Clocks need to to constantly stay in nonequilibrium.


Global circulation of the atmosphere due to the forcing of the sun.
Nonequilibrium steady state. Cyclic. Alive.

In the entropy balance equation in this slide:

> 0 nonequilibrium (life)
= 0 equilibrium (death)


But wasn’t entropy subjective?
Does “reality” depend on the observer?
What is the observer is no Ph.D. in physics?

And here comes my own result: If you change perspective on entropy the second law stays the same. Physical laws are in the end invariant under this symmetry: the symmetry of changing prior beliefs. Yet breaking this symmetry, and “spending” some prejudices is necessary to be able to do physics (akin to choice of reference frames or measure units). The important fact is that physics gives us a dictionary to translate between different perspectives.


What is this symmetry of physics? How we assign probabilities to things. What is the probability of The Die? Doesn’t make sense. It makes sense talking of the probability of rolling a dice. Not the object, but the process. So what is the probability of rolling the dice? Is it 1/6? Well but maybe we could have some prejudices on the die. For example we might think it is loaded. Now the entropy of the system will depend on this prejudice. And also what we will learn when we roll the die (we might learn something = flux of information = flux of heat).  Prejudice and learning will change according to prior beliefs (entropy and entropy flux). But the die will roll independently of what we think of it!

That’s what science is about: having prejudices and confronting them with facts, and updating our prejudices to newer ones. It is based on the assumption that “reality” is independent of how we describe it. Physics is not about “reality”, it is about how we measure it.


It doesn’t make sense to talk about “the entropy of an object”, or the heat released. It makes sense to talk about in that particular context, with respect to a certain experimental apparatus, “how much entropy is measured”. For example, I prefer “The Higgs mechanism” to “The Higgs boson”. Language is how we describe reality. The language of physics today is still too oriented towards objects. Native American Indians had a natural language that adapted to processes. “The passage of a cloud”, rather than “a cloud”.

“All things physical are information-theoretic in origin and this is a participatory universe. Observer participancy gives rise to information; and information gives rise to physics.” J. A. Wheeler.


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