I wanted to mention Verlinde’s paper but stopped because I don’t quite understand it. It seems to require all kinds of physics background knowledge that I don’t have. Maybe you could give a purely mathematical explanation of the underlying datastructure? Something along the lines of “R^4 with Lorentz transformations” or “Hilbert space with Hermitian operators”?
The fragments that I understand don’t come from the paper, which is over my head, but from Verlinde’s blog.
In particular, he says: “The starting point is a microscopic theory that knows about time, energy and number of states. That is all, nothing more. This is sufficient to introduce thermodynamics. From the number of states one can construct a canonical partition function, and the 1st law of thermodynamics can be derived. No other input is needed, certainly not Newtonian mechanics. TIme translation symmetry gives by Noether’s theorem a conserved quantity. This defines energy. Hence, the notion of energy is already there when there is just time, no space is needed.
Temperature is defined as the conjugate variable to energy. Geometrically it can be identified with the periodicity of euclidean time that is obtained after analytic continuation. Again there is nothing needed about space. Temperature exists if there is only time.”
So I think the underlying structure might be something like Kauffman’s random boolean networks, or even simpler—a set of states and a transition function from states to states—the “microscopic dynamics”.
I wanted to mention Verlinde’s paper but stopped because I don’t quite understand it. It seems to require all kinds of physics background knowledge that I don’t have. Maybe you could give a purely mathematical explanation of the underlying datastructure? Something along the lines of “R^4 with Lorentz transformations” or “Hilbert space with Hermitian operators”?
The fragments that I understand don’t come from the paper, which is over my head, but from Verlinde’s blog.
In particular, he says: “The starting point is a microscopic theory that knows about time, energy and number of states. That is all, nothing more. This is sufficient to introduce thermodynamics. From the number of states one can construct a canonical partition function, and the 1st law of thermodynamics can be derived. No other input is needed, certainly not Newtonian mechanics. TIme translation symmetry gives by Noether’s theorem a conserved quantity. This defines energy. Hence, the notion of energy is already there when there is just time, no space is needed.
Temperature is defined as the conjugate variable to energy. Geometrically it can be identified with the periodicity of euclidean time that is obtained after analytic continuation. Again there is nothing needed about space. Temperature exists if there is only time.”
So I think the underlying structure might be something like Kauffman’s random boolean networks, or even simpler—a set of states and a transition function from states to states—the “microscopic dynamics”.