Everyday entropy: conformal fields

Conformal Photography

Conformal field theory helps to describe the entropy or physical configuration of a system in terms of its own information.  This may seem obvious, but the way in which the fundamental units of information like spin, charge, frequency, etc., relate to the macroscopic observables like magnetism or phase critical point are obscure.  Dr.Simmons-Duffin explains in this remarkable seminar how second order phase transitions like the critical points of water and ferromagnets have identical entropy profiles.  This says something about the consistency of entropy, but I’m not sure what.

Abstract: “Conformal Field Theory (CFT) describes the long-distance dynamics of numerous quantum and statistical many-body systems. The long-distance limit of a many-body system is often so complicated that it is hard to do precise calculations. However, powerful new techniques for understanding CFTs have emerged in the last few years, based on the idea of the Conformal Bootstrap. I will explain how the Bootstrap lets us calculate critical exponents in the 3d Ising Model to world-record precision, how it explains striking relations between magnets and boiling water, and how it can be applied to questions across theoretical physics.”

This is also an example of how to talk about entropy without mentioning entropy.  It  describes how physical configuration affects the information (spin, magnetism, heat, compression) in a system and in particular the phase transition dynamics.  The bootstrapping technique means finding constraints for the answer and then looking at the shape of the possibilities rather than calculating the answer from first principles and initial conditions.  It’s extremely useful in unmeasurable situations where Feynman diagrams and other perturbative techniques fail.  The bootstrapping method looks a lot like Einstein’s special relativity paper: what could the phenomenon look like for the result to fit the outcome as we observe it?  Conformal bootstrapping almost certainly has applications in the phase shift from urban to rural development patterns and the phase shift from just getting by to being filthy rich.

Conformal field theory also seems related to the Navier–Stokes existence and smoothness problem, which concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics (such as with turbulence). These equations describe the motion of a fluid (that is, a liquid or a gas) in space. Solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.

The relative simplicity of the 2d Ising model versus 3d also seems to point to the fundamental flatness of information and its essential opposition to mass in 3 dimensional space and time.  Information is flat and orthogonal, so its existence in 3 dimensions is a convoluted knot of two dimensional topologies, which makes it incredibly difficult to track information in 3d.

The difference between knowledge and understanding becomes apparent in the turbulence.  Where an event can’t be predicted from the set of preconditions, knowledge falls apart and you have to understand the general contours of how things work.  The unknown and unknowable lives in these pockets between the equations and formulae, and the danger of knowing the math without understanding the phenomena is that you are either baffled by the pockets of entropy or you ignore them.  Physicists seem to have largely abandoned real words in favour of verbal representations of equations, which creates a gulf between them and society that makes it impossible for them to have an influence, even when their knowledge is really important, as with climate change, deforestation and refugee crises.

Note: The Monte Carlo technique is another technique for solving uncertainties.

3d Ising Model

General Ising Model

Entropy is, by definition, that part of a thing that is not definable.  At its maximum, entropy equal energy distributed randomly in a volume of space.  In information and thermodynamics, entropy can be thought of as the measure of uncertainty.  Information un-communicated or energy unavailable.  This un-ness of entropy is somehow related to vacuum space, which we know is full of fields and energy and information, but can’t be defined in terms of anything we can experience.  Space isn’t like a solid, liquid, gas or plasma.  It is unknown in the way that entropy is uncertain.  But, while they are profoundly foreign, they behave incredibly consistently.  So entropy is usually defined in terms of what it does: it allows certain reactions and prohibits others; it increases.  Likewise, space transmits electromagnetic excitations; it bends; it pushes things together to make gravity.  In the end, entropy has no satisfactory definition in part because of its relationship with space in the physical sense, which is unknown, and uncertainty in the informational sense.  This is why the bootstrap method is so important, both mathematically and philosophically.  Entropy is consistent, and it is possible to define a lot of what it isn’t, so you can look at what is left over, the form of what is left undefined, and say the shape of that void is entropy.



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