3 ms·

I absolutely adore that we can predict these systems very well in infinite dimensions and some cutting-edge theories can perform well down to only 8 dimensions,

by WJW 4y ago

I absolutely adore that we can predict these systems very well in infinite dimensions and some cutting-edge theories can perform well down to only 8 dimensions, but theories describing the behavior of 3-dimensional systems are apparently still some way off and may be impossible. This reminds me of one of my favorite mathematician jokes: A doctor is invited to a maths conference by their mathematician friend. After they come out of one particularly difficult talk about 12-dimensional string theory they ask their friend: "It was very interesting but how can you possibly visualize a 12-dimensional system???!?". The friend replies: "Oh it's quite simple really: you just visualize an N-dimensional systems and then set N=12!".

zitterbewegung 4y ago
Honestly it’s best to visualize a n dimensional system by a bunch of sliders that can have various values. Like in 4d to visualize that system you have a rotation slider similar to a movie but the axis isn’t time.
- hobs 4y ago
  Why not just a spreadsheet? I guess that might not be visualization...
- PaulHoule 4y ago
  Why visualize it when you can just compute an integral? https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Statistical_Mechanics_(Styer)/10%3A_Appendices/10.04%3A_D-_Volume_of_a_Sphere_in_d_Dimensions https://phys.libretexts.org/Bookshelves/Thermodynamics_and_S...
PaulHoule 4y ago
Generically in the theory of phase transitions and critical phenomena the infinite dimensional case is easy because mean field theory works, but there is some dimension at which that breaks, frequently N=6, and you can write an asymptotic expansion in N-ε for the critical exponents which converges enough for most things that you can get the right answers in 3 dimensions. We worked through this book when I was in grad school... https://www.amazon.com/Theory-Critical-Phenomena-Introduction-Renormalization/dp/0198513933 https://www.amazon.com/Theory-Critical-Phenomena-Introductio...
onos 4y ago
Problem with these systems is that you can’t study their equilibrium with simulations. That’s cause their equilibrium time scale gets longer and longer as you lower temp and it becomes too expensive to simulate long enough to get there. So it’s all theory and there are no models yet that can be fully characterized analytically.
modeless 4y ago
Geoff Hinton gave this wisdom in one of the lectures of his legendary Coursera deep learning course: "To deal with hyper-planes in a 14-dimensional space, visualize a 3D space and say 'fourteen' to yourself very loudly. Everyone does it."
carlob 4y ago
To be fair the mean field, or infinite dimensional version of a lattice system is fairly easy to visualize, because it's just the complete graph.
- klyrs 4y ago
  Oh, it's way more interesting than that. Each dimension is linear. Any two dimensions form a grid. There are infinitely many orthogonal directions along which you can go an arbitrary distance. In the complete graph, any two points are equidistant. It's rather boring. Also, (and you probably couldn't pay a physicist to care about this, but) you can have an infinite complete graph, with countably infinite nodes and edges. The infinite dimensional grid, on the other hand, has uncountably infinite nodes and edges even if the dimensions are countably infinite.
  - carlob 4y ago
    That's not how you do the mean field approximation in practice. In your Hamiltonian you have N particles or nodes, which you send to infinity in the thermodynamic limit. In the mean field approximation you are just letting everybody interact with everybody and then scaling the interaction strength as 1/N so that energy scales as N. This is the core difference between the Edwards-Anderson and the Sherrington-Kirkpatrick models: instead of interacting with a finite set of neighbors you interact with everybody else (or an extensive number of neighbors).

by WJW 4y ago

zitterbewegung 4y ago

Honestly it’s best to visualize a n dimensional system by a bunch of sliders that can have various values. Like in 4d to visualize that system you have a rotation slider similar to a movie but the axis isn’t time.

hobs 4y ago
Why not just a spreadsheet? I guess that might not be visualization...
PaulHoule 4y ago
Why visualize it when you can just compute an integral? https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Statistical_Mechanics_(Styer)/10%3A_Appendices/10.04%3A_D-_Volume_of_a_Sphere_in_d_Dimensions https://phys.libretexts.org/Bookshelves/Thermodynamics_and_S...

PaulHoule 4y ago

Generically in the theory of phase transitions and critical phenomena the infinite dimensional case is easy because mean field theory works, but there is some dimension at which that breaks, frequently N=6, and you can write an asymptotic expansion in N-ε for the critical exponents which converges enough for most things that you can get the right answers in 3 dimensions. We worked through this book when I was in grad school... https://www.amazon.com/Theory-Critical-Phenomena-Introduction-Renormalization/dp/0198513933 https://www.amazon.com/Theory-Critical-Phenomena-Introductio...

onos 4y ago

Problem with these systems is that you can’t study their equilibrium with simulations. That’s cause their equilibrium time scale gets longer and longer as you lower temp and it becomes too expensive to simulate long enough to get there. So it’s all theory and there are no models yet that can be fully characterized analytically.

modeless 4y ago

Geoff Hinton gave this wisdom in one of the lectures of his legendary Coursera deep learning course: "To deal with hyper-planes in a 14-dimensional space, visualize a 3D space and say 'fourteen' to yourself very loudly. Everyone does it."

carlob 4y ago

To be fair the mean field, or infinite dimensional version of a lattice system is fairly easy to visualize, because it's just the complete graph.

klyrs 4y ago
Oh, it's way more interesting than that. Each dimension is linear. Any two dimensions form a grid. There are infinitely many orthogonal directions along which you can go an arbitrary distance. In the complete graph, any two points are equidistant. It's rather boring. Also, (and you probably couldn't pay a physicist to care about this, but) you can have an infinite complete graph, with countably infinite nodes and edges. The infinite dimensional grid, on the other hand, has uncountably infinite nodes and edges even if the dimensions are countably infinite.
- carlob 4y ago
  That's not how you do the mean field approximation in practice. In your Hamiltonian you have N particles or nodes, which you send to infinity in the thermodynamic limit. In the mean field approximation you are just letting everybody interact with everybody and then scaling the interaction strength as 1/N so that energy scales as N. This is the core difference between the Edwards-Anderson and the Sherrington-Kirkpatrick models: instead of interacting with a finite set of neighbors you interact with everybody else (or an extensive number of neighbors).