4 ms·

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).