This has disconnected (“energy surfaces”) based on a count of As and Bs ; it is ergodic within these ensembles
In[]:=
ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB"},ResourceFunction["StringTuples"]["AB",5],8,"StatesGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB","A"->"B","B"->"A"},ResourceFunction["StringTuples"]["AB",3],8,"StatesGraph"]
Out[]=
In[]:=
VertexList
Out[]=
{AAB,AAA,ABA,BAA,ABB,BAB,BBA,BBB}
This changes only one bit at a time:
In[]:=
ResourceFunction["MultiwaySystem"][{"A"->"B","B"->"A"},ResourceFunction["StringTuples"]["AB",3],8,"StatesGraph"]
Out[]=
In[]:=
VertexList
Out[]=
{AAB,AAA,ABA,BAA,ABB,BAB,BBA,BBB}
Causal Graphs
Causal Graphs
In[]:=
ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB","A"->"B","B"->"A"},ResourceFunction["StringTuples"]["AB",3],4,"CausalGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB"},ResourceFunction["StringTuples"]["AB",3],2,"CausalGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB","A"->"B","B"->"A"},ResourceFunction["StringTuples"]["AB",3],3,"CausalGraph"]
Out[]=
In[]:=
ResourceFunction["MultiwaySystem"][{"AB"->"BA","BA"->"AB","A"->"B","B"->"A"},ResourceFunction["StringTuples"]["AB",3],2,"EvolutionCausalGraph"]
Out[]=
From Mano’s code:
From Mano’s code:
“Amplitude” is the number of ways to get from initial state to final state
Transition weight depends on the number of updates allowed .... because you can keep flipping back and forth
Enumerate “paths” to get from string to another
Enumerate “paths” to get from string to another
Conservation laws allow only cases with the same A count and B count to be transformed into each other
Elements of a braid group ;; in the end we are constructing a permutations
Enumerate sorting networks going from initial state to final state
Renormalizability
Renormalizability
There’s only one kind of loop; We can get rid of repeated loops with counterterms
The content of renormalizability is that then we can compare different scattering processing finitely
The content of renormalizability is that then we can compare different scattering processing finitely
Subtracting path counting between different scattering amplitudes we’ll get finite results
Roadmap
Roadmap
Generate possible actual evolution sequences for flip rules.
Can do it between all possible states (vacuum generating function)
or between particular states
Can do it between all possible states (vacuum generating function)
or between particular states
Then find the causal graphs corresponding to evolution sequences.
Because of causal invariance many evolution sequences will give the same causal graph
(but then we’ll have weights for distinct causal graphs, that correspond to the number of evolution graphs that equivalence into these)
Because of causal invariance many evolution sequences will give the same causal graph
(but then we’ll have weights for distinct causal graphs, that correspond to the number of evolution graphs that equivalence into these)
There will be an infinite number of evolution graphs. Reduce with counterterms. Then evaluate differences to get renormalized amplitudes.
[[ This is all zero dimensional field theory ]]
Going to space
Going to space
In the end, the features we’re looking will be interwoven into a hypergraph rewriting setup
From Nik: propagator
From Nik: propagator
The analog of propagator for graphs is the inverse of Laplacian [Laplacian] (L = D - A), which plays the role of the 2nd order kinetic term (diffusion, random walks, maze solvers, etc.).
The second order is because it connects two vertices (the adjacency matrix has a row and a column, which is rank 2).
Inverting it is like reverse-engineering the rules to infer the past state from effect to its cause.
Operating with [Laplacian] seems to correspond to an updating event and a multiway edge in WPP.
Hypergraphs presumably should generalize this to higher-order kinetic terms and theories with bilaplacian [Laplacian]^2 (biharmonic operator) in its equations.
I think it doesn’t mean that WolframModel is biharmonic because it’s not about the state but its dynamics (multiway system), so [Laplacian]^2 operation should correspond to a higher-order multiway edge, turning the multiway system into a hypergraph and somehow computing Feynman Sum over hypergraph paths instead.
The second order is because it connects two vertices (the adjacency matrix has a row and a column, which is rank 2).
Inverting it is like reverse-engineering the rules to infer the past state from effect to its cause.
Operating with [Laplacian] seems to correspond to an updating event and a multiway edge in WPP.
Hypergraphs presumably should generalize this to higher-order kinetic terms and theories with bilaplacian [Laplacian]^2 (biharmonic operator) in its equations.
I think it doesn’t mean that WolframModel is biharmonic because it’s not about the state but its dynamics (multiway system), so [Laplacian]^2 operation should correspond to a higher-order multiway edge, turning the multiway system into a hypergraph and somehow computing Feynman Sum over hypergraph paths instead.
At every step, distribute probability equally to all connected nodes. Claim: after t steps, the result is the result of applying a 2nd order finite difference order
Two nodes separated on the graph by distance s ... get a certain probability density
Multiway graph with transpositions
Multiway graph with transpositions
These graphs don’t include loops
What is the multiway graph going from one state to another?
Foliate this graph and draw branchial graphs