In[]:=
With[{b=2},GatherBy[Tuples[{0,1},6],Total/@Partition[#,b]&]]
Out[]=
{{{0,0,0,0,0,0}},{{0,0,0,0,0,1},{0,0,0,0,1,0}},{{0,0,0,0,1,1}},{{0,0,0,1,0,0},{0,0,1,0,0,0}},{{0,0,0,1,0,1},{0,0,0,1,1,0},{0,0,1,0,0,1},{0,0,1,0,1,0}},{{0,0,0,1,1,1},{0,0,1,0,1,1}},{{0,0,1,1,0,0}},{{0,0,1,1,0,1},{0,0,1,1,1,0}},{{0,0,1,1,1,1}},{{0,1,0,0,0,0},{1,0,0,0,0,0}},{{0,1,0,0,0,1},{0,1,0,0,1,0},{1,0,0,0,0,1},{1,0,0,0,1,0}},{{0,1,0,0,1,1},{1,0,0,0,1,1}},{{0,1,0,1,0,0},{0,1,1,0,0,0},{1,0,0,1,0,0},{1,0,1,0,0,0}},{{0,1,0,1,0,1},{0,1,0,1,1,0},{0,1,1,0,0,1},{0,1,1,0,1,0},{1,0,0,1,0,1},{1,0,0,1,1,0},{1,0,1,0,0,1},{1,0,1,0,1,0}},{{0,1,0,1,1,1},{0,1,1,0,1,1},{1,0,0,1,1,1},{1,0,1,0,1,1}},{{0,1,1,1,0,0},{1,0,1,1,0,0}},{{0,1,1,1,0,1},{0,1,1,1,1,0},{1,0,1,1,0,1},{1,0,1,1,1,0}},{{0,1,1,1,1,1},{1,0,1,1,1,1}},{{1,1,0,0,0,0}},{{1,1,0,0,0,1},{1,1,0,0,1,0}},{{1,1,0,0,1,1}},{{1,1,0,1,0,0},{1,1,1,0,0,0}},{{1,1,0,1,0,1},{1,1,0,1,1,0},{1,1,1,0,0,1},{1,1,1,0,1,0}},{{1,1,0,1,1,1},{1,1,1,0,1,1}},{{1,1,1,1,0,0}},{{1,1,1,1,0,1},{1,1,1,1,1,0}},{{1,1,1,1,1,1}}}
In[]:=
With[{b=2},Map[ArrayPlot[{#},Mesh->True,ImageSize->{Automatic,15},Epilog->Table[Style[Line[{{i,0},{i,1}}],Thick,Red],{i,0,6,2}]]&,GatherBy[Tuples[{0,1},6],Total/@Partition[#,b]&],{2}]]
Out[]=
{{
},{
,
},{
},{
,
},{
,
,
,
},{
,
},{
},{
,
},{
},{
,
},{
,
,
,
},{
,
},{
,
,
,
},{
,
,
,
,
,
,
,
},{
,
,
,
},{
,
},{
,
,
,
},{
,
},{
},{
,
},{
},{
,
},{
,
,
,
},{
,
},{
},{
,
},{
}}
In[]:=
Divisors[8]
Out[]=
{1,2,4,8}
In[]:=
Graph[GraphUnion@@(Graph[TwoWayRule@@@Partition[Append[#,First[#]],2,1]]&/@With[{b=2},Map[ArrayPlot[{#},Mesh->True,ImageSize->{Automatic,12},Epilog->Table[Style[Line[{{i,0},{i,1}}],Thick,Red],{i,0,6,2}]]&,GatherBy[Tuples[{0,1},6],Total/@Partition[#,b]&],{2}]]),VertexLabels->{x_:>Placed[x,Center]},PerformanceGoal->"Quality",EdgeStyle->LightGray]
Out[]=
In[]:=
PathGraph[{
,
,
}]
Out[]=
PathGraph[{
,
,
}]
In[]:=
PathGraph[{a,b,c}]
Out[]=
PathGraph
In[]:=
Graph[{TwoWayRule[a,a]}]
Out[]=
#->BCAOperator[{2,{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}}}][#]&/@Tuples[{Tuples[Range[0,2],4],{0,1}}]
Out[]=
In[]:=
Graph[With[{b=2},Map[(Total/@Partition[Sign[First[#]],b])&,#->BCAOperator[{2,{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}}}][#]&/@Tuples[{Tuples[Range[0,2],4],{0,1}}],{2}]]]
Out[]=
Not conserving particle number:
Particular state has a particular path....

Start from a particular initial condition; or start from a class

It quickly visits high entropy states...

Claim: evolution from an ensemble of initial states is similar to time evolution from a single state

From Brad:

Powers of 3 in Base 2

Single Initial State over time visits many coarse-grained bins

Instead of looking at the whole state, look at a binned version
Entropy of a particular binned configuration is the number of microstates consistent with that binning.

Spatial Entropy

How many of each possible block occurs?

Coarse-Grained Entropy

1. Ensembles [start with an ensemble of possible initial conditions, that are consistent with a given constraint (e.g. particles only on the left) ]
​
2. Time averages [over time, does it visit coarse-grained states with typical probabilities?]

Big effect: we evolve to a state whose statistics is “typical” [i.e. the state is “random”]

Multiway version

Start with a pure STG. Then add state equivalences.
(Totaling in regions is the main state equivalence)

Orderless case