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7
Jon Lederman
[WSS20] Weyl's law and The Wolfram Model of physics
Jon Lederman, Spinor
Posted
2 years ago
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Weyl's law and The Wolfram Model of physics
by
Jon Lederman
Introduction
The goal of this work is explore Weyl’s Law as it may relate to the graphs underlying the Wolfram Model of Physics. Weyl’s law describes the nature of the asymptotic growth of the eigenvalues of the Laplacian operator on bounded domains having Dirichlet and Neumann boundary conditions. The Laplace-Beltrami operator is a generalization of the Laplacian for operating on Riemannian manifolds. In the context of the Wolfram Model of physics, an ongoing area of investigation is whether the principles of Einstein’s theory of General Relativity thereby arise as emergent phenomena. As General Relativity is described in the language of Riemannian Geometry, the question of whether properties of Riemannian manifolds also hold on these graphs. Weyl’s law is a statement regarding the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with DIrichlet or Neumann boundary conditions. The law has been extended to non-Euclidean space through a generalization of the Laplace operator called the Laplace-Beltrami operator. This investigation of Weyl’s law in the context of the Wolfram Model posits a number of fundamental questions at the intersection of Riemannian geometry and graph theory, among which a principal question is what the appropriate analogue for the Laplace-Beltrami operator is for digraphs. In particular, does there exist a generalization of the graph Laplacian that encodes geometric information in the Wolfram model graphs analogous to Laplace-Beltrami operator on Riemannian manifolds.
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:
1
)
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;
2
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c
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;
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x
,
w
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t
.
Discretizing the Heat Equation
How can we map the heat equation to a graph? Formally, a graph comprises a set of vertices connected via a series of edges. Our end goal is to study the behavior of the Laplacian and it’s generalization to non-Euclidean spaces, the Laplace-Beltrami operator with respect to graphs generated via the evolutionary application of rules as defined in the Wolfram Model. For now, however, Imagine a simple planar graph such as the following :
I
n
[
]
:
=
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[
{
1
6
,
1
6
}
,
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[
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=
Each vertex may associated with a temperature, such that the composite set of temperatures defines a heat distribution on the graph. We can define a temperature or heat distribution function
ϕ
(v) that maps a temperature to each vertex. For visualization purposes, we can indicate the temperature on a graph using color. To do so, let’s define some functions to color our graph based upon the temperature at each vertex. As our immediate goal is to model the temporal evolution of heat flow based upon the heat equation, We will use these coloring functions later visualizing the heat flow on more complex graphs such as those arising in the Wolfram Model.
I
n
[
]
:
=
(
*
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[
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=
{
H
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}
;
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[
g
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[
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=
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;
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G
[
g
_
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t
m
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:
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o
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[
{
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=
g
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[
{
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]
=
t
m
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;
n
g
]
;
(
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*
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e
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[
g
_
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[
g
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]
;
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[
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[
]
,
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x
C
o
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[
g
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]
]
;
(
*
C
o
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s
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=
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[
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,
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a
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[
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,
{
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e
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L
i
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t
[
g
]
,
M
a
p
[
C
o
l
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a
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[
"
R
a
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b
o
w
"
]
,
t
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m
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t
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s
]
}
]
]
;
(
*
C
o
l
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r
s
a
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r
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p
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b
y
i
t
s
i
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(
f
r
o
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x
W
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s
)
*
)
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n
a
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[
g
_
]
:
=
C
o
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B
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p
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[
g
,
G
e
t
T
e
m
p
e
r
a
t
u
r
e
s
[
g
]
]
;
Let' s apply a random temperature distribution to our simple grid graph for illustrative purposes :
I
n
[
]
:
=
C
o
l
o
r
B
y
R
a
n
d
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[
s
i
m
p
l
e
G
r
i
d
G
r
a
p
h
]
O
u
t
[
]
=
We are using the Rainbow color scheme as defined in Mathematica, but are free to choose other schemes. The Rainbow scheme maps purple to the coldest temperature and red to the hottest :
I
n
[
]
:
=
C
o
l
o
r
D
a
t
a
[
"
R
a
i
n
b
o
w
"
]
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u
t
[
]
=
C
o
l
o
r
D
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F
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t
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N
a
m
e
:
R
a
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b
o
w
G
r
a
d
i
e
n
t
:
We can further imagine that a vertex coupling two vertices allows heat to flow directly between the two vertices. On the other hand, if two vertices are not directly connected no heat can directly flow, but instead may flow through some other path if the vertices are connected through some other set of vertices. Since we are seeking to model the temporal evolution of heat diffusion, we will necessarily define some initial conditions on our graph. Although the random distribution looks pretty, it' s not particularly realistic for what we might expect of our initial conditions. Instead, we seek something like a delta function wherein our heat distribution is characterized by a concentrated heat source. In order to do that, we will define some functionality to generate a a rough delta or point source distribution on our graph. While we’re at it, let’s also define some functionality to normalize temperatures between 0-1 so that we can conviently apply a color map.
I
n
[
]
:
=
(
*
S
e
t
s
a
d
e
l
t
a
f
u
n
c
t
i
o
n
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*
)
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o
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[
g
_
,
v
e
r
t
e
x
_
,
d
i
s
t
a
n
c
e
_
]
:
=
M
o
d
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l
e
[
{
n
g
=
g
}
,
(
A
n
n
o
t
a
t
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V
a
l
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[
{
n
g
,
#
}
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]
=
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t
a
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t
A
r
r
a
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[
1
,
L
e
n
g
t
h
[
#
]
]
)
&
@
V
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t
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x
L
i
s
t
[
N
e
i
g
h
b
o
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r
a
p
h
[
g
,
v
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t
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x
,
d
i
s
t
a
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e
]
]
;
n
g
]
;
(
*
N
o
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m
a
l
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s
t
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m
p
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a
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p
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b
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t
w
e
e
n
0
a
n
d
1
*
)
A
t
t
r
i
b
u
t
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s
[
N
o
r
m
a
l
i
z
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T
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m
p
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r
a
t
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s
]
=
{
H
o
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d
F
i
r
s
t
}
;
N
o
r
m
a
l
i
z
e
T
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m
p
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r
a
t
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r
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s
[
g
_
]
:
=
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e
t
T
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m
p
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r
a
t
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r
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s
[
g
,
N
[
#
/
M
a
x
[
#
]
]
]
&
@
G
e
t
T
e
m
p
e
r
a
t
u
r
e
s
[
g
]
;
(
*
N
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r
m
a
l
i
z
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s
a
v
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c
t
o
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o
f
i
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a
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c
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n
d
i
t
i
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s
*
)
N
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m
a
l
i
z
e
C
o
n
d
i
t
i
o
n
s
[
i
c
_
]
:
=
N
[
i
c
/
M
a
x
[
i
c
]
]
;
(
*
Z
e
r
o
s
o
u
t
t
h
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t
e
m
p
e
r
a
t
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d
i
s
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i
b
u
t
i
o
n
*
)
M
a
k
e
U
n
i
f
o
r
m
l
y
C
o
l
d
[
g
_
]
:
=
S
e
t
T
e
m
p
e
r
a
t
u
r
e
s
[
g
,
C
o
n
s
t
a
n
t
A
r
r
a
y
[
0
,
V
e
r
t
e
x
C
o
u
n
t
[
g
]
]
]
;
Let' s now clear our temperature distribution on simpleGridGraph and instead impose a delta function like initial condition:
I
n
[
]
:
=
S
e
t
T
e
m
p
e
r
a
t
u
r
e
s
[
s
i
m
p
l
e
G
r
i
d
G
r
a
p
h
,
C
o
n
s
t
a
n
t
A
r
r
a
y
[
0
,
2
5
6
]
]
;
C
o
l
o
r
B
y
I
n
t
e
r
n
a
l
T
e
m
p
e
r
a
t
u
r
e
[
s
i
m
p
l
e
G
r
i
d
G
r
a
p
h
]
;
d
e
l
t
a
G
r
i
d
G
r
a
p
h
=
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Now we are set up to represent temperatures on our graph and define delta - like initial conditions. But, we are yet to attack the most important question of how to perform temporal evolution of the heat equation on the graph. In short, we will discretize over space but leave the time dependence as a continuous variable. In order to do that, we need to define the adjacency matrix A for our graph. If we define a matrix in which each row index corresponds to a vertex in the graph and similarly, each column, we define the entry for each row as 0 if an edge exists between the respective vertices indexed by the row and column. Thus, if we have N vertices in the graph, the adjacency matrix is an N*N matrix of 1’s and 0’s indicating whether particular vertices are connected to one another.
The final step in encoding the heat equation PDE to a graph is to figure out how to model the Laplacian operator itself. A simple approach is to recognize physically that the rate of heat diffusion at a given vertex with respect an adjacent vertex by Newton’s law of cooling is proportional to the difference in temperature between the adjacent vertices assuming the temperature differential is small. Alternatively, we could have evaluated the discrete difference approximation to the second derivative using a numerical approximation scheme. This approach would yield a central difference approximation with a dependence upon the inverse square of the discrete distance between adjacent nodes. In our scheme, adjacent vertices are assumed to have a unit distance. However, because of complications involving applying the central difference approximation to graphs, we choose the former approach.
We are now ready to write down our discretized heat equation on a graph. In words, the time derivative of the temperature of each vertex equals the linear superposition of temperature differences between ALL adjacent nodes.
For purposes of this discussion, we will normalize
α
to 1. We can now manipulate this equation as follows. The operator ' deg' shown below calculates the number of outgoing or equivalently incoming vertices to a given vertex. In the end we arrive with a discrete operator equation for a new operator L, which represents the discrete Laplacian operator for our graph. After some derivation as follows, the Laplacian matrix emerges as the appropriate operator for the heat equation on a graph:
Here D is a diagonal matrix wherein each entry on the diagonal counts the number of outgoing (or equivalently incoming) edges for a vertex. In essence, we are left with a simple one-dimensional homogeneous ordinary differential equation in time.
It is understood here that
ϕ
is a vector of temperature coefficients defining the heat distribution on the graph. In our case, we seek to solve the this equation in which we define some initial conditions specifying the temperature distribution at time 0. As noted previously, this equation can be solved by standard techniques for ODEs by projecting the initial conditions onto the eigenspace, performing the temporal evolution there and projecting back again to temperature space. And indeed that is what we shall do as codified in the following Wolfram Language code. First, let’s calculate the graph associated matrices such as the adjacency matrix and the Laplacian matrix:
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[
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:
=
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The next set of code performs the heat kernel dynamics using the above described approach of eigenvector decomposition.
I
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[
]
:
=
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;
Let' s try out this code on our simpleGridGraph to see how a delta - function initial condition propagates in time by visualizing the temperature evolution at several different time advances starting at 0.
I
n
[
]
:
=
S
e
t
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,
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s
t
a
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A
r
r
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y
[
0
,
2
5
6
]
]
;
C
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1
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=
Advance by 0.1 seconds :
I
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[
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:
=
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=
Advance again by another 0.15 seconds :
I
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[
]
:
=
T
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[
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h
,
0
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1
5
]
;
C
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=
Advance by another 0.25 seconds :
I
n
[
]
:
=
T
i
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[
d
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0
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2
5
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;
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=
Advance one more time by 0.5 seconds :
I
n
[
]
:
=
T
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0
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;
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[
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=
This looks reasonable! We can see the heat diffusion over time. However, for visualization purposes perhaps an automated animation would provide some insight for visualization of the temporal evolution of the heat equation. The following set of code generates animation frames over a range of time values and sets up the appropriate animations.
I
n
[
]
:
=
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#
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;
Before applying this code, let' s define a more realistic graph in which vertices are coupled to all nearest neighbors including diagonal directions. We also will define some initial conditions on our graph.
I
n
[
]
:
=
(
*
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[
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;
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;
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[
[
1
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]
;
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]
;
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;
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d
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s
[
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;
I
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[
]
:
=
a
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=
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,
0
,
5
,
0
.
2
,
5
0
0
,
7
]
O
u
t
[
]
=
Next, we will apply this to one of the Wolfram Models. Specifically, we will look at 7714 from the Registry of Notable Universe Models. Before doing so, however, a fundamental question arises. Wolfram Model graphs are digraphs. However, the Laplacian matrix for digraphs is not well defined and in general will not be a symmetric positive-semidefinite matrix. One solution, which we adopt here, is to discard the directedness information and convert the graph to an undirected graph. Nonetheless, the validity of discarding the structure of the digraph is a question that warrants further consideration. In any case, we proceed with the approach of converting to an undirected graph.
I
n
[
]
:
=
(
*
W
o
l
f
r
a
m
M
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)
W
M
=
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[
"
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M
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"
]
;
H
G
2
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=
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[
"
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]
;
U
n
i
v
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s
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7
7
1
4
=
W
M
[
{
{
{
1
,
2
,
2
}
,
{
3
,
1
,
4
}
}
{
{
2
,
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,
2
}
,
{
2
,
3
,
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}
,
{
4
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}
}
}
,
{
{
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}
}
,
7
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]
;
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7
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=
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O
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[
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=
I
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:
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2. Weyl' s Law
Background
Weyl’s law has a rich history and diverse field of application. In general, it is a statement regarding the asymptotic growth of the eigenvalues of the Laplacian on bounded domains with DIrichlet or Neumann boundary conditions. The subject has roots in acoustics, specifically relating to the asymptotic behavior of the overtones of notes on instruments in a room, for example. In physics, the impetus for the development of Weyl’s law grew out of the black body radiation problem. Weyl' s law has been generalized to address non-Euclidean spaces such as d-dimensional manifolds with an associated Riemannian metric. In this case, the Laplacian must be replaced with the corresponding operator in these spaces, which is referred to as the Laplace-Beltrami operator. In the context of a manifold can be expressed as follows :
Here
Ω⊂
d
is a bounded domain with an associated volume vol (
Ω
).
ω
d
is the volume of a unit ball in
d
and N(
λ
) is the number of eigenvalues of the Laplace-Beltrami operator less than
λ
.
A simple example illustrating Weyl’s law for a two-dimensional Dirichlet problem is derived by Matt Stevenson and proceeds as follows. We are interested in solving the following eigenvalue problem:
on the bounded domain
Ω
.
Take the boundary :
In two dimensions this becomes :
This equation can be solved using standard techniques such as separation of variables yielding the following separable solution :
The eigenvalues are :
Then, define an eigenvalue counting function :
This equation is suggestive of the coefficients of an ellipse equation. We can define an ellipse and the associated area in the first quadrant cut out by the ellipse by dividing by lambda and re-introducing the continuous variables x and y :
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For each eigenvalue pair, we can define a unit - area square :
The counting function can now be expressed as follows :
Exploration of Weyl’s Law WIth Respect to Wolfram Model Graphs Using The Laplacian Operator
As noted, a central issue concerns the appropriate analogue of the Laplace-Beltrami operator for d-dimensional manifolds in the context of graphs. This is an open question on the forefront of current research and will be discussed below in section 3. As a starting point, it will be instructive to attempt to naively apply the standard Laplacian operator to Wolfram Model graphs and examine whether the relationship holds in any form. Let us continue with Universe 7714. First, we will estimate it’s dimension as defined in the Wolfram Physics Project.
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