the percolation threshold for a honeycomb lattice
TRANSCRIPT
The Percolation Threshold for a Honeycomb Lattice
The Percolation Threshold for a Honeycomb Lattice
PercolationPercolation
Way of studying lattice disorder
Ignores physical and chemical properties
Way of defining geometric constants
Way of studying lattice disorder
Ignores physical and chemical properties
Way of defining geometric constants
Types of PercolationTypes of Percolation
Site Percolation - if two adjacent sites exist, then the bond between them exists.
Bond Percolation - If a bond exists, both sites must exist.
Site Percolation - if two adjacent sites exist, then the bond between them exists.
Bond Percolation - If a bond exists, both sites must exist.
ExampleThe Honeycomb LatticeExampleThe Honeycomb Lattice
Types of PercolationTypes of Percolation
Type of percolation will effect the probability associated with generating a specific cluster size
Our focus will be on site percolation.
Type of percolation will effect the probability associated with generating a specific cluster size
Our focus will be on site percolation.
Infinite ClustersInfinite Clusters
Infinite cluster is defined as a cluster that spans from one end to the other of an infinite lattice.
Not all probabilities will give this.
Define the percolation threshold pc
Infinite cluster is defined as a cluster that spans from one end to the other of an infinite lattice.
Not all probabilities will give this.
Define the percolation threshold pc
Percolation ThresholdPercolation Threshold
The percolation threshold is defined as the probability of a site existing, where we see an infinite cluster for the first time.
Few exact results. Most turn to numeric methods.
The percolation threshold is defined as the probability of a site existing, where we see an infinite cluster for the first time.
Few exact results. Most turn to numeric methods.
Concerns with Monte CarloConcerns with Monte Carlo
Mean value? Min Value?
Several conflicting results
Seems we can only do as good as an upper and lower bound.
Mean value? Min Value?
Several conflicting results
Seems we can only do as good as an upper and lower bound.
A Method to Calculate PcA Method to Calculate Pc
If we simply had a line of a sites
stretching to infinity, the probability of
getting an infinite cluster would be given
by…
P = pcL
pc must be 1 to see an infinite cluster
If we simply had a line of a sites
stretching to infinity, the probability of
getting an infinite cluster would be given
by…
P = pcL
pc must be 1 to see an infinite cluster
A Method to Calculate PcA Method to Calculate Pc
If we consider lattices that have more
connectivity, we would need to consider
the number of ways a particular L could
be realized.
P = N1pc1L1 + N2pc2
L2 + …
If we consider lattices that have more
connectivity, we would need to consider
the number of ways a particular L could
be realized.
P = N1pc1L1 + N2pc2
L2 + …
A Method to Calculate PcA Method to Calculate Pc
To start, lets consider only the shortest length as the would give the smallest value of pc.
By careful enumerating and counting we can come up with N and L as a function of m.
To start, lets consider only the shortest length as the would give the smallest value of pc.
By careful enumerating and counting we can come up with N and L as a function of m.
A Method to Calculate PcA Method to Calculate Pc
Then L=2m-1 and N=m2m.
So we get
P = m2mpc(2m-1) = m(2pc
2)m / pc
If 2pc2 < 1 then P=0
Then L=2m-1 and N=m2m.
So we get
P = m2mpc(2m-1) = m(2pc
2)m / pc
If 2pc2 < 1 then P=0
A Method to Calculate PcA Method to Calculate Pc
Thus, when this is equal to 1, we first get an infinite cluster which is the definition of pc.
Hence, Pc = 2(-1/2) = .7071
Which is consistent with the Monte Carlo
Thus, when this is equal to 1, we first get an infinite cluster which is the definition of pc.
Hence, Pc = 2(-1/2) = .7071
Which is consistent with the Monte Carlo
Triangle latticeTriangle lattice
If you go through the same method, but use the triangular lattice, you get the exact result of .5 which is the excepted result!
If you go through the same method, but use the triangular lattice, you get the exact result of .5 which is the excepted result!
Issues With other latticeIssues With other lattice
Square lattice doesn’t work because the shortest distance is a straight line.
Inverted honeycomb doesn’t work because it is also a straight line.
Generalization will need to include more probabilities.
Square lattice doesn’t work because the shortest distance is a straight line.
Inverted honeycomb doesn’t work because it is also a straight line.
Generalization will need to include more probabilities.
For the FutureFor the Future
Try to generalize into a broad equation.
Use to solve harder lattices.
Can we relate bond percolation? Perhaps a correlation between the two?
Try to generalize into a broad equation.
Use to solve harder lattices.
Can we relate bond percolation? Perhaps a correlation between the two?
ThanksThanks
Dedicated to the memory of
Carlos Busser
Aug 2007 - Dec 2007
Dedicated to the memory of
Carlos Busser
Aug 2007 - Dec 2007