programming assessed work 2
DESCRIPTION
University of BathTRANSCRIPT
Programming Skills Assessed Work 2
Candidate Number: 20825
April 29, 2015
1 Diffusion in One Dimension
This program will perform M = 100 random walks through L = 10 sites,labelled from i = 0, . . . , L− 1, and print the average path length, which is 45.02in this case
1.1 C Code
#include <stdio.h>
int ranInt(int max); // returns a random integer from 0 to max -1
int rHop(int curSite , int numSites); // increments current site
with periodic boundaries
int lHop(int curSite , int numSites); // decrements current site
with periodic boundaries
main()
{
// VARIABLE DECLARATIONS
int i=0; // current site , allowed from 0 to L-1
int L=10; // number of sites
int pos[L]; // array indexing the sites from 0 to L-1,
incrementing each time that site is visited
int visit =0; // increments each time a site is visited for the
first time
int tempRan; // temporary storage of random integer
int j; // for -loop index
int k; // averaging index
float M=100; // number of paths to be averaged
float tot=0; // running total for averaging
int seed =3; // sets up random number generator
srandom(seed);
for(k=0;k<M;k++) // averaging loop
{
// INITIALISE
for(j=0;j<L;j++)
{
pos[j]=0;
}
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visit =0;
// RANDOM WALK
for(j=0;visit <L;j++) // loop terminates when particle has
visited all L sites
{
pos[i]++;
if(pos[i]==1) visit ++;
tempRan=ranInt (2); // the particle can move in two directions
if (tempRan ==0) i=lHop(i,L);
if (tempRan ==1) i=rHop(i,L);
}
tot+=j-1; // j is the number of hops in total , including the
hop after the last site was visited
}
printf("%f\n", tot/M);
}
// FUNCTION DEFINITIONS
int ranInt(int max) // returns a random integer from 0 to max -1
{
return random () % max;
}
int rHop(int curSite , int numSites) // increments current site
with periodic boundaries
{
if(curSite ==numSites -1)
{
curSite =0;
}
else
{
curSite ++;
}
return curSite;
}
int lHop(int curSite , int numSites) // decrements current site
with periodic boundaries
{
if(curSite ==0)
{
curSite=numSites -1;
}
else
{
curSite --;
}
return curSite;
}
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Figure 1: a line graph showing length of path taken for different environments
1.2 Discussion and Interpretation
Figure 1 shows a few key properties expected in this problem. Firstly, the graphintercepts at the origin, because it would take no time to explore an environmentof size zero
Secondly, there is an exponential relationship between the number of sitesL and the path length
Proof. To demonstrate why this must be true, consider how the mean pathlength in this problem would increase as the environment grew larger
For example, between L1 = 10 and L2 = 20, the shortest possible pathsdouble, and therefore the mean length must increase by more than double
This means that it is reasonable to suggest that the path length J will satisfy
dJ
dL∝ ∆L.
This is a property satisfied uniquely by the exponential function
These results were generated using a similar program that looped over Lfrom 1 to 50 and printed the results to a text file, which was then turned intoa graph using the online service Plotly c© 2015
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2 Diffusion in Three Dimensions
Here the average path length is 10261.4 to explore the all 103 sites
2.1 C Code
#include <stdio.h>
int ranInt(int max); // returns a random integer from 0 to max -1
int rHop(int curSite , int numSites); // increments current site
with periodic boundaries
int lHop(int curSite , int numSites); // decrements current site
with periodic boundaries
main()
{
// VARIABLE DECLARATIONS
int i1=0, i2=0, i3=0; // coordinates of current site , allowed
from 0 to L-1
int L=10; // number of sites per coordinate axis
int pos[L][L][L]; // array indexing the sites from 0 to L-1,
incrementing each time that site is visited
int visit =0; // increments each time a site is visited for the
first time
int tempRan; // temporary storage of random integer
int j, j1, j2, j3; // for -loop indices
int k; // averaging index
float M=100; // number of paths to be averaged
float tot=0; // running total for averaging
int seed =3; // sets up random number generator
srandom(seed);
for(k=0;k<M;k++) // averaging loop
{
// INITIALISE
for(j1=0;j1 <L;j1++)
{
for(j2=0;j2 <L;j2++)
{
for(j3=0;j3 <L;j3++)
{
pos[j1][j2][j3]=0;
}
}
}
visit =0;
// RANDOM WALK
for(j=0;visit <L*L*L;j++) // loop terminates when particle has
visited all L^3 sites
{
pos[i1][i2][i3]++;
if(pos[i1][i2][i3]==1)
{
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visit ++;
}
tempRan=ranInt (6); // the particle can travel in six
directions
if (tempRan ==0) i1=lHop(i1,L);
if (tempRan ==1) i1=rHop(i1,L);
if (tempRan ==2) i2=lHop(i2,L);
if (tempRan ==3) i2=rHop(i2,L);
if (tempRan ==4) i3=lHop(i3,L);
if (tempRan ==5) i3=rHop(i3,L);
}
tot+=j-1; // j is the number of hops in total , including the
hop after the last site was visited
}
printf("%f\n", tot/M);
}
// FUNCTION DEFINITIONS
int ranInt(int max) // returns a random integer between 1 and
max
{
return random () % max;
}
int rHop(int curSite , int numSites) // increments current site
with periodic boundaries
{
if(curSite ==numSites -1)
{
curSite =0;
}
else
{
curSite ++;
}
return curSite;
}
int lHop(int curSite , int numSites) // decrements curent site
with periodic boundaries
{
if(curSite ==0)
{
curSite=numSites -1;
}
else
{
curSite --;
}
return curSite;
}
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Figure 2: a line graph showing length of path taken for different environments
2.2 Discussion and Interpretation
Figure 2 has the same shape as Figure 1, but with a much steeper gradient,because of the additional dimensions
The average path length for L = 3 sites per axis (giving 33 = 27 sites intotal) is 119. The corresponding value in the previous question, for a line oflength L = 27 was 354
The corresponding values are lower at every point in this question becausethere are more directions to choose from, so the space is explored more quickly
A periodic line (a 1D object) can be happily represented in 2D as a circle,but a periodic 3D cube cannot be visualised without intersecting itself
Despite this, it is certainly true that every point on the cube of side lengthL can be mapped to a corresponding line of length L3, which could be answeredcompletely using the tools in the previous question, provided that all six possibleneighbours are handled appropriately at each site.
Therefore, it is reasonable to expect that these questions would have similarresults
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3 Diffusion on a Network
There are B sites in each of the A clusters in this network. The clusters areconnected in a periodic line, that can be moved along only when in sites i = 0and i = 1 in each cluster, and only left and right respectively. Each site isconnected to every other within the same cluster
3.1 C Code
#include <stdio.h>
int ranInt(int max); // returns a random integer from 0 to max -1
int rHop(int curCluster , int numClusters); // increments current
cluster with periodic boundaries
int lHop(int curCluster , int numClusters); // decrements current
cluster with periodic boundaries
int siteHop(int curSite , int numSites); // returns a random new
site that is different to the old site
main()
{
// VARIABLE DECLARATIONS
int c=0; // current cluster , allowed from 0 to A-1
int i=0; // current site within cluster , allowed from 0 to B-1
int A=5; // number of clusters
int B=4; // number of sites in a cluster
int pos[A][B]; // indexes sites and their clusters
int visit =0; // increments each time a site in a cluster is
visited for the first time
int tempRan; // temporary storage of random integer
int j, a, b; // for -loop indices
int k; // averaging index
float tot=0; // running total for averaging
float M=100;
int seed =3; // sets up random number generator
srandom(seed);
for(k=0;k<M;k++) // averaging loop
{
// INITIALISE
for(a=0;a<A;a++)
{
for(b=0;b<B;b++)
{
pos[a][b]=0;
}
}
visit =0;
// RANDOM WALK
for(j=0;visit <A*B;j++)
{
pos[c][i]++;
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if(pos[c][i]==1)
{
visit ++;
}
if(i>1) // then particle cannot change cluster
{
tempRan =0;
}
else // particle will either change cluster or hop within
cluster
{
tempRan=ranInt (2); // if current site is 0 or 1, particle
either changes cluster or hops within cluster , with
equal chance
}
if(tempRan ==0) // then particle hops to site from 0 to B-1
within cluster
{
i=siteHop(i,B);
}
if(tempRan ==1) // then particle hops between clusters
{
if(i==0)
{
c=lHop(c, A);
}
if(i==1)
{
c=rHop(c, A);
}
}
}
tot+=j-1; // j is the number of hops in total , including the
hop after the last site was visited
}
printf("%f\n", tot/M);
}
// FUNCTION DEFINITIONS
int ranInt(int max) // returns a random integer between 1 and
max
{
return random () % max;
}
int rHop(int curCluster , int numClusters) // increments current
cluster with periodic boundaries
{
if(curCluster == numClusters -1)
{
curCluster =0;
}
else
{
curCluster ++;
}
return curCluster;
}
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int lHop(int curCluster , int numClusters) // decrements current
cluster with periodic boundaries
{
if(curCluster ==0)
{
curCluster=numClusters -1;
}
else
{
curCluster --;
}
return curCluster;
}
int siteHop(int oldSite , int numSites) // returns a random new
site that is different to the old site
{
int newSite=ranInt(numSites -1);
if(newSite >= oldSite) // then shift up the new site so that the
particle doesn ’t hop to it’s old site
{
newSite ++;
}
return newSite;
}
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Figure 3: a heat map showing length of path taken for different environments
3.2 Discussion and Interpretation
Figure 3 shows how both the number of clusters, A, and the number of sites percluster, B, determine the path length
Both quantities demonstrate the exponential pattern already observed, al-though examining the corners of the heat map shows that B contributes morethan A to the length of path taken
However, surprisingly, the graph is almost symmetric about the diagonal,meaning that A and B contribute a very similar amount
B contributes more than A because, as B grows larger, it becomes less likelyat any moment for the particle to be able to change cluster, and hence inevitablythe full environment
On the other hand, A acts like L in question one, only there is a randomamount of time between hops
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