The pivot algorithm is a dynamic method for generating d-dimensional canonical ensembles by modifying chain dimensions through pivot moves in a Markov chain. It involves selecting a pivot point and rotating coordinates while checking for self-avoiding walks to determine accepted moves. Analysis includes calculating the scaling exponent through the square average radius of gyration for different dimensions.

1. PIVOT ALGORITHM
Vineet Kumar Doshi
11010759
CL 622

2. What Is Pivot Algorithm?
• The Pivot algorithm is a very efficient ‘dynamic’ algorithm for
generating d-dimensional canonical ensemble.[1][2][3]
• It drastically modifies the chain dimension.

3. How It Works?
The pivot algorithm uses pivot moves as the transitions in
Markov chain which proceeds as follows.
• Choose the pivot point.
• Transform the coordinates of either side by rotating about the
pivot point.
• If the resulting walk is self-avoiding the move is accepted,
otherwise the move is rejected and the original walk is
retained.

4. How To Rotate About Pivot?
For rotating the chain about its pivot one must have
knowledge about coordinate transformation.
Following are the common rotation matrices for 2D system:

5. Example
A*(1,2)
Pivot P(1,1) O
A(2,1)
If we have a line segment joining P (1, 1) and A (2, 1), and we want the line segment to rotate
90° counterclockwise about point P (P is pivot).
So the x and y coordinate of the new point A:
1.
2.
3.
4.
Hence, the new coordinates after 90° counterclockwise rotation is (1, 2).

6. Rotation Matrix for 3D System

7. Analysis – Scaling Exponent
Scaling exponent can be computed for the system using
Square Average Radius of Gyration.

8. Analysis – Scaling Exponent (contd.)
< Rg2
> ∝ (N) 2ν
For 2D system: ν = 0.75
For 3D system: ν = 0.588

9. References
• [1] M. Kroger, Introduction to Computational Physics,
Lecture Notes; ETH Zurich (2008).
• [2] Joachim P. Wittmer et. al., Monte Carlo Simulation of
Polymers, NIC Series 23, 83-140 (2001).
• [3] Nathan Clisby, Efficient Implementation of pivot
algorithm for self-avoiding walks, ARC, (2010).