Return to Question

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up, uniformly select an unoccupied site at most a distance $d$ from both of the stone's nearest neighbors which does not disturb the stone ordering along the interval (the stone's original position can be reselected), and place the stone back down at this site. If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?

Update (due to a suggestion by Douglas Zare) - The $k$ stones must be kept in order, and the distance between a stone and its two nearest-neighbors along the interval (or one nearest-neighbor if the stone occupies the left-most or right-most position of the chain) can be at most $d$.

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up and uniformly select an unoccupied site to place it back down which satisfies the dual criterion of: (a) being at most a distance $d$ from both of the stone's nearest neighbors, and (b) not disturbing the initial ordering of the stones along the integer interval (note that the stone's original position can be reselected). If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?

Update (due to a suggestion by Douglas Zare) - The $k$ stones must be kept in order, and the distance between a stone and its two nearest-neighbors along the interval (or one nearest-neighbor if the stone occupies the left-most or right-most position of the chain) can be at most $d$.

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up, uniformly select an unoccupied site at most a distance $d$ from the stone's original position (including the original position), and place the stone back down at this selected site. If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?

Update (due to a suggestion by Douglas Zare) - The $k$ stones must be kept in order, and the distance between a stone and its two nearest-neighbors along the interval (or one nearest-neighbor if the stone occupies the left-most or right-most position of the chain) can be at most $d$.

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up, uniformly select an unoccupied site at most a distance $d$ from both of the stone's nearest neighbors which does not disturb the stone ordering along the interval (the stone's original position can be reselected), and place the stone back down at this site. If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?

Update (due to a suggestion by Douglas Zare) - The $k$ stones must be kept in order, and the distance between a stone and its two nearest-neighbors along the interval (or one nearest-neighbor if the stone occupies the left-most or right-most position of the chain) can be at most $d$.

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up, uniformly select an unoccupied site at most a distance $d$ from the stone's original position (including the original position), and place the stone back down at this selected site. If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?

Imagine I place $k$ stones on an infinite one-dimensional integer interval $Z$ s.t. no stone is more than some distance $d$ from any other stone. For example, if $d=1$ and $k = 5$, we might place the five stones are positions $(-2,-1,0,1,2)$.

I then proceed to do the following:

For each of some arbitrary number of discrete time steps, I select one of the $k$ stones with uniform probability. I then lift the stone up, uniformly select an unoccupied site at most a distance $d$ from the stone's original position (including the original position), and place the stone back down at this selected site. If $d=1$, no stone can be moved from its original position and the center of mass of the stones/system, $C_m$, will be immobile. However, if $d=2$, starting from the state $(-2,-1,0,1,2)$, the center of mass will move to either the right or left with some per step probability of $\frac{1}{k}$ (until we leave this initial system state).

My question is - provided some number of stones $k$, and some "leashing distance" $d$, how can we characterize the dynamics of the random walk taken by the above system's center of mass, $C_m$? What is the average step time and size, and, in terms of Euclidean distance, what mean square displacement can we expect after some number of steps, $T$?

Update (due to a suggestion by Douglas Zare) - The $k$ stones must be kept in order, and the distance between a stone and its two nearest-neighbors along the interval (or one nearest-neighbor if the stone occupies the left-most or right-most position of the chain) can be at most $d$.