Analysis of Flexibility

Consider two systems, $A$ and $B$, each consists of $m$ sites, and each site is adjacent to $Z$ other sites. When $n$ separate elements are placed into system $A$, the total number of possible configurations is $W_A = {C}_m^n$. The phase space of $A$ is denoted by $\Omega_A$. When placing a linear chain of $n$ elements into system $B$, the total number of possible configurations is $W_B = mZ\left(n-2 \right)\left(Z-1 \right)$. The phase space of $B$ is denoted by $\Omega_B$.

 

When $n \ll m$, $W_A > W_B$;when $n = m$, $W_A < W_B$.

 

There should exist a value of $n$ and a corresponding topological structure of the system containing $m$ sites of $Z$ adjacency, which maximizes the total number of possible configurations $W$ of the system.

 

The flexibility of a system is defined as

$\Upsilon = -\dfrac{1}{\delta H} \dfrac{d\Omega}{dt}$,

where $H$ is the energy of the system.

 

The flexibity of an element in this system is defined by

$\upsilon = \dfrac{1}{E \langle t \rangle}$,

where $E$ is the kinetic energy of the element and $\langle t \rangle$ the average time to go from one site to any another site.

 

The optimization of $\Upsilon$ and $\upsilon$ should find applications in both physics and biology.