Two-State Equilibrium Modulated by an External Field

As the final example of a system consisting of two-state particles, we consider a linear string of polymers. Each polymer can be in one of two states, long (A) or short (B) with corresponding lengths I a and /, respectively. We denote by Qa and the internal PF of the polymer in the A and B states and by Waa, bb, and Wab the corresponding interaction between nearest pairs. The total length of the system is 

Note also that we assume that JVab= ba- If the two ends of the polymer are not identical, or similar, then we must distinguish between PVab and IVba- 

Since N = Na + Nb is fixed, then fixing L also fixes Na, by (4.2.62). Therefore the canonical PF for this system is 

The tension r is defined as positive when the system is stretched. In the three-dimensional system, the pressure is defined as positive when the system is compressed. Therefore we have the factor exp( — PPV) inihe T, P, N ensemble, whereas here we have exp[j8rL]. 

With these definitions we can perform the sum in (4.2.64) simply by writing one specific configuration and summing over all possible configurations. The sum over all L values is the same as the sum over all possible vectors s of the system. The result after cyclization is 

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