Phase Transitions in One Dimension and the Helix—Coil Transition in Polyamino Acids

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Abstract
Phase transitions in one dimension are discussed from the point of view of order—disorder transitions in linear polymers using the formalism of sequence generating functions due to Lifson. If the statistical weight vj of an ordered sequence of j units has the form (lnvj)/j=a—bj−α, then a phase transition occurs when 0<α<1. As demonstrated by Fisher, this will occur, for example, if certain many-body interactions of short range are introduced. This behavior is obtained if we consider long-range pair potentials of the form 1/r1+α, for 0<α<1, occurring between all the units of an ordered sequence. An exponential potential, e−γx, gives a phase transition in the limit γ→0 if interactions are restricted to the units in an ordered sequence. The occurence of a phase transition arises from the convergence of the sequence generating function and its first derivative at the value of the unit partition function equal to the statistical weight of the ordered unit. This gives rise to a bend in the curve of the unit partition function as a function of temperature and, hence, a discontinuity in the population of ordered states. End effects in an ordered sequence in one dimension (the analog of surface effects in higher dimensions) are equivalent to the case of α=1; hence, as in polypeptides, one-dimensional systems with end effects show no true discontinuities.

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