Can ideal polymer chain possess non-Gaussian fluctuations?

Nechaev Sergei

Abstract

We are habitually accustomed that the concept of an "ideal polymer chain" of length L is the synonym of a random walk with Gaussian fluctuations controlled by the exponent ï? = 1/2 for the mean-square displacement r2 ~ L2. In the talk I demonstrate that it is not true when ideal polymer is pushed by external geometric constraints to the subset of configurational space which typically is highly improbable. As an example, I consider an ensemble of 2D random paths of length L stretched over a forbidden void (semicircle of radius R). Such a stretching forces random paths to stay in the vicinity of the semicircle boundary, which influences drastically the typical path’s span, d, above the semicircle. Stretching is ensured by the condition L ~ R. The resulting paths’ conformations are "atypical" since their realizations is highly improbable in the ensemble of unconstrained Gaussian trajectories. Statistics in such a tiny subset of the Gaussian ensemble is controlled by collective behavior of correlated modes, which results in a scaling for fluctuations, different from Gaussian: at large R we have d ~ R with  = 1/3. Simple dimensional analysis and direct analytic computations support this result. There are many examples of correlated one-dimensional stochastic processes, the standard deviation of which is characterized by a critical exponent  = 1/3, but not  = 1/2 (as for the distribution of independent random variables). Such processes include ballistic aggregation, traffic models, stochastic growth etc. The behavior of these models is related to the solutions of so-called Kardar-Parisi-Zhang (KPZ) equation. The goal of my talk is to demonstrate that the one-dimensional KPZ scaling  = 1/3 can occur in the model of simple two-dimensional stretched wandering above an impenetrable curved void.