Polymer extension under flow: Some statistical properties of the work distribution function.

Abstract

In an extension of earlier studies from this group on the application of the Jarzynski equality to the determination of the elastic properties of a finitely extensible Rouse model of polymers under flow [A. Ghosal and B. J. Cherayil, J. Chem. Phys. 144, 214902 (2016)], we derive several new theoretical results in this paper on the nature of the distribution function P(w) that governs the long-time limit t>>1 of the fluctuations in the work w performed by the polymer during flow-induced stretching. In particular, we show that an expression for the average of the nth power of the work, ⟨wn(t)⟩, can be obtained in closed form in this limit, making it possible to exactly calculate three important statistical measures of P(w): the mean μ, the skewness γ1, and the kurtosis γ2 (apart from the variance σ2). We find, for instance, that to leading order in t, the mean grows linearly with t at a constant value of the dimensionless flow rate Wi and that the slope of the μ-t curve increases with increasing Wi. These observations are in complete qualitative agreement with data from Brownian dynamics simulations of flow-driven double-stranded DNA by Latinwo and Schroeder [Macromolecules 46, 8345 (2013)]. We also find that the skewness γ1 exhibits an interesting inversion of sign as a function of Wi, starting off at positive values at low Wi and changing to negative values at larger Wi. The inversion takes place in the vicinity of what we interpret as a coil-stretch transition. Again, the finding exactly reproduces behavior seen in other numerical and experimental work by the above group Latinwo et al. [J. Chem. Phys. 141, 174903 (2014)]. Additionally, at essentially the same value of Wi at which this sign inversion takes place, we observe that the kurtosis reaches a minimum, close to 1, providing further evidence of the existence of a coil-stretch transition at this location. Our calculations reproduce another numerical finding: a power law dependence on Wi of the rate of work production that is characterized by two distinct regimes, one lying below the putative coil-stretch transition, where the exponent assumes one value, and the other above, where it assumes a second.