Polymers critical point originates Brownian non-Gaussian diffusion

We demonstrate that size fluctuations close to polymers critical point originate the non-Gaussian diffusion of their center of mass. Static universal exponents $\gamma$ and $\nu$—depending on the polymer topology, on the dimension of the embedding space, and on equilibrium phase—concur to determine the potential divergency of a dynamic response, epitomized by the center-of-mass kurtosis. Prospects in experiments and stochastic modeling brought about by this result are briefly outlined.

Figure 1

Metric and entropic exponents of different polymer network topologies (i.e., homeomorphism types) in $d=2$ and $d=3$ under good solvent conditions. The first row refers to linear self-avoiding polymers: values in $d=2$ are exact and were first computed via Coulomb gas [40]; $d=3$ values have been originally obtained by Wilson-Fisher expansion in dimension $d=\epsilon -4$ for the ${\phi }^4$ field theory of the $O(n\to 0)$ model with $\epsilon =1$ [41] (first number), and later estimated with high precision Monte Carlo simulations [42, 43] (second number). Exponents from the second to the fourth row refer to star polymers with $\mathcal{R}$ arms [36], ring polymers, and watermelon graphs with $\mathcal{L}$ independent loops [36], respectively. Since topology is kept fixed, the $\nu$ exponent does not vary with respect to the linear case. The $\gamma$ exponent is obtained filling proper values for $\nu , d, \mathcal{L}, n_L$ in Eq. (2), together with ${\sigma }_L=(2-L)(9L+2)/64$ (exact, $d=2$) or ${\sigma }_L=\epsilon (2-L)L/16+O\left({\epsilon }^2\right)$ ($d=3$). For instance, with star polymers one has $\mathcal{L}=0, n_1=\mathcal{R}$, and $n_{\mathcal{R}}=1$ (the $n_2$ monomers with functionality $L=2$ do not contribute as ${\sigma }_2=0$); note that the two-arms topology ($n_1=2, \mathcal{R}=2$) corresponds to linear polymer. Similarly, with watermelon graphs $\mathcal{L}\ge 1, n_{\mathcal{L}+1}=2$. The last row refers instead to lattice animals, i.e., polymers in which the number of loops and branches can vary. In this case critical exponents in $d$ dimensions are related to the Lee-Yang edge singularity [38] through the relations ${\nu }_{d+2}=({\beta }_d+1)/2$ and ${\theta }_{d+2}={\beta }_d+2$, where ${\beta }_d$ is the exponent controlling magnetization near the edge singularity. As the latter is exactly solvable in $d=0$ and $d=1$ with ${\beta }_0=-1$ and ${\beta }_1=-1/2$, respectively, one gets ${\gamma }_2=1-{\theta }_2=0, {\gamma }_3=1-{\theta }_3=-1/2$, and ${\nu }_3=3/2$. The exact expression for ${\nu }_{d+2}$, however, breaks down with $d=0$ and for ${\nu }_2$ one has to rely on numerical estimates [44], such as the one reported in the table. Note that with $d\ge 3$ exponents $\gamma$ and $\nu$ are not independent but follow the relation $\gamma =1-(d-2)\nu$.

Figure 2

Unit-variance initial $x$ PDF for the c.m. of a linear polymer close to criticality. For comparison purposes, a unit-variance Gaussian PDF is also plotted in a red dash-dotted line.

Figure 3

Unit-variance initial $x$ PDF for the c.m. of a star polymer in $d=3$ with different numbers of arms $\mathcal{R}$ close to criticality.