Generalized Langevin equation formulation for anomalous diffusion in the Ising model at the critical temperature

Dynamical critical behavior of the two-dimensional three-state Potts model

We investigate the dynamical critical behavior of the two-dimensional three-state Potts model with single spin-flip dynamics in equilibrium. We focus on the mean-squared deviation of the magnetization M (MSD_{M}) as a function of time, as well as on the autocorrelation function of M. Our simulations reveal the existence of two crossover behaviors at times τ_{1}∼L^{z_{1}} and τ_{2}∼L^{z_{2}}, separating three dynamical regimes. MSD_{M} appears to shift from ordinary diffusion in the first regime, to anomalous diffusion in the second, and finally to be constant in the third regime. The magnetization autocorrelation function, on the other hand, is found to fluctuate between exponential decay, stretched-exponential decay, and then again exponential decay along these three regimes. This behavior is in agreement with the one reported recently for the two-dimensional Ising ferromagnet [Phys. Rev. E 108, 034118 (2023)2470-004510.1103/PhysRevE.108.034118], indicating that the existence of two dynamic critical exponents is not a peculiarity of the Ising model itself. A comparison of both MSD_{M} and the magnetization's autocorrelation function suggests that within our numerical accuracy the exponents z_{1} and z_{2} are shared between the Ising and three-state Potts models at least for the particular case of single spin-flip dynamics studied here, even though their equilibrium universality classes are clearly distinct. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}, in which α is the anomalous exponent in the intermediate regime. Since the ratio γ/ν is not shared between the two models, it follows that α is not shared either, an aspect well verified in our simulations. Finally, we also discuss the relevance of our main findings using another useful observable, namely the line magnetization M_{l}.

The relaxation dynamics of single flow-stretched polymers in semidilute to concentrated solutions

Recent experiments on the return to equilibrium of solutions of entangled polymers stretched by extensional flows [Zhou and Schroeder, Phys. Rev. Lett. 120, 267801 (2018)] have highlighted the possible role of the tube model's two-step mechanism in the process of chain relaxation. In this paper, motivated by these findings, we use a generalized Langevin equation (GLE) to study the time evolution, under linear mixed flow, of the linear dimensions of a single finitely extensible Rouse polymer in a solution of other polymers. Approximating the memory function of the GLE, which contains the details of the interactions of the Rouse polymer with its surroundings, by a power law defined by two parameters, we show that the decay of the chain's fractional extension in the steady state can be expressed in terms of a linear combination of Mittag-Leffler and generalized Mittag-Leffler functions. For the special cases of elongational flow and steady shear flow, and after adjustment of the parameters in the memory function, our calculated decay curves provide satisfactory fits to the experimental decay curves from the work of Zhou and Schroeder and earlier work of Teixeira et al. [Macromolecules 40, 2461 (2007)]. The non-exponential character of the Mittag-Leffler functions and the consequent absence of characteristic decay constants suggest that melt relaxation may proceed by a sequence of steps with an essentially continuous, rather than discrete, spectrum of timescales.

Super slowing down in the bond-diluted Ising model

In models in statistical physics, the dynamics often slows down tremendously near the critical point. Usually, the correlation time τ at the critical point increases with system size L in power-law fashion: τ∼L^{z}, which defines the critical dynamical exponent z. We show that this also holds for the two-dimensional bond-diluted Ising model in the regime p>p_{c}, where p is the parameter denoting the bond concentration, but with a dynamical critical exponent z(p) which shows a strong p dependence. Moreover, we show numerically that z(p), as obtained from the autocorrelation of the total magnetization, diverges when the percolation threshold p_{c}=1/2 is approached: z(p)-z(1)∼(p-p_{c})^{-2}. We refer to this observed extremely fast increase of the correlation time with size as super slowing down. Independent measurement data from the mean-square deviation of the total magnetization, which exhibits anomalous diffusion at the critical point, support this result.

Approximate dynamical eigenmodes of the Ising model with local spin-exchange moves

We establish that the Fourier modes of the magnetization serve as the dynamical eigenmodes for the two-dimensional Ising model at the critical temperature with local spin-exchange moves, i.e., Kawasaki dynamics. We obtain the dynamical scaling properties for these modes and use them to calculate the time evolution of two dynamical quantities for the system, namely, the autocorrelation function and the mean-square deviation of the line magnetizations. At intermediate times 1≲t≲L^{z_{c}}, where z_{c}=4-η=15/4 is the dynamical critical exponent of the model, we find that the line magnetization undergoes anomalous diffusion. Following our recent work on anomalous diffusion in spin models, we demonstrate that the generalized Langevin equation with a memory kernel consistently describes the anomalous diffusion, verifying the corresponding fluctuation-dissipation theorem with the calculation of the force autocorrelation function.