Unraveling anomalous relaxation effects in the thermodynamic limit

Abstract

We address two central open problems in the theory of anomalous Mpemba-like relaxations: their extension beyond one spatial dimension and their consistent formulation in the thermodynamic limit. Our framework is the antiferromagnetic Ising model on a square lattice under an externally applied magnetic field, which enables us to work in the presence of a phase transition. The rich phase diagram contains two control parameters: temperature and magnetic field. We demonstrate that the standard assumption of relaxation dominated by a single leading exponential is inconsistent for intensive observables exhibiting standard fluctuations. Instead, as the system size increases, a continuous spectrum of time scales emerges. Nevertheless, we make the ansatz that, in the vicinity of the phase transition, the spectral projector onto the slowest time scales can be effectively characterized in terms of an equilibrium thermodynamic quantity: the susceptibility associated with the order parameter of the metastable phase. Combined with the richness of the phase diagram, this ansatz yields qualitative and semi-quantitative predictions for optimal protocols leading to a variety of anomalous relaxation phenomena involving simultaneous variations of temperature and magnetic field. These include direct and inverse Mpemba effects, cooling-heating asymmetries, and faster heating induced by precooling. Careful Monte Carlo simulations validate our theoretical predictions. Furthermore, minimal post-optimization suffices to convert our analytically guided protocols into fully optimal ones that display anomalous relaxations in their most pronounced form.

Figures (12)

• Figure 1: (a) Phase map of our 2D AFM Ising model showing the critical line from reference WK97. Our working points A to E are summarized in table \ref{['tab:working_points']}. (b) $\chi_{\mathrm{st}}$ as a function of the temperature $T_b$ for lattices $N = 256\times256$ (purple) and $N = 512\times512$ (green), which appear almost superimposed. For points A, B, C and E, $h=4.01$ (point A has a cross marker to aid visibility). For point D (triangle marker), $h=3.9$. The horizontal dashed line shows the upper bound on $\chi_{\mathrm{st}}$ found for the 1D model.
• Figure 2: Difference $\Delta=\chi_{\mathrm{st}}(L=256) - \chi_{\mathrm{st}}(L=512)$ as a function of the temperature $T$.
• Figure 3: Relaxation to equilibrium of observables for lattice sizes $N = 256\times256$ (purple) and $N= 512\times512$ (green). Note that the green dots occlude the purple ones. The inset shows their difference. (a) Exchange energy per spin as a function of time, 1-temperature jump protocol. (b) Magnetization per spin as a function of time, 1-temperature jump protocol. (c) Exchange energy per spin as a function of time, 2-temperature jump protocol. (d) Magnetization per spin as a function of time, 2-temperature jump protocol.
• Figure 4: Relaxation curves for (a) $e_J^\perp$ and (b) $m_\text{u}^\perp$ as a function of time in semi-logarithmic scale---data for $N= 512\times512$ and $T=2.5$ (point E), green for staggered protocol and blue for the random protocol. See section \ref{['sec:effective_description']} for the description of both protocols.
• Figure 5: Fits to simulation data using the two functional forms (\ref{['eq:ansatz-1']}) and (\ref{['eq:ansatz-2']}) for ${\langle{e_{J}^{\perp}}\rangle}_t$ at point B ($T=0.067, h=4.01$), obtained over the fitting window [$t_\mathrm{min}$, $t_\mathrm{max}$]. (a) Comparison of fits (blue and green curves) to data (purple dots). The dots are bigger than the error to aid visualization. (b) Simulation error as a function of time. (c) Difference between the two ansatzse normalized by the simulation error.