Finite-size scaling of quasi-stationary-state temperature

Antonio Rodríguez, Fernando D. Nobre, and Constantino Tsallis

Phys. Rev. E 105, 044111 – Published 7 April 2022

Abstract

We numerically study, from first principles, the temperature $T_{QSS}$ and duration $t_{QSS}$ of the longstanding initial quasi-stationary state of the isolated $d$ -dimensional classical inertial $α − X Y$ ferromagnet with two-body interactions decaying as $1 / r_{i j}^{α}$ $(α \geq 0)$ . It is shown that this temperature $T_{QSS}$ (defined proportional to the kinetic energy per particle) depends, for the long-range regime $0 \leq α / d \leq 1$ , on $(α, d, U, N)$ with numerically negligible changes for dimensions $d = 1, 2, 3$ , with $U$ the energy per particle and $N$ the number of particles. We verify the finite-size scaling $T_{QSS} - T_{\infty} \propto 1 / N^{β}$ where $T_{\infty} \equiv 2 U - 1$ for $U ≲ U_{c}$ , and $β$ appears to depend sensibly only on $α / d$ . Our numerical results indicate that neither the scaling with $N$ of $T_{QSS}$ nor that of $t_{QSS}$ depend on $U$ .

Received 17 January 2022
Accepted 18 March 2022

DOI:https://doi.org/10.1103/PhysRevE.105.044111

Physics Subject Headings (PhySH)

Classical statistical mechanics Magnetic phase transitions Scaling laws of complex systems

Statistical Physics & Thermodynamics

Authors & Affiliations

Antonio Rodríguez ¹, Fernando D. Nobre², and Constantino Tsallis ^2,3,4

¹GISC, Departamento de Matemática Aplicada a la Ingeniería Aeroespacial, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros s/n, 28040 Madrid, Spain
²Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
³Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
⁴Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria

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Vol. 105, Iss. 4 — April 2022

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Images

Figure 1
The caloric curve of the $α − X Y$ model [cf. Eq. (2.6)] applies to the whole long-range-interaction regime ( $0 \leq α / d \leq 1$ ), and exhibits a ferromagnetic-paramagnetic continuous phase transition at the critical values $T_{c} = 1 / 2$ and $U_{c} = 3 / 4$ . The blue dashed line in the ferromagnetic phase ( $U ≲ U_{c}$ ) represents the set of kinetic temperatures $T_{\infty}$ , associated with the QSSs in the limit $N \to \infty$ [cf. Eq. (1.1)]. By monitoring numerically the time evolution of the average kinetic temperature $T (t)$ , one finds a plateau associated with $T_{\infty}$ , followed by a later state whose temperature $T_{BG}$ ( $T_{BG} > T_{\infty}$ ) coincides with the one predicted by BG statistical mechanics; these two temperatures are illustrated for an energy per particle $U = 0.69$ .
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Figure 2
(a) The durations of the QSSs for the $α − X Y$ model $α / d = 0.9$ ] are represented conveniently vs the total number of rotators in a log-log scale, showing numerical data for $N = l^{6}$ ( $l = 5, 6, 7, 8, 9, 10$ ), dimensions $d = 1, 2, 3$ , and typical values of the energy per particle (decreasing values of $U$ from top to bottom). Straight-line fittings lead to the scaling of Eq. (2.9), where $A (0.9) = 0.52 \pm 0.02$ . (b) The durations of the QSSs for the $α − X Y$ model ( $N = 6^{6} = 46 656$ ) are exhibited vs ${(α / d)}^{2}$ ( $0 \leq α / d \leq 0.9$ ) in a log-linear scale, for $d = 1, 2, 3$ , and two typical values of the energy per particle: $U = 0.69$ (upper straight line) and $U = 0.65$ (lower straight line); the straight-line fittings follow the scaling of Eq. (2.10), where $B (46 656) = 9.40 \pm 0.12$ . In both panels, data were obtained considering the initial condition $M (0) ≃ 0$ ; the factor $d^{ρ}$ becomes important as one approaches the short-range-interaction threshold, so that $ρ = 0$ ( $α / d \leq 0.7$ ) and $ρ = 0.31$ ( $α / d = 0.8, 0.9$ ), and one expects that $ρ \to 0$ in the thermodynamic limit, for all $0 \leq α / d \leq 1$ [29]. For the values of $U$ considered, the exponent $A (α / d)$ [Eq. (2.9)], as well as the coefficient $B (N)$ [Eq. (2.10)], are essentially independent of $U$ , within error bars.
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Figure 3
Numerical data obtained from simulations of the inertial $α − X Y$ model in a closed ring ( $d = 1$ ), $α = 0.9$ , total energy per rotator $U = 0.69$ , and an initial total magnetization $M (0) ≃ 0$ , are represented vs time in linear-log scales. (a) The total magnetization squared $M^{2} (t)$ (lower black line), as well as the difference of the kinetic temperature with respect to its value in the limit $N \to \infty$ , $T (t) - T_{\infty}$ (upper red line), are plotted for a total number of rotators $N = 6^{6} = 46 656$ . The results show that the agreement of the equation of state [cf. Eq. (2.6)] along its paramagnetic branch ( $M = 0$ ), with $T_{\infty} \equiv 2 U - 1$ [Eq. (1.1)], is strictly valid throughout the QSS only for $N \to \infty$ , but it is slightly violated for finite $N$ . (b) The kinetic temperature $T (t)$ is plotted for different number of rotators, $N = l^{6}$ ( $l = 5, 6, 7, 8, 9, 10$ ); the inset on the left displays an enlargement of a time window illustrating the limit of Eq. (1.1). According to Fig. 1, the associated characteristic temperatures are $T_{\infty} = 0.38$ and $T_{BG} = 0.475$ .
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Figure 4
The kinetic temperature $T (t)$ is represented vs time in linear-log plots, for the inertial $α − X Y$ model with an energy per rotator $U = 0.69$ and initial magnetization $M (0) ≃ 0$ . (a) Data for a total number of rotators $N = 6^{6} = 46 656$ along a closed ring ( $d = 1$ ) and several values of $α$ ( $0 \leq α \leq 1$ ). In the upper inset we show a time window highlighting the changes from the QSSs to the second plateau in linear-log plots, emphasizing the increase of $T_{QSS}$ with $α$ ; the corresponding differences $T_{QSS} - T_{\infty}$ are represented vs $α$ in the lower inset. (b) Data for a total number of rotators $N = 10^{6}$ , $α / d = 0.9$ , and dimensions $d = 1, 2, 3$ , showing that the duration $t_{QSS}$ increases with $d$ [29]. The changes from the QSSs to the second plateau are highlighted in the inset in a linear-log plot, showing that $T_{QSS}$ presents a slight dependence on $d$ , i.e., decreasing as $d$ increases. Throughout the whole long-range regime ( $0 \leq α / d \leq 1$ ), the corresponding characteristic temperatures are $T_{\infty} = 0.38$ and $T_{BG} = 0.475$ .
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Figure 5
The differences $T_{QSS} - T_{\infty}$ are represented vs $N$ (log-log scales) for the inertial $α − X Y$ model, considering typical values of $α$ ( $0 \leq α \leq 1$ ), dimensions $d = 1$ (circles), $d = 2$ (squares), and $d = 3$ (triangles), initial magnetization $M (0) ≃ 0$ , and different values of the total energy per particle: (a) $U = 0.65$ ; (b) $U = 0.69$ ; (c) $U = 0.72$ . Let us emphasize that the data for $T_{QSS}$ are more sensible to changes in $α$ , than to those in $d$ , as pointed out in Fig. 4; due to this, in (d) similar analyses are carried only for $d = 1$ . By considering the scaling of Eq. (1.2), the corresponding values for the exponent $β$ are estimated in each case, and the results are plotted in the lower insets ( $β$ vs $α$ ). In all cases considered, one notices that $T_{QSS} \to T_{\infty}$ as $N \to \infty$ [following Eq. (1.2)], and the exponent $β$ essentially remains unchanged (within numerical error bars) for ( $0 \leq α \leq 0.5$ ) and appears to decay linearly for $0.5 < α \leq 1.0$ .
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Figure 6
Data from Fig. 5 (total energy per particle $U = 0.69$ and dimensions $d = 1, 2, 3$ ) are collapsed for each value of $α$ ( $α = 0.6, 0.7, 0.8, 0.9$ from bottom to top) by introducing a factor $f (α, d)$ in the ordinate (see inset). As one approaches the short-range-interaction threshold (e.g., $α / d = 0.9)$ this factor recovers $f (d) = d^{- ρ}$ , used in Ref. [29] for the analyses of the duration $t_{QSS}$ .
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