Strange scaling and relaxation of finite-size fluctuation in thermal equilibrium

We numerically exhibit two strange phenomena of finite-size fluctuation in thermal equilibrium of a paradigmatic long-range interacting system having a second-order phase transition. One is a nonclassical finite-size scaling at the critical point, which differs from the prediction by statistical mechanics. With the aid of this strange scaling, the scaling theory for infinite-range models conjectures the nonclassical values of critical exponents for the correlation length. The other is relaxation of the fluctuation strength from one level to another in spite of being in thermal equilibrium. A scenario is proposed to explain these phenomena from the viewpoint of the Casimir invariants and their nonexactness in finite-size systems, where the Casimir invariants are conserved in the Vlasov dynamics describing the long-range interacting systems in the limit of large population. This scenario suggests appearance of the reported phenomena in a wide class of isolated long-range interacting systems.

Figure 1

Comparison between the finite-size fluctuation $N{V}_M/T$ (points) and the zero-field susceptibility $\epsilon {\chi }_{xx}^{\mathrm{V}}$ obtained in the Vlasov dynamics, whose level is reported by the black solid lines. The dot-dashed gray and the dashed light-blue lines in the ordered phase ($T<T_{\mathrm{c}}=1/2$) are isothermal ${\chi }_{xx}^{\mathrm{T}}$ and isoentropic ${\chi }_{xx}^{\mathrm{E}}$ zero-field susceptibilities [26], respectively, while they coincide with the Vlasov one in the disordered phase. Points are averages over 100 realizations. Averaging time $t_{\mathrm{av}}$ for one realization is $t_{\mathrm{av}}=100$, and $t_{\mathrm{ini}}=0$. $N={10}^2$ (red triangles), ${10}^3$ (blue diamonds), ${10}^4$ (magenta inverse triangles), and ${10}^5$ (orange circles). (a) $V_M$ is defined by Eq. (6). (Inset) Double-logarithmic graph in the ordered phase with the horizontal axis of the reduced temperature $\tau$. Slopes of the two green dotted guide lines are $-1$ and $-1/4$ from top to bottom, where the former corresponds to ${\gamma }_{-}=1$ for ${\chi }_{xx}^{\mathrm{T}}$ and ${\chi }_{xx}^{\mathrm{E}}$ and the latter corresponds to ${\gamma }_{-}=1/4$ for ${\chi }_{xx}^{\mathrm{V}}$. (b) $V_M$ is defined by the second line of Eq. (6) for both the ordered and the disordered phases with $\epsilon =1$.

Figure 2

Finite-size fluctuation $V_M$ at the critical point $T=T_{\mathrm{c}}$. Points are averages over 100 realizations. Averaging times $t_{\mathrm{av}}$ for one realization are ${10}^2$ (red squares), ${10}^3$ (blue triangles), ${10}^4$ (magenta circles), and ${10}^5$ (black inverse triangles). $t_{\mathrm{ini}}=0$. The solid lines are computed by the least-mean square method in the displayed intervals, and their slopes are $-0.779, -0.770$, and $-0.779$ from bottom to top. The upper and the lower black dashed lines have the slopes $-1/2$ and $-1$, respectively, for comparison. (Inset) The vertical axis is $V_M{N}^{4/5}$ to enhance difference among $t_{\mathrm{av}}$.

Figure 3

Fluctuation $N{V}_M/T$ as a function of $t_{\mathrm{av}}$. $t_{\mathrm{ini}}=0$. (a) $T=0.45$; (b) $T=0.40$. In both panels $N=100$ (red triangles) and 1000 (blue diamonds). The black and the light-blue horizontal lines represent levels of the Vlasov and the isoentropic susceptibilities.

Figure 4

The same with Fig. 3, but $t_{\mathrm{ini}}={10}^5$.

Figure 5

Power spectra of ${\mathit{M}}^2\left(t\right)$ for $T=0.40$ (red solid line), $0.45$ (blue dashed line), $0.49$ (magenta dotted line), and $0.50$ (black dot-dashed line) with $N=1000$. Power spectra for $T>T_{\mathrm{c}}$ are also computed up to $T=0.60$ (not shown), and they are similar with for $T=0.50$ except for slight changes of slopes in a small $f$ region.

Figure 6

Examinations of the scaling theory. (a) Disordered phase; (b) ordered phase. (Insets) ${\chi }_{xx}$ for the Vlasov dynamics (blue dashed lower lines) and $1/D$ (red solid upper lines). The vertical black segments mark the computing interval of $\tau$.