Progressive quenching: Globally coupled model

We study the processes in which fluctuating elements of a system are progressively fixed (quenched) while keeping the interaction with the remaining unfixed elements. If the interaction is global among Ising spin elements and if the unfixed part is reequilibrated each time after fixing an element, the evolution of a large system is martingale about the equilibrium spin value of the unfixed spins. Due to this property the system starting from the critical point yields the final magnetization, whose distribution shows non-Gaussian and slow transient behavior with the system size.

Figure 1

Among $N_0=9$ spins (vertices) the three ($T=3$) spins on the blue (filled) dots have been fixed. These spins apply a field $h$ on the remaining six ($N=6$) spins on the red (open) circles, which obey canonical equilibrium. Each edge denotes the coupling, $j_0/N_0,$ and those couplings among the fixed spins have been omitted.

Figure 2

(a) Markovian biased random walk corresponding to the present model of progressive quenching. From each node [blue (thick) dot] in the transition network, $(T,M),$ the possible branched transitions, $M\to M\pm 1,$ occur with probabilities $(1\pm m_{N_0-T,(j_0/N_0)M}^{\left(\mathrm{eq}\right)})/2,$ which corresponds, respectively, to fixing $s_{T+1}$ at $\pm 1.$ (b) Three sample histories with $j_0=1.5$ (curves near the diagonals) and three others with $j_0=0$ (curves near the horizontal axis) are shown by different colors (brightness) for the system with the total size $N_0=256.$ (c) The six sample histories [curves of different colors (brightness)] with $j_0=j_{0,c}(\simeq 1.030),$ the “critical coupling” with the size $N_0=256,$ are superposed on the contour plot of $m_{N_0-T,h_T}^{\left(\mathrm{eq}\right)}$ for the same $j_0$ [almost straight lines inside the triangle with gradient of color (brightness)]. The value of $m_{N_0-T,h_T}^{\left(\mathrm{eq}\right)}$ is positive (negative), respectively, above (below) the horizontal axis.

Figure 3

(a, b) Probability distributions of the mean spin value, $M_T/T$, in the quenched part at different stages, $T=2^k$ with integers $k=4–8$ with the fixed system size, $N_0=2^8=256.$ The initial conditions are (a) $M_0=0$ and (b) $M_1=1$, respectively. In both (a) and (b) the increment of $T$ is indicated by the thick red arrows. (c) Probability density of the magnetization in the final state, $M_{N_0},$ normalized by the maximizer of the probability, $M_{N_0}/M_{N_0}^{\left(\max \right)}.$ The system size, $N_0=2^k,$ is varied with integers $k=5–10$. The increment in size, $N_0,$ is indicated by the thick red arrows. Inset: The same data but as a function of $M_{N_0}/N_0.$ The increment in size, $N_0,$ is indicated by the thick red arrow. (d) Log-log plot of $M_{N_0}^{\left(\max \right)}/N_0$ versus $N_0$ (thick dots). The solid line is a linear fitting (i.e., $M_{N_0}^{\left(\max \right)}/N_0\sim {N_0}^{\alpha }$) with the slope $\alpha \simeq -0.45$, that is, $M_{N_0}^{\left(\max \right)}\sim N_0^{1-0.45}.$ Thin dots with the dashed line (the fitted slope $\alpha \simeq -0.40$) show the similar results under the condition that the first fixed spin is $s_1=+1$ [see Fig. 3].