Inverse problem for multispecies ferromagneticlike mean-field models in phase space with many states

In this paper we solve the inverse problem for the Curie-Weiss model and its multispecies version when multiple thermodynamic states are present as in the low temperature phase where the phase space is clustered. The inverse problem consists of reconstructing the model parameters starting from configuration data generated according to the distribution of the model. We demonstrate that, without taking into account the presence of many states, the application of the inversion procedure produces very poor inference results. To overcome this problem, we use the clustering algorithm. When the system has two symmetric states of positive and negative magnetizations, the parameter reconstruction can also be obtained with smaller computational effort simply by flipping the sign of the magnetizations from positive to negative (or vice versa). The parameter reconstruction fails when the system undergoes a phase transition: In that case we give the correct inversion formulas for the Curie-Weiss model and we show that they can be used to measure how close the system gets to being critical.

Figure 1

Boltzmann-Gibbs distribution of the total magnetization $m_N\left(\sigma \right)$ for $J=1.5,h=0.22$, and different values of the number of spins $N$. The distribution is given by the dashed blue line for $N=200$, by the dot-dashed red line for $N=1000$, and by the continuous green line for $N=8000$. The peak of the distribution is centered around the solution $m^{*}=0.922$ of the consistency equation (2).

Figure 2

${\overline{J}}_{\exp }$ (upper panel) and ${\overline{h}}_{\exp }$ (lower panel) as a function of $N$ for $J=1.5,h=0.22$, and $M=1000$. The error bars are standard deviations on 20 different $M$ samples of configurations of the same system (see the text for the details of the simulation). The horizontal lines correspond to the exact values of $J=1.5$ (upper panel) and $h=0.22$ (lower panel).

Figure 3

${\overline{J}}_{\exp }$ (left panel) and ${\overline{h}}_{\exp }$ (right panel) as a function of $J$ for $h=0.22$ and $N=1000$. The error bars are standard deviations on 20 different $M$ samples of configurations of the same system (see the text for the details of the simulation). The continuous red line in the left panel represents ${\overline{J}}_{\exp }=J$, and the horizontal red line in the right panel corresponds to the exact value of the magnetic field $h=0.22$.

Figure 4

Boltzmann-Gibbs distribution of the total magnetization for $J=1.5,h=0$, and different values of the number of spins $N$. The distribution is given by the dashed red line for $N=1000$ and by the continuous green line for $N=8000$. The peaks of the distribution are centered around the two symmetric solutions $\pm m^{*}$ of the consistency equation (2) with $m^{*}=0.8586$.

Figure 5

Values of ${\overline{J}}_{\exp }$ (upper panels) and ${\overline{h}}_{\exp }$ (lower panels) as a function of $N$ for $J=1.5$ and $h=0$ obtained by changing the sign of the negative magnetizations before to apply the inversion procedure (left panels) and with the clustering algorithm (right panels). The error bars are standard deviations on 20 different independent $M$ samples of the same system ($M=1000$—see the text for the details of the simulation). The horizontal lines correspond to the exact values of $J=1.5$ (upper panels) and $h=0$ (lower panels).

Figure 6

Boltzmann-Gibbs distribution of the total magnetization for $J=1.5,h=0.001$, and different values of the number of spins $N$. The distribution is given by the dot-dashed red line for $N=1000$ and by the continuous green line for $N=8000$. Note that the peaks of the distribution are centered around the two solutions of the consistency equation (2), $m_1^{*}=0.85899$, the stable one, and $m_2^{*}=-0.85812$, the metastable one, whose probability vanishes as $N$ goes to infinity (continuous green curve).

Figure 7

Boltzmann-Gibbs distribution of the total magnetization for $J=1.02,h=0.001$, and different values of the number of spins $N$. The distribution is given by the dot-dashed red line for $N=1000$, by the dashed blue line for $N=3000$, and by the continuous green line for $N=11000$. Note that the peaks of the distribution are centered around the two solutions of the consistency equation (2), $m_1^{*}=0.261727$, the stable one, and $m_2^{*}=-0.211086$, the metastable one, whose probability vanishes as $N$ goes to infinity (continuous green curve).

Figure 8

Values of ${\overline{J}}_{\exp }$ (left panel) and ${\overline{h}}_{\exp }$ (right panel) as a function of $J$ for $h=0.001$ and $N=1000$. The error bars are standard deviations on 20 different $M$ samples of configurations of the same system (see the text for the details of the simulation). The continuous red line in the left panel represents ${\overline{J}}_{\exp }=J$, the horizontal red line in the right panel corresponds to the exact value of the magnetic field $h=0.001$.

Figure 9

Values of ${\overline{J}}_{\exp }$ (upper panels) and ${\overline{h}}_{exp}$ (lower panels) as a function of $N$ for $J=1.02$ and $h=0.001$ obtained by changing the sign of the negative magnetizations before applying the inversion procedure (left panels) and with the clustering algorithm (right panels). The error bars are standard deviations on 20 different $M$ samples of the same system ($M=1000$—see the text for the details of the simulation). The horizontal lines correspond to the exact values of $J=1.02$ (upper panels) and $h=0.001$ (lower panels).

Figure 10

$J=1,h=0$. Left panels: Values of ${\overline{J}}_{\exp }$ (upper panel) and ${\overline{h}}_{\exp }$ (lower panel) as a function of $N$ obtained with the standard inversion equations (9) and (10). Right panels: Values of ${\overline{J}}_{\mathrm{crit}}$ (upper panel) and ${\overline{h}}_{\mathrm{crit}}$ (lower panel) as a function of $N$ obtained with Eqs. (11) for $J_{\mathrm{crit}}$ and (12) for $h_{\mathrm{crit}}$. For all panels: Error bars are standard deviations on 20 different $M$ samples of the same system ($M=1000$—see the text for the details of the simulation); the horizontal lines correspond to the exact values of $J=1$ (upper panels) and $h=0$ (lower panels).

Figure 11

$J=0.999,h=0.0001$. Left panels: Values of ${\overline{J}}_{\exp }$ (upper panel) and ${\overline{h}}_{\exp }$ (lower panel) as a function of $N$ obtained with the standard inversion equations (9) and (10). Right panels: Values of ${\overline{J}}_{\mathrm{crit}}$ (upper panel) and ${\overline{h}}_{\mathrm{crit}}$ (lower panel) as a function of $N$ obtained with Eqs. (11) for $J_{\mathrm{crit}}$ and (12) for $h_{\mathrm{crit}}$. For all panels: The error bars are standard deviations on 20 different $M$ samples of the same system ($M=1000$—see the text for the details of the simulation); the horizontal lines correspond to the exact values of $J=0.999$ (upper panels) and $h=0.0001$ (lower panels).

Figure 12

Boltzmann-Gibbs distribution of the total magnetization for $J_{11}=J_{22}=1.4,J_{12}=0.98,h_1=0.001,h_2=0.002$, and $N_1=N_2=N/2=100$ spins. The system (15) admits three solutions $(m_1(\mathbf{J},\mathbf{h}),m_2(\mathbf{J},\mathbf{h}))$, corresponding to two maxima and a minimum of the pressure function.

Figure 13

Elements of the matrix ${\overline{\mathbf{J}}}_{\exp }$ as a function of $N$ for $J_{11}=J_{22}=1.4,J_{12}=0.98,h_1=0.001$, and $h_2=0.002$. The values of ${\overline{J}}_{11}^{\exp }$ (crosses), ${\overline{J}}_{12}^{\exp }$ (dots), and ${\overline{J}}_{22}^{\exp }$ (squares) in the left panels are obtained with the sign flip, and those of the right panels are obtained with the clustering algorithm. The horizontal lines correspond to the exact values of the elements of the matrix $\mathbf{J}$.

Figure 14

Elements of ${\overline{\mathbf{h}}}_{\exp }$ as a function of $N$ for $J_{11}=J_{22}=1.4,J_{12}=0.98,h_1=0.001$, and $h_2=0.002$. The values of ${\overline{h}}_1^{\exp }$ (crosses) and ${\overline{h}}_2^{\exp }$ (dots) in the left panels are obtained with the sign flip, and those of the right panels are obtained with the clustering algorithm. The horizontal lines correspond to the exact values of the elements of $\mathbf{h}$.

Figure 15

Boltzmann-Gibbs distribution of the total magnetization for $J_{11}=J_{22}=2.8,J_{12}=0.7$, $h_1=-h_2=0.32$, and $N_1=N_2=N/2=100$ spins. The system (15) admits five solutions $(m_1(\mathbf{J},\mathbf{h}),m_2(\mathbf{J},\mathbf{h}))$, three of which are maxima of the pressure function.

Figure 16

Elements of the matrix ${\overline{\mathbf{J}}}_{\exp }$ as a function of $N$ for $J_{11}=J_{22}=2.8,J_{12}=0.7$, and $h_1=-h_2=0.32$ obtained with clustering algorithm. The horizontal lines correspond to the exact values of the elements of the matrix $\mathbf{J}$.

Figure 17

Elements of ${\overline{\mathbf{h}}}_{\exp }$ as a function of $N$ for $J_{11}=J_{22}=2.8,J_{12}=0.7$, and $h_1=-h_2=0.32$ obtained with a clustering algorithm. The horizontal lines correspond to the exact values of the elements of $\mathbf{h}$.