Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou model. II. Phenomenology of quantum eigenstates (2024)

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  Figure 1

  Classical Poincaré surface of sectionfor $\alpha$-FPUT with $\alpha =1$ at two energies, $E=0.14$ (a)and $E=0.16$ (b), and for the general case with $\lambda =1/16$ at $E=1/3$ the saddle energy (c), generated by one single chaotic orbit.

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  Figure 2

  Examples of Husimi functions of six eigenstates in the energy interval $[E-\delta E/2,E+\delta E/2]$ at $E=0.14$ with $\delta E/E\simeq 0.0014$, for the singlet of $\alpha$-FPUT with $\alpha =1$. Here, the irreducible Hilbert space is of size ${\mathcal{N}}_S=60301$, with the cutoff of $n$ set to $N=600$ and the Planck constant $\hslash =1\times {10}^{-3}$. Panels (a1)–(a6): Husimi QSOS given by Eq.(25) plotted in the logarithmic scale, the darkest blues show the area where ${\overline{\mathcal{Q}}}_k\left({\alpha }_2\right)<{10}^{-11}$. Panels (b1)–(b6) are plots of the completely projected Husimi functions ${\mathcal{P}}_k\left({\alpha }_2\right)$ of Eq.(26) and panels (c1)–(c6) are the projected Husimi functions from the classical energy shell ${\tilde{\mathcal{P}}}_k\left({\alpha }_2\right)$ given by Eq.(27). Panels (d1)–(d6): projected Husimi functions in configuration space ${\tilde{\mathcal{P}}}_k\left(\mathbf{q}\right)$ of Eq.(28). The darker colors in panels (b1)–(d6) for projected Husimi functions indicate larger density, where the color scale on the right encodes the relative amplitude. Panels from the same column correspond to the same eigenstate.

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  Figure 3

  Husimi QSOS plotted in the logarithmic scale, of the same eigenstate as the one shown in Fig.2, but with much higher resolution about $\sim 1\times {10}^6$ grid points. It corresponds to SOS of Fig.1.

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  Figure 4

  Examples of Husimi functions of six eigenstates in the energy interval $[E-\delta E/2,E+\delta E/2]$ at $E=1/3$ with $\delta E/E\simeq 0.002$, for the singlet of general FPUT with $\lambda =1/16$. Here, the cutoff of $n$ is set to $N=600$ and the Planck constant $\hslash =5\times {10}^{-3}$. Panels (a1)–(a6): Husimi QSOS given by Eq.(25) plotted in the logarithmic scale. Panels (b1)–(b6): projected Husimi functions in configuration space ${\tilde{\mathcal{P}}}_k\left(\mathbf{q}\right)$ given by Eq.(28) plotted with the color scale on the right encoding the relative amplitude. Panels in the same column correspond to the same eigenstate.

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  Figure 5

  (a)The histogram of distribution of the overlap index $M$ for an ensemble of approximately 1200 eigenstates, in the energy interval $[E-\delta E/2,E+\delta E/2]$ at $E=0.14$ with $\delta E/E\simeq 0.01$, for the singlet of $\alpha$-FPUT, where the irreducible Hilbert space is of size ${\mathcal{N}}_S\sim 3.75\times {10}^5$. Here, the cutoff $N=1500$ and the Planck constant $\hslash =(4\pm 0.016)\times {10}^{-4}$. (b)Regions of different (logarithmic) values of the SALI on the classical SOS for $E=0.14$ the same as Fig.1, at $t=1000$: the initial conditions colored dark blue correspond to chaotic orbits, the yellowish color indicates ordered motion, and the intermediate color suggests sticky orbits. Panels (b1)–(b4): Husimi QSOS plotted in the logarithmic scale, for four eigenstates selected from the ensemble, with different values of $M$ index, from left to right $M\simeq -1,-0.5,0.5,1$, respectively, where the darkest blue indicates area where ${\overline{\mathcal{Q}}}_k\left({\alpha }_2\right)<{10}^{-11}$.

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  Figure 6

  The histogram of distribution of overlap index $M$ for an ensemble of approximately 1200 eigenstates from the singlet of $\alpha$-FPUT, in the energy interval $[E-\delta E/2,E+\delta E/2]$ with $\delta E/E\simeq 0.01$, for energies $E=0.14$ (top panels), and $E=0.16$ (bottom panels). The Planck constant is $\hslash ={\hslash }_0\pm j{\hslash }_0/1000$, where from left to right ${\hslash }_0=1\times {10}^{-3}(\text{with}j=0,1,2,3,4,5),7\times {10}^{-4}(\text{with}j=0,2,4),4\times {10}^{-4}(\text{with}j=4)$.

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  Figure 7

  Decay of the fraction of mixed eigenstates ${\chi }_M(E,\hslash )$ with respect to the Planck constant $\hslash$, at two energies $E=0.14$ (squares) and $E=0.16$ (circles), where $\delta E/E\simeq 0.01$ and $N(E,\delta E,\hslash )\approx 1200$. The dash-dotted lines and dashed lines show the power law ${\chi }_M(E,\hslash )\sim {\hslash }^{\xi }$, where (a)is for mixed eigenstates with $-0.8\le M\le 0$, and in $\text{(b)}-0.8\le M\le 0.8$.

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  Figure 8

  (a1) The variance of momentum ${\sigma }_p^2=\langle p^2\rangle -{\langle p\rangle }^2$ vs $t$ for $\alpha$-FPUT with $\alpha =1$ at $E=0.14$, where ${\sigma }_p^2$ is calculated with 4000 initial conditions from the classical SOS that are uniformly distributed in the chaotic region with $q_2\in [-0.25,-0.15]$ and $p_2=0$. The red dashed horizontal line indicates $\overline{{\sigma }_p^2}$, the longtime averaged value of ${\sigma }_p^2$. (a2) The temporal fluctuation of ${\sigma }_p^2$, as a function of time, across the time windows $[t-\mathrm{\Delta }t/2,t+\mathrm{\Delta }t/2]$, with $\mathrm{\Delta }t=30$, and the blue dashed horizontal line indicates the threshold $2\%$ of the peak value of ${\mu }_p^2$. (b1), (b2) Analogous data as (a1), (a2) respectively, at $E=0.16$. (c)Classical transport time $t_T$ as a function of energy $E$ in $\alpha$-FPUT (Hénon-Heiles), using two different thresholds. $\text{(d)}{t}_T$ vs $E$ in general FPUT ($2\%$ threshold).

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  Figure 9

  Scaled semiclassical DOS $2\pi {\hslash }^{2}g\left(E\right)$ for $\alpha$-FPUT (a)and the general case (b), where the red dashed vertical lines denote the saddle energies (see paper I). (c)Ratio ${\alpha }_{\mathcal{L}}$ as a function of $1/\hslash$, where black dashed line is for $E=0.14$ and blue dash-dotted line for $E=0.16$, in $\alpha$-FPUT, with respect to the bottom $x$ axis (colored in black). On the lines the black and blue dot indicate ${\alpha }_{\mathcal{L}}$ at $\hslash =4\times {10}^{-4}$. The red solid line is for $E=1/3$ in general FPUT, with respect to the top $x$ axis in red color, where the red dot indicates the ratio at $\hslash =2\times {10}^{-3}$.

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  Figure 10

  Distributions of the $\alpha =1$ and $\alpha =2$ Rényi-Wehrl ELMs, for the $M\ge 0.8$ eigenstates, as a function of the Planck constant $\hslash$ (indicated at the top of each column, the Planck constant remains constant across each respective column), at two energies for the singlet of $\alpha$-FPUT. The main figuresshow the histogram and its best-fitting $\beta$ distribution, while the insets show the corresponding cumulative distribution. (Colors online: black are data, red is the best-fitting). The vertical dash lines indicate ${\mathcal{L}}_{|\mathcal{R}\rangle }^1\approx 0.66$ for $\alpha =1$ cases and ${\mathcal{L}}_{|\mathcal{R}\rangle }^2=0.5$ for $\alpha =2$, from random pure states. Parameters of the best fit $\beta$ distribution for $P\left({\mathcal{L}}^1\right)$ from left to right: at $E=0.14$ are (6.81,2.32), (7.91,2.80), (7.49,2.26), (7.17,2.35) and at $E=0.16$ are (8.87,2.33), (10.49,3.02), (11.27,3.62), (13.17,4.55) with ${\mathcal{L}}_0^1=0.76$, where for $P\left({\mathcal{L}}^2\right)$ here we set ${\mathcal{L}}_0^2=0.64$. Eigenstates for the statistics in each panel, are selected from an ensemble of $N(E,\delta E,\hslash )\approx 1200$ states with $\delta E/E\simeq 0.01$.

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  Figure 11

  The parameter ${\beta }_a$ extracted from the best-fitting $\beta$ distribution, versus the controlling ratio ${\alpha }_{\mathcal{L}}$, for the singlet of $\alpha$-FPUT with various Planck constant $\hslash$, at two energies $E=0.14$ (squares) and $E=0.16$ (circles). The (triangle) data point indicated by the arrow corresponds to the general case with $\lambda =1/16$ at $E=1/3$, is obtained from the distribution of ELMs shown in Fig.12. The inset displays the coefficient of determination, $R^2$, plotted against ${\alpha }_{\mathcal{L}}$, which serves as a measure of the goodness of fit.

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  Figure 12

  Top panels: distributions of the $\alpha =1$ and $\alpha =2$ Rényi-Wehrl ELMs of $N(E,\delta E,\hslash )\simeq 1200$ eigenstates, from the singlet of general FPUT with $\lambda =1/16$, at energy $E=1/3$ with $\delta E/E\simeq 0.01$ and $\hslash =2\times {10}^{-3}$. Parameters of the best-fitting $\beta$ distribution for $P\left({\mathcal{L}}^1\right)$ are $(11.60,2.55),{\mathcal{L}}_0^1=0.72$ and ${\mathcal{L}}_0^2=0.59$. The vertical dashed lines indicate ${\mathcal{L}}_{|\mathcal{R}\rangle }^{\alpha }$. Middle panels: Husimi QSOSs given by Eq.(25) (plotted in the logarithmic scale) of the maximally localized and extended states, which are pointed out by red arrows in the left top panel. Bottom panels: Projected Husimi functions in configuration space ${\tilde{\mathcal{P}}}_k\left(\mathbf{q}\right)$ given by Eq.(28) (the color scale encodes the relative amplitude), for the same two states plotted previously in the middle panels.

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  Figure 13

  A flowchart illustrating the relation between the projected Husimi function in configuration space and the configuration-space probability density $|{\psi }_k{\left(\mathbf{q}\right)|}^2$, linked through the Wigner function.

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