Density of states and dynamical crossover in a dense fluid revealed by exponential mode analysis of the velocity autocorrelation function

Extending a preceding study of the velocity autocorrelation function (VAF) in a simulated Lennard-Jones fluid [Phys. Rev. E 92, 042166 (2015)] to cover higher-density and lower-temperature states, we show that the recently demonstrated multiexponential expansion method allows for a full account and understanding of the basic dynamical processes encompassed by a fundamental quantity as the VAF. In particular, besides obtaining evidence of a persisting long-time tail, we assign specific and unambiguous physical meanings to groups of exponential modes related to the longitudinal and transverse collective dynamics, respectively. We have made this possible by consistently introducing the interpretation of the VAF frequency spectrum as a global density of states in fluids, generalizing a solid-state concept, and by giving to specific spectral components, obtained through the VAF exponential expansion, the corresponding meaning of partial densities of states relative to specific dynamical processes. The clear identification of a high-frequency oscillation of the VAF with the near-top excitation frequency in the dispersion curve of acoustic waves is a neat example of the power of the method. As for the transverse mode contribution, its analysis turns out to be particularly important, because the multiexponential expansion reveals a transition marking the onset of propagating excitations when the density is increased beyond a threshold value. While this finding agrees with the recent literature debating the issue of dynamical crossover boundaries, such as the one identified with the Frenkel line, we can add detailed information on the modes involved in this specific process in the domains of both time and frequency. This will help obtain a still missing full account of transverse dynamics, in both its nonpropagating and propagating aspects which are linked through dynamical transitions depending on both the thermodynamic states and the excitation wave vectors.

Figure 1

Simulated LJ states in the $({\rho }^{*},T^{*})$ plane. Black open circles indicate the points investigated in Ref. [22]. The other points refer to this work: blue dots are higher density states on the same $T^{*}=1.35$ isotherm, while pink squares are at lower temperatures along the ${\rho }^{*}=0.80$ isochore. Also shown are the liquid-vapor coexistence line (red, full line) [35], and the two boundaries of the solid-fluid coexistence region (green, dot-dashes) [56].

Figure 2

Simulated VAF for $t^{*}\le 1$. Only the part of $Z\left(t\right)$ close to the time axis is shown, and the steep decrease from the initial value is omitted for clarity. (a) States along the $T^{*}=1.35$ isotherm at the indicated densities. Density increases from top to bottom. (b) States along the ${\rho }^{*}=0.80$ isochore at the indicated temperatures. Temperature decreases from top to bottom. The middle line in (a) and the top line in (b) are the same curve.

Figure 3

Log-log plots of the VAF from simulation (black solid lines) at $t^{*}\ge 0.1$ for the states along the $T^{*}=1.35$ isotherm. The $A{t}^{-3/2}$ behavior (red straight lines) is calculated with the theoretical values of the coefficient $A$ from Eq. (2). Curves are plotted in order of increasing density from top (${\rho }^{*}=0.65$) to bottom (${\rho }^{*}=0.95$), and for clarity each curve is shifted downwards by a factor of 10 with respect to the preceding one. The higher density curves are interrupted in the time intervals where $Z\left(t\right)$ takes negative values. The vertical green bars mark the values of $t_{\mathrm{R}}^{*}$. See text for details.

Figure 4

Same as in Fig. 3 for the thermodynamic states along the ${\rho }^{*}=0.80$ isochore. Curves are plotted in order of decreasing temperature from top ($T^{*}=1.35$) to bottom ($T^{*}=0.90$), with each curve shifted downwards by a factor of 10 with respect to the preceding one. The top curve is also displayed in Fig. 3.

Figure 5

$Z\left(t\right)$ and $\widehat{Z}\left(\omega \right)$ at four thermodynamic states. In the left frames, MD data for the VAF (symbols) and the multiexponential fit (red solid line through the points) are displayed at short times. The various components of the fit function, grouped together as indicated by the top right labels, are also displayed separately. In the right panels, the corresponding FT's are shown. For graphical clarity, not all available data points have been displayed.

Figure 6

Detail of $Z\left(t\right)$ close to the time axis at three thermodynamic states indicated by the labels. Symbols, lines, and colors are as in Fig. 5.

Figure 7

Contribution of the sum of the R3 and R4 terms to the VAF time dependence. Solid curves refer to thermodynamic states at $T^{*}=1.35$ and are plotted in order of increasing density from top (${\rho }^{*}=0.65$) to bottom (${\rho }^{*}=0.95$). Dotted curves refer to states at ${\rho }^{*}=0.80$, plotted in order of decreasing temperature from top ($T^{*}=1.20$) to bottom ($T^{*}=0.90$) and have been divided by a factor of 20 for graphical clarity. All curves are plotted for $0.5\le t^{*}\le t_{\mathrm{R}}^{*}$. The dashed line is the ${\left(t^{*}\right)}^{-3/2}$ function, arbitrarily scaled.

Figure 8

Simulated $S(Q,\omega )$ (dots) at two densities and two $Q$ values indicated by the labels. Also shown are the VE fits (solid lines through the data points) and the partial contributions to the fit of the sum of the two central modes and of the two sound modes.

Figure 9

(a) The curves joining symbols with error bars are the dispersion curves ${\omega }_{\mathrm{s}}^{*}\left(Q^{*}\right)$ of the longitudinal acoustic mode along the $T^{*}=1.35$ isotherm, in order of increasing density from bottom (${\rho }^{*}=0.65$) to top (${\rho }^{*}=0.95$). Dashed lines, in the same order, mark the corresponding values of the C2 oscillation frequency. The hydrodynamic dispersion straight lines $c_{\mathrm{s}}^{*}{Q}^{*}$ are also shown (dash-dots) for the highest and the lowest density. (b) Damping $z_{\mathrm{s}}^{*}\left(Q_{\mathrm{m}}^{*}\right)$ of the longitudinal acoustic mode (dots with error bar) at $Q^{*}=Q_{\mathrm{m}}^{*}$, i.e., where the respective dispersion curve reaches its maximum value in (a), and damping $-\mathrm{Re}{\left(z_{\mathrm{C}2}\right)}^{*}$ of the complex pair C2 (crosses). The color code is the same in the two frames.

Figure 10

Oscillation frequency of the C2 exponential (full squares) as a function of density at $T^{*}=1.35$ including the low-density states of Ref. [22]. The dashed line depicts the $3.5c_{\mathrm{s}}^{*}$ behavior explained in the text.

Figure 11

(a) Time dependence of $Z_{\mathrm{T}}\left(t\right)$, i.e., sum of the fitted exponentials belonging to the low-frequency group. Curves are ordered for increasing density from top (${\rho }^{*}=0.60$) to bottom (${\rho }^{*}=0.95$) in steps of 0.05. (b) Frequency spectra of the curves in (a), normalized to their respective zero frequency value. Here lower densities correspond to lower curves (at small frequency). (c) Frequency spectra of the transverse current autocorrelation function at a selected $Q$ value (see text) and for the same densities as in (a) and (b). Curves are ordered for increasing density from highest to lowest peak height. The color code is the same in all three frames.

Figure 12

The lines joining symbols with error bars are the transverse dispersion curves obtained as the positions of maxima of $C_{\mathrm{T}}(Q,\omega )$. Data for the $T^{*}=1.35$ states are shown in order of increasing density from bottom (${\rho }^{*}=0.80$) to top (${\rho }^{*}=0.95$). Dashed lines, in the same order, mark the corresponding position of maxima of ${\widehat{Z}}_{\mathrm{T}}\left(\omega \right)$.