Probing the existence of phase transitions in one-dimensional fluids of penetrable particles

Phase transitions in one-dimensional classical fluids are usually ruled out by using van Hove's theorem. A way to circumvent the conclusions of the theorem is to consider an interparticle potential that is everywhere bounded. Such is the case of, e.g., the generalized exponential model of index 4 (GEM-4 potential), which in three dimensions gives a reasonable description of the effective repulsion between flexible dendrimers in a solution. An extensive Monte Carlo simulation of the one-dimensional GEM-4 model [S. Prestipino, Phys. Rev. E 90, 042306 (2014)] has recently provided evidence of an infinite sequence of low-temperature cluster phases, however, also suggesting that upon pushing the simulation forward what seemed a true transition may eventually prove to be only a sharp crossover. We hereby investigate this problem theoretically by use of three different and increasingly sophisticated approaches (i.e., a mean-field theory, the transfer matrix of a lattice model of clusters, and the exact treatment of a system of point clusters in the continuum) to conclude that the alleged transitions of the one-dimensional GEM-4 system are likely just crossovers.

Figure 1

One-dimensional GEM-4, mean-field results for six temperatures, $T=0,0.01,...,0.05$, marked as black, blue, cyan, green, magenta, and red lines, respectively. We here show results deriving from the optimization of $\overline{n}$ only (top: best $\overline{n}$ and $\overline{F}/L$ values; bottom: lattice spacing). Each inset shows a magnification of the $d$ interval from 1.3 to 1.7 (top: $\overline{n}$; bottom: lattice spacing).

Figure 2

One-dimensional GEM-4, mean-field results for the same temperatures considered in Fig. 1. We here plot the density $\rho$ (top) and the chemical potential $\mu$ (bottom) as a function of the pressure $P$, as resulting from the optimization of both $\overline{n}$ and $d$ (only the first three transitions are shown). In each inset, we show a magnification of the $P$ interval from 2.3 to 2.7, which makes us better appreciate the effect of heating the system: While the transition from the two-cluster to the three-cluster phase is still first order for $T=0.03$, it has by now become a sharp (but smooth) crossover for $T=0.04$.

Figure 3

One-dimensional GEM-4, transfer-matrix results for various temperatures ($T=0,0.01,...,0.05$, same colors as in Figs. 1 and 2). In the top panel, the average NN distance, the correlation length, and the connected NN correlation are plotted as a function of pressure. In the bottom panel, the average cluster size $\langle n_1\rangle$, the average squared occupancy, and the density are shown.

Figure 4

One-dimensional GEM-4, transfer-matrix results for various temperatures ($T=0.01,0.02,0.025,0.026,...,0.034,0.035,0.04,0.05$; for the sake of clarity, the line colors are alternatively black and blue, except for $T=0.028$ (red curves) where the density shows two distinct jumps at two nearby pressures): average cluster size (top panel) and number density (bottom panel) near the transition from the one-cluster to the two-cluster phase. In the inset, we show a magnification of the density for $T=0.028$, which clearly signals the existence of two distinct steps (the raw data are the dots, joined by straight-line segments to help the eye).

Figure 5

One-dimensional GEM-4, transfer-matrix results for various temperatures (the same as in Fig. 4): Correlation length $\xi$ near the transition from the one-cluster to the two-cluster phase.

Figure 6

One-dimensional GEM-4 phase diagram on the $P-T$ plane: Exact transfer-matrix results (open blue circles and squares) vs mean-field results (black dots). The exact Widom line is also shown (dotted red line).

Figure 7

One-dimensional GEM-4 phase diagram on the $\rho -T$ plane: Exact transfer-matrix results (open blue circles and squares) vs mean-field results (black dots).

Figure 8

One-dimensional $\text{GEM-}\infty$ (PSM), transfer-matrix results for various temperatures ($T=0,0.01,...,0.05$, same colors as in Figs. 1, 2, and 3): average cluster size $\langle n_1\rangle$ (which coincides with the density) and average squared size $\langle n_1^2\rangle$. In the $\alpha \to \infty$ limit, the discontinuous jumps of the density at the lowest temperatures have disappeared, leaving a sharp but continuous crossover at all nonzero temperatures.

Figure 9

One-dimensional GEM-4, comparison between $\mu$ and $\rho$ results for the continuous model (Sec. 4) and for the lattice model (Sec. 3) in the 1-2 transition region (for $X\equiv h/\sqrt{2\pi m\epsilon {\sigma }^2}=0.01$ and $a=1$). Top: Continuous model ($T=0.002,0.003,0.004,0.005$, black; $T=0.01$, blue; $T=0.015$, cyan; $T=0.02$, green; $T=0.025$, magenta; $T=0.03$, red). Observe that even for such a low temperature as $T=0.002$ the density varies smoothly across the “transition.” In the same panel, the black dots are $T=0.02$ density data from Ref. [32]. Bottom: Lattice model ($T=0.01$, blue; $T=0.02$, green; $T=0.025$, magenta; $T=0.03$, red).