Thermodynamics of a one-dimensional self-gravitating gas with periodic boundary conditions

We study the thermodynamic properties of a one-dimensional gas with one-dimensional gravitational interactions. Periodic boundary conditions are implemented as a modification of the potential consisting of a sum over mirror images (Ewald sum), regularized with an exponential cutoff. As a consequence, each particle carries with it its own background density. Using mean-field theory, we show that the system has a phase transition at a critical temperature. Above the critical temperature the gas density is uniform, while below the critical point the system becomes inhomogeneous. Numerical simulations of the model, which include the caloric curve, the equation of state, the radial distribution function, and the largest Lyapunov exponent, confirm the existence of the phase transition, and they are in good agreement with the theoretical predictions.

Figure 1

Plot of the interaction energy $LV(x,y)$ vs $\frac{1}{L}|x-y|$ in the Miller-Rouet model.

Figure 2

Plot of the function $F\left(y_0\right)$ defined in (37) for $y_0<0$ vs $|y_0|$.

Figure 3

Plots of the density $\rho \left(x\right)$ vs $x$ for several values of ${\alpha }^2=2\beta g^2$ (solid curves). The dashed curves show the approximation (41), which is valid for $\alpha \sim 2\pi$. Left: $\alpha =6.3$, right: $\alpha =7$.

Figure 4

Plot of the specific heat per particle $\frac{1}{3}[c_V\left(T\right)-\frac{1}{2}]$ vs $T/T_{c1}$. This is the excess of the specific heat over the constant value $\frac{1}{2}$, which it takes in the homogeneous phase (times $1/3$). The specific heat has a finite jump at $T_{c1}$. The solid curve shows the exact result (53) and the dashed curve shows the approximation (58) valid near the critical point.

Figure 5

Illustration of convergence of kinetic energy per particle in simulation for the first 100 time units out of a total time of 1200. The numbers in the legend represent the value of $\frac{U}{N}(\times {10}^{-4})$. The upper pair of plots represent temperatures above the critical point while the lower pair represents temperatures below it.

Figure 6

Per-particle kinetic energy plotted against per-particle energy for varying number of particles.

Figure 7

Per-particle kinetic energy vs per-particle energy for $N=160$. A clear-cut jump is evident in the first derivative of the caloric curve.

Figure 8

$\mu$-space snapshots (left column) and time-averaged values of the radial distribution function (right column) for $U/N=2.08\times {10}^{-4}$ and $N=160$ at different instants of time. Corresponding elapsed time for each row is mentioned in the $g\left(r\right)$ plot.

Figure 9

$\mu$-space snapshots (left column) and time-averaged values of the radial distribution function (right column) for $U/N=4.17\times {10}^{-4}$ and $N=160$ at different instants of time. Corresponding elapsed time for each row is mentioned in the $g\left(r\right)$ plot.

Figure 10

Time-averaged pressure vs per-particle kinetic energy for $N=160$.

Figure 11

Largest Lyapunov characteristic exponent vs per-particle kinetic energy for $N=160$.

Figure 12

Caloric curve: theoretical prediction vs simulation results for $N=160$.

Figure 13

Plot of the caloric curve $(U_{\mathrm{sim}}/N,E_{\text{kin}}/N)$ for $N=160$. Dashed curve: approximative result using the approximation $K_1\left(\alpha \right)=\pi \alpha y_0^2$.