Triangle-Well and Ramp Interactions in One-Dimensional Fluids: A Fully Analytic Exact Solution

Abstract

The exact statistical-mechanical solution for the equilibrium properties, both thermodynamic and structural, of one-dimensional fluids of particles interacting via the triangle-well and the ramp potentials is worked out. In contrast to previous studies, where the radial distribution function g(r) was obtained numerically from the structure factor by Fourier inversion, we provide a fully analytic representation of g(r) up to any desired distance. The solution is employed to perform an extensive study of the equation of state, the excess internal energy per particle, the residual multiparticle entropy, the structure factor, the radial distribution function, and the direct correlation function. In addition, scatter plots of the bridge function versus the indirect correlation function are used to gauge the reliability of the hypernetted-chain, Percus–Yevick, and Martynov–Sarkisov closures. Finally, the Fisher–Widom and Widom lines are obtained in the case of the triangle-well model.

Appendices

A: Inverse Laplace Transform of $F_{\ell \ell _1}(s)$

Let us start by considering the following mathematical identity,

$$\begin{aligned} 1=x^{\ell _1}\sum _{j=0}^{\ell -1}\left( {\begin{array}{c}\ell _1+j-1\\ \ell _1-1\end{array}}\right) (1-x)^{j}+(1-x)^\ell \sum _{j_1=0}^{\ell _1-1}\left( {\begin{array}{c}\ell +j_1-1\\ \ell -1\end{array}}\right) x^{j_1}, \quad \ell ,\ell _1\ge 1, \end{aligned}$$

(A.1)

which can be proved by induction. Dividing both sides by $(1-x)^\ell x^{\ell _1}$, Eq. (A.1) becomes

$$\begin{aligned} \frac{1}{(1-x)^\ell x^{\ell _1}}=\sum _{j=1}^{\ell }\frac{\left( {\begin{array}{c}\ell +\ell _1-j-1\\ \ell _1-1\end{array}}\right) }{(1-x)^{j}}+\sum _{j_1=1}^{\ell _1} \frac{\left( {\begin{array}{c}\ell +\ell _1-j_1-1\\ \ell -1\end{array}}\right) }{x^{j_1}}. \end{aligned}$$

(A.2)

Next, the change of variable $s=-a_\beta x-\beta p$ yields

$$\begin{aligned} F_{\ell \ell _1}(s)= \sum _{j=1}^{\ell }\frac{\left( {\begin{array}{c}\ell +\ell _1-j-1\\ \ell _1-1\end{array}}\right) }{a_\beta ^{\ell +\ell _1-j}}\frac{(-1)^{\ell _1}}{(a_\beta +\beta p+s)^{j}}+\sum _{j_1=1}^{\ell _1}\frac{\left( {\begin{array}{c}\ell +\ell _1-j_1-1\\ \ell -1\end{array}}\right) }{a_\beta ^{\ell +\ell _1-j_1}}\frac{(-1)^{\ell _1-j_1}}{(\beta p+s)^{j_1}}, \end{aligned}$$

(A.3)

where $F_{\ell \ell _1}(s)$ is defined in Eq. (3.10). Making use of the laplace relation $\mathscr {L}^{-1}\{(a+s)^{-\ell }\}=\varTheta (r) \mathrm {e}^{-a r}r^{\ell -1}/(\ell -1)!$, where $\varTheta (\cdots )$ is the Heaviside step function, the inverse Laplace transform of $F_{\ell \ell _1}(s)$ is

$$\begin{aligned} f_{\ell \ell _1}(r)\equiv&\mathscr {L}^{-1}\left\{ F_{\ell \ell _1}(s)\right\} \nonumber \\ =&\frac{(-1)^{\ell _1}}{a_\beta ^{\ell +\ell _1}}\mathrm {e}^{-\beta pr}\varTheta (r)\left[ \mathrm {e}^{-a_\beta r} \sum _{j=1}^{\ell }\frac{\left( {\begin{array}{c}\ell +\ell _1-j-1\\ \ell _1-1\end{array}}\right) a_\beta ^{j}}{(j-1)!}r^{j-1}+\sum _{j_1=1}^{\ell _1}\frac{\left( {\begin{array}{c}\ell +\ell _1-j_1-1\\ \ell -1\end{array}}\right) (-a_\beta )^{j_1}}{(j_1-1)!}r^{j_1-1} \right] . \end{aligned}$$

(A.4)

In the special case $\ell _1=0$, one simply has

$$\begin{aligned} f_{\ell 0}(r)=\mathrm {e}^{-(a_\beta +\beta p)r} \frac{r^{\ell -1}}{(\ell -1)!}\varTheta (r). \end{aligned}$$

(A.5)

B: Widom Line in the Low-Temperature Limit

The Widom line is determined by the two conditions

$$\begin{aligned} \varOmega _\beta (-\kappa +\beta p)=\varOmega _\beta (\beta p),\quad \varOmega '_\beta (-\kappa +\beta p)=\varOmega '_\beta (\beta p). \end{aligned}$$

(B.1)

The first equality is the condition for $s=-\kappa$ to be a pole of $\mathscr {G}(s)$, in agreement with Eq. (2.5). The second equality is the extremum condition $\left( \partial \kappa /\partial {\beta p}\right) _{\beta }=0$.

From an inspection of Eqs. (3.1a) and (3.1b) one can check that, in the low-temperature limit ($\beta \epsilon \gg 1$), the solutions to Eq. (B.1) scale as

$$\begin{aligned} \beta p \rightarrow \sqrt{x\frac{a_\beta }{X_\beta }},\quad \kappa \rightarrow y \beta p, \end{aligned}$$

(B.2)

where the parameters x and y are pure numbers to be determined. Insertion of the scaling laws (B.2) yields

$$\begin{aligned} \lim _{\beta \epsilon \rightarrow \infty } \beta p\left[ \varOmega _\beta (-\kappa +\beta p)-\varOmega _\beta (\beta p)\right] =y\left( x-\frac{1}{y-1}\right) , \end{aligned}$$

(B.3a)

$$\begin{aligned} \lim _{\beta \epsilon \rightarrow \infty } (\beta p)^2\left[ \varOmega _\beta '(-\kappa +\beta p)-\varOmega _\beta '(\beta p)\right] =1-\frac{1}{(y-1)^2}. \end{aligned}$$

(B.3b)

Then, the conditions (B.1) imply $x=1$, $y=2$.

Next, inserting $\beta p\rightarrow \sqrt{a_\beta /X_\beta }$ into Eq. (3.3b) and taking the limit $\beta \epsilon \rightarrow \infty$, one obtains $n^*\rightarrow \frac{1}{2}$. It is worth noticing that in the case of a one-dimensional square-well fluid of range $\lambda$, the low-temperature limit of the Widom line is described by $\beta p \rightarrow \sqrt{2/(\lambda ^2-1)X_\beta }$, $\kappa \rightarrow 2 \beta p$, and $n^*\rightarrow 1/(\lambda +1)$.