Revised Enskog equation for hard rods

In the absence of a trapping potential, Percus derived [19] a kinetic equation for hard rods by starting from the microscopic equation of motion

$$\begin{align} \partial_t \varrho_v + v \partial_x \varrho_v & = \int_{-\infty}^v {\mathrm{d}}v^{^{\prime}} \, \left(v-v^{^{\prime}}\right)\left[\varrho_v\left(x-a\right)\varrho_{v^{^{\prime}}}\left(x\right) - \varrho_v\left(x\right)\varrho_{v^{^{\prime}}}\left(x+a\right)\right]\nonumber\\ & \quad + \int_{v}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right)\left[\varrho_{v^{^{\prime}}}\left(x\right)\varrho_v\left(x+a\right) - \varrho_{v^{^{\prime}}}\left(x-a\right)\varrho_v\left(x\right)\right], \end{align} \tag{ 1 }$$

where

$$\begin{align} \varrho_v\left(x,t\right) = \sum_{i = 1}^N \delta\left(x-x_i\left(t\right)\right)\delta\left(v-v_i\left(t\right)\right) \end{align} \tag{ 2 }$$

denotes the microscopic phase-space distribution function. In order to do this, he used the Enskog form of Boltzmann's molecular chaos assumption

$$\begin{align} \langle \varrho_v\left(x\right)\varrho_{v^{^{\prime}}}\left(x+a\right) \rangle \approx \rho_v\left(x\right) \rho_{v^{^{\prime}}}\left(x+a\right) g\left(x+a/2\right), \end{align} \tag{ 3 }$$

where the brackets indicate ensemble averaging with respect to an initial local generalized Gibbs state as is appropriate when considering the hydrodynamics of an integrable system [12, 13], $\rho_v(x) = \langle \varrho_v(x)\rangle$ denotes the single-particle distribution function and $g(x+a/2)$ denotes the pair correlation function of a generalized Gibbs state with the same density of rods as at the midpoint of the rod centers at the instant of collision, given by [28, 29]

$$\begin{align} g\left(x+a/2\right) = \frac{1}{1-n\left(x+a/2\right)a}, \end{align} \tag{ 4 }$$

where n(x) is the density of rods at position x. This yields the Enskog equation

$$\begin{align} \partial_t \rho_v + v \partial_x \rho_v& = \int_{-\infty}^v {\mathrm{d}}v^{^{\prime}} \, \left(v-v^{^{\prime}}\right)\left[g\left(x-a/2\right)\rho_v\left(x-a\right)\rho_{v^{^{\prime}}}\left(x\right) - g\left(x+a/2\right) \rho_v\left(x\right)\rho_{v^{^{\prime}}}\left(x+a\right)\right] \nonumber\\ & + \int_{v}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right)\left[g\left(x+a/2\right)\rho_{v^{^{\prime}}}\left(x\right)\rho_v\left(x+a\right) - g\left(x-a/2\right)\rho_{v^{^{\prime}}}\left(x-a\right)\rho_v\left(x\right)\right]. \end{align} \tag{ 5 }$$

We note that any translation-invariant state of the form

$$\begin{align} \rho_v\left(x\right) = f\left(v\right), \end{align} \tag{ 6 }$$

where $f : \mathbb{R} \to \mathbb{R}$ is a non-negative function, lies in the kernel of this collision integral for each v'; such states are trivially stationary under the hard-rod dynamics and define global generalized Gibbs ensembles for the untrapped hard rod gas [20, 30].

Truncating equation (5) at the ballistic (Euler) and diffusive (Navier–Stokes) scales yields kinetic equations that are known to be exact in the appropriate scaling limits [19–21], respectively

$$\begin{align} \partial_t \rho_v + \partial_x \left(\frac{v - a j}{1-an}\rho_v\right) = 0 \end{align} \tag{ 7 }$$

at the ballistic scale, where

$$\begin{align} n = \int_{-\infty}^\infty {\mathrm{d}}v^{^{\prime}} \, \rho_{v^{^{\prime}}}, \quad j = \int_{-\infty}^\infty {\mathrm{d}}v^{^{\prime}} \, v^{^{\prime}}\rho_{v^{^{\prime}}}, \end{align} \tag{ 8 }$$

and

$$\begin{align} \partial_t \rho_v + \partial_x \left(\frac{v - a j}{1-an}\rho_v\right) = \frac{1}{2}a^2 \partial_x\left(\frac{1}{1-an} \int_{-\infty}^{\infty} {\mathrm{d}}v^{^{\prime}} |v^{^{\prime}}-v|\left(\partial_x \rho_v \rho_{v^{^{\prime}}} - \rho_v \partial_x \rho_{v^{^{\prime}}}\right)\right) \end{align} \tag{ 9 }$$

if diffusive corrections are included. Both these equations preserve the global generalized Gibbs states (6). It is important to note that the mathematical derivation [21] of the diffusive-scale equation (9) requires a stronger notion of local equilibrium than is needed [20] to derive equation (7). The former assumption does not always seem to be justified in practice [5]. Thus there is a precise sense in which the diffusive-scale kinetic equation (9) is less accurate than the ballistic-scale kinetic equation (7).

The currently accepted [11, 14] diffusive-scale kinetic theory for hard rods in an external trapping potential V(x) consists solely of adding a standard Boltzmann forcing term to the above equations. At the ballistic scale, this yields (m = 1)

$$\begin{align} \partial_t \rho_v + \partial_x \left(\frac{v - a j}{1-an}\rho_v\right) - V^{^{\prime}}\left(x\right) \partial_v \rho_v = 0, \end{align} \tag{ 10 }$$

while including diffusive corrections, we have

$$\begin{align} & \partial_t \rho_v + \partial_x \left(\frac{v - a j}{1-an}\rho_v\right) - V^{^{\prime}}\left(x\right) \partial_v \rho_v\nonumber\\ & \quad = \frac{1}{2}a^2 \partial_x\left(\frac{1}{1-an} \int_{-\infty}^{\infty} {\mathrm{d}}v^{^{\prime}} |v^{^{\prime}}-v|\left(\partial_x \rho_v \rho_{v^{^{\prime}}} - \rho_v \partial_x \rho_{v^{^{\prime}}}\right)\right). \end{align} \tag{ 11 }$$

In previous work, we both verified numerically that equation (10) remained valid until a state-dependent 'time to chaos' at which diffusive corrections became important, and pointed out that this equation could only be deduced from equation (5) if the trapping potential did not modify the Enskog approximation to the pair correlation function of the gas [5]. We will confirm below that revising the Enskog approximation yields no modifications to these equations at diffusive order [23] but to provide some additional motivation for this discussion, let us first consider the stationary states of equations (10) and (11). To understand stationary states of the ballistic scale dynamics (10), it is easiest to change variables to the 'free density' [19, 30]

$$\begin{align} \theta_v = \rho_v/\left(1-an\right), \end{align} \tag{ 12 }$$

to yield

$$\begin{align} \partial_t \theta_v + \left(\frac{v-aj}{1-an}\right) \partial_x \theta_v - V^{^{\prime}}\left(x\right) \partial_v \theta_v = 0, \end{align} \tag{ 13 }$$

so that stationary states satisfy

$$\begin{align} \left(\frac{v-aj}{1-an}\right) \partial_x \theta_v = V^{^{\prime}}\left(x\right) \partial_v \theta_v. \end{align} \tag{ 14 }$$

The possibility of infinitely many stationary states solving this equation was raised in previous work [5, 14]. Naively such states correspond to generalized Gibbs ensembles, but such states are not expected to survive the integrability-breaking effect of general trapping potentials. It has nevertheless been observed numerically that for hard rods in harmonic traps [5, 6] and rational Calogero particles in arbitrary traps [10], non-Maxwellian stationary states persist for all accessible times.

Our first contribution below is an explicit construction of stationary states solving equation (14) for any differentiable trapping potential V(x). To this end, let $F:\mathbb{R} \to \mathbb{R}$ be a differentiable, non-decreasing function with $F^{^{\prime}}(x) \unicode{x2A7E} 0$ for all $x \in \mathbb{R}$, such that there exists a solution $\epsilon_v(x)$ to the integral equation

$$\begin{align} \epsilon_v = \frac{v^2}{2} + V\left(x\right) - \mu - a \int_{-\infty}^{\infty} {\mathrm{d}}v^{^{\prime}}\,F\left(\epsilon_{v^{^{\prime}}}\right). \end{align} \tag{ 15 }$$

Then $\theta_v = F^{^{\prime}}(\epsilon_v)$ solves equation (14), since in this state, j = 0 by evenness of ε_v in v and one can show that

$$\begin{align} \partial_v \epsilon_v = v, \quad \partial_x \epsilon_v = \left(1-an\right)V^{^{\prime}}\left(x\right). \end{align} \tag{ 16 }$$

Our construction, which is parameterized by the functional degree of freedom F, extends straightforwardly to the Euler-scale hydrodynamics of other quantum and classical integrable systems. For classical hard rods, we can compare this solution directly against exact results for the grand canonical thermal state in a trap [22]. In the local density approximation (whose errors will be treated systematically below), the latter recovers equation (15) with the specific choice

$$\begin{align} F\left(\epsilon\right) = - \frac{1}{\beta} e^{-\beta \epsilon}. \end{align} \tag{ 17 }$$

The solutions corresponding to more general choices of F can be viewed as maximizers of classical entropies that are not the usual Boltzmann–Gibbs entropy, which reflects the freedom to choose an entropy function in ballistic-scale hydrodynamics [30]. The ballistic-scale hydrodynamics in a trap conserves an infinite family of such generalized entropies [5].

It is interesting to consider whether these stationary states remain stationary when diffusive corrections are included. It was argued in previous work [11] that only thermal states in the above sense, satisfying $\theta_v = e^{-\beta \epsilon_v}$ and

$$\begin{align} \epsilon_v = \frac{v^2}{2} + V\left(x\right) - \mu + \frac{a}{\beta} \int_{-\infty}^{\infty} {\mathrm{d}}v^{^{\prime}}\,e^{-\beta \epsilon_{v^{^{\prime}}}} \end{align} \tag{ 18 }$$

are stationary under the full dissipative evolution equation (11). We can see this directly by imposing vanishing of the dissipative part of the phase-space current in equation (11) for general stationary states $\theta_v = F(\epsilon_v)$ of the ballistic scale equation, which yields the condition

$$\begin{align} \frac{F^{^{\prime\prime}}\left(\epsilon_{v^{^{\prime}}}\right)}{F^{^{\prime}}\left(\epsilon_{v^{^{\prime}}}\right)} = \frac{F^{^{\prime\prime}}\left(\epsilon_v\right)}{F^{^{\prime}}\left(\epsilon_v\right)}, \quad v^{^{\prime}} \neq v, \end{align} \tag{ 19 }$$

on F. This implies that $F(\epsilon)$ is exponential in ε, so that the corresponding stationary state ρ_v is always separable in x and v and therefore Maxwellian in v by equation (15).

We now explain why the collision integral proposed by Percus [19] does not preserve thermal states in an external trapping potential. To our knowledge this observation has not been made before, although as we shall discuss below, it is consistent with van Beijeren and Ernst's critique [23] of Enskog's prescription and van Beijeren's results for three-dimensional hard spheres [24].

The single-particle distribution function for trapped thermal states of hard rods at inverse temperature β and chemical potential µ is given by [22]

$$\begin{align} \rho_v\left(x\right) = \sqrt{\frac{\beta}{2\pi}}e^{-\beta v^2/2}n\left(x\right), \end{align} \tag{ 20 }$$

where the local particle density n(x) satisfies the nonlocal integral equation

$$\begin{align} \log{n\left(x\right)} + \beta \left(V\left(x\right) - \mu\right) - \log{\sqrt{\frac{2\pi}{\beta}}} = \log{\left(1-\int_{x-a}^x {\mathrm{d}}y \, n\left(y\right)\right)} - \int_x^{x+a} {\mathrm{d}}y\, \frac{n\left(y\right)}{1-\int_{y-a}^y {\mathrm{d}}z \, n\left(z\right)}. \end{align} \tag{ 21 }$$

This result is exact in the grand canonical ensemble [22], and by equivalence of ensembles should be exact in the canonical ensemble in the thermodynamic limit.

Let us start by interpreting equations (20) and (21) for the exact thermal state in a trap. To relate these expressions to the thermodynamic-Bethe-ansatz-like equation (18), we define a nonlocally corrected version of the free density $\theta_v(x)$ in a trap, namely

$$\begin{align} \theta_v\left(x\right) = \frac{\rho_v\left(x\right)}{1-\int_{x-a}^x {\mathrm{d}}y \, n\left(y\right)}. \end{align} \tag{ 22 }$$

Writing $\theta_v(x) = e^{-\beta \epsilon_v(x)}$ as above, an exact and nonlocally corrected integral equation for ε_v is given by

$$\begin{align} \epsilon_v\left(x\right) = \frac{v^2}{2} + V\left(x\right) - \mu + \frac{1}{\beta} \int_x^{x+a} {\mathrm{d}}y\, \int_{-\infty}^{\infty} {\mathrm{d}}v^{^{\prime}}\,e^{-\beta \epsilon_{v^{^{\prime}}}\left(y\right)}. \end{align} \tag{ 23 }$$

This integral equation (or indeed Percus's original treatment [22]) implies corrections to the local density approximation equation (18) for trapped hard rods in thermal equilibrium [11, 31] at all orders in the rod length a. We expect that such nonlocal terms are generically present for equilibrium states of interacting particles in integrability-breaking traps, and thus correct predictions [11] invoking the local density approximation, with the size of these corrections determined by the scattering length of the interactions.

We are now in a position to correct equation (5). Let us suppose that the Enskog prescription equation (4) remains valid in the presence of a trapping potential. This predicts the kinetic equation

$$\begin{align} & \partial_t \rho_v + v \partial_x \rho_v - V^{^{\prime}}\left(x\right) \partial_v \rho_v\nonumber\\ & \quad = \int_{-\infty}^v {\mathrm{d}}v^{^{\prime}} \, \left(v-v^{^{\prime}}\right)\left[g\left(x-a/2\right)\rho_v\left(x-a\right)\rho_{v^{^{\prime}}}\left(x\right) - g\left(x+a/2\right) \rho_v\left(x\right)\rho_{v^{^{\prime}}}\left(x+a\right)\right] \nonumber\\ & \qquad + \int_{v}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right)\left[g\left(x+a/2\right)\rho_{v^{^{\prime}}}\left(x\right)\rho_v\left(x+a\right) - g\left(x-a/2\right)\rho_{v^{^{\prime}}}\left(x-a\right)\rho_v\left(x\right)\right]. \end{align} \tag{ 24 }$$

However, upon substituting the exact equilibrium state equation (20) into this equation, the left-hand side yields

$$\begin{align} \mathrm{L. \, H.\,S} = v\rho_v\left(\frac{n\left(x-a\right)}{1-\int_{x-a}^x {\mathrm{d}}y \, n\left(y\right)} - \frac{n\left(x+a\right)}{1-\int_{x+a}^x {\mathrm{d}}y \, n\left(y\right)}\right), \end{align} \tag{ 25 }$$

while the right-hand side yields

$$\begin{align} \mathrm{R.\,H.\,S} = v\rho_v \left(\frac{n\left(x-a\right)}{1-an\left(x-a/2\right)} - \frac{n\left(x+a\right)}{1-an\left(x+a/2\right)}\right). \end{align} \tag{ 26 }$$

The problem is now apparent: in an inhomogeneous trapping potential, Percus's proposed collision integral will not preserve the exact thermal state equation (20). To preserve this thermal state to all orders in the rod length a, we find that it suffices to replace the Enskog prescription equation (4) by the approximation

$$\begin{align} \langle \varrho_v\left(x\right)\varrho_{v^{^{\prime}}}\left(x+a\right) \rangle \approx \rho_v\left(x\right) \rho_{v^{^{\prime}}}\left(x+a\right) g^{\left(2\right)}\left(x,x+a\right), \end{align} \tag{ 27 }$$

where $g^{(2)}(x,x+a)$ denotes the exact nonuniform equilibrium contact pair correlation function of the hard rod gas. In the appendix, we show that for hard rods on an infinite line, in any potential that is confining as $|x| \to \infty$,

$$\begin{align} g^{\left(2\right)}\left(x,x+a\right) = \frac{1}{1-\int_{x}^{x+a}{\mathrm{d}}y \, n\left(y\right)}. \end{align} \tag{ 28 }$$

We emphasize that equation (27) is a strict equality in thermal equilibrium within a trap, unlike the naive extrapolation from equilibrium without a trap [28, 29] represented by equation (4). Moreover, this correction should persist even in the absence of a trap, which follows by considering arbitrarily weak but still confining potentials $V(x) \to 0$.

We note that 'revising' Enskog's prescription by using the exact nonuniform equilibrium pair correlation function at the instant of collision was previously advocated by van Beijeren and Ernst [23, 24], who argued that the resulting 'revised Enskog equation' should be exact at sufficiently short times. For the specific case of trapped, three-dimensional hard spheres, van Beijeren further noted that revising Enskog's prescription was necessary in order to obtain thermal stationary states from kinetic theory [24]. For trapped hard rods, the revised Enskog equation reads

$$\begin{align} & \partial_t \rho_v + v \partial_x \rho_v - V^{^{\prime}}\left(x\right) \partial_v \rho_v \nonumber\\ & \quad = \int_{-\infty}^v {\mathrm{d}}v^{^{\prime}} \, \left(v-v^{^{\prime}}\right)\left[g^{\left(2\right)}\left(x-a,x\right)\rho_v\left(x-a\right)\rho_{v^{^{\prime}}}\left(x\right) - g^{\left(2\right)}\left(x,x+a\right) \rho_v\left(x\right)\rho_{v^{^{\prime}}}\left(x+a\right)\right]\nonumber\\ & \qquad + \int_{v}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right)\left[g^{\left(2\right)}\left(x,x+a\right)\rho_{v^{^{\prime}}}\left(x\right)\rho_v\left(x+a\right) - g^{\left(2\right)}\left(x-a,x\right)\rho_{v^{^{\prime}}}\left(x-a\right)\rho_v\left(x\right)\right] \end{align} \tag{ 29 }$$

with $g^{(2)}(x,x+a)$ given by equation (28), and one can verify directly that this preserves the thermal state (20). The revised Enskog equation also respects the appropriate microscopic conservation laws, namely conservation of total particle number and total energy in a generic trap and conservation of center-of-mass energy in a harmonic trap.

We now characterize some of the stationary states of equation (29) for differentiable potentials V(x). Specifically, we will show that any such potential for which arbitrary generalized Gibbs ensembles (GGEs) are stationary in equation (29) is uniform, and that all separable, time-reversal symmetric stationary states of equation (29) are thermal.

3.2.1. Stationary GGEs imply a uniform potential.

Generalized Gibbs states of the untrapped hard rod gas take the form of any spatially uniform profile

$$\begin{align} \rho_{v}\left(x\right) = f\left(v\right), \end{align} \tag{ 30 }$$

for some non-negative function f(v). Let us suppose that for some differentiable choice of trap V(x), the kinetic equation (29) is stationary with respect to all such $\rho_v(x)$. Since the collision integral vanishes for any translation-invariant state, we deduce that

$$\begin{align} -V^{^{\prime}}\left(x\right) f^{^{\prime}}\left(v\right) = 0 \end{align} \tag{ 31 }$$

for all non-negative f, which implies that

$$\begin{align} V^{^{\prime}}\left(x\right) = 0, \end{align} \tag{ 32 }$$

i.e. the potential is uniform.

3.2.2. Separable stationary states are thermal.

Let us now suppose that equation (29) has a separable, time-reversal symmetric stationary state of the form

$$\begin{align} \rho_v\left(x\right) = n\left(x\right) f\left(v\right) \end{align} \tag{ 33 }$$

with f(v) even in v. Without loss of generality we assume that $\int_{-\infty}^{\infty} {\mathrm{d}}v \, f(v) = 1$, so that n(x) is the local particle density. In general, the condition for stationarity can be written as

$$\begin{align} & v \partial_x \rho_v - V^{^{\prime}}\left(x\right) \partial_v \rho_v \nonumber\\ & \quad = g^{\left(2\right)}\left(x,x+a\right) \int_{-\infty}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right) \left(\unicode{x1D7D9}_{v^{^{\prime}}\lt v} \rho_v\left(x\right)\rho_{v^{^{\prime}}}\left(x+a\right) + \unicode{x1D7D9}_{v^{^{\prime}}\gt v} \rho_{v^{^{\prime}}}\left(x\right)\rho_{v}\left(x+a\right)\right)\nonumber\\ & \qquad -g^{\left(2\right)}\left(x-a,x\right) \int_{-\infty}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right)\left(\unicode{x1D7D9}_{v^{^{\prime}}\lt v}\rho_{v}\left(x-a\right)\rho_{v^{^{\prime}}}\left(x\right)+\unicode{x1D7D9}_{v^{^{\prime}}\gt v}\rho_{v^{^{\prime}}}\left(x-a\right)\rho_{v}\left(x\right)\right). \end{align} \tag{ 34 }$$

For separable states equation (33), this reduces to

$$\begin{align} & v f\left(v\right) \partial_x \log{n}\left(x\right) - V^{^{\prime}}\left(x\right) f^{^{\prime}}\left(v\right)\nonumber\\ & \quad = \left(\frac{n\left(x+a\right)}{1-\int_{x}^{x+a}{\mathrm{d}}y\, n\left(y\right)} -\frac{n\left(x-a\right)}{1-\int_{x-a}^x {\mathrm{d}}y \, n\left(y\right)}\right) \int_{-\infty}^\infty {\mathrm{d}}v^{^{\prime}} \, \left(v^{^{\prime}}-v\right)f\left(v\right)f\left(v^{^{\prime}}\right). \end{align}$$

Using normalization and time-reversal symmetry of f, we can write this as

$$\begin{align} \partial_x \log{n\left(x\right)} + \frac{n\left(x+a\right)}{1-\int_{x}^{x+a}{\mathrm{d}}y\, n\left(y\right)} -\frac{n\left(x-a\right)}{1-\int_{x-a}^x {\mathrm{d}}y \, n\left(y\right)} = V^{^{\prime}}\left(x\right) \frac{\partial_v \log{f\left(v\right)}}{v}. \end{align} \tag{ 35 }$$

In order that the right hand side is independent of v, we require that

$$\begin{align} \partial_v \log f\left(v\right) = -\beta v \end{align} \tag{ 36 }$$

for some constant β. By normalization of f(v), this implies that

$$\begin{align} f\left(v\right) = \sqrt{\frac{\beta}{2\pi}}e^{-\beta v^2/2}, \end{align} \tag{ 37 }$$

i.e. f(v) is Maxwellian. (Note that this constraint is only imposed if $V^{^{\prime}}(x) \neq 0$ for some x, so there is no contradiction with the previous result.) In particular, the density n(x) must be related to the trapping potential via the equation

$$\begin{align} \partial_x \log{n\left(x\right)} + \beta V^{^{\prime}}\left(x\right) = \frac{n\left(x-a\right)}{1-\int_{x-a}^x {\mathrm{d}}y \, n\left(y\right)} - \frac{n\left(x+a\right)}{1-\int_{x}^{x+a}{\mathrm{d}}y\, n\left(y\right) }, \end{align} \tag{ 38 }$$

which is precisely the spatial derivative of equation (21). Thus n(x) coincides with the thermal state in a trap up to an integration constant that we identify as the chemical potential.

Consider N hard rods on an infinite line in an external trapping potential V(x) in thermal equilibrium at inverse temperature β. The canonical partition function can be written as

$$\begin{align} Z = Z_{k,N} Z_{N} \end{align} \tag{ A1 }$$

where the 'kinetic' partition function

$$\begin{align} Z_{k,N} = \prod_{i = 1}^N \int_{-\infty}^\infty {\mathrm{d}}p_i\, e^{-\beta p_i^2/2} = \left(\frac{2\pi}{\beta}\right)^{N/2} \end{align} \tag{ A2 }$$

and the 'configurational' partition function

$$\begin{align} Z_{N} & = \int_{-\infty}^\infty {\mathrm{d}}x_1 \int_{x_1+a}^{\infty} {\mathrm{d}}x_2 \ldots \int_{x_{N-1}+a}^{\infty} {\mathrm{d}}x_N \, e^{-\beta \sum_{j = 1}^N V\left(x_j\right)} \end{align} \tag{ A3 }$$

$$\begin{align} & = \int_{-\infty}^\infty {\mathrm{d}}x_N \int_{-\infty}^{x_N-a} {\mathrm{d}}x_{N-1} \ldots \int_{-\infty}^{x_{2}-a} {\mathrm{d}}x_1 \, e^{-\beta \sum_{j = 1}^N V\left(x_j\right)} \end{align} \tag{ A4 }$$

can be written in two different ways depending on whether the rods are enumerated from left to right or right to left respectively. Then the one-particle density can be written as

$$\begin{align} n\left(x\right) = e^{-\beta V\left(x\right)} \frac{1}{Z_N} \sum_{M = 0}^{N-1} Z_{M}\left(-\infty,x\right)Z_{N-1-M}\left(x,\infty\right) \end{align} \tag{ A5 }$$

where the sum is over all possible arrangements of M rods to the left and $N-1-M$ rods to the right of the rod centered at x and the one-sided partition functions

$$\begin{align} Z_{M}\left(-\infty,x\right) = \int_{-\infty}^{x-a} {\mathrm{d}}x_M \int_{-\infty}^{x_M-a} {\mathrm{d}}x_{M-1} \ldots \int_{-\infty}^{x_{2}-a} {\mathrm{d}}x_1 \, e^{-\beta \sum_{j = 1}^M V\left(x_j\right)} \end{align} \tag{ A6 }$$

and

$$\begin{align} Z_{M}\left(x,\infty\right) = \int_{x+a}^{\infty} {\mathrm{d}}x_1 \int_{x_1+a}^{\infty} {\mathrm{d}}x_2 \ldots \int_{x_{M-1}+a}^{\infty} {\mathrm{d}}x_M \, e^{-\beta \sum_{j = 1}^M V\left(x_j\right)} \end{align} \tag{ A7 }$$

weight these arrangements appropriately, with the convention that $Z_{0}(-\infty,x) = Z_{0}(x,\infty) = 1$. Similarly, the contact two-particle distribution function can be written as

$$\begin{align} n^{\left(2\right)}\left(x,x+a\right) = e^{-\beta V\left(x\right)}e^{-\beta V\left(x+a\right)} \frac{1}{Z_N} \sum_{M = 0}^{N-2} Z_{M}\left(-\infty,x\right)Z_{N-2-M}\left(x+a,\infty\right). \end{align} \tag{ A8 }$$

In general, these expressions are difficult to extract predictions from as $N \to \infty$, but their form is suggestive of multiplying power series with coefficients Z_M . This in turn suggests that dramatic simplifications could occur in the grand canonical ensemble, which indeed turns out to be the case.

Let us now consider the grand canonical partition for hard rods on an infinite line, following Percus [22]. This can be written as

$$\begin{align} \Xi = \sum_{N = 0}^\infty z^N Z_{k,N} Z_N. \end{align} \tag{ A9 }$$

where the fugacity $z = e^{\beta \mu}$. Introducing an effective fugacity

$$\begin{align} \tilde{z} = z \sqrt{\frac{2\pi}{\beta}}, \end{align} \tag{ A10 }$$

we can write this as

$$\begin{align} \Xi = \sum_{N = 0}^\infty \tilde{z}^N Z_N. \end{align} \tag{ A11 }$$

Then, introducing the one-sided grand canonical partition functions

$$\begin{align} \Xi\left(-\infty,x\right) = \sum_{N = 0}^{\infty} \tilde{z}^N Z_{N}\left(-\infty,x\right) \end{align} \tag{ A12 }$$

and

$$\begin{align} \Xi\left(x,\infty\right) = \sum_{N = 0}^{\infty} \tilde{z}^N Z_{N}\left(x,\infty\right), \end{align} \tag{ A13 }$$

we can write the one-particle density as [22]

$$\begin{align} n\left(x\right) = \frac{1}{\Xi} \tilde{z} e^{-\beta V\left(x\right)} \Xi\left(-\infty,x\right)\Xi\left(x,\infty\right) \end{align} \tag{ A14 }$$

and the contact two-particle distribution function as

$$\begin{align} n^{\left(2\right)}\left(x,x+a\right) = \frac{1}{\Xi} \tilde{z}^2 e^{-\beta V\left(x\right)} e^{-\beta V\left(x+a\right)}\Xi\left(-\infty,x\right)\Xi\left(x+a,\infty\right). \end{align} \tag{ A15 }$$

This implies a simple formula for the contact pair correlation function, namely

$$\begin{align} g^{\left(2\right)}\left(x,x+a\right) = \frac{n^{\left(2\right)}\left(x,x+a\right)}{n\left(x\right)n\left(x+a\right)} = \frac{\Xi}{\Xi\left(-\infty,x+a\right)\Xi\left(x,\infty\right)}. \end{align} \tag{ A16 }$$

Then, using the formula [22]

$$\begin{align} \Xi\left(-\infty,x+a\right)\Xi\left(x,\infty\right) = \Xi\left(1-\int_x^{x+a} {\mathrm{d}}y \, n\left(y\right)\right), \end{align} \tag{ A17 }$$

we deduce that the exact contact pair correlation function for trapped hard rods in thermal equilibrium is given by

$$\begin{align} g^{\left(2\right)}\left(x,x+a\right) = \frac{1}{1-\int_x^{x+a} {\mathrm{d}}y \, n\left(y\right)}. \end{align} \tag{ A18 }$$

We emphasize that this 'equation of state' for the pair correlation function holds for even for generalized-Gibbs-type ensembles with an arbitrary probability distribution in momentum space (although these are not strictly stationary except in the limit that $V(x) = 0$). For such ensembles, the configurational partition function is unchanged while the kinetic partition function is modified to

$$\begin{align} Z_{k,N} = \prod_{i = 1}^N \int_{-\infty}^\infty {\mathrm{d}}p_i\, e^{-f\left(p_i\right)} = \left(\int_{-\infty}^\infty {\mathrm{d}}p \, e^{-f\left(p\right)}\right)^{N} \end{align} \tag{ A19 }$$

where the function f uniquely determines the generalized Gibbs ensemble in question. In the treatment above, this change only modifies the effective fugacity $\tilde{z}$ and therefore does not affect equation (A18).