1-dimensional hard rods

A 1-dimensional system having hard sphere interactions. The statistical mechanics of this system can be solved exactly (see Ref. 1).

Canonical Ensemble: Configuration Integral

Consider a system of length $L$ defined in the range $[0,L]$.

Our aim is to compute the partition function of a system of $N$ hard rods of length $\sigma$.

Model:

• External Potential; the whole length of the rod must be inside the range:

$V_0(x_i)=\{\begin{array}{ccc}0& ;& \sigma /2<x<L-\sigma /2\\ \mathrm{\infty }& ;& .\end{array}$

• Intermolecular pair potential:

$\mathrm{\Phi }(x_i,x_j)=\{\begin{array}{ccc}0& ;& |x_i-x_j|>\sigma \\ \mathrm{\infty }& ;& |x_i-x_j|<\sigma \end{array}$

where $x_k$ is the position of the center of the k-th rod.

Consider that the particles are ordered according to their label: $x_0<x_1<x_2<\cdots <x_{N-1}$; taking into account the pair potential we can write the canonical partition function (configuration integral) of a system of $N$ particles as:

$\frac{Z(N,L)}{N!}=\underset{\sigma /2}{\overset{L+\sigma /2-N\sigma }{\int }}d{x}_0\underset{x_0+\sigma }{\overset{L+\sigma /2-N\sigma +\sigma }{\int }}d{x}_1\cdots \underset{x_{i-1}+\sigma }{\overset{L+\sigma /2-N\sigma +i\sigma }{\int }}d{x}_i\cdots \underset{x_{N-2}+\sigma }{\overset{L+\sigma /2-N\sigma +(N-1)\sigma }{\int }}d{x}_{N-1}.$

Variable change: ${\omega }_k=x_k-(k+\frac{1}{2})\sigma$ ; we get:

$\frac{Z(N,L)}{N!}=\underset{0}{\overset{L-N\sigma }{\int }}d{\omega }_0\underset{{\omega }_0}{\overset{L-N\sigma }{\int }}d{\omega }_1\cdots \underset{{\omega }_{i-1}}{\overset{L-N\sigma }{\int }}d{\omega }_i\cdots \underset{{\omega }_{N-2}}{\overset{L-N\sigma }{\int }}d{\omega }_{N-1}.$

Therefore:

$\frac{Z(N,L)}{N!}=\frac{(L-N\sigma {)}^N}{N!}.$

$Q(N,L)=\frac{(L-N\sigma {)}^N}{{\mathrm{\Lambda }}^{N}N!}.$

Thermodynamics

Helmholtz energy function

$A(N,L,T)=-k_{B}T\log Q$

In the thermodynamic limit (i.e. $N\to \mathrm{\infty };L\to \mathrm{\infty }$ with $\rho =\frac{N}{L}$, remaining finite):

$A(N,L,T)=N{k}_{B}T[\log (\frac{N\mathrm{\Lambda }}{L-N\sigma })-1].$

Equation of state

Using the thermodynamic relations, the pressure (linear tension in this case) $p$ can be written as:

$p=-{(\frac{\partial A}{\partial L})}_{N,T}=\frac{N{k}_{B}T}{L-N\sigma };$

$Z=\frac{pL}{N{k}_{B}T}=\frac{1}{1-\eta },$

where $\eta \equiv \frac{N\sigma }{L}$; is the fraction of volume (i.e. length) occupied by the rods.

References

1. Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review 50 pp. 955- (1936)
2. L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, 15 pp. 951-961 (1949)
3. L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, 16 pp. 137-143 (1950)