# Difference between revisions of "1-dimensional hard rods"

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− | + | '''1-dimensional hard rods''' are basically [[hard sphere model | hard spheres]] confined to 1 dimension (not to be confused with [[3-dimensional hard rods]]). The model is given by the [[intermolecular pair potential]]: | |

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: <math> \Phi (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\ | : <math> \Phi (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\ | ||

\infty &; & |x_i-x_j| < \sigma \end{array} \right. </math> | \infty &; & |x_i-x_j| < \sigma \end{array} \right. </math> | ||

− | where <math> \left. x_k \right. </math> is the position of the center of the k-th rod | + | where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential; the whole length of the rod must be inside the range: |

+ | : <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\ | ||

+ | \infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math> | ||

+ | == Canonical Ensemble: Configuration Integral == | ||

+ | The [[statistical mechanics]] of this system can be solved exactly (see Ref. 1). | ||

+ | Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>. The aim is to compute the [[partition function]] of a system of <math> \left. N \right. </math> hard rods of length <math> \left. \sigma \right. </math>. | ||

Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>; | Consider that the particles are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math>; | ||

taking into account the pair potential we can write the canonical partition function | taking into account the pair potential we can write the canonical partition function | ||

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: <math> \left. A(N,L,T) = - k_B T \log Q \right. </math> | : <math> \left. A(N,L,T) = - k_B T \log Q \right. </math> | ||

− | In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>, remaining finite): | + | In the [[thermodynamic limit]] (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>, remaining finite): |

:<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math> | :<math> A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. </math> |

## Revision as of 12:24, 20 February 2008

**1-dimensional hard rods** are basically hard spheres confined to 1 dimension (not to be confused with 3-dimensional hard rods). The model is given by the intermolecular pair potential:

where is the position of the center of the k-th rod, along with an external potential; the whole length of the rod must be inside the range:

## Contents

## Canonical Ensemble: Configuration Integral

The statistical mechanics of this system can be solved exactly (see Ref. 1). Consider a system of length defined in the range . The aim is to compute the partition function of a system of hard rods of length . Consider that the particles are ordered according to their label: ; taking into account the pair potential we can write the canonical partition function (configuration integral) of a system of particles as:

Variable change: ; we get:

Therefore:

## Thermodynamics

In the thermodynamic limit (i.e. with , remaining finite):

## Equation of state

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

where ; is the fraction of volume (i.e. length) occupied by the rods.

## References

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