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

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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 parttion function (configuration integral) of a system of <math> N </math> particles as: | + | :taking into account the pair potential we can write the canonical parttion function (configuration integral) of a system of <math> N </math> particles as: |

: <math> | : <math> | ||

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Variable change: <math> \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. </math> ; we get: | Variable change: <math> \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. </math> ; we get: | ||

− | + | <math> | |

\frac{ Z \left( N,L \right)}{N!} = \int_{0}^{L-N\sigma} d \omega_0 | \frac{ Z \left( N,L \right)}{N!} = \int_{0}^{L-N\sigma} d \omega_0 | ||

\int_{\omega_0}^{L-N\sigma} d \omega_1 \cdots | \int_{\omega_0}^{L-N\sigma} d \omega_1 \cdots | ||

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Q(N,L) = \frac{ (V-N)^N}{\Lambda^N N!}. | Q(N,L) = \frac{ (V-N)^N}{\Lambda^N N!}. | ||

</math> | </math> | ||

+ | == Thermodynamics == | ||

+ | |||

+ | [[Helmholz energy function]] | ||

+ | : <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 = N/L </math> remaining finite: | ||

==References== | ==References== |

## Revision as of 11:47, 27 February 2007

Hard Rods, 1-dimensional system with hard sphere interactions.

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

## Canonical Ensemble: Configuration Integral

This part could require further improvements

Consider a system of length defined in the range .

Our aim is to compute the partition function of a system of hard rods of length .

Model:

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

- Pair Potential:

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

Consider that the particles are ordered according to their label: ;

- taking into account the pair potential we can write the canonical parttion function (configuration integral) of a system of particles as:

Variable change: ; we get:

Therefore:

## Thermodynamics

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

## 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)