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

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

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

+ | |||

+ | Consider a system of length <math> \left. L \right. </math> defined in the range <math> \left[ 0, L \right] </math>. | ||

+ | |||

+ | Our 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>. | ||

+ | |||

+ | Model: | ||

+ | |||

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

+ | |||

+ | * Pair Potential: | ||

+ | |||

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

+ | \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. | ||

+ | |||

+ | 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: | ||

+ | |||

+ | : <math> | ||

+ | \frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} d x_0 | ||

+ | \int_{x_0+\sigma}^{L+\sigma/2-N\sigma+\sigma} d x_1 \cdots | ||

+ | \int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i \sigma} d x_i \cdots | ||

+ | \int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma} d x_{N-1}. | ||

+ | </math> | ||

+ | |||

+ | 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 | ||

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

+ | \int_{\omega_{i-1}}^{L-N\sigma} d \omega_i \cdots | ||

+ | \int_{\omega_{N-2}}^{L-N\sigma} d \omega_{N-1}. | ||

+ | </math> | ||

+ | |||

+ | Therefore: | ||

+ | <math> | ||

+ | \frac{ Z \left( N,L \right)}{N!} = \frac{ (V-N)^{N} }{N!}. | ||

+ | </math> | ||

+ | |||

+ | : <math> | ||

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

+ | </math> | ||

+ | |||

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

#[http://dx.doi.org/10.1103/PhysRev.50.955 Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review '''50''' pp. 955- (1936)] | #[http://dx.doi.org/10.1103/PhysRev.50.955 Lewi Tonks "The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres", Physical Review '''50''' pp. 955- (1936)] |

## Revision as of 16:27, 26 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

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: