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 \\
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\infty &; & elsewhere. \end{array} \right. </math>
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 +
* 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>
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 +
where <math> \left. x_k \right. </math> is the position of the center of the k-th rod.
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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>;
 +
 +
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
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\int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma} d x_{N-1}.
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</math>
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Variable change: <math> \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. </math> ; we get:
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 +
: <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}.
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</math>
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 +
Therefore:
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<math>
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\frac{ Z \left( N,L \right)}{N!} =  \frac{ (V-N)^{N} }{N!}.
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</math>
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 +
: <math>
 +
Q(N,L) = \frac{ (V-N)^N}{\Lambda^N N!}.
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</math>
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==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  \left. L \right. defined in the range  \left[ 0, L \right] .

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

Model:

  • External Potential; the whole length of the rod must be inside the range:
 V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\
\infty &; & elsewhere. \end{array} \right.
  • Pair Potential:
 V (x_i,x_j) = \left\{ \begin{array}{lll} 0 & ; & |x_i-x_j| > \sigma \\
\infty &; & |x_i-x_j| < \sigma \end{array} \right.

where  \left. x_k \right. 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 parttion function (configuration integral) of a system of  N particles as:


\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}.

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


\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}.

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


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

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)