1-dimensional hard rods

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

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Consider a system of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. L \right. } defined in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 0, L \right] } .

Our aim is to compute the partition function of a system of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. N \right. } hard rods of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \sigma \right. } .

Model:

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\ \infty &; & elsewhere. \end{array} \right. }

Pair Potential:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } particles as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \omega_k = x_k - (k+\frac{1}{2}) \sigma \right. } ; we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 {(L-N\sigma )^{N}}{N!}}.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(N,L) = \frac{ (L-N \sigma )^N}{\Lambda^N N!}. }

Thermodynamics

Helmholz energy function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. A(N,L,T) = - k_B T \log Q \right. }

In the thermodynamic limit (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \rightarrow \infty; L \rightarrow \infty} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \frac{N}{L} } , remaining finite):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \left( N,L,T \right) = - N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right) - 1 \right]. }

References

1-dimensional hard rods

Canonical Ensemble: Configuration Integral

Thermodynamics

References

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