1-dimensional hard rods - Revision history

87.8.62.54 at 08:42, 24 April 2021

2021-04-24T08:42:57Z

← Older revision		Revision as of 10:42, 24 April 2021
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	f(x_2-x_1)		f(x_2-x_1)
	\cdots		\cdots
	f(L-x_{N-1}),		f(x_0+L-x_{N-1}),
	</math>		</math>
	where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions		where <math>N!</math> does not appear one would have <math>N!</math> analogous expressions

79.25.58.27 at 21:49, 5 April 2021

2021-04-05T21:49:00Z

← Older revision		Revision as of 23:49, 5 April 2021
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	\cdots		\cdots
	\int_0^{\infty} d y_{N-1}		\int_0^{\infty} d y_{N-1}
			f(y_0)
	f(y_1)		f(y_1)
	~~f(y_2)~~
	\cdots		\cdots
	f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right):		f(y_{N-1}) \delta \left( \sum_{i=0}^{N-1} y_i-L \right):
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	\cdots		\cdots
	\int_0^{\infty} d y_{N-1}		\int_0^{\infty} d y_{N-1}
			f(y_0)
	f(y_1)		f(y_1)
	~~f(y_2)~~
	\cdots		\cdots
	f(y_{N-1})		f(y_{N-1})

Carl McBride: /* Canonical Ensemble: Configuration Integral */ Removed a dead external link

2012-11-22T15:40:39Z

Canonical Ensemble: Configuration Integral: Removed a dead external link

← Older revision		Revision as of 17:40, 22 November 2012
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	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 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
	~~([http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral])~~
	of a system of <math> N </math> particles as:		of a system of <math> N </math> particles as:

Carl McBride: /* References */ Added a publication

2012-10-22T11:02:18Z

References: Added a publication

← Older revision		Revision as of 13:02, 22 October 2012
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	*[http://dx.doi.org/10.1063/1.1706788 Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids '''6''' 609 (1963)]		*[http://dx.doi.org/10.1063/1.1706788 Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids '''6''' 609 (1963)]
	*[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)]		*[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)]
			*[http://dx.doi.org/10.1063/1.475640 Gerardo Soto-Campos, David S. Corti, and Howard Reiss "A small system grand ensemble method for the study of hard-particle systems", Journal of Chemical Physics '''108''' pp. 2563-2570 (1998)]
	*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)]		*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)]

	[[Category:Models]]		[[Category:Models]]
	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Carl McBride: /* Chemical potential */ Added an internal link

2012-08-15T15:56:05Z

Chemical potential: Added an internal link

← Older revision		Revision as of 17:56, 15 August 2012
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	== Chemical potential ==		== Chemical potential ==
	The chemical potential is given by		The [[chemical potential]] is given by

	:<math>		:<math>

Carl McBride: /* Equation of state */ Added an internal link

2012-08-15T15:55:23Z

Equation of state: Added an internal link

← Older revision		Revision as of 17:55, 15 August 2012
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	</math>		</math>

	The compressibility factor is		The [[compressibility factor]] is

	:<math>		:<math>

Franzl aus tirol: chemical potential

2012-08-15T15:45:44Z

chemical potential

← Older revision		Revision as of 17:45, 15 August 2012
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	p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma};		p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} = \frac{ N k_B T}{L - N \sigma};
	</math>		</math>

			The compressibility factor is

	:<math>		:<math>
	Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta},		Z = \frac{p L}{N k_B T} = \frac{1}{ 1 - \eta} = \underbrace{1}_{Z_{\mathrm{id}}}+\underbrace{\frac{\eta}{1-\eta}}_{Z_{\mathrm{ex}}},
	</math>		</math>

	where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods.		where <math> \eta \equiv \frac{ N \sigma}{L} </math>; is the fraction of volume (i.e. length) occupied by the rods. 'id' labels the ideal and 'ex' the excess part.

	It was shown by van Hove <ref>[http://dx.doi.org/10.1016/0031-8914(50)90072-3 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, '''16''' pp. 137-143 (1950)]</ref> that there is no [[Solid-liquid phase transitions \|fluid-solid phase transition]] for this system (hence the designation ''Tonks gas'').		It was shown by van Hove <ref>[http://dx.doi.org/10.1016/0031-8914(50)90072-3 L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, '''16''' pp. 137-143 (1950)]</ref> that there is no [[Solid-liquid phase transitions \|fluid-solid phase transition]] for this system (hence the designation ''Tonks gas'').

			== Chemical potential ==
			The chemical potential is given by

			:<math>
			\mu=\left(\frac{\partial A}{\partial N}\right)_{L,T}=k_{B}T\left(\ln\frac{\rho\Lambda}{1-\rho\sigma}+\frac{\rho\sigma}{1-\rho\sigma}\right)=k_{B}T\left(\ln\frac{\rho\Lambda}{1-\eta}+\frac{\eta}{1-\eta}\right)
			</math>

			with ideal and excess part separated:

			:<math>
			\beta\mu=\underbrace{\ln(\rho\Lambda)}_{\beta\mu_{\mathrm{id}}}+\underbrace{\ln\frac{1}{1-\eta}+\frac{\eta}{1-\eta}}_{\beta\mu_{\mathrm{ex}}}
			</math>

	== Isobaric ensemble: an alternative derivation ==		== Isobaric ensemble: an alternative derivation ==
	Adapted from Reference <ref>J. M. Ziman ''Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems'', Cambridge University Press (1979) ISBN 0521292808</ref>. If the rods are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math> the canonical [[partition function]] can also be written as:		Adapted from Reference <ref>J. M. Ziman ''Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems'', Cambridge University Press (1979) ISBN 0521292808</ref>. If the rods are ordered according to their label: <math> x_0 < x_1 < x_2 < \cdots < x_{N-1} </math> the canonical [[partition function]] can also be written as:

Carl McBride: Added an internal link

2012-08-15T10:26:44Z

Added an internal link

← Older revision		Revision as of 12:26, 15 August 2012
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	0 & ; & \|x_{i}-x_{j}\|>\sigma\\ \infty & ; & \|x_{i}-x_{j}\|<\sigma \end{array}\right. </math>		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, along with an external potential. Thus, the Boltzmann factor is		where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential. Thus, the [[Boltzmann factor]] is

	: <math>e_{ij}:=e^{-\beta\Phi_{12}(x_{i},x_{j})}=\Theta(\|x_{i}-x_{j}\|-\sigma)=\left\{ \begin{array}{lll} 1 & ; & \|x_{i}-x_{j}\|>\sigma\\ 0 & ; & \|x_{i}-x_{j}\|<\sigma \end{array}\right. </math>		: <math>e_{ij}:=e^{-\beta\Phi_{12}(x_{i},x_{j})}=\Theta(\|x_{i}-x_{j}\|-\sigma)=\left\{ \begin{array}{lll} 1 & ; & \|x_{i}-x_{j}\|>\sigma\\ 0 & ; & \|x_{i}-x_{j}\|<\sigma \end{array}\right. </math>

Franzl aus tirol: Boltzmann factor, Configuration Integral

2012-08-15T09:39:12Z

Boltzmann factor, Configuration Integral

← Older revision		Revision as of 11:39, 15 August 2012
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	'''1-dimensional hard rods''' (sometimes known as a ''Tonks gas'' <ref>[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)]</ref>) consist of non-overlapping line segments of length <math>\sigma</math> who all occupy the same line which has length <math>L</math>. One could also think of this model as being a string of [[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]]:		'''1-dimensional hard rods''' (sometimes known as a ''Tonks gas'' <ref>[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)]</ref>) consist of non-overlapping line segments of length <math>\sigma</math> who all occupy the same line which has length <math>L</math>. One could also think of this model as being a string of [[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]]:

	: <math> \Phi_{12} (~~x_i~~,~~x_j~~) = \left\{ \begin{array}{lll} 0 & ; & \|~~x_i~~-~~x_j~~\| > \sigma \\		: <math> \Phi_{12}(x_{i},x_{j})=\left\{ \begin{array}{lll}
	\infty &; & \|~~x_i~~-~~x_j~~\| < \sigma \end{array} \right. </math>		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, along with an external potential; the ~~whole length of the rod must be inside the range:~~		where <math> \left. x_k \right. </math> is the position of the center of the k-th rod, along with an external potential. Thus, the Boltzmann factor is

	: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x < L - \sigma/2 \\		: <math>e_{ij}:=e^{-\beta\Phi_{12}(x_{i},x_{j})}=\Theta(\|x_{i}-x_{j}\|-\sigma)=\left\{ \begin{array}{lll} 1 & ; & \|x_{i}-x_{j}\|>\sigma\\ 0 & ; & \|x_{i}-x_{j}\|<\sigma \end{array}\right. </math>

			The whole length of the rod must be inside the range:

			: <math> V_{0}(x_i) = \left\{ \begin{array}{lll} 0 & ; & \sigma/2 < x_i < L - \sigma/2 \\
	\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math>		\infty &; & {\mathrm {elsewhere}}. \end{array} \right. </math>

	== Canonical Ensemble: Configuration Integral ==		== Canonical Ensemble: Configuration Integral ==
	The [[statistical mechanics]] of this system can be solved exactly.		The [[statistical mechanics]] of this system can be solved exactly.
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	of a system of <math> N </math> particles as:		of a system of <math> N </math> particles as:

	: <math>		:<math>\begin{align}
	\frac{ Z \left( N,L \right)}{N!} = \int_{\sigma/2}^{L+\sigma/2-N\sigma} ~~d x_0~~		\frac{Z\left(N,L\right)}{N!} & =\int_{\sigma/2}^{L-\sigma/2}dx_{0}\int_{\sigma/2}^{L-\sigma/2}dx_{1}\cdots\int_{\sigma/2}^{L-\sigma/2}dx_{N-1}\prod_{i=1}^{N-1}e_{i-1,i}\\
	\int_{~~x_0~~+\sigma}^{L+\sigma/2-N\sigma+\sigma} ~~d x_1~~ \cdots		& =\int_{\sigma/2}^{L+\sigma/2-N\sigma}dx_{0}\int_{x_{0}+\sigma}^{L+\sigma/2-N\sigma+\sigma}dx_{1}\cdots\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i\sigma}dx_{i}\cdots\int_{x_{N-2}+\sigma}^{L+\sigma/2-N\sigma+(N-1)\sigma}dx_{N-1}.
	\int_{x_{i-1}+\sigma}^{L+\sigma/2-N\sigma+i \sigma} ~~d x_i~~ \cdots		\end{align}</math>
	\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 - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:		Variable change: <math> \left. \omega_k = x_k - \left(k+\frac{1}{2}\right) \sigma \right. </math> ; we get:

	:<math>		:<math>\begin{align}
	\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_{i-1}}^{L-N\sigma}d\omega_{i}\cdots\int_{\omega_{N-2}}^{L-N\sigma}d\omega_{N-1}\\
	\int_{\~~omega_0~~}^{L-N\sigma} d \~~omega_1~~ \cdots		& =\int_{0}^{L-N\sigma}d\omega_{0}\cdots\int_{\omega_{i-1}}^{L-N\sigma}d\omega_{i}\frac{(L-N\sigma-\omega_{i})^{N-1-i}}{(N-1-i)!}=\int_{0}^{L-N\sigma}d\omega_{0}\frac{(L-N\sigma-\omega_{0})^{N-1}}{(N-1)!}
	\int_{\omega_{i-1}}^{L-N\sigma} d \~~omega_i~~ \cdots		\end{align}</math>
	\int_{\omega_{N-2}}^{L-N\sigma} d \omega_{N-1}.
	</math>

	Therefore:		Therefore:

Carl McBride: /* References */ Added a publication

2010-11-25T14:49:47Z

References: Added a publication

← Older revision		Revision as of 16:49, 25 November 2010
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	*[http://dx.doi.org/10.1063/1.1699116 Zevi W. Salsburg, Robert W. Zwanzig, and John G. Kirkwood "Molecular Distribution Functions in a One-Dimensional Fluid", Journal of Chemical Physics '''21''' pp. 1098-1107 (1953)]		*[http://dx.doi.org/10.1063/1.1699116 Zevi W. Salsburg, Robert W. Zwanzig, and John G. Kirkwood "Molecular Distribution Functions in a One-Dimensional Fluid", Journal of Chemical Physics '''21''' pp. 1098-1107 (1953)]
	*[http://dx.doi.org/10.1063/1.1699263 Robert L. Sells, C. W. Harris, and Eugene Guth "The Pair Distribution Function for a One-Dimensional Gas", Journal of Chemical Physics '''21''' pp. 1422-1423 (1953)]		*[http://dx.doi.org/10.1063/1.1699263 Robert L. Sells, C. W. Harris, and Eugene Guth "The Pair Distribution Function for a One-Dimensional Gas", Journal of Chemical Physics '''21''' pp. 1422-1423 (1953)]
			*[http://dx.doi.org/10.1063/1.1706788 Donald Koppel "Partition Function for a Generalized Tonks' Gas", Physics of Fluids '''6''' 609 (1963)]
	*[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)]		*[http://dx.doi.org/10.1103/PhysRev.171.224 J. L. Lebowitz, J. K. Percus and J. Sykes "Time Evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods", Physical Review '''171''' pp. 224-235 (1968)]
	*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)]		*[http://dx.doi.org/10.3390/e10030248 Paolo V. Giaquinta "Entropy and Ordering of Hard Rods in One Dimension", Entropy '''10''' pp. 248-260 (2008)]