1-dimensional hard rods: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
Line 58: Line 58:
In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>,  remaining finite):
In the thermodynamic limit (i.e. <math> N \rightarrow \infty; L \rightarrow \infty</math> with <math> \rho = \frac{N}{L} </math>,  remaining finite):


:<math>  A \left( N,L,T \right) = - N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right)  - 1 \right]. </math>
:<math>  A \left( N,L,T \right) = N k_B T \left[ \log \left( \frac{ N \Lambda} { L - N \sigma }\right)  - 1 \right]. </math>
 
== Equation of state ==
 
From the basic thermodynamics, the pressure  [''linear tension in this case''] <math> \left. p \right. </math> can
be written as:
 
<math>
p = - \left( \frac{ \partial A}{\partial L} \right)_{N,T} =
</math>


==References==
==References==

Revision as of 12:00, 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

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

Helmholtz energy function

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

Equation of state

From the basic thermodynamics, the pressure [linear tension in this case] can be written as:

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)
  2. L. van Hove "Quelques Propriétés Générales De L'intégrale De Configuration D'un Système De Particules Avec Interaction", Physica, 15 pp. 951-961 (1949)
  3. L. van Hove, "Sur L'intégrale de Configuration Pour Les Systèmes De Particules À Une Dimension", Physica, 16 pp. 137-143 (1950)