Abstract
In the previous chapter we have learned that analytic calculations on the basis of microscopic interactions can become very difficult or even impossible. In such cases computer simulations are helpful.And even though thermodynamics is not the theory of many particle systems based on microscopic interactions, Statistical Mechanics is this theory, it possesses noteworthy ties to computer simulation. In the following we want to discuss some of them.
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Notes
- 1.
Every other conceivable distribution can replace this choice if desired.
- 2.
This works and is simple, but not necessarily efficient. It bypasses the importance sampling capability of the Metropolis MC mentioned above.
- 3.
Probably you have noticed the similarity to our discussion of the universal van der Waals equation (4.12), where we also use the gas-liquid critical point expressed via the parameters \(a\) and \(b\) to map the results of the universal theory onto specific systems. There as well as here we can also use experimental data for the second virial coefficient to fit \(a\) and \(b\) or \(\epsilon \) and \(\sigma \). These values again will differ to some extend from the ones obtained via the critical parameters.
- 4.
These particular quantities were calculated for system of 108 particles using the Molecular Dynamics technique (R. Hentschke, E. M. Aydt, B. Fodi, E. Stöckelmann Molekulares Modellieren mit Kraftfeldern., http://constanze.materials.uni-wuppertal.de), but Metropolis Monte Carlo could have used instead.
- 5.
What we just have described is known as Gibbs-Ensemble Monte Carlo originally invented by A. Z. Panagiotopoulos Determination of phase coexistence properties of fluids by direct Monte Carlo simulation in a new ensemble. Mol. Phys. 61, 813 (1987).
- 6.
The effect is concentrated near the interface or, because above \(T_c\) the interface vanishes, where the interface develops upon cooling. The rotation of the pressure chamber used here ensures greater homogeneity.
- 7.
We use \( \rho _{liq}^* -\rho _{gas}^* = A_o t^{\beta } - A_1 t^{\beta + \Delta }\) and \( \rho _{liq}^* +\rho _{gas}^* = 2 \rho _c^* + D_o t^{1-\alpha }\) with \(t=T_c^* -T^*\) (M. Ley-Koo, M. S. Green Consequences of the renormalization group for the thermodynamics of fluids near the critical point. Phys. Rev. A 23, 2650 (1981).). The (3D Ising) critical exponent values are \(\beta =0.326\), \(\alpha =0.11\), and \(\Delta =0.52\) (cf. our discussion of critical exponents and scaling beginning on p. 138).
- 8.
- 9.
from S. Schreiber, R. Hentschke Monte Carlo simulation of osmotic equilibria. J. Chem. Phys. 135, 134106 (2011).
- 10.
In praxis choosing the method largely depends on the experience of the person doing the calculation.
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Hentschke, R. (2014). Thermodynamics and Molecular Simulation. In: Thermodynamics. Undergraduate Lecture Notes in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36711-3_6
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DOI: https://doi.org/10.1007/978-3-642-36711-3_6
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