Microcanonical ensemble: Difference between revisions
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'''Microcanonical ensemble''' | |||
== Ensemble variables == | == Ensemble variables == | ||
(One component system, 3-dimensional system, ... ): | (One component system, 3-dimensional system, ... ): | ||
* <math> \left. N \right. </math>: | * <math> \left. N \right. </math>: number of particles | ||
* <math> \left. V \right. </math>: | * <math> \left. V \right. </math>: is the volume | ||
* <math> \left. E \right. </math>: | * <math> \left. E \right. </math>: is the [[internal energy]] (kinetic + potential) | ||
== Partition function == | == Partition function == | ||
:<math> Q_{NVE} = \frac{1}{h^{3N} N!} \ | :<math> Q_{NVE} = \frac{1}{h^{3N} N!} \iint d (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E). | ||
</math> | </math> | ||
| Line 17: | Line 18: | ||
*<math> \left. h \right. </math> is the [[Planck constant]] | *<math> \left. h \right. </math> is the [[Planck constant]] | ||
*<math> \left( q \right)^{ | *<math> \left( q \right)^{3N} </math> represents the 3N Cartesian position coordinates. | ||
*<math> \left( p \right)^{ | *<math> \left( p \right)^{3N} </math> represents the 3N momenta. | ||
* <math> H \left(p,q\right) </math> | * <math> H \left(p,q\right) </math> represents the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta. | ||
*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]] | *<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]] | ||
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where: | where: | ||
*<math> \left. S \right. </math> is the entropy | *<math> \left. S \right. </math> is the [[Entropy|entropy]]. | ||
*<math> \left. k_B \right. </math> is the [[Boltzmann constant]] | *<math> \left. k_B \right. </math> is the [[Boltzmann constant]] | ||
== References == | == References == | ||
<references/> | |||
;Related reading | |||
* D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press | |||
* [http://dx.doi.org/10.1063/1.4931484 Philipp Schierz, Johannes Zierenberg and Wolfhard Janke "Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques", Journal of Chemical Physics '''143''' 134114 (2015)] | |||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] | ||
Latest revision as of 13:27, 13 November 2015
Microcanonical ensemble
Ensemble variables[edit]
(One component system, 3-dimensional system, ... ):
- : number of particles
- : is the volume
- : is the internal energy (kinetic + potential)
Partition function[edit]
where:
- is the Planck constant
- represents the 3N Cartesian position coordinates.
- represents the 3N momenta.
- represents the Hamiltonian, i.e. the total energy of the system as a function of coordinates and momenta.
- is the Dirac delta distribution
Thermodynamics[edit]
where:
- is the entropy.
- is the Boltzmann constant
References[edit]
- Related reading
- D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press
- Philipp Schierz, Johannes Zierenberg and Wolfhard Janke "Molecular Dynamics and Monte Carlo simulations in the microcanonical ensemble: Quantitative comparison and reweighting techniques", Journal of Chemical Physics 143 134114 (2015)