Microcanonical ensemble: Difference between revisions

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Microcanonical Ensemble (Clasical statistics):
'''Microcanonical ensemble'''
 
== Ensemble variables ==  
== Ensemble variables ==  
(One component system, 3-dimensional system, ... ):
(One component system, 3-dimensional system, ... ):


* <math> \left. N \right. </math>: Number of Particles
* <math> \left. N \right. </math>: number of particles


* <math> \left. V \right. </math>: Volumne
* <math> \left. V \right. </math>: is the volume


* <math> \left. E \right. </math>: Internal enerrgy (kinetic + potential)
* <math> \left. E \right. </math>: is the [[internal energy]] (kinetic + potential)


== Partition function ==  
== Partition function ==  


<math> Q_{NVE} = \frac{1}{h^{3N} N!} \int \int d  (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).
:<math> Q_{NVE} = \frac{1}{h^{3N} N!} \iint d  (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).
</math>
</math>


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*<math>  \left. h \right. </math> is  the [[Planck constant]]
*<math>  \left. h \right. </math> is  the [[Planck constant]]


*<math> \left( q \right)^{3n} </math> represents the 3N Cartesian position coordinates.
*<math> \left( q \right)^{3N} </math> represents the 3N Cartesian position coordinates.
 
*<math> \left( p \right)^{3N} </math> represents the 3N momenta.
 
* <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]]
 
== Thermodynamics ==
 
:<math> \left. S = k_B \log Q_{NVE} \right. </math>


*<math> \left( p \right)^{3n} </math> represents the 3N momenta.
where:


* <math> H \left(p,q\right) </math> represent the Hamiltonian, i.e. the total energy of the system as a function of coordinates and momenta.
*<math> \left. S \right. </math> is the [[Entropy|entropy]].


*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution|Dirac delta function]]
*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]


== References ==
== References ==
# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Alogrithms to Applications", Academic Press
<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]]

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:

  • 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.

Thermodynamics[edit]

where:

  • is the entropy.

References[edit]

Related reading