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Partition function (Read 5794 times)
Gerrit-Jan Linker
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Partition function
28.06.10 at 16:46:47
 
Partition function
 
In statistical mechanics, the partition function Z encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives.
 
The partition function is a function of the temperature T and the microstate energies E1, E2, E3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
 
The canonical partition function is denoted by the letter Z (Zustandssumme; "sum over states"):
Z = s=1N exp(-βEs) ; exp(-βEs) is called the Bolzmann factor
β=1/kBT; β is the inverse temperature
 
The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability Ps that the system occupies microstate s is
Ps = (1/Z) exp(-βEs)
 
s Ps = 1 ; this is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies.  
 
See also
Microstate, Macrostate
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1296416531/0#0
 
Source:
http://en.wikipedia.org/wiki/Partition_function_%28statistical_mechanics%29
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« Last Edit: 30.01.11 at 20:43:14 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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Gerrit-Jan Linker
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Vibrational Partition function
Reply #1 - 08.02.12 at 07:19:58
 
Vibrational Partition function
 
The vibrational partition function traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system.  
 
The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom.
 
Approximations: Quantum Harmonic Oscillator
 
Vibrational eigenmodes or vibrational normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. It is a first order approximation to the partition function.
It allows one to calculate the contribution of the vibrational degree of freedom of molecules towards thermodynamic variables. A quantum harmonic oscillator has an energy spectrum characterized by:
 
Eji = ħωj(i+)
 
where j is an index representing the vibrational mode, and i is the quantum number for each energy level of the jth vibrational mode. The vibrational partition function is then calculated as:
 
Qvib = ∏ji e^(-Eji / kT )
 
See also:
Vibrational temperature
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1328681507/0#0
Zero point vibration / zero point energy
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1237471881
 
Source:
http://en.wikipedia.org/wiki/Vibrational_partition_function
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« Last Edit: 08.02.12 at 07:21:55 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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