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Ensemble average (Read 3655 times)
Gerrit-Jan Linker
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Ensemble average
30.05.10 at 13:21:22
 
Ensemble average
 
An ensemble is a large number of replications of a system, e.g. all configurations calculated in a simulation.
 
For a closed system, the ensemble average of some property X of the system is:
 
<X(t)> = ∑s X(s)P(s,t) / ∑sP(s,t)
 
or in integral notation:
 
<X(t)> = ∫XP(s,t) ds / ∫ P(s,t) ds
 
where,
<X> is the average of property X
the sum is over all possible accessible states s
X(s) is the value of the property in state s
P(s) is the probability of finding the system in state s (see Partition function http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1277736407)
 
For a system with a large number of degrees of freedom, the differences of actual values and the expectationvalue are usually quite small. Further, under equilibrium conditions, P is not a function of t so that X is independent of time and that the expectation value is a time average:
 
<X> = 0T X(t)dt / T
 
See also:
Ensemble
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1277718050
Partition function
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1277736407/0#0
 
Source:
Kittel, Thermal Physics
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« Last Edit: 28.06.10 at 17:04:55 by Gerrit-Jan Linker »  

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