4.2 Canonical ensemble
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University of Vienna
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[PDF]Handout 8. Canonical Ensemble - Stanford University
micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf
[PDF]Handout 5. Microcanonical Ensemble - Stanford University
micro.stanford.edu/~caiwei/.../Chap5_Microcanonical_Ensemble_v04.pd...
Why uniform ensemble and microcanonical ensemble are ...
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Stack Exchange
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On completeness of description of an equilibrium canonical ...
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ResearchGate
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https://books.google.com/books?isbn=3319003275
Wolfgang Paul, Jörg Baschnagel - 2013 - Mathematics
... 211 Weierstrass random walk, 137, 138 Sharpe ratio, 205 Short position, ... 20 kinetic theory of gases, 2 microcanonical ensemble, 27 partition function, 20, 28 ...[PDF]staff teaching assignment - The University of Hong Kong
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University of Hong Kong
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[PDF]Canonical Ensemble: the Boltzmann Factor
www.cmp.caltech.edu/~mcc/BNU/Notes2_3.pdf
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[PDF]Ensembles - UMD
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University of Maryland, College Park
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[PDF]Microcanonical Simulation of Complex Actions: The Wess ...
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arXiv
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by AL Jonsson - 2009 - Cited by 16 - Related articles
(41) with the microcanonical ensemble (NVE, or constant number of particles, volume, ... S3° is the ratio of number of contacts made in the TS structures to the ..... Sharpe T., Jonsson A.L., Rutherford T.J., Daggett V., Fersht A.R. The role of the ...The Effect of Context on the Folding of β-Hairpins
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by AL Jonsson - 2011 - Cited by 1 - Related articles
Aug 6, 2011 - A ratio of the effect on the rate versus stability generally take values between 0 ... WW domain simulations (Petrovich et al., 2006; Sharpe et al., 2007), a new 298 ... The microcanonical ensemble (constant number of particles, ...Entropy
Entropy is a key concept in both thermodynamics and statistical mechanics, as well as in
information theory (a measure of uncertainty or lack of information). In information theory,
if an experiment has N possible outcomes with equal probability, then the entropy is
S = kB logN (35)
In microcanonical ensemble,
S(N; V;E) = kB log
2
4
number of microscopic states between
the constant energy surfaces:
E H( ) E + E
3
5 (36)
For an ideal gas,
S(N; V;E) = kB log
~
(E + E) ~
(E)
N! h3N (37)
The numerator inside the log is the volume of the phase space between the two constant
energy surfaces. h is Planck's constant, which is the fundamental constant from quantum
mechanics.
Yes, even though we only discuss classical equilibrium statistical mechanics, a bare minimum
of quantum mechanical concepts is required to x some problems in classical mechanics.
We can view this as another evidence that classical mechanics is really just an approximation
and quantum mechanics is a more accurate description of our physical world. Fortunately,
these two terms can be intuitively understandable without working with quantum mechanics
equations. The following are the justi cations of the two terms in the denominator.
1. N! term: Quantum mechanics says that the gas molecules are all identical or indistinguishable.
Even though we would like to label molecules as 1; 2; :::;N there is really
no way for us to tell which one is which! Therefore, two molecular structures with
coordinates: x1 = (1; 2; 3), x2 = (4; 5; 6) and x1 = (4; 5; 6), x2 = (1; 2; 3) are indistinguishable
from each other.
Swapping the location between two molecules does not give a new microscopic state.
2. h3N term: h = 6:626 1034 J s is Planck's constant.
The numerator, ~
(E), is the phase space volume and has the unit of (momentum
distance)3N.
The term inside log has to be dimensionless, otherwise, the magnitude of entropy would
depend on our choices for the units of length, time, mass, and etc, which would be
clearly absurd.
h has the unit of momentum distance. Therefore h3N has exactly the right unit to
make the entire term inside the log dimensionless.
13
The uncertainty principle in quantum mechanics states that we cannot measure both
the position and the momentum of any particle to in nite accuracy. Instead, their
error bar must satisfy the relation:
qi pi h for any i = 1; ; 3N (38)
Therefore, ~
(E)=(N! h3N) gives us the number of distinguishable states contained inside
a phase space volume of
(~E).
We can show that the entropy expression for the ideal gas in microcanonical ensemble is
S(N; V;E) = NkB
"
log
V
N
4 mE
3Nh2
3=2
!
+
5
2
#
(Sackur-Tetrode formula) (39)
We will derive the Sackur-Tetrode formula later. (Stirling's formula is used to derive it.)
De ne number density = N
V , and de Broglie wavelength
=
h p
4 mE=3N
=
h
p
2 mkBT
(40)
then
S(N; V;E) = NkB
5
2
log( 3)
(41)
In molecular simulations, the microcanonical ensemble is usually referred to as the
NV E ensemble.
the position and the momentum of any particle to in nite accuracy. Instead, their
error bar must satisfy the relation:
qi pi h for any i = 1; ; 3N (38)
Therefore, ~
(E)=(N! h3N) gives us the number of distinguishable states contained inside
a phase space volume of
(~E).
We can show that the entropy expression for the ideal gas in microcanonical ensemble is
S(N; V;E) = NkB
"
log
V
N
4 mE
3Nh2
3=2
!
+
5
2
#
(Sackur-Tetrode formula) (39)
We will derive the Sackur-Tetrode formula later. (Stirling's formula is used to derive it.)
De ne number density = N
V , and de Broglie wavelength
=
h p
4 mE=3N
=
h
p
2 mkBT
(40)
then
S(N; V;E) = NkB
5
2
log( 3)
(41)
In molecular simulations, the microcanonical ensemble is usually referred to as the
NV E ensemble.
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