An Introduction to Monte Carlo Methods in Statistical Physics.
An Introduction to Monte Carlo Methods in Statistical Physics.
An Introduction to Monte Carlo Methods in Statistical Physics.
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Simple Sampl<strong>in</strong>g <strong>Monte</strong> <strong>Carlo</strong><br />
Simple Sampl<strong>in</strong>g <strong>Monte</strong> <strong>Carlo</strong><br />
The Partition function conta<strong>in</strong>s all <strong>in</strong>formation about a system<br />
Z =<br />
all<br />
∑<br />
e<br />
states<br />
−<br />
H /k<br />
T<br />
B ≈ e<br />
∑<br />
M states<br />
−<br />
H /k<br />
B<br />
T<br />
Example: N Is<strong>in</strong>g sp<strong>in</strong>s on a square lattice<br />
H<br />
= − J<br />
i<br />
∑<br />
,<br />
j<br />
σ<br />
i<br />
σ<br />
j<br />
, σ<br />
i<br />
= ± 1<br />
At low temperature, only two states contribute very much<br />
(i.e. all sp<strong>in</strong>s up or all sp<strong>in</strong>s down). Simple sampl<strong>in</strong>g is very<br />
<strong>in</strong>efficient s<strong>in</strong>ce it is very unlikely <strong>to</strong> generate these two<br />
states! Use importance sampl<strong>in</strong>g <strong>in</strong>stead.<br />
Remember, for an N=10,000 Is<strong>in</strong>g model Z has 2 10,000 terms!
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