Talk:Percolation analysis

2009-09-24T08:09:01Z

147.96.6.43: New page: Muy interesante. There must be something wrong with the energetic definition of bonds since max(0,exp(u)) = exp(u). Posiblemente b=min(1,exp(-u)) ? lgmac

Muy interesante.

There must be something wrong with the energetic definition of bonds
since max(0,exp(u)) = exp(u). Posiblemente b=min(1,exp(-u)) ?

lgmac

1-dimensional Ising model

2009-02-19T17:05:06Z

147.96.6.43: minor index change

The '''1-dimensional Ising model''' is an [[Ising Models| Ising model]] that consists of a system with <math> N </math> spins in a row. The energy of the system is given by

:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>,

where each variable <math> S_j </math> can be either -1 or +1.

The [[partition function]] of the system will be:

:<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>,

where <math> \Omega^N </math> represents the possible configuration of the N ''spins'' of the system,
and <math> K = J/k_B T </math>

:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}}\sum_{S_{N}} e^{K S_{N-1} S_{N} }
</math>

Performing the sum of the possible values of <math> S_{N} </math> we get:

:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right]
</math>

Taking into account that <math> \cosh(K) = \cosh(-K) </math>

:<math> Q_{N} = \sum_{S_1} \sum_{S_2} e^{K S_1S_2} \sum_{S_3} e^{K S_2 S_3} \cdots \sum_{S_{N-1}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K ) \right]
</math>

Therefore:

:<math> Q_{N} = \left( 2 \cosh K \right) Q_{N-1} </math>

:<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </math>

The [[Helmholtz energy function]] in the [[thermodynamic limit]] will be

:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math>
==References==
# Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 2 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])
[[Category: Models]]

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Talk:Percolation analysis

1-dimensional Ising model