1-dimensional Ising model: Difference between revisions
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Consider a system with <math> N </math> spins in a row. The energy of the system will be given by | |||
Consider a system with <math> N </math> spins in a row. | |||
The energy of the system will be given by | |||
:<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>, | :<math> U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} </math>, | ||
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where each variable <math> S_j </math> can be either -1 or +1. | where each variable <math> S_j </math> can be either -1 or +1. | ||
The partition function of the system will be: | 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>, | :<math> Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]</math>, | ||
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:<math> Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N </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 | The [[Helmholtz energy function]] in the [[thermodynamic limit]] will be | ||
:<math> A = - N k_B T \log \left( 2 \cosh K \right) </math> | :<math> A = - N k_B T \log \left( 2 \cosh K \right) </math> | ||
==References== | |||
[[Category: Models]] | [[Category: Models]] | ||
Revision as of 12:08, 28 May 2007
Consider a system with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N } spins in a row. The energy of the system will be given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = -J \sum_{i=1}^{N-1} S_{i} S_{i+1} } ,
where each variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_j } can be either -1 or +1.
The partition function of the system will be:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_N = \sum_{\Omega^N } \exp \left[ K \sum_{i=1}^{N-1} S_i S_{i+1} \right]} ,
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega^N }
represents the possible configuration of the N spins of the system,
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K = J/k_B T }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} } }
Performing the sum of the possible values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{N} } we get:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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-2}} e^{K S_{N-2} S_{N-1}} \left[ 2 \cosh ( K S_{N-1} ) \right] }
Taking into account that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cosh(K) = \cosh(-K) }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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] }
Therefore:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_{N} = \left( 2 \cosh K \right) Q_{N-1} }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_N = 2^{N} \left( \cosh K \right)^{N-1} \approx ( 2 \cosh K )^N }
The Helmholtz energy function in the thermodynamic limit will be
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = - N k_B T \log \left( 2 \cosh K \right) }