| … | … | @@ -185,6 +185,67 @@ So here we go. |
| 185 | 185 | Ising Model |
| 186 | 186 | \end_layout |
| 187 | 187 | |
| 188 | \begin_layout Standard |
| 189 | The part of the Ising Model that is interesting here is the Hamiltonian. |
| 190 | It is a sum of pair–wise interactions of nearest neighbors: |
| 191 | \begin_inset Formula |
| 192 | \[ |
| 193 | \mathcal{H}=-\sum_{ij}J_{ij}\sigma_{i}\sigma_{j}-\mu\sum_{j}h_{j}\sigma_{j} |
| 194 | \] |
| 195 | |
| 196 | \end_inset |
| 197 | |
| 198 | with |
| 199 | \begin_inset Formula $\sigma_{i}$ |
| 200 | \end_inset |
| 201 | |
| 202 | the spins. |
| 203 | I won't explain this. |
| 204 | I copied it from Wikipedia. |
| 205 | Read that article if you don't know what this is. |
| 206 | \end_layout |
| 207 | |
| 208 | \begin_layout Section* |
| 209 | Link Grammar |
| 210 | \end_layout |
| 211 | |
| 212 | \begin_layout Standard |
| 213 | I won't explain what Link Grammar (LG) is, or how it works. |
| 214 | Read the docs. |
| 215 | However, I want to write down a a Hamiltonian for it, which is something |
| 216 | that the Link Grammar docs do not do. |
| 217 | It is analogous to the Ising Model Hamiltonian: |
| 218 | \begin_inset Formula |
| 219 | \[ |
| 220 | \mathcal{H}=-\sum_{ijk\cdots}C_{ijk\cdots}w_{i}w_{j}w_{k}\cdots |
| 221 | \] |
| 222 | |
| 223 | \end_inset |
| 224 | |
| 225 | The |
| 226 | \begin_inset Formula $w_{i}$ |
| 227 | \end_inset |
| 228 | |
| 229 | are the words in the sentence. |
| 230 | Written out here, in order to resemble the Ising Model. |
| 231 | The |
| 232 | \begin_inset Formula $C_{ijk\cdots}$ |
| 233 | \end_inset |
| 234 | |
| 235 | is meant to be the LG cost, a single real–valued floating point number, |
| 236 | of the the LG disjunct |
| 237 | \begin_inset Formula $ijk\cdots$ |
| 238 | \end_inset |
| 239 | |
| 240 | . |
| 241 | \end_layout |
| 242 | |
| 243 | \begin_layout Standard |
| 244 | The analogy here is strained, and the notation is inadequate. |
| 245 | This is a brainstorm idea, so we'll fix the problems here later. |
| 246 | If the idea pans out. |
| 247 | \end_layout |
| 248 | |
| 188 | 249 | \begin_layout Section* |
| 189 | 250 | The End |
| 190 | 251 | \end_layout |