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grand_canonical_expectation_value [2013/10/13 05:24] nikolaj |
grand_canonical_expectation_value [2013/10/13 16:22] nikolaj |
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| ==== Discussion ==== | ==== Discussion ==== | ||
| - | We adopt the names of observables in canonical ensamble for the grand canonical ensamble. For example, if the internal energy in the canonical ensamble is defined as $U=\langle H\rangle$, then the grand canonical expectation value of the energy is denoted by $U$ as well and if formed from the sequence of Hamiltonians $H_N$. | + | We adopt the names of observables in canonical ensamble for the grand canonical ensamble. For example, if the internal energy in the canonical ensamble is defined as $U=\langle H\rangle$, then the grand canonical expectation value of the energy is denoted by $U$ as well and if formed from the sequence of all the $N$-particle Hamiltonians $H_N$. |
| - | Now we also introduce the sequence $\hat N$ which gives us the particle number of each canonical ensamble, i.e. $\langle\hat N_N\rangle_N=N$. Then the expected particle number of the grand canonical ensemble is given by $\langle\hat N\rangle=\sum_{N=0}^\infty w_N\ N$. (Notice that $\hat N$ is sometimes denoted by $N$, which can get a little confusing.) | + | We also extend functions $f$ of classical canonical observables to such sequences. I.e. if $A$ has $A_N$, then $f(A)$ has entries $f(A_N)$. |
| ==== Parents ==== | ==== Parents ==== | ||
| === Requirements === | === Requirements === | ||
| [[Grand canonical weight]] | [[Grand canonical weight]] | ||
