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grand_canonical_expectation_value [2013/10/13 15:14] nikolaj |
grand_canonical_expectation_value [2014/03/21 11:11] (current) |
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| ===== Grand canonical expectation value === | ===== Grand canonical expectation value === | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ w $ ... grand canonical weight | | + | | @#55CCEE: context | @#55CCEE: $ w $ ... grand canonical weight | |
| - | | @#FFBB00: $\langle A\rangle:=\sum_{N=0}^\infty w_N\cdot \langle A_N\rangle_N$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\langle A\rangle:=\sum_{N=0}^\infty w_N\cdot \langle A_N\rangle_N$ | |
| The functional $\langle \cdot\rangle_N$ denotes the expectation in the canonical ensamble of particle number $N$. So the grand canonical expectation value $\langle \cdot\rangle$ takes sequences of observables to a real. | The functional $\langle \cdot\rangle_N$ denotes the expectation in the canonical ensamble of particle number $N$. So the grand canonical expectation value $\langle \cdot\rangle$ takes sequences of observables to a real. | ||
| ==== 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 $H_N$. |
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| + | 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 === | + | === Context === |
| [[Grand canonical weight]] | [[Grand canonical weight]] | ||
