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 grand_canonical_expectation_value [2013/10/13 16:22]nikolaj grand_canonical_expectation_value [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/10/13 16:22 nikolaj 2013/10/13 16:22 nikolaj 2013/10/13 16:18 nikolaj 2013/10/13 15:14 nikolaj 2013/10/13 05:24 nikolaj 2013/10/13 05:23 nikolaj 2013/10/13 02:40 nikolaj 2013/10/13 02:39 nikolaj 2013/10/13 00:57 nikolaj 2013/10/13 00:56 nikolaj 2013/10/13 00:55 nikolaj 2013/10/13 00:51 nikolaj 2013/10/13 00:50 nikolaj 2013/10/13 00:43 nikolaj 2013/10/13 00:43 nikolaj 2013/10/13 00:40 nikolaj 2013/10/13 00:38 nikolaj 2013/10/13 00:32 nikolaj 2013/10/13 00:23 nikolaj old revision restored (2013/09/12 21:30) Next revision Previous revision 2013/10/13 16:22 nikolaj 2013/10/13 16:22 nikolaj 2013/10/13 16:18 nikolaj 2013/10/13 15:14 nikolaj 2013/10/13 05:24 nikolaj 2013/10/13 05:23 nikolaj 2013/10/13 02:40 nikolaj 2013/10/13 02:39 nikolaj 2013/10/13 00:57 nikolaj 2013/10/13 00:56 nikolaj 2013/10/13 00:55 nikolaj 2013/10/13 00:51 nikolaj 2013/10/13 00:50 nikolaj 2013/10/13 00:43 nikolaj 2013/10/13 00:43 nikolaj 2013/10/13 00:40 nikolaj 2013/10/13 00:38 nikolaj 2013/10/13 00:32 nikolaj 2013/10/13 00:23 nikolaj old revision restored (2013/09/12 21:30) Line 1: Line 1: ===== 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. Line 10: Line 10: 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 $H_N$. 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 $H_N$. - We also extend ​function ​of classical canonical observables to such sequences. I.e. if $A$ has $A_N$, then $f(A)$ has entries $f(A_N)$. + 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]]