Editing Grand canonical Monte Carlo
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probability), while <math>f(o)</math> is the probability density of state <math>o</math>. | probability), while <math>f(o)</math> is the probability density of state <math>o</math>. | ||
As noted by Norman and Filinov <ref>G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)</ref>, evaluation of the proper acceptance rules requires very | As noted by Norman and Filinov <ref>G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)</ref>, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) grand canonical probability density: | ||
:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math> | :<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math> | ||
where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the [[De | where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]]. | ||
The sub-index ''L'' makes emphasis on a particular definition of [[microstate]] in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by: | The sub-index ''L'' makes emphasis on a particular definition of [[microstate]] in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by: | ||
:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P | :<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P | ||
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math> | |||
where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in | where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in parentheis indicate that the positions are not atributted to any particular choice of labelling and the sum runs over all posible particle label permutations. | ||
Upon trial insertion of an extra particle, one obtains: | Upon trial insertion of an extra particle, one obtains: | ||
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The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal. | The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal. | ||
== References == | == References == | ||
<references/> | <references/> |