Ergodic properties of the quantum ideal gas in the Maxwell{endash}Boltzmann statistics
- Dipartimento di Matematica, Universita di Bologna, 40127 Bologna (Italy)
It is proved that the q uantization of the Volkovyski{endash}Sinai model of ideal gas (in the Maxwell{endash}Boltzmann statistics) enjoys at the thermodynamical limit the property of quantum mixing in the following sense: lim{sub {vert_bar}{ital t}{vert_bar}{r_arrow}{infinity}}lim{sub {sub {sub m}/{ital L}{r_arrow}{rho}}{sup {ital m},{ital L}{r_arrow}{infinity}}}{omega}{sub {beta} ,L}{sup {ital m}}({ital e}{sup {ital iH}{sub {ital m}}t/{h_bar}}{times}{ital Ae}{sup {minus}{ital iH}{sub {ital mt}}/ mC}{ital B})=lim{sub {sub {ital m}}}/{ital L}{r_arrow}{rho}{sup {ital m},{ital L}{r_arrow}{infinity}}{omega}{sub {beta},{ital L}}{sup {ital m}}( {ital A}){center_dot}lim{sub {sub {sub m}/{ital L}{r_arrow}{rho}}{sup {ital m},{ital L}{r_arrow}{infinity}}}{omega}{sub {beta},{ital L}}{sup {ital m}}({ital B} ). Here {ital H}{sub {ital m}} is the Schr{umlt o}dinger operator of {ital m} free particles moving on a circle of length {ital L}; {ital A} and {ital B} are the Weyl quantization of two classical observables {ital a} and {ital b}; {omega}{sup {ital m}}{sub {beta},{ital L}}({ital A}) is the corresponding quantum Gibbs state. Moreover, one has lim{sub {sub {ital m}}}/{ital L}{r_arrow}{rho}{sup {ital m},{ital L}{r_arrow}{infinity}}{omega}{sub {beta},{ital m}}({ital A})={ital P}{sub {rho},{beta} ({ital a})}, where {ital P}{sub {rho},{beta}}({ital a}) is the classical Gibbs measure. The consequent notion of quantum ergodicity is also independently proven. {copyright} {ital 1996 American Institute of Physics.}
- OSTI ID:
- 387538
- Journal Information:
- Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 37; ISSN JMAPAQ; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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