Canonical versus grand canonical occupation numbers for simple systems
An ideal gas of N indistinguishable particles is described by a canonical ensemble (c.e.) and also by a grand canonical ensemble (g.c.e.) which has N as the mean total number of particles, the temperature and volume being the same in both cases. Exact mean occupation numbers n/sub u/((N) are found if the system has only two states 1 and 2 of energies E/sub 2/ greater than or equal to E/sub 1/. This should apply to quantum wells and similar simple systems. For systems which have captured one particle, the theory gives the simplest answers, and one finds a maximum discrepancy of 17% between the two ensembles for the fermion case. It occurs when E/sub 2/-E/sub 1/ approx. 53 meV at room temperature. For N = 1 the mean occupation number for the c.e. is identical for fermions and for bosons, being in both cases given by n/sub 2/(1) = /exp(E/sub 2/-E/sub 1/)/kT) + 1//sup -1/, n/sub 1/(1) = 1 - N/sub 2/(1). For large N one reverts to the usual situation and the discrepancy between the ensembles becomes small.
- Research Organization:
- Univ. of Florida, Gainesville (USA)
- OSTI ID:
- 6280409
- Journal Information:
- J. Stat. Phys.; (United States), Vol. 53:1-2
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
SUPERCONDUCTIVITY AND SUPERFLUIDITY
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOSONS
OCCUPATION NUMBER
FERMIONS
BOLTZMANN STATISTICS
ENERGY GAP
ENERGY LEVELS
ENTROPY
EQUILIBRIUM
GASES
POTENTIALS
QUANTUM MECHANICS
STATISTICAL MECHANICS
THERMODYNAMICS
FLUIDS
MECHANICS
PHYSICAL PROPERTIES
THERMODYNAMIC PROPERTIES
656002* - Condensed Matter Physics- General Techniques in Condensed Matter- (1987-)
657002 - Theoretical & Mathematical Physics- Classical & Quantum Mechanics