Thermodynamic properties of harmonic oscillator system in noncommutative phase space

In the last 15 years, noncommutative effects have received much attention and have been extensively studied in the fields of quantum mechanics, field theory, condensed matter physics, and astrophysics. The aim of this paper is to investigate the thermodynamic properties of a harmonic oscillator system in noncommutative phase space. For an example, the effects of noncommutativity between positions and that between momenta in the phase space on the thermodynamic properties of two-and three-dimensional harmonic oscillator system are studied by a statistical method. First, in the commutative phase space, the thermodynamic state functions are obtained from the partition functions of the harmonic oscillator system which satisfies Boltzmann statistics. Then, in the noncomummutative phase space, both noncommutative positions and noncommutative momenta are represented in terms of the commutative positions and momenta of the usual quantum mechanics by linear transformation method. Meanwhile, the other physical quantities such as the volume element, the number of microstates, and partition function in the noncommutative phase space are represented in terms of commutative positions and momenta. Finally, the thermodynamic and statistical state functions for the system in the noncommutative phase space are derived from the partition function, and the thermodynamic state functions in noncummutative and commutative phase spaces are compared with each other. The results show that the noncommutative effect changes the values of microscopic functions such as the partition function and entropy with the correction terms including noncummutative parameters. As the noncommutative parameters vanishes, i.e., reaches the commutative limit, the partition and entropy functions of the system coincide with the results of usual thermodynamics and statistical physics. Moreover, the macroscopic state functions such as the internal energy and heat capacity, remain constant. The results imply that the correction terms in the partition function and entropy may result from the corrections of the number of microstates and potential energy of the system by noncommutativity of the position and momentum. In conclusion, the method used in the paper is corresponding to the classical system that satisfies Boltzmann statistics, and the results derived here can provide a starting point for further studying the quantum system that satisfies Fermi-Dirac and Bose-Einstein statistics.