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2000年以来, 有关非对易空间的各种物理问题一直是研究的热点, 并在量子力学、场论、凝聚态物理、天体物理等各领域中已被广泛地探讨. 采用统计物理方法讨论非对易效应对谐振子体系热力学性质的影响. 先以对易相空间中确定二维和三维谐振子的配分函数求出谐振子体系的热力学函数; 非对易相空间中的坐标和动量通过坐标-坐标和动量-动量之间的线性变换而以对易相空间中的坐标和动量来表示; 最终以非对易相空间中求出配分函数来讨论非对易效应对谐振子体系热力学性质的影响. 结果显示, 在非对易相空间中谐振子体系的配分函数和熵表达式均包含因非对易引起的修正项. 从分析结果得出如下结论: 非对易效应对谐振子的配分函数和熵函数等微观状态函数有一定的影响, 但对谐振子体系的内能、热容量等宏观热力学函数没有影响. 研究结果只是对应于满足玻尔兹曼统计的经典体系, 对于满足费米-狄拉克和玻色-爱因斯坦统计的量子体系需进一步推广研究.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 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.
[1] Snyder H 1947 Phys. Rev. 71 38
[2] Banks T, Fischler W, Shenker S H, Susskind L 1997 Phys. Rev. D 55 5112
[3] Anwar A, Dulat S 2012 J. Xinjiang Univ. (Nat. Sci. Ed.) 29 448 (in Chinese) [阿布都外力·艾尼瓦尔, 沙依甫加马力·达吾来提 2012 新疆大学学报(自然科学版) 29 448]
[4] Dulat S, Li K 2009 Eur. Phys. J. C 60 163
[5] Ma K, Dulat S 2011 Phys. Rev. A 84 012104
[6] Masum H, Dulat S, Ma K 2012 J. Xinjiang Univ. (Nat. Sci. Edi) 29 318 (in Chinese) [玉苏音·买苏木, 沙依甫加马力·达吾来提, 马凯 2012 新疆大学学报(自然科学版) 29 318]
[7] Li K, Chamoun N 2006 Chin. Phys. Lett. 23 1122
[8] Yakup R, Dulat S, and Obulkasim A 2012 Coll. Phys. 31 1 (in Chinese) [热依木阿吉·亚克甫, 沙依甫加马力·达吾来提, 阿斯叶古丽·吾布力卡丝木 2012 大学物理 31 1]
[9] Luo Y H, Ge Z M 2006 Commun. Theor. Phys. 46 967
[10] Zhang X L, Liu H, Yu H J, Zhang W H 2011 Acta Phys. Sin. 60 040303 (in Chinese) [张秀兰, 刘恒, 余海军, 张文海 2011 物理学报 60 040303]
[11] Wei G F, Long C Y, Long Z W, Qin S J, Fu Q 2008 Chin. Phys. C 32 338
[12] Sun Y Q, Long S M, Huang C J, Zhang K 2008 J. Sichuan Nor. Univ. (Nat. Sci. Ed.) 31 342 (in Chinese) [孙彦清,龙姝明,黄朝军,张锴 2008 四川师范大学学报(自然科学版) 31 342]
[13] Mamat M, Dulat S, Wupur Y 2014 Coll. Phys. 313 11 (in Chinese) [买买提热夏提·买买提, 沙依甫加马力·达吾来提, 亚森江·吾普尔, 买买吐尔逊·巴卡吉 2014 大学物理 33 11]
[14] Zhou S W, Liu W B 2007 Acta Phys. Sin. 56 6767 (in Chinese) [周史薇, 刘文彪 2007 物理学报 56 6767]
[15] Huang J H, Sheng Z M 2010 Chin. Phys. B 19 010316
[16] Bastos C, Bernardini A E, Bertolami O, Dias N C, Prata J N 2014 Phys. Rev. D 90 045023
[17] Samary D O 2014 Int. J. Math. Anal. 8 1285
[18] Panella O, Roy P 2014 Phys. Rev. A 90 042111
[19] Santos V, Maluf R V, Almeida C A S 2014 Ann. Phys. 349 402
[20] Han Y W, Hong Y 2014 Chin. Phy. B 23 100401
[21] Belhaj A, Chabab M, Moumni H E, Sedra M B 2013 Afr. Rev. Phys. 8 105
[22] Liang J, Liu Y C, Zhu Q 2014 Chin. Phys. C 38 025101
[23] Wang Z C 2010 Thermodynamics and Statistical Physics (Beijing: Higher Education Press) p190 (in Chinese) [王志诚 2010 热力学·统计物理(北京: 高等教育出版社) 第190页]
[24] Li K, Wang J H, Chen C Y 2005 Mod. Phys. Lett. A 20 2165
[25] Seiberg N, Witten E 1994 Nucl. Phys. B 426 19
[26] Wang J H, Li K, Liu P 2006 HEP & NP 30 387 (in Chinese) [王剑华, 李康, 刘鹏 2006 高能物理与核物理 30 387]
[27] Bertolami O, Rosa J G, de Aragao C M L, Castorina P, Zappala D 2005 Phys. Rev. D 72 025010
[28] Mojtaba N, Mehdi S 2013 Chin. J. Phys. 51 94
[29] Chaichian M, Sheikh Jabbari M M, Tureanu A 2001 Phys. Rev. Lett. 86 2716
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[1] Snyder H 1947 Phys. Rev. 71 38
[2] Banks T, Fischler W, Shenker S H, Susskind L 1997 Phys. Rev. D 55 5112
[3] Anwar A, Dulat S 2012 J. Xinjiang Univ. (Nat. Sci. Ed.) 29 448 (in Chinese) [阿布都外力·艾尼瓦尔, 沙依甫加马力·达吾来提 2012 新疆大学学报(自然科学版) 29 448]
[4] Dulat S, Li K 2009 Eur. Phys. J. C 60 163
[5] Ma K, Dulat S 2011 Phys. Rev. A 84 012104
[6] Masum H, Dulat S, Ma K 2012 J. Xinjiang Univ. (Nat. Sci. Edi) 29 318 (in Chinese) [玉苏音·买苏木, 沙依甫加马力·达吾来提, 马凯 2012 新疆大学学报(自然科学版) 29 318]
[7] Li K, Chamoun N 2006 Chin. Phys. Lett. 23 1122
[8] Yakup R, Dulat S, and Obulkasim A 2012 Coll. Phys. 31 1 (in Chinese) [热依木阿吉·亚克甫, 沙依甫加马力·达吾来提, 阿斯叶古丽·吾布力卡丝木 2012 大学物理 31 1]
[9] Luo Y H, Ge Z M 2006 Commun. Theor. Phys. 46 967
[10] Zhang X L, Liu H, Yu H J, Zhang W H 2011 Acta Phys. Sin. 60 040303 (in Chinese) [张秀兰, 刘恒, 余海军, 张文海 2011 物理学报 60 040303]
[11] Wei G F, Long C Y, Long Z W, Qin S J, Fu Q 2008 Chin. Phys. C 32 338
[12] Sun Y Q, Long S M, Huang C J, Zhang K 2008 J. Sichuan Nor. Univ. (Nat. Sci. Ed.) 31 342 (in Chinese) [孙彦清,龙姝明,黄朝军,张锴 2008 四川师范大学学报(自然科学版) 31 342]
[13] Mamat M, Dulat S, Wupur Y 2014 Coll. Phys. 313 11 (in Chinese) [买买提热夏提·买买提, 沙依甫加马力·达吾来提, 亚森江·吾普尔, 买买吐尔逊·巴卡吉 2014 大学物理 33 11]
[14] Zhou S W, Liu W B 2007 Acta Phys. Sin. 56 6767 (in Chinese) [周史薇, 刘文彪 2007 物理学报 56 6767]
[15] Huang J H, Sheng Z M 2010 Chin. Phys. B 19 010316
[16] Bastos C, Bernardini A E, Bertolami O, Dias N C, Prata J N 2014 Phys. Rev. D 90 045023
[17] Samary D O 2014 Int. J. Math. Anal. 8 1285
[18] Panella O, Roy P 2014 Phys. Rev. A 90 042111
[19] Santos V, Maluf R V, Almeida C A S 2014 Ann. Phys. 349 402
[20] Han Y W, Hong Y 2014 Chin. Phy. B 23 100401
[21] Belhaj A, Chabab M, Moumni H E, Sedra M B 2013 Afr. Rev. Phys. 8 105
[22] Liang J, Liu Y C, Zhu Q 2014 Chin. Phys. C 38 025101
[23] Wang Z C 2010 Thermodynamics and Statistical Physics (Beijing: Higher Education Press) p190 (in Chinese) [王志诚 2010 热力学·统计物理(北京: 高等教育出版社) 第190页]
[24] Li K, Wang J H, Chen C Y 2005 Mod. Phys. Lett. A 20 2165
[25] Seiberg N, Witten E 1994 Nucl. Phys. B 426 19
[26] Wang J H, Li K, Liu P 2006 HEP & NP 30 387 (in Chinese) [王剑华, 李康, 刘鹏 2006 高能物理与核物理 30 387]
[27] Bertolami O, Rosa J G, de Aragao C M L, Castorina P, Zappala D 2005 Phys. Rev. D 72 025010
[28] Mojtaba N, Mehdi S 2013 Chin. J. Phys. 51 94
[29] Chaichian M, Sheikh Jabbari M M, Tureanu A 2001 Phys. Rev. Lett. 86 2716
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