当系统含有[tex=0.857x1.0]HcQeTeQtUqN73yUJqDRZkQ==[/tex]个玻色子，[tex=1.143x1.143]apbWnliuup7Dz3/V3Pvpyg==[/tex]个费米子，总能量为[tex=0.786x1.0]XvHgf70VtK2FH5G93l0k3g==[/tex]，体积为[tex=0.643x1.0]SW0o8G0GHsmLXldwnq7xKg==[/tex]时，粒子的分布[tex=1.786x1.357]Ge5fo4gkj7anlWAj5xWX4uJ33ZnyRBCKeq+piYDdR+o=[/tex]和[tex=1.786x1.429]BJeJaB2M3wORRIjEqCnX716mwRS3fFRPhwGUJYfnM2U=[/tex]必须满足条件[tex=8.571x8.643]Uhu2/1bhRl3dw2iQmc0iyX6b3TwKYp2i4uiaWmx5hYUJeo7u0KeQ5eC9lVqjnUYYgIqXVNB+SHWmIyfQKnSVGGahMYe0hULsENN3YXgXhUNkgWPdt9yTNZTZhN625NDr01fQE3HPC02w/piwUeynjP9sIHiUPD9a3vQ6e2IqH0iUGI8pS0B9LzVvFbW0eoXs+TceORTuFiBvVVyfoeVrejf9JKXcmrYzxK5n4iMPN1s=[/tex]（1）才有可能实现.玻色子处在分布[tex=1.786x1.357]Ge5fo4gkj7anlWAj5xWX4uJ33ZnyRBCKeq+piYDdR+o=[/tex] ，费米子处在分布[tex=1.786x1.429]BJeJaB2M3wORRIjEqCnX716mwRS3fFRPhwGUJYfnM2U=[/tex] 时，其微观状态数分别为[tex=8.429x6.071]djZnQh2W/VbJAxy3qceBGEorZp30pXC44DzXaaWU79T9a6GsGu845VYAH++Cr8rUY6SattxBA+guI97Ma8nesZsPX+xLoLEx6feyCtFT1Rz/3C+fmn22N03hXxwFpCiiP/Dsk6djUAup7rU5YBzXI4bQnvxnTx3K4vJJFRc/6rtNYq+6drKsbWXj/AgCyRGAsrTfgTHKrG6rv5xRAKotDndbTQjKLjEZ/v+N/OzAd+e/Ik1nkYzrIbRFg6aGyFsv6Rt4WURmz5E5Bb8hvnv0wgOAwy8IHjlVod51ZAqxvDE=[/tex]系统的微观状态数[tex=1.714x1.286]ec1ynY+XKsExF48WR29f/Q==[/tex]为[tex=4.429x1.286]agsVjskzaUaUYTQzUgJz5fMgqbFm5F+5JDzghCKIMiSaLB4oQNhYiabdx7s/9KM8[/tex]（3）平衡状态下系统的最概然分布是在满足式（1）条件下使[tex=1.714x1.286]ec1ynY+XKsExF48WR29f/Q==[/tex]或[tex=2.714x1.286]Bs1DrPXXycqyMbYkFgaf6GojFQBCqQtZvADxXcAkAOc=[/tex]为极大的分布. 将式（2）和式（3）取对数，利用斯特令公式可得[tex=38.571x2.714]Bs1DrPXXycqyMbYkFgaf6HFGYxYfXwoV8g2QVgq2knRiOTX7MpsVn7bSjQFRlIzXNiE9Lpg9pOiYsvdg5huXQOuXrQceMdPe6FrH4DMg01Bic8ue9b0gGuapbpLCEFLFC2joZ+CmItpoKYRiOodknAY/KRrmYBXfmP7kY/w/z92+z221htb4rGjkxkGzdwxv4GjFYpn9+NGPEcoOTqUqBGSoXFSbT6rLOOTgPO34pXUS7irNE7UO/TN1WJzBhLUMEU58VjUl0W3f3ikyEPrdruELRDo7kHnrLn+XGjHqiotA4rCBEAVTWGW6JAFQVvh+ufeG2cEgwxGSDNGr9eIYKPHTN5LOmgGISk6VEEEWLGIdsqYmYmMETL+jQMhX2mNwTogBiO/TFmfhwYbCFTBlNgui3ToPJJr2DWb9hyqhjvPtTa1rB/Lo9uia59FELPVEj+2uGWUKH1rYMn1DWUJJvw==[/tex]令各[tex=0.786x1.0]kCsj31iX1lO8IT30KSoDQw==[/tex]和[tex=0.786x1.429]qhTMWQqsSGGgiSArbcKa+w==[/tex]有[tex=1.214x1.214]wm3EUIURT9vwiiAXx6hjLg==[/tex] 和[tex=1.214x1.429]f+s4dnzfiIbM9ZWFx+8MAibXMugo0EacncJUl2alGYA=[/tex]的变化，[tex=2.714x1.286]Bs1DrPXXycqyMbYkFgaf6GojFQBCqQtZvADxXcAkAOc=[/tex]将因而有[tex=3.286x1.286]+hyTI2SgACgb5spvrf/7PE2nZXvMcYIwfExFGA+NZCs=[/tex]的变化，使用权[tex=2.714x1.286]Bs1DrPXXycqyMbYkFgaf6GojFQBCqQtZvADxXcAkAOc=[/tex]为极大的分布[tex=1.786x1.357]Ge5fo4gkj7anlWAj5xWX4uJ33ZnyRBCKeq+piYDdR+o=[/tex]和[tex=1.786x1.429]BJeJaB2M3wORRIjEqCnX716mwRS3fFRPhwGUJYfnM2U=[/tex]必使[tex=4.571x1.286]+hyTI2SgACgb5spvrf/7PJPiZDKGH3edlQJ2VdjRwUs=[/tex]即[tex=21.214x3.0]+hyTI2SgACgb5spvrf/7PATe29GHx2tVlQLCIpSmu+C7yMUNAD/1aLxehCpB7qcWQ7TkHPIcAcsgZBOcXCTTPZIq8Wwn+OdiTx3HeaQaZKh/N2flQYZGvWAb0pI5ttAGqZkDYkz87pOBezbIDg5/BXpVTLGwFLf0nsfaL+gJClQR5mRQTjzr9aAwiIxbRKai7p85Vw0VVdSfFUN6c6VURAidTtgSM9KVh9ib0QPmMkPZm5olAXsY3+1SJsM8rg2y49FYwl1CqA0QFUp5pwnCug==[/tex]但这此致[tex=1.214x1.214]wm3EUIURT9vwiiAXx6hjLg==[/tex]和[tex=1.214x1.429]f+s4dnzfiIbM9ZWFx+8MAibXMugo0EacncJUl2alGYA=[/tex]不完全是独立的，它们必须满足条件[tex=11.357x8.286]m5QOqfmGUNhj1/OyRTqRQRvWxvhiRwipfKTO8/73Vn90JEpM73x56hakfnzbaV/KL3uV8Na4n1Cnb51mPOcfHKW5rkigkywLzjbVzEHlzGjTWjBlnFUvMdhxgWpz6vvMrEoi3Sf4eF2F5vDAaz8PBDWHqKe6kOIMB0mIPf0fu/XyF26QNggnQ0/5v30BkyY6msYFTHQsN/GAcOGh5y7TNYiLK1L0R7xC36KTd6x1NDsFsQD8LF3ymqkt8cUUXrduiQ75dJkjsGpAyPTHhld9Rg==[/tex]用拉氏乘子[tex=0.643x0.786]SPoVA3bJlgfP9Ek9O4AbuA==[/tex] ，[tex=0.929x1.143]fIk50ncZ+Y/yRB3Y06DyEg==[/tex]和[tex=0.571x1.214]DXE2qJe9QayJDT2HOCKrUg==[/tex]分别乘这三个式子并从[tex=3.286x1.286]+hyTI2SgACgb5spvrf/7PE2nZXvMcYIwfExFGA+NZCs=[/tex]中减去，得[tex=40.786x3.0]+hyTI2SgACgb5spvrf/7PPzWQNnwJaqzghYyiyCQYT3ci1Q3iqLLrk6eHsxjpa2zNEc7ttsfPf0M8BCAh8lqf6hXdYhToTFL7l5to05aJjnUceo7WhE1gIJuDjpgFIpZTx/tlnkFiOVB4/KMQpWBXWz0MN/cTcNzMJg4aE571yhj/HjVaHgHt/iMWXceTKS6lMjQt3TivcsvtYoYjmf4p2+N6jlSbKhnV19BUBjrzTM/eg3xaxWGccG/zd9u1YPTKJAX3M4Nz1SqtJTSgIJuSwlzC/E1znyINEHbOUnnnNZFI6gVJ9C3YmcuqBoVebKjD9YYebbaA/5OoJ+IekDPDKYf+vrtlD+3VV8YGVpuB89/HuIKSQN1+V06Uemo+0KXjtWB53rUDNYBVO8wsEzpayDPomrJ22Rq0tsPU0U9QJfkGgsJZWJzk5APVl5tcBcrvLtIeen06Q8bSZFuAUQJ+D+G8MbTNS1/QOk04Vig3gM=[/tex]根据拉氏乘子法原理，每个[tex=1.214x1.214]wm3EUIURT9vwiiAXx6hjLg==[/tex]和[tex=1.214x1.429]f+s4dnzfiIbM9ZWFx+8MAibXMugo0EacncJUl2alGYA=[/tex]的系数都等于零，所以得[tex=8.857x5.286]NJTJfFcpj6dI9kAEx0FYEj1BWWJLJzTIqFanILD/e0R1YHfdhD3j8eCOuvtiWoW+ej/6rFMPSuTi7aapSWEUM0+gQuA0gKP+n5ULmIQNgEVFX7lGdkiqMc+XN5qxawnxd9bQ6iu2eTWZAap48YhhiyWc2c2wVMSCzCwHZ5HSxj5Q1xj+d6xoguAt2lt0I6MFsiqZtDHURl4IdrBHBF+Po4TeojUalx6sEH9pr1ZqAYSh06EvzgBwHEg+2V9r5IG6[/tex]即[tex=6.786x4.929]feEkz2fzZP9G20yNHe6f5EeRMtz9E+WdGwNbQzZ+G6y5paxVhXxh6rSO+u4HL5F+uHzq9a/5JlvdZCRFVq2D8lV26VO2pU1sTZttBOZKW29ktd0VoC09I7X3oHxwzt193Rf2nDlWW2VIz98phLFo4lhpkaeCp2Mu7eQspbl/BO/zKPV65mpWhCBPODGYHn2A+jsiKdSyNVmPmElv49Ega6XZyNz76khOupu4WLztLKo=[/tex]（4）拉氏乘子[tex=0.643x0.786]SPoVA3bJlgfP9Ek9O4AbuA==[/tex]， [tex=0.929x1.143]fIk50ncZ+Y/yRB3Y06DyEg==[/tex]和[tex=0.571x1.214]DXE2qJe9QayJDT2HOCKrUg==[/tex]由条件（1）确定. 式（4）表明，两种粒子分别遵从玻色分布和费米分布，其中[tex=0.643x0.786]SPoVA3bJlgfP9Ek9O4AbuA==[/tex]和[tex=0.929x1.143]fIk50ncZ+Y/yRB3Y06DyEg==[/tex]不同，但[tex=0.571x1.214]DXE2qJe9QayJDT2HOCKrUg==[/tex]相等。