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→Calculation methods
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.
一旦计算出一个系统的特征状态函数，该系统就“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全解决了，但是最一般的(和现实的)情况对于一个精确的解决方案来说太复杂了。存在各种方法来近似真实的总体，并允许计算平均量。
一旦计算出一个系统的特征状态函数，该系统就“解决”了(宏观观测量可以从特征状态函数中提取)。然而，计算热力学系综的特征状态函数并不一定是一项简单的工作，因为它涉及到考虑系统的每一种可能状态。虽然一些假设的系统已经被完全求解了，但是最一般的(和现实的)情况对于一个精确的解来说太复杂了。存在各种方法来近似真实的系综，并且计算平均量。
* A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few [[toy model]]s.<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].
* A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few [[toy model]]s.<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].
* 对于非常小的微观系统，可以通过简单地列举系统所有可能状态（利用量子力学中的严格对角化，或者经典力学中对所有相空间积分）来直接得到系综。
* 对于包含很多分离的微观系统的宏观系统，每个子系统可以单独分析。尤其是粒子间无相互作用的理想气体具有这种性质，从而可以精确地得到[麦克斯韦–玻尔兹曼统计]]， [[费米-狄拉克统计]]，和[[波色-爱因斯坦统计]]。<ref name="tolman"/>
* 某些存在相互作用的宏观系统也存在精确解。通过运用微妙的数学技巧，已经找到了几个[[玩具模型]]的精确解。<ref>{{cite book | isbn = 9780120831807 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. | pages = }}</ref> Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.
One approximate approach that is particularly well suited to computers is the Monte Carlo method, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.
一个特别适合于计算机的近似方法是蒙特卡罗方法，它只随机选择(具有相当的权重)系统的几个可能状态进行检查。只要这些状态构成系统全部状态集的代表样本，就可以得到近似的特征函数。随着随机样本数量的增加，误差可以降低到任意低的水平。
* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.
* [[Path integral Monte Carlo]], also used to sample the canonical ensemble.
* [[Metropolis–Hastings 算法]] 是一种经典的蒙特卡罗算法，最初用于正则系综的采样。
* [[路径积分蒙特卡罗方法]] 也可以用于正则系综的采样。