Staniscia F, Chavanis P H, De Ninno G
Dipartimento di Fisica, Università di Trieste, Trieste, Italy.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 May;83(5 Pt 1):051111. doi: 10.1103/PhysRevE.83.051111. Epub 2011 May 10.
We provide a detailed discussion of out-of-equilibrium phase transitions in the Hamiltonian mean-field (HMF) model in the framework of Lynden-Bell's statistical theory of the Vlasov equation. For two-level initial conditions, the caloric curve β(E) only depends on the initial value f(0) of the distribution function. We evidence different regions in the parameter space where the nature of the phase transitions between magnetized and nonmagnetized states changes: (i) For f(0)>0.10965, the system displays a second-order phase transition; (ii) for 0.109497<f(0)<0.10965, the system displays a second-order phase transition and a first-order phase transition; (iii) for 0.10947<f(0)<0.109497, the system displays two second-order phase transitions; and (iv) for f(0)<0.1047, there is no phase transition. The passage from a first-order to a second-order phase transition corresponds to a tricritical point. The sudden appearance of two second-order phase transitions from nothing corresponds to a second-order azeotropy. This is associated with a phenomenon of phase reentrance. When metastable states are taken into account, the problem becomes even richer. In particular, we find another situation of phase reentrance. We consider both microcanonical and canonical ensembles and report the existence of a tiny region of ensemble inequivalence. We also explain why the use of the initial magnetization M(0) as an external parameter, instead of the phase level f(0), may lead to inconsistencies in the thermodynamical analysis. Finally, we mention different causes of incomplete relaxation that could be a limitation to the application of Lynden-Bell's theory.
我们在林登 - 贝尔关于弗拉索夫方程的统计理论框架下，对哈密顿平均场（HMF）模型中的非平衡相变进行了详细讨论。对于两能级初始条件，热曲线β(E)仅取决于分布函数的初始值f(0)。我们证明了参数空间中不同的区域，其中磁化态和非磁化态之间相变的性质发生变化：（i）对于f(0)>0.10965，系统呈现二阶相变；（ii）对于0.109497<f(0)<0.10965，系统呈现二阶相变和一阶相变；（iii）对于0.10947<f(0)<0.109497，系统呈现两个二阶相变；（iv）对于f(0)<0.1047，不存在相变。从一阶相变到二阶相变的转变对应于一个三临界点。从无到有突然出现两个二阶相变对应于一个二阶共沸现象。这与相重入现象相关。当考虑亚稳态时，问题变得更加丰富。特别是，我们发现了另一种相重入情况。我们考虑了微正则系综和正则系综，并报告了存在一个微小的系综不等价区域。我们还解释了为什么使用初始磁化强度M(0)作为外部参数，而不是相水平f(0)，可能会导致热力学分析中的不一致。最后，我们提到了不完全弛豫的不同原因，这可能是林登 - 贝尔理论应用的一个限制。
