Microcanonical Thermodynamics: Phase Transitions in Small Systems | Statistical Physics Research & Nanoscale Applications" 优化说明： 1. 保持核心术语"Microcanonical Thermodynamics"和"Phase Transitions"以突出学术主题 2. 移除单引号使标题更规范 3. 补充关键词"Statistical Physics"增强搜索相关性 4. 添加应用场景"Nanoscale Applications"说明其在纳米系统研究中的价值 5. 整体符合学术文献的SEO标题结构（核心理论+研究对象+应用领域）

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Boltzmann's formula S = In[W(E)] defines the microcanonical ensemble. The usual textbooks on statistical mechanics start with the microensemble but rather quickly switch to the canonical ensemble introduced by Gibbs. This has the main advantage of easier analytical calculations, but there is a price to pay -- for example, phase transitions can only be defined in the thermodynamic limit of infinite system size. The question how phase transitions show up from systems with, say, 100 particles with an increasing number towards the bulk can only be answered when one finds a way to define and classify phase transitions in small systems. This is all possible within Boltzmann's original definition of the microcanonical ensemble.Starting from Boltzmann's formula, the book formulates the microcanonical thermodynamics entirely within the frame of mechanics. This way the thermodynamic limit is avoided and the formalism applies to small as well to other nonextensive systems like gravitational ones. Phase transitions of first order, continuous transitions, critical lines and multicritical points can be unambiguously defined by the curvature of the entropy S(E, N). Special attention is given to the fragmentation of nuclei and atomic clusters as a peculiar phase transition of small systems controlled, among others, by angular momentum.The dependence of the liquid-gas transition of small atomic clusters under prescribed pressure is treated. Thus the analogue to the bulk transition can be studied. The book also describes the microcanonical statistics of the collapse of a self-gravitating system under large angular momentum.