Topology and Phase Transitions: The Case of the Short Range Spherical Model

We characterize the topology of the phase space of the Berlin-Kac spherical model in the context of the so called Topological Hypothesis, for spins lying in hypercubic lattices of dimension d. For zero external field we are able to characterize the topology exactly, up to homology. We find that, even though there is a continuum of changes in the topology of the corresponding manifolds, for d ≥ 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level. We show however that these changes do not coincide with the phase transitions and conversely, that no topological discontinuity can be associated to the points where the phase transitions take place. At variance with what happens in the Mean Field version of this same model, we show that these abrupt topological changes are accessible thermodynamically. We conclude that, even in short range systems, the topological mechanism does not seem to be responsible for the triggering of a phase transition. We also analyze the case of spins connected to a macroscopic number of (but not all) neighbors, and find that, similar to the results found for the fully connected version, in this case the topological hypothesis seems to hold: the phase transition coincides with an accumulation point of the topological changes present in configuration space. The question of the ensemble equivalence in the short range spherical model is also considered.     

On the Distribution of the Wave Function for Systems in Thermal Equilibrium

For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ?. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ?, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ? with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρ_β = (1/Z) exp (?β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ? are often used.     

The Kraichnan–Kazantsev Dynamo

We investigate the dynamo effect generated by an incompressible, helicity-free flow drawn from the Kraichnan statistical ensemble. The quantum formalism introduced by Kazantsev [A. P. Kazantsev, Sov. Phys. JETP 26, 1031–1034 (1968)] is shown to yield the growth rate and the spatial structure of the magnetic field. Their dependences on the magnetic Reynolds number and the Prandtl number are analyzed. The growth rate is found to be controlled by the largest between the diffusive and the viscous characteristic times. The same holds for the magnetic field correlation length and the corresponding spatial scales.     

集合预报物理基础的探讨

将集合预报中的每次积分算程视为非平衡统计物理理论中的准粒子轨迹，由此对Lorenz模型 进行了数值试验，计算了初值位于不同性质平衡态附近时准粒子数处于基态和第一激发态随 时间的演化.结果证明：(1)若动力系统在整个相空间内存在稳定的平衡态，在稳定的平衡态 附近，系统随时间长期演化行为是可预测的.(2)若动力系统在整个相空间内不存在任何稳定 的平衡态，初值位于远离非稳定的平衡态，则在1—2周内准粒子多数分布在低能量态，即预 报是最可几率的.(3)若初始状态位于非稳定平衡态附近，系统随时间的演化几乎是不可预测 的.这从理论上说明了作大量积分算程的集合预报其效果会比单一初值的单程积分要好.这就 从物理上对集合预报能提高准确率提供了一种解释. 关键词： 集合预报 Lorenz模型 正则分布 概率密度分布     

Colligative Properties of Solutions: I. Fixed Concentrations

Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based model of a solvent--solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing-point depression. The limit of infinitesimal concentrations is described in a subsequent paper.     

On Large Densities in Fermi Systems

In the present paper, we present rigorous results on the behavior of Fermi systems for large densities.     

Coherent modes in an ensemble of quantum dots generated by dynamic dipole–dipole interactions

The generation and stabilization of coherent modes induced by electric dipole–dipole interactions in a quantum dot ensemble are verified, by means of a standard interaction Hamiltonian scheme, and also by way of a full quantum mechanical path integral on the coherent state basis.     

利用扩展系综法计算水的自由能

利用扩展系综法得到了正则系综下水的TIP4P模型的自由能值为-21．485±0．035kJ/mol， 并与其他方法所得的结果作了比较.提出了选择该方法中关键参数(平衡因子)的有效方法， 并讨论了该方法的可移植性. 关键词： 自由能 TIP4P水模型 扩展系综 分子动力学模拟 水分子团     

非简谐振子自由能的微扰展开

推导出正则系统的特性函数－自由能分别在经典情况及量子情况下的生扰表达式，并对非简谐振子自由能的微扰展开进行了具体计算。     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.