The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy s(e,m) is obtained as a function of the energy e and the magnetization vector m in the thermodynamic limit. Since, for this model, e is uniquely determined by m, the same information can be encoded either in s(m) or s(e,m1,m2). Although these two entropies correspond to the same physical setting of fixed e and m, their concavity properties differ. The entropy s_h(u), describing the model at fixed total energy u and in a homogeneous external magnetic field h of arbitrary direction, is obtained by reduction from the nonconcave entropy s(e,m1,m2). In doing so, concavity, and therefore equivalence of ensembles, is restored. s_h(u) has nonanalyticities on surfaces of co-dimension 1 in the (u,h)-space. Projecting these surfaces into lower-dimensional phase diagrams, we observe that the resulting phase transition lines are situated in the positive-temperature region for some parameter values, and in the negative-temperature region for others. In the canonical setting of a system coupled to a heat bath of positive temperatures, the nonanalyticities in the microcanonical negative-temperature region cannot be observed, and this leads to a situation of effective nonequivalence even when formal equivalence holds.
The Curie-Weiss model is an exactly soluble model of ferromagnetism that allows one to study in detail the thermodynamic functions, in particular their properties in the neighbourhood of the critical temperature. In this model every magnetic moment interacts with every other magnetic moment. Because of its simplicity and because of the correctness of at least of some of its predictions, the Curie-Weiss model occupies an important place in the statistical mechanics literature and its application to information theory. It is frequently presented as an introduction to the Ising model or to spin g…
The object of this study is a cell model with Curie-Weiss interaction potential. We have already proved the possibility of a mathematically rigorous transition from a continuous system of interacting particles to such a model and made an accurate calculation of its grand partition function. In the present research, we derive an explicit accurate equation of state of the cell model with Curie-Weiss potential. It turned out that this model has a sequence of first-order phase transitions at temperatures below the critical. We analyzed the mechanism of these transitions based on the behavior of th…
Phase transitions are typically accompanied by non-analytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real-axis as the system size is increased. Recent experiments have shown that partition function zeros are not just a theoretical concept. They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size. Motivated by this progress, we investigate here the partition function zeros f…
The coherence length of the medium-range order (MRO) in metallic liquids is known to display a Curie-Weiss temperature dependence; its inverse is linearly related to temperature, and when extrapolated from temperatures above the glass transition, the coherence length diverges at a negative temperature with a critical exponent of unity. We propose a mean-field pseudospin model that explains this behavior. Specifically, we model the atoms and their local environment as Ising spins with antiferromagnetic exchange interactions. We further superimpose an exchange interaction between dynamical heter…
Among the Markov chains breaking detailed-balance that have been proposed in the field of Monte-Carlo sampling in order to accelerate the convergence towards the steady state with respect to the detailed-balance dynamics, the idea of 'Lifting' consists in duplicating the configuration space into two copies $σ=\pm$ and in imposing directed flows in each copy in order to explore the configuration space more efficiently. The skew-detailed-balance Lifted-Markov-chain introduced by K. S. Turitsyn, M. Chertkov and M. Vucelja [Physica D Nonlinear Phenomena 240 , 410 (2011)] is revisited for the Curie…
We consider an application of probabilistic coupling techniques which provides explicit estimates for comparison of local expectation values between label permutation invariant states, for instance, between certain microcanonical, canonical, and grand canonical ensemble expectations. A particular goal is to obtain good bounds for how such errors will decay with increasing system size. As explicit examples, we focus on two well-studied mean-field models: the discrete model of a paramagnet and the mean-field spherical model of a continuum field, both of which are related to the Curie-Weiss model…
Topological Sector Fluctuations and Curie Law Crossover in Spin Ice
在低温下，自旋冰进入库仑相-一种具有代数相关性和拓扑约束自旋配置的状态。在Ho 2 Ti 2 O 7，我们已经观察到实验，这个过程是伴随着一个非标准的温度演变的波矢量依赖的磁化率，测量中子散射。分析和数值方法揭示了两个居里定律之间的交叉，一个表征高温顺磁制度，和其他的低温拓扑约束制度，我们称之为自旋液体居里定律的签名。该理论与中子散射实验结果吻合得很好。在更一般的基础上，i）两个居里定律的存在似乎是经典自旋液体的突现规范场的一般性质，ii）揭示了在阻挫材料中测量精确的居里-外斯温度的实验困难;（三）规范自由度和自旋自由度之间的映射意味着有限波矢处的磁化率可以用作波动的局部探针，拓扑扇区
We introduce a mean-field model for analysing the dynamics of human consciousness. In particular, inspired by the Giulio Tononi's Integrated Information Theory and by the Max Tegmark's representation of consciousness, we study order-disorder phase transitions on Curie-Weiss models generated by processing EEG signals. The latter have been recorded on healthy individuals undergoing deep sedation. Then, we implement a machine learning tool for classifying mental states using, as input, the critical temperatures computed in the Curie-Weiss models. Results show that, by the proposed method, it is p…
