Lee-Yang and Fisher zeros are crucial for the study of phase transitions in the grand canonical and the canonical ensembles, respectively. However, these powerful methods do not cover the isothermal-isobaric ensemble (NPT ensemble), which reflects the conditions of many experiments. In this work we present a theory of the phase transitions in terms of the zeros of the NPT-ensemble partition functions in the complex plane. The proposed theory provides an approach to calculate all the partition function zeros in the NPT ensemble, which form certain curves in the thermodynamic limit. To verify the theory we consider Tonks gas and van der Waals fluid in the NPT ensemble. In the case of Tonks gas, similarly to the Lee-Yang circle theorem, we obtain an exact equation for the zero limit curve. We also derive an approximated limit curve equation for van der Waals fluid in terms of the Szegö curve. This curve fits numerically calculated zeros and correctly describes how the phenomenon of phase transition depends on the temperature.
Theory of solutions in energy representation (ER method) developed by Matubayasi and Nakahara provides with an approximate way of calculating solvation free energies (or, identically, the excess chemical potentials) from atomistic simulations. In this document we provide some derivation details of this, to our opinion, theoretically involved method, which will help a non-specialist to follow. There are three points which differ this document from a regular textbook on statistical mechanics or research articles: 1) Derivation is detailed and all approximations are explicitly stated; 2) Statisti…
A methodology to sample the isothermal-isobaric ensemble using Langevin dynamics is proposed, which combines novel features of geometric integrators for the equations of motion. By employing the Trotter expansion, the methodology generates a robust, symmetric and accurate numerical algorithm. In order to show that the proposed method correctly samples the phase-space, simulations in the isotropic NPT ensemble were carried out for two analytical examples. Also this method permits to study a solid-solid phase transition, by conducting a fully flexible-cell molecular dynamics simulation. Addition…
We obtain exact closed-form expressions for the partition function of the one-dimensional Ising model in the fixed-$M$ ensemble, for three commonly-used boundary conditions: periodic, antiperiodic and Dirichlet. These expressions allow for the determination of fluctuation-induced forces in the canonical ensemble, which we term Helmholtz forces. The thermodynamic expressions and the calculations flowing from them should provide insights into the nature and behavior of fluctuation induced forces in interesting and as-yet unexplored regimes.
Statistics of distinguishable particles has become relevant in systems of colloidal particles and in the context of applications of statistical mechanics to complex networks. When studying these type of systems with the standard textbook formalism, non-physical results such as non-extensive entropies are obtained. In this paper, we will show that the commonly used expression for the partition function of a system of distinguishable particles leads to huge fluctuations of the number of particles in the grand canonical ensemble and, consequently, to non-equivalence of statistical ensembles. We w…
The microcanonical effective partition function, constructed from a Feynman-Hibbs potential, is derived using generalized ensemble theory. The form of the effective Hamiltonian is amenable to Monte Carlo simulation techniques and the relevant Metropolis function is presented. Using the derived expression for the microcanonical effective partition function, the low-temperature entropy of a proton in an anharmonic potential is numerically evaluated and compared with the exact quantum mechanical canonical result.
University of Houston（休斯顿大学）；Texas Center for Superconductivity, University of Houston（休斯顿大学德克萨斯超导中心）
We introduce an ensemble learning scheme for community detection in complex networks. The scheme uses a Machine Learning algorithmic paradigm we call Extremal Ensemble Learning. It uses iterative extremal updating of an ensemble of network partitions, which can be found by a conventional base algorithm, to find a node partition that maximizes modularity. At each iteration, core groups of nodes that are in the same community in every ensemble partition are identified and used to form a reduced network. Partitions of the reduced network are then found and used to update the ensemble. The smaller…
Models of non-interacting fermions coupled to auxilliary classical degrees of freedom are relevant to the understanding of a wide variety of problems in many body physics, {\it e.g.} the description of manganites, diluted magnetic semiconductors or strongly interacting electrons on lattices. Monte Carlo sampling over the classical fields is a powerful, yet notoriously challenging, method for this class of problems -- it requires the solution of the fermion problem for each classical field configuration. Conventional Monte Carlo methods minimally utilize the information content of these solutio…
Fixing the number of particles $N$, the quantum canonical ensemble imposes a constraint on the occupation numbers of single-particle states. The constraint particularly hampers the systematic calculation of the partition function and any relevant thermodynamic expectation value for arbitrary $N$ since, unlike the case of the grand-canonical ensemble, traces in the $N$-particle Hilbert space fail to factorize into simple traces over single-particle states. In this paper we introduce a projection operator that enables a constraint-free computation of the partition function and its derived quanti…
The analytical canonical NLT partition function of a quasi-one dimensional (q1D) system of hard disks [J. Chem Phys. v.153, 144111 (2020)] provides a direct analytical description of the thermodynamics and ordering in this system (a pore) as a function of the linear density N/L both for finite and infinite systems. It is alternative to the well-developed transfer matrix method based on the isobaric NPT ensemble. We derive the analytical formulae for the actual distance dependence of the translational pair distribution function (PDF) and the DF of distances between next neighbor disks, and then…
In the previous paper of this series [JCTC 2020, 16, 3757], we presented a theoretical and algorithmic framework based on a localized representation of the occupied space that exploits the inherent sparsity in the real-space evaluation of the EXX interaction in finite-gap systems. This was accompanied by a detailed description of exx, a massively parallel hybrid MPI/OpenMP implementation of this approach in Quantum ESPRESSO that enables linear-scaling hybrid DFT based AIMD in the NVE/NVT ensembles of condensed-phase systems containing 500--1000 atoms (in fixed orthorhombic cells) with a wall t…
For realistic crystals, the free energy strictly formulated in ensemble theory can hardly be obtained because of the difficulty in solving the high-dimension integral of the partition function, the dilemma of which makes it even a doubt if the rigorous ensemble theory is applicable to phase transitions of condensed matters. In the present work, the partition function of crystal vanadium under compression up to $320$ GPa at room temperature is solved by an approach developed very recently, and the derived equation of state is in a good agreement with all the experimental measurements, especiall…
Fluctuation statistics of mesoscopic Bose-Einstein condensate: reconciling the master equation with the partition function to revisit the Uhlenbeck-Einstein dilemma
从几个不同的角度研究了理想介观玻色-爱因斯坦凝聚体的原子涨落统计。通过推广巨正则分析（适用于正则系综问题），我们得到了一个自洽的平均凝聚粒子数方程，该方程与激光主方程方法计算的微观结果相吻合。对于一个谐波陷阱的情况下，我们得到一个解析表达式的凝聚粒子数是非常准确的，在所有温度下，与数值正则系综结果相比。应用类似的广义巨正则处理的方差，我们得到一个准确的结果，只有低于临界温度。采用随机路径积分的形式主义，具有很好的精度的波动分布的所有高阶矩的分析结果。我们进一步讨论了一种混合处理，它将主方程/随机路径积分分析与基于正则系综准粒子形式主义获得的结果结合在一起[V. V. Kocharovsky等人，Phys.Rev.A61，053606（2000）]，在所有温度下产生与数值模拟基本上完美的一致。
We discuss partition function of 2d CFTs decorated by higher qKdV charges in the thermodynamic limit when the size of the spatial circle goes to infinity. In this limit the saddle point approximation is exact and at infinite central charge generalized partition function can be calculated explicitly. We show that leading 1/c corrections to free energy can be reformulated as a sum over Young tableaux which we calculate for the first two qKdV charges. Next, we compare generalized ensemble with the "eigenstate ensemble" that consists of a single primary state. At infinite central charge the ensemb…
Partition functions of a canonical ensemble of non-interacting bound electrons are a key ingredient of the super-transition-array approach to the computation of radiative opacity. A few years ago, we published a robust and stable recursion relation for the calculation of such partition functions. In this paper, we propose an optimization of the latter method and explain how to implement it in practice. The formalism relies on the evaluation of elementary symmetric polynomials, which opens the way to further improvements.
The Generalized Gibbs Ensemble (GGE) is relevant to understand the thermalization of quantum systems with an infinite set of conserved charges. In this work, we analyze the GGE partition function of 2D Conformal Field Theories (CFTs) with a U(1) charge and quantum Benjamin-Ono$_{2}$ (qBO$_{2}$) hierarchy charges. We use the Alday-Gaiotto-Tachikawa (AGT) correspondence to express the thermal trace in terms of the Alba-Fateev-Litvinov-Tarnopolskiy (AFLT) basis of descendants, which diagonalizes all charges. We analyze the GGE partition function in the thermodynamic semiclassical limit, including…
A lower bound on the grand partition function of a classical charge-symmetric system is adapted to the neutral grand canonical ensemble, in which the system is constrained to have zero total charge. This constraint permits us to consider two-body potentials that are only conditionally positive definite.
