Squeezed ensemble for systems with first-order phase transitions

Abstract

All ensembles of statistical mechanics are equivalent in the sense that they give the equivalent thermodynamic functions in the thermodynamic limit. However, when investigating microscopic structures in the first-order phase transition region, one must choose an appropriate statistical ensemble. The appropriate choice is particularly important when one investigates finite systems, for which even the equivalence of ensembles does not hold. We propose a class of statistical ensembles, which always give the correct equilibrium state even in the first-order phase transition region. We derive various formulas for this class of ensembles, including the one by which temperature is obtained directly from energy without knowing entropy. Moreover, these ensembles are convenient for practical calculations because of good analytic properties. We also derive formulas which relate statistical-mechanical quantities of different ensembles, including the conventional ones, for finite systems. The formulas are useful for obtaining results with smaller finite size effects, and for improving the computational efficiency. The advantages of the squeezed ensembles are confirmed by applying them to the Heisenberg model and the frustrated Ising model. arXiv:1903.04111v2 [cond-mat.stat-mech]

Thermal pure quantum matrix product states: a simple numerical protocol for finite temperature

“…Here, the k-th TPQ-MPS state has a micricaninical temperature approximated as k B T (k) = N (l − u * k )/2k (with higher order correction given in Refs. [17,30]). Its microcanonical energy is given as u * k = ⟨k| ĥ|k⟩/⟨k|k⟩.…”

Statistical ensembles for phase coexistence states specified by noncommutative additive observables

“…The statistical ensemble is a fundamental concept in equilibrium statistical mechanics [1][2][3], and various types of ensembles have been devised [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The thermodynamic functions of the macroscopic systems with shortrange interactions are independent of the ensemble used in the calculation [20].…”

“…For example, when the periodic boundary conditions are imposed, the grand canonical ensemble (GCE) gives either a single phase state or a statistical mixture of single phases, and it cannot give a phase coexistence state (see [19]; we will also give examples in Sec. V).…”

“…For example, in finite systems which undergo first-order phase transitions (in the thermodynamic limit) the concavity of the Boltzmann entropy is broken generally, even with short-range interactions (see Appendix C of Ref. [19]). The concavity breaking leads to thermodynamic anomalies such as negative specific heat [29][30][31][32][33][34][35], which was indeed observed experimentally [36][37][38].…”

“…In this paper, we propose a new class of ensembles which are free from the above problems of the conventional ensembles, by generalizing the squeezed ensemble (SE) [19]. The SE was introduced as a class of ensembles with macroscopically well-defined energy.…”