De Carlo Leonardo, Wick William D
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy.
Department of Economics and Finance, Luiss Guido Carli, Viale Romania, 32, 00197 Rome, Italy.
Entropy (Basel). 2023 Mar 25;25(4):564. doi: 10.3390/e25040564.
In a wavefunction-only philosophy, thermodynamics must be recast in terms of an ensemble of wavefunctions. In this perspective we study how to construct Gibbs ensembles for magnetic quantum spin models. We show that with free boundary conditions and distinguishable "spins" there are no finite-temperature phase transitions because of high dimensionality of the phase space. Then we focus on the simplest case, namely the mean-field (Curie-Weiss) model, in order to discover whether phase transitions are even possible in this model class. This strategy at least diminishes the dimensionality of the problem. We found that, even assuming exchange symmetry in the wavefunctions, no finite-temperature phase transitions appear when the Hamiltonian is given by the usual energy expression of quantum mechanics (in this case the analytical argument is not totally satisfactory and we relied partly on a computer analysis). However, a variant model with additional "" does have a phase transition to a magnetized state. (With respect to dynamics, which we do not consider here, wavefunction energy induces a non-linearity which nevertheless preserves norm and energy. This non-linearity becomes significant only at the macroscopic level.) The three results together suggest that magnetization in large wavefunction spin chains appears if and only if we consider indistinguishable particles and block macroscopic dispersion (i.e., macroscopic superpositions) by energy conservation. Our principle technique involves transforming the problem to one in probability theory, then applying results from large deviations, particularly the Gärtner-Ellis Theorem. Finally, we discuss Gibbs vs. Boltzmann/Einstein entropy in the choice of the quantum thermodynamic ensemble, as well as open problems.
在仅基于波函数的理念中，热力学必须依据波函数的系综来重新构建。从这个角度出发，我们研究如何为磁性量子自旋模型构建吉布斯系综。我们表明，在自由边界条件和可区分的“自旋”情况下，由于相空间的高维性，不存在有限温度的相变。然后我们聚焦于最简单的情况，即平均场（居里 - 外斯）模型，以探究在这类模型中是否甚至有可能出现相变。这种策略至少降低了问题的维度。我们发现，即使假设波函数具有交换对称性，当哈密顿量由量子力学的常规能量表达式给出时（在这种情况下，解析论证并不完全令人满意，我们部分依赖于计算机分析），也不会出现有限温度的相变。然而，一个具有额外“ ”的变体模型确实有向磁化态的相变。（关于动力学，我们在此不考虑，波函数能量会诱导一种非线性，尽管如此它仍保持范数和能量。这种非线性仅在宏观层面变得显著。）这三个结果共同表明，在大波函数自旋链中，当且仅当我们考虑不可区分的粒子并通过能量守恒阻止宏观色散（即宏观叠加）时，才会出现磁化现象。我们的主要技术涉及将问题转化为概率论中的问题，然后应用大偏差的结果，特别是加特纳 - 埃利斯定理。最后，我们讨论在量子热力学系综的选择中吉布斯熵与玻尔兹曼/爱因斯坦熵的问题，以及开放性问题。
