Camiola Vito Dario, Luca Liliana, Romano Vittorio
Department of Mathematics and Computer Science, University of Catania, 95125 Catania, Italy.
Entropy (Basel). 2020 Sep 13;22(9):1023. doi: 10.3390/e22091023.
The approach based on the Wigner function is considered as a viable model of quantum transport which allows, in analogy with the semiclassical Boltzmann equation, to restore a description in the phase-space. A crucial point is the determination of the Wigner function at the equilibrium which stems from the equilibrium density function. The latter is obtained by a constrained maximization of the entropy whose formulation in a quantum context is a controversial issue. The standard expression due to Von Neumann, although it looks a natural generalization of the classical Boltzmann one, presents two important drawbacks: it is conserved under unitary evolution time operators, and therefore cannot take into account irreversibility; it does not include neither the Bose nor the Fermi statistics. Recently a diagonal form of the quantum entropy, which incorporates also the correct statistics, has been proposed in Snoke et al. (2012) and Polkovnikov (2011). Here, by adopting such a form of entropy, with an approach based on the Bloch equation, the general condition that must be satisfied by the equilibrium Wigner function is obtained for general energy dispersion relations, both for fermions and bosons. Exact solutions are found in particular cases. They represent a modulation of the solution in the non degenerate situation.
基于维格纳函数的方法被视为量子输运的一种可行模型，该模型类似于半经典玻尔兹曼方程，能够恢复相空间中的描述。一个关键点是确定源于平衡密度函数的平衡态维格纳函数。后者是通过对熵进行约束最大化得到的，而在量子背景下熵的公式化是一个有争议的问题。冯·诺依曼给出的标准表达式虽然看起来是经典玻尔兹曼表达式的自然推广，但存在两个重要缺点：它在幺正演化时间算符下是守恒的，因此无法考虑不可逆性；它既不包含玻色统计也不包含费米统计。最近，斯诺克等人（2012年）和波尔科夫尼科夫（2011年）提出了一种量子熵的对角形式，该形式也包含了正确的统计。在此，通过采用这种熵的形式，利用基于布洛赫方程的方法，针对费米子和玻色子的一般能量色散关系，得到了平衡态维格纳函数必须满足的一般条件。在特定情况下找到了精确解。它们表示非简并情形下解的一种调制。
