Thermogalvanic cells (or thermo-electrochemical cells), which are composed of a liquid electrolyte with hot and cold electrodes, are capable of efficiently converting thermal energy into electricity [2, 3]. However, a microscopic mechanism that dominates the performance of thermogalvanic cells is yet to be fully understood, hindering their performance improvement and widespread deployment.

Fig. 1. Schematic illustration of a thermogalvanic cell. Determining the descriptors of the Seebeck coefficient of a redox couple is primarily important to achieve efficient waste-energy harvesting.

The research group led by Moritomo [4] identified the scaling relation between the Seebeck coefficient (*α*) of [Fe^{III}(CN)_{6}]^{3-}/[Fe^{II}(CN)_{6}]^{4-}
redox couple and the viscosity (*η*) of solvents, where *α* ranges from 0.14 mV/K (glycerin, *η* = 1412 mPa s) to 3.6 mV/K (acetone, *η*
= 0.3 mPa s) as a function of *η*^{‒0.4}. X-ray absorption near-edge structure (XANES) showed the deformation of Fe^{2+}
coordination environment with decreasing *η*, suggesting that a low-*η*
solvent weakly interacts with a solute to provide a large configuration-entropy change upon redox, and thus large *α*.

To clarify the origin for this scaling relation, the microscopic understandings of both *η* and *α*
are required. Large-scale quantum mechanics/molecular mechanics (QM/MM)
calculation of electrolytes could provide accurate electrolyte
structures, shedding light on the atomistic mechanism [5]. In addition,
considering a relatively low coefficient of determination (*R*^{2}) for the scaling relation, *α*
is also affected by other macroscopic/microscopic descriptors.
Following systematic data collection, state-of-the-art machine learning
analyses could determine discriminating features [6], and the resulting
accurate performance prediction could enable the development of advanced
thermogalvanic cells for sustainability.

[1] J. He and T. M. Tritt, Science 357, eaak9997 (2017).

[2] T. I. Quickenden and Y. Mua, J. Electrochem. Soc. **142**, 3985 (1995).

[3] H. J. V. Tyrrell, D. A. Taylor, and C. M. Williams, Nature **177**, 668 (1956).

[4] D. Inoue, H. Niwa, H. Nitani, and Y. Moritomo, J. Phys. Soc. Jpn. **90**, 033602 (2021).

[5] Y. Tateyama, J. Blumberger, M. Sprik, and I. Tavernelli, J. Chem. Phys. **122**, 234505 (2005).

[6] A. Chen, X. Zhang, and Z. Zhou, InfoMat **2**, 553 (2020).

Recently, Tsunetsugu et al. [4] showed a rich phase diagram consisting of the ferro- and antiferroelectric quadrupole orderings for Pr-based compounds. By analyzing a minimal model for an fcc lattice system [Fig. 1(a)] under the mean-field approximation, they obtained a variety of electric quadrupole orderings in the ground state [Fig. 1(b)]. The resultant quadrupole configurations are shown in Fig. 1(c). Interestingly, they also identified exotic partial orders of quadrupoles, which are denoted as *zyox* and *zoxy*, at finite temperatures, as shown in Fig. 1(d). These partial orders are characterized by the coexistence of the quadrupole order in the three-sublattice network and the disordered state in the remaining sublattice. Notably, they showed that the third-order term with respect to the ordered parameter in the free energy plays an important role in stabilizing the aforementioned partial quadrupole orders. This emergent third-order term in the free energy, which does not appear in the case of the conventional magnetic orderings, is characteristic of the electric quadrupole ordering. Meanwhile, this third-order term is relevant for the nematic phase in liquid crystals [5], which indicates that the obtained quadrupole orderings correspond to the nematic states of electronic origin. Their results are set to inspire future studies not only on relevant materials, such as PrMgNi_{4} [6], but also further exploration of multipole-induced exotic electronic orderings.

Fig. 1 (a) Fcc lattice. (b) Ground-state phase diagram in the plane of the isotropic interaction, *J*, and the anisotropic interaction, *K*, expressed in units of the energy difference, *E*_{1}, between the Γ_{3} and Γ_{1} levels. (c) Schematics of the two- and four-sublattice quadrupole orderings. The yellow lines represent the symmetry axes 3*z*^{2}-*r*^{2}, 3*x*^{2}-*r*^{2}, and 3*y*^{2}-*r*^{2}. (d) Finite-temperature (*T*) phase diagram. [These figures have been taken from Figs. 1(a), 2(c), 3, and 5(d) of Ref. 4].

[1] Y. Kuramoto, H. Kusunose, and A. Kiss, J. Phys. Soc. Jpn. **78**, 072001 (2009).

[2] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Rev. Mod. Phys. **81**, 807 (2009).

[3] T. Onimaru and H. Kusunose, J. Phys. Soc. Jpn. **85**, 082002 (2016).

[4] H. Tsunetsugu, T. Ishitobi, and K. Hattori, J. Phys. Soc. Jpn. **90**, 043701 (2021).

[5] P. G. de Gennes, *The Physics of Liquid Crystals* (Clarendon, Oxford, 1974).

[6] Y. Kusanose, T. Onimaru, G.-B. Park, Y. Yamane, K. Umeo, T. Takabatake, N. Kawata, and T. Mizuta, J. Phys. Soc. Jpn. **88**, 083703 (2019).

While classical and quantum regimes are typically regarded as distinct, classical dynamics can, in fact, be used to determine the time evolution of a quantum many-body system under non-equilibrium, thus doing away with a full-blown quantum machinery. Essentially, the classical equation of motion for the phase space distribution can provide an approximate solution of the quantum equation of motion for the density matrix (matrix describing a quantum system’s statistical state), if the initial classical analogue of the quantum distribution function is known. This beneficial feature of classical dynamics can even be extended to the case of field theories, where, instead of discrete degrees of freedom, we deal with continuous degrees of freedom.

However, there is a caveat: the classical dynamics approach is less useful when describing equilibrium behavior of quantum systems. For instance, in the case of black-body cavity radiation, classical considerations lead to the Rayleigh-Jeans divergence. Thus, a framework that can help us reach quantum statistical equilibrium following a classical dynamical evolution is highly desired.

To that end, the authors propose a novel framework called “the replica evolution method” in which quantum statistics is realized by solving classical equations of motion. To start with, they consider a set of classical field configurations (Φ_{xτ},π_{xτ}_{}) in (3+1) dimensions (three spatial dimensions, one replica index (τ) dimension), called “replicas,” where each replica interacts with its nearest neighbors through a τ-derivative term that eventually leads to their thermalization. The time evolution of the replicas is governed by the classical Hamilton’s equation of motion.

The most interesting feature of their framework is that the replica index ‘τ’ can be regarded as the imaginary time index in finite temperature quantum field theory, so that the replica evolution is technically similar to the fictitious time evolution in a hybrid Monte Carlo simulation. This, in turn, implies that, given a distribution of initial replica Hamiltonian values, the replicas should relax to the equilibrium quantum distributions following a long-time evolution. Moreover, the τ-averaged field variables obey classical equations of motion when the fluctuations among replicas are small. From these two aspects, one can expect that replica evolution would describe the real time evolution of quantum systems. Actually, the time-correlation function in the replica evolution is found to reproduce that in quantum field theory to a good approximation. This framework can thus allow for more precise predictions of relaxation processes in quantum systems and possibly help unify the background field and the particle degrees of freedom, paving the way for an integrated transport theory.

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