Abstract
In this chapter, we shall discuss entropy in the context of the steady-state quantum transport problem. The work presented here deals with noninteracting fermions. We consider situations where we have a few reservoirs which exchange particles and energy with each other through a central scattering region. The distribution within the reservoirs, specified by their temperature and chemical potential, set the boundary conditions for the scattering problem. The scattering basis provides the most natural framework for the analysis of this problem. We derive the exact local entropy in the scattering basis and show that it is additive over subspaces of the one-body Hilbert space. We systematically develop the entropies that would be inferred by a local observer with access to varying degrees of information about the system. We prove inequalities connecting these entropy measures and find that the least knowledgeable formulation leads to the greatest entropy. We also prove statements of the third law of thermodynamics for open quantum systems in equilibrium and in nonequilibrium steady states. Finally, appropriately normalized (per-state) local entropies are defined and are used to quantify the departure from local equilibrium. We provide exact results in the absence of many-body interactions but only a working ansatz in their presence.
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Notes
- 1.
It is usual to include a prefactor of k B which we here set to unity.
- 2.
For clarity of notation, Fock-space operators are written with a hat, while matrices defined in the one-body Hilbert space of the system are written without a hat.
- 3.
As mentioned previously, we use the term matrices to highlight the fact that they are not operators in the Fock space but are defined on the one-body Hilbert space of the system.
- 4.
\(f_{\mathcal {A}}\) is a generalization to any subspace \(\mathcal {A}\) of the nonequilibrium distribution function f s(ω), as introduced in Chap. 2, which selects the subspace defined by the probe-system coupling Γp.
- 5.
Furthermore, in their attempt to extend the notion of heat to the nonequilibrium setting, Esposito et al. [26] seem to be under the misapprehension that heat is a state function in standard thermodynamics.
- 6.
Although we use the position-local subspace in this section, the results of course hold for any subspace of the one-body Hilbert space \(\mathcal {H}\) of the system. In this section, we also drop the subscript in \(\mathcal {H}_{1}\), for brevity of notation, to mean the one-body Hilbert space of the system.
- 7.
The mean local occupancy is in fact the particle density and similarly the mean local energy is the energy density since, in the position-local basis, the projection operator P(x) obeys Eq. (5.65).
- 8.
The y-axis in Fig. 5.3 indicates S∕k B, where k B is the Boltzmann constant. It has been set to unity k B = 1 in all the definitions of the entropies appearing in this chapter including in Sect. 5.7. However, whenever we cite numerical values for the temperatures in [K] (e.g., in Fig. 5.4), it is understood that its conversion to the appropriate energy units ([eV] for our systems) is accompanied by the appropriate numerical value of k B.
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Shastry, A. (2019). Entropy. In: Theory of Thermodynamic Measurements of Quantum Systems Far from Equilibrium. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-33574-8_5
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