Green-Kubo formulas with symmetrized correlation functions for quantum systems in steady states: the shear viscosity of a fluid in a steady shear flow

Abstract: For a quantum system in a steady state with a constant current of heat or particles driven by a temperature or chemical potential difference between two reservoirs attached to the system, the fluctuation theorem for the current was previously shown to lead to the Green-Kubo formula for the linear response coefficient for the current expressed in terms of the symmetrized correlation function of the current density operator. In this article, we show that for a quantum system in a steady state with a constant rate of work done on the system, the fluctuation theorem for a quantity induced in the system also leads to the Green-Kubo formula expressed in terms of the symmetrized correlation function of the induced quantity. As an example, we consider a fluid in a steady shear flow driven by a constant velocity of a solid plate moving above the fluid.

• The paper establishes that fluctuation theorems yield symmetrized Green-Kubo formulas for quantum transport coefficients.
• It rigorously derives shear viscosity for a quantum fluid under steady shear flow using microreversibility and ensemble theory.
• It provides a framework applicable to both current-driven and work-driven quantum steady states in non-equilibrium scenarios.

Symmetrized Green-Kubo Formulas and Fluctuation Theorems in Quantum Steady States

Introduction and Theoretical Context

This work rigorously establishes the generality of Green-Kubo formulas with symmetrized time-correlation functions for linear response coefficients in quantum systems maintained in non-equilibrium steady states by external drivings, either via boundary-induced currents or sustained work input. The Green-Kubo formalism, traditionally applied within equilibrium linear response theory, is critically extended to cases where stationary non-equilibrium is maintained, leading to generalized fluctuation-dissipation relations. The formal derivations are explicitly connected to fluctuation theorems, particularly emphasizing quantum extensions of Crooks-type relations and microreversibility. The primary demonstration is provided for shear viscosity in a quantum fluid under steady shear flow, with generalization to thermal transport in steady heat conduction.

Extension of Fluctuation Theorems and Symmetrized Green-Kubo Formulas

The paper identifies two primary classes of quantum steady states:

1. Current-Driven Steady States: Characterized by a constant current (heat or particles) between reservoirs at different thermodynamic parameters (e.g., temperature, chemical potential).
2. Work-Driven Steady States: Characterized by a steady rate of work input, such as that exerted by moving mechanical boundaries.

For both classes, it is shown that fluctuation theorems for the induced quantities (current or work) yield Green-Kubo expressions where transport coefficients are given by integrals of symmetrized quantum correlation functions of the relevant operators evaluated in equilibrium. This is a nontrivial generalization, requiring quantum microreversibility, correct treatment of the system plus reservoirs, and statistical ensemble considerations.

Shear Viscosity in Quantum Steady Shear Flow

An explicit model is constructed: a quantum fluid confined between two plates, with the upper plate driven at constant velocity, creating a steady shear profile. The derivation involves:

• Construction of the total Hamiltonian for fluid, plates, and thermal reservoir, including interaction and time-dependence due to the boundary motion.
• Careful initial and final state preparation in accordance with quantum measurement and ensemble theory.
• Utilization of the principle of microreversibility and anti-unitary time reversal symmetry to relate forward and backward processes.
• Definition of cumulant generating functions for the fluctuating shear work, and the associated symmetry as a quantum analog of classical fluctuation theorems.

Applying these constructs, a fluctuation theorem for the total shear work is derived, which directly provides, via the second derivative at zero counting field, a symmetrized correlation function expression for the shear viscosity:

$$\eta = \frac{V}{2 k_B T} \lim_{\tau \to \infty} \frac{1}{\tau} \int_0^\tau dt_1 \int_0^\tau dt_2 \, C_\eta(t_1, t_2)$$

where $C_\eta$ is the symmetrized two-time correlation function of the shear stress operator in the Heisenberg picture, evaluated in equilibrium.

Green-Kubo Formula for Thermal Conductivity (Appendix)

A parallel derivation is provided for heat conduction in quantum systems between reservoirs at different temperatures. Using a weak-coupling Hamiltonian and quantum initial/final eigenstate sampling, the cumulant generating function for integrated heat transfer is analyzed. The fluctuation theorem is again shown to lead to the Green-Kubo expression for the thermal conductivity, involving the symmetrized correlation function of the heat current operator.

Implications and Theoretical Impact

The framework consolidates the understanding that symmetrized quantum correlation functions describe linear response (transport) in both equilibrium and certain classes of non-equilibrium steady states. These results clarify the precise meaning of quantum fluctuation-dissipation relations and show how microreversibility and time-reversal symmetry encode symmetry properties of response functions (including their connection to fluctuation theorems and large deviation functionals).

The findings have significant implications:

• Theoretical: The approach shows that the Green-Kubo formula, commonly derived via Kubo's linear response theory in equilibrium, also naturally emerges from fluctuation theorems in non-equilibrium quantum steady states, provided the appropriate symmetrization.
• Practical: The results provide a recipe for calculating nonequilibrium quantum transport coefficients—in particular, viscosity and conductivity—from equilibrium simulations or analytic calculations, using symmetrized correlation functions.
• Foundational: The extension to quantum systems amplifies the unification of classical and quantum non-equilibrium statistical physics, and suggests protocols for experimental verification, particularly in systems where quantum coherence and dissipation compete.

Potential Future Directions

These findings set the stage for further investigations, including:

• Applications to systems with strong interactions, non-Markovian environments, or explicit quantum coherence effects.
• Extension to higher-order (nonlinear) response regimes and time-dependent driving protocols beyond steady states.
• Numerical and experimental evaluation of symmetrized correlators in quantum materials, cold atom systems, and mesoscopic conductors.
• Analysis of how quantum integrability, topological effects, or many-body localization modify the symmetry and fluctuation properties exploited here.

Conclusion

This work rigorously establishes that fluctuation theorems for induced quantities in quantum steady states universally yield Green-Kubo formulas for corresponding transport coefficients, provided symmetrized quantum correlation functions are used. This insight unifies fluctuation-dissipation concepts across classical and quantum domains and clarifies the theoretical status of Green-Kubo relations outside strict equilibrium. The explicit illustration for shear viscosity in steady quantum shear flow provides a concrete paradigm, with wide-ranging implications for theory and quantum transport applications ["Green-Kubo formulas with symmetrized correlation functions for quantum systems in steady states: the shear viscosity of a fluid in a steady shear flow" (1204.5533)].