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.

Appendix: Green-Kubo Formula for Thermal Conductivity

1.1 A.1 Thermal Conductivity

Consider a system driven to a steady heat conduction state by a temperature difference between two heat reservoirs, A and B. We assume that their temperatures, T ^A and T ^B, satisfy T ^A>T ^B. The inverse temperatures associated with T ^A and T ^B are defined by β ^A≡1/(k _B T ^A) and β ^B≡1/(k _B T ^B), respectively. We assume that the system and the reservoirs are subject to a constant magnetic field B.

We define the thermal conductivity κ of the system by the following linear response relation for the average heat current $\overline{J}_{Q}(\mathbf{B})$:

(A.1)

where A and L are the cross-sectional area and the length of the system. Δβ is defined by

(A.2)

where $\overline{T}$ is defined by

(A.3)

1.2 A.2 Total Hamiltonian

The total Hamiltonian for the system and the reservoirs is given by

(A.4)

where $H_{\mathbf{B}}^{\mathrm{s}}$ and $H_{\mathbf{B}}^{k}$ are the Hamiltonians for the system and the k-th reservoir (k=A,B) while H ^int is a weak coupling between the system and the reservoirs. Before the initial time t=0 and after the final time t=τ, we set H ^int=0 so that the system is detached from the reservoirs.

We assume that H _B satisfies

(A.5)

so that

(A.6)

1.3 A.3 Initial Eigenstates

Just before t=0, the system is detached from the reservoirs and through a measurement of the energy in the system, we find the system to be in an eigenstate $\vert {i^{\mathrm{s}}_{\mathbf{B}}}\rangle$ of the system Hamiltonian $H^{\mathrm{s}}_{\mathbf{B}}$ with energy eigenvalue $E^{\mathrm{s}}_{\mathbf{B}}(i^{\mathrm{s}}_{\mathbf{B}})$. Before t=0, we assume that the system is in an equilibrium state at the inverse temperature $\overline{\beta}=(\beta ^{\mathrm{A}}+\beta ^{\mathrm{B}})/2$ so that the initial eigenstate $\vert {i^{\mathrm{s}}_{\mathbf{B}}}\rangle$ is selected by the following canonical ensemble distribution:

(A.7)

where $Z^{\mathrm{s}}_{\mathbf{B}}$ is the partition function for the initial canonical ensemble for the system. We choose this value $\overline{\beta}$ for the initial inverse temperature for the system so that we can later show the fluctuation theorem for heat current. We assume that after a long time interval, the steady state for the system should become independent of its initial inverse temperature so that we can choose its value almost freely as long as the value is not so different from β ^A or β ^B.

Just before t=0, through a measurement of the energy in each reservoir, we find the k-th reservoir (k=A,B) to be in an eigenstate $\vert {i^{k}_{\mathbf{B}}}\rangle$ of the reservoir Hamiltonian $H^{k}_{\mathbf{B}}$ with energy eigenvalue $E^{k}_{\mathbf{B}}(i^{k}_{\mathbf{B}})$. Before t=0, we assume that the reservoir is in an equilibrium state at inverse temperature β ^k so that the initial eigenstate $\vert {i^{k}_{\mathbf{B}}}\rangle$ is selected by the following canonical ensemble distribution:

(A.8)

where $Z^{k}_{\mathbf{B}}$ is the partition function for the initial canonical ensemble for the reservoir.

The initial eigenstate |i _B 〉 for the total system is then $\vert {i_{\mathbf{B}}}\rangle \equiv \vert {i^{\mathrm{s}}_{\mathbf{B}}}\rangle \otimes \vert {i^{\mathrm{A}}_{\mathbf{B}}}\rangle \otimes \vert {i^{\mathrm{B}}_{\mathbf{B}}}\rangle$ and the initial canonical ensemble distribution for the total system is

(A.9)

According to (2.20), $\vert {{}^{\varTheta }i^{\mathrm{s}}_{\mathbf{B}}}\rangle \equiv \varTheta \vert {i^{\mathrm{s}}_{\mathbf{B}}}\rangle$ is an eigenstate of $H^{\mathrm{s}}_{-\mathbf{B}}={}^{\varTheta }H^{\mathrm{s}}_{\mathbf{B}}$ with the energy eigenvalue $E^{\mathrm{s}}_{\mathbf{B}}(i^{\mathrm{s}}_{\mathbf{B}})$ so that

(A.10)

and $\vert {{}^{\varTheta }i^{k}_{\mathbf{B}}}\rangle \equiv\varTheta \vert { i^{k}_{\mathbf{B}}}\rangle$ is an eigenstate of $H^{k}_{-\mathbf{B}}={}^{\varTheta }H^{k}_{\mathbf{B}}$ with the energy eigenvalue $E^{k}_{\mathbf{B}}(i^{k}_{\mathbf{B}})$ so that

(A.11)

1.4 A.4 Final Eigenstates

Just after t=τ, the system is detached from the reservoirs and through a measurement of the energy in the system and those in the reservoirs, we find the system to be in an eigenstate $\vert {f^{\mathrm{s}}_{\mathbf{B}}}\rangle$ of the system Hamiltonian $H^{\mathrm{s}}_{\mathbf{B}}$ with energy eigenvalue $E^{\mathrm{s}}_{\mathbf{B}}(f^{\mathrm{s}}_{\mathbf{B}})$ while we find the k-th reservoir to be in an eigenstate $\vert {f^{k}_{\mathbf{B}}}\rangle$ of the reservoir Hamiltonian $H^{k}_{\mathbf{B}}$ with energy eigenvalue $E^{k}_{\mathbf{B}}(f^{k}_{\mathbf{B}})$. The final eigenstate |f _B 〉 for the total system is then $\vert {f_{\mathbf{B}}}\rangle \equiv \vert {f^{\mathrm{s}}_{\mathbf{B}}}\rangle \otimes \vert {f^{\mathrm{A}}_{\mathbf{B}}}\rangle \otimes \vert {f^{\mathrm{B}}_{\mathbf{B}}}\rangle$. Note that the set of all the initial eigenstates for the total system is the same as the set of all the final eigenstates: {|i _B 〉}={|f _B 〉}.

According to (2.20), $\vert {{}^{\varTheta }i^{\mathrm{s}}_{\mathbf{B}}}\rangle \equiv\varTheta \vert {i^{\mathrm{s}}_{\mathbf{B}}}\rangle$ is an eigenstate of $H^{\mathrm{s}}_{-\mathbf{B}}={}^{\varTheta }H^{\mathrm{s}}_{\mathbf{B}}$ with the energy eigenvalue $E^{\mathrm{s}}_{\mathbf{B}}(i^{\mathrm{s}}_{\mathbf{B}})$ so that

(A.12)

and $\vert {{}^{\varTheta }f^{k}_{\mathbf{B}}}\rangle \equiv\varTheta \vert {{}^{\varTheta }f^{k}_{\mathbf{B}}}\rangle$ is an eigenstate of $H^{k}_{-\mathbf{B}}={}^{\varTheta }H^{k}_{\mathbf{B}}$ with the energy eigenvalue $E^{k}_{\mathbf{B}}(f^{k}_{\mathbf{B}})$ so that

(A.13)

1.5 A.5 Cumulant Generating Function for Heat Current

By applying the first-order time-dependent perturbation theory [23], where we assume H ^int to be weak, we find that |〈f _B |U _B |i _B 〉|² is appreciable only when

(A.14)

For sufficiently long τ, we can then assume that

(A.15)

where $\Delta E^{\mathrm{s}}_{\mathbf{B}}$ for the system is defined by

(A.16)

and $\Delta E^{k}_{\mathbf{B}}$ for the k-th reservoir is defined by

(A.17)

For a forward process for the total system from an initial eigenstate |i _B 〉 to a final eigenstate |f _B 〉, we define the heat transferred from the k-th reservoir into the system by

(A.18)

and the average heat current through the system by

(A.19)

We then define the cumulant generating function for the heat current by

(A.20)

where the transient process average is defined by

(A.21)

In the following, we will assume that the limit in (A.20) exists. Using the cumulant generating function, we can then obtain the average heat current by

(A.22)

and the thermal conductivity by

(A.23)

$\langle {\!\langle {J^{Q}_{\mathbf{B}}}\rangle \!}\rangle$ can also be written as

(A.24)

where $\tilde {\rho}_{\mathbf{B}}$, the density matrix corresponding to the initial canonical ensemble distribution for the total system (A.9) is defined by

(A.25)

We have also defined the heat current operator $\tilde {J}^{Q}_{\mathbf{B},\mathrm{F}}$ in the Heisenberg picture by

(A.26)

where $H^{k}_{\mathbf{B},\mathrm{F}}(t)\equiv U_{\mathbf{B}}(t)^{\dag}H^{k}_{\mathbf{B}}U_{\mathbf{B}}(t)$ with U _B (t) being the time evolution operator for the total system at t and

(A.27)

so that

(A.28)

Recalling $Q^{k}_{\mathbf{B}}=- \{ E^{k}_{\mathbf{B}}(f^{k}_{\mathbf{B}})-E^{k}_{\mathbf{B}}(i^{k}_{\mathbf{B}}) \}$ (A.18) and using $|\langle {f_{\mathbf{B}}}\vert U_{\mathbf{B}}\vert {i_{\mathbf{B}}}\rangle |^{2}=\langle {i_{\mathbf{B}}}\vert U_{\mathbf{B}} ^{\dag} \vert {f_{\mathbf{B}}}\rangle \*\langle {f_{\mathbf{B}}}\vert U_{\mathbf{B}}\vert {i_{\mathbf{B}}}\rangle$, $\sum_{f_{\mathbf{B}}}\vert {f_{\mathbf{B}}}\rangle \langle {f_{\mathbf{B}}}\vert =1$, and $\tilde {\rho}_{\mathbf{B}}\vert {i_{\mathbf{B}}}\rangle =\rho_{\mathbf{B}}(i_{\mathbf{B}})\vert {i_{\mathbf{B}}}\rangle$, we can show (A.24):

(A.29)

1.6 A.6 Fluctuation Theorem for the Heat Current

We can show the fluctuation theorem for the heat current,

(A.30)

where

(A.31)

The fluctuation theorem follows from

(A.32)

which we can show as follows.

(A.33)

In this derivation, we have used a consequence of the principle of microreversibility,

(A.34)

and

(A.35)

where (A.11) and (A.13) are used. We have also used

(A.36)

where we have used (A.15), $\Delta E^{\mathrm{s}}_{\mathbf{B}}+ \Delta E^{\mathrm{A}}_{\mathbf{B}}+\Delta E^{\mathrm{B}}_{\mathbf{B}}$, as well as

(A.37)

and

(A.38)

We have also used

(A.39)

which follows from (A.12) and (A.13).

(A.32) can be written as

(A.40)

which is similar to (2.41) for the shear stress.

1.7 A.7 Green-Kubo Formula for the Thermal Conductivity

Using the fluctuation theorem for the heat current (A.30), we find

(A.41)

Using this equation, we can derive the Green-Kubo relation,

(A.42)

which follows from

(A.43)

where we have used (A.20) and (A.23).

Recalling $Q^{k}_{\mathbf{B}}=- \{ E^{k}_{\mathbf{B}}(f^{k}_{\mathbf{B}})-E^{k}_{\mathbf{B}}(i^{k}_{\mathbf{B}}) \}$ (A.18) and using $[\tilde {\rho}_{\mathbf{B}},H^{k}_{\mathbf{B}}]=0$ and $[H^{\mathrm{A}}_{\mathbf{B}},H^{\mathrm{B}}_{\mathbf{B}}]=[H^{\mathrm{A}}_{\mathbf{B},\mathrm{F}}(\tau),H^{\mathrm{B}}_{\mathbf{B},\mathrm{F}}(\tau)]=0$, we can then show

(A.44)

Using this equation in the Green-Kubo relation (A.42), we obtain

(A.45)

where V=AL and we have defined the heat current density operator $\tilde {j}^{Q}_{\mathbf{B},\mathrm{F}}(t)$ in the Heisenberg picture by

(A.46)

Following steps similar to those for the shear viscosity, we then obtain the Green-Kubo formula for κ,

(A.47)

where the symmetrized correlation function of the heat current density operator is defined by

(A.48)

The subscript “eq” indicates that the statistical average is taken when the temperature difference between the reservoirs is kept at zero so that Δβ=0 and the system and the reservoirs are all in equilibrium at the same temperature $\overline{T}$.