Clausius inequality beyond the weak-coupling limit: The quantum Brownian oscillator

We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is noncanonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature $T_{\text{eff}}^{\star }$ with an effective mass and an effective spring constant. We derive an effective Clausius inequality $\delta {\mathcal{Q}}_{\text{eff}}^{\star }\le T_{\text{eff}}^{\star }dS$, where $\delta {\mathcal{Q}}_{\text{eff}}^{\star }$ is the heat exchanged between the effective (weakly coupled) oscillator and the bath, and $S$ represents a thermal entropy of the effective oscillator, being identical to the von-Neumann entropy of the coupled oscillator. Using this inequality (for a cyclic process in terms of a variation of the coupling strength) we confirm the validity of the second law. For a fixed coupling strength this inequality can also be tested for a process in terms of a variation of either the oscillator mass or its spring constant. Then it is never violated. The properly defined Clausius inequality is thus more robust than assumed previously.

Figure 1

(Color online) $y=k_0/k_{\text{eff}}^{\star }=2/\left(1+\left(\Omega +\gamma \right){⟨{\widehat{p}}^2⟩}_{\beta }/\left\{\Omega {\left(M{\mathbf{w}}_0\right)}^2{⟨{\widehat{q}}^2⟩}_{\beta }\right\}\right)$ versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature), where ${\mathbf{w}}_0$ is the renormalized eigen frequency of the oscillator ${\widehat{H}}_s$ [cf. Eq. (9)]; for $k_{\text{eff}}^{\star }$ refer to Eq. (39). From bottom to top, (blue solid: damping parameter $\gamma =10$, overdamped), (black dash: $\gamma =4$, overdamped), (green solid: $\gamma =3/2$, underdamped) and (red dash: $\gamma =1/2$, underdamped). We have $y\le 1$ so that $M/M_{\text{eff}}^{\star }=1/\left(2-y\right)\le 1$. Here $\hslash =k_B={\mathbf{w}}_0=\Omega =M=1$.

Figure 2

(Color online) $y=S_N$ (von-Neumann entropy) versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature); for $S_N$ refer to Eq. (41). From top to bottom, (blue solid: $\gamma =10$), (black dash: $\gamma =4$), (green solid: $\gamma =3/2$) and (red dash: $\gamma =1/2$). As $\gamma$ decreases, then $S_N$ decreases. Here $\hslash =k_B={\mathbf{w}}_0=\Omega =1$.

Figure 3

(Color online) $y=10\cdot \left(\partial {\mathcal{Q}}_s/\partial \gamma -T\partial S_N/\partial \gamma \right)/\hslash$ (dimensionless) versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature); for $y$ refer to Eqs. (50, 51). From bottom to top (at $T=0$), (blue solid: $\gamma =10$), (black dash: $\gamma =4$), (green solid: $\gamma =3/2$) and (red dash: $\gamma =1/2$). Here $\hslash =k_B={\mathbf{w}}_0=\Omega =M=1$.

Figure 4

(Color online) $y=100\cdot \left(\partial {\mathcal{W}}_{\text{eff}}^{\star }/\partial \gamma \right)/\hslash$ (dimensionless) versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature); for $y$ refer to Eq. (59). From top to bottom (at $T=0.5$), (blue dash: $\gamma =10$), (red solid: $\gamma =1/2$), (black dash: $\gamma =4$) and (green solid: $\gamma =3/2$). Here $\hslash =k_B={\mathbf{w}}_0=\Omega =M=1$.

Figure 5

(Color online) $y=\left(\partial {\mathcal{Q}}_s/\partial M-T_{\text{eff}}^{\star }\partial S_N/\partial M\right)/\left(\hslash {\mathbf{w}}_0/M\right)$ (dimensionless) versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature); for $y$ refer to Eq. (71). From top to bottom, (blue solid: $\gamma =10$), (black dash: $\gamma =4$), (green solid: $\gamma =3/2$) and (red dash: $\gamma =1/2$). As $\gamma$ decreases, then $y$ decreases. Here $\hslash =k_B={\mathbf{w}}_0=\Omega =M=1$.

Figure 6

(Color online) $y=\left(\partial {\mathcal{Q}}_s/\partial k_0-T_{\text{eff}}^{\star }\partial S_N/\partial k_0\right)/\left(\hslash /\left(M{\mathbf{w}}_0\right)\right)$ (dimensionless) versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature); for $y$ refer to Eq. (72). From top to bottom, (blue solid: $\gamma =10$), (black dash: $\gamma =4$), (green solid: $\gamma =3/2$) and (red dash: $\gamma =1/2$). As $\gamma$ decreases, then $y$ decreases. Here $\hslash =k_B={\mathbf{w}}_0=\Omega =M=1$.

Figure 7

(Color online) $y=\left(\partial {\mathcal{Q}}_s/\partial M-T\partial S_N/\partial M\right)/\left(\hslash {\mathbf{w}}_0/M\right)$ (dimensionless) versus $x=k_{B}T/\hslash {\mathbf{w}}_0$ (dimensionless temperature); for $y$ refer to Eq. (73). From top to bottom, (blue solid: $\gamma =10$), (black dash: $\gamma =4$), (green solid: $\gamma =3/2$) and (red dash: $\gamma =1/2$). As $\gamma$ decreases, then $y$ decreases. Here $\hslash =k_B={\mathbf{w}}_0=\Omega =M=1$.