Jarzynski equality and the second law of thermodynamics beyond the weak-coupling limit: The quantum Brownian oscillator

Since it was introduced, the Jarzynski equality (JE) has been attracting a great deal of interest due to its remarkable attribute. It explicitly states that if a given system, initially prepared in thermal equilibrium, is driven far from this equilibrium by an external perturbation, then this non-equilibrium process satisfies JAR97

where the symbol $W(t_{\text{\scriptsize f}})$ denotes the microscopic work performed on the system in a single run for a variation of the external parameter $\{\lambda(t)\,|\,0\leq t\leq t_{\text{\scriptsize f}}\}$ according to the pre-determined protocol, and the symbol $P(W)$ is the probability distribution of the work value $W$; the symbol $\beta=1/(k_{\mbox{\tiny B}}T)$ denotes the inverse temperature of the environment, and $\Delta F$ is the Helmholtz free energy change of the system between the initial and final states, equivalent to the average reversible work $\langle W\rangle_{\text{\scriptsize rev}}$ in the corresponding isothermal process CAL85 . As such, the average $\langle\cdots\rangle$ is evaluated over a large number of runs. Then, applying the Jensen inequality to the JE (1), we can easily obtain $\langle W\rangle\geq\Delta F$ as an expression of the second law of the thermodynamics JAR08 ; BOK10 . Further, as a generalization of JE, Crooks introduced the fluctuation theorem given by CRO98 ; CRO99

where the symbols $P_{\text{\scriptsize f}}(W)$ and $P_{\text{\scriptsize r}}(W)$ are the probability densities of performing the work $W$ when its protocol runs in the forward and reverse directions, respectively.

The JE has been verified in experiments with small-scale systems, where the fluctuations of work values are sufficiently observable JAR02 , e.g., in determination of the free energy change between the unfolded and folded conformations of a single RNA molecule LIP02 ; BUS05 . However, its strict validity is still in dispute, especially in association with the second law beyond the weak coupling between system and bath (see, e.g., an instructive critique in COH04 as well as Jarzynski’s reply in JAR04 ).

The JE has also been discussed in the quantum domain, where no notion of trajectory in the phase space is available and so instead the spectrum information has to be used for determination of the work probability distribution $P(W)$. Most of those attempts were made within isolated or weakly-coupled systems TAS00 ; MUK03 ; ROE04 ; ESP06 ; TAL07 ; MAH07 ; ENG07 ; CRO08 ; AND08 ; DEF08 ; TAK09 ; MAH10 ; MAH11 ; NGO12 . However, the finite coupling strength between system and bath in small-scale devices gives rise to some quantum subtleties which can no longer be neglected for studying the thermodynamic properties of such devices SHE02 ; MAH04 ; CAP05 .

In CAM09 , the fluctuation theorem (2), immediately reproducing the JE, was discussed for arbitrary open quantum systems with no restriction on the coupling strength. Its key idea was such that if the total system (i.e., open quantum system plus bath) is initially in a thermal canonical state, but otherwise isolated, and an external perturbation $\lambda(t)$ acts solely on the open system, then the work performed on the total system may be interpreted as the work performed on the open system alone. Then the final result was explicitly obtained [cf. (17)-(17a)] by mimicking the classical case such that the work on the system-of-interest $H_{s}(x,\lambda)$ beyond the weak-coupling limit is directly related to the free energy change of a potential of mean force JAR04

where the symbols $H_{b}(y)$ and $h_{\text{\tiny int}}(x,y)$ denote the bath and interaction Hamiltonians, respectively; note here that on the right-hand side the $\lambda$-dependency exists only in $H_{s}(x,\lambda)$ [cf. (16)]. This quantum result was subsequently applied to a solvable model (e.g., TAL09 ).

However, as will be discussed below, it is not clear if this quantum fluctuation theorem is associated directly with the system-of-interest $\hat{H}_{s}(t)$ alone, beyond the weak-coupling limit, being the quantum description of $H_{s}(x,\lambda)$ in (1); in fact, an attempt to relate the quantum description of ${\mathcal{H}}_{s}^{*}(x,\lambda)$ to the second law of thermodynamics within the coupled system $\hat{H}_{s}(t)$ would even lead to a violation of this law. In this paper we intend to resolve this issue, within a time-dependent quantum Brownian oscillator as a mathematically manageable scheme, by introducing explicitly a generalized Jarzynski equality (GJE) directly associated with the open system $\hat{H}_{s}(t)$ with no restriction on the coupling strength, and then discussing the resultant second law with no violation. The general layout of this paper is the following. In Sect. 2 we briefly review the results known from the references and useful for our discussions. In Sect. 3 we discuss the second law beyond the weak-coupling regime, which shows that the JE in its known form can be associated with this law only in the vanishingly small coupling regime. In Sect. 4 we introduce the GJE consistent with the second law, valid at an arbitrary coupling strength. Finally we give the concluding remarks of this paper in Sect. 5.

The quantum Brownian oscillator in consideration is given by the Caldeira-Leggett model Hamiltonian ING98 ; WEI08

where a system linear oscillator, a bath, and a system-bath interaction are given by

respectively. Here the angular frequency $y(t)>0$ varies in the time interval $[0,t_{\text{\scriptsize f}}]$ according to an arbitrary but pre-determined protocol (for the sake of convenience, let $y_{0}=y(0)$ in what follows), and the constants $c_{j}$ denote the coupling strengths. The total system $\hat{H}(0)$ initially prepared is in a canonical thermal equilibrium state $\hat{\rho}_{\beta}=e^{-\beta\hat{H}(0)}/Z_{\beta}(y_{0})$ with the initial partition function $Z_{\beta}(y_{0})$. Then the initial internal energy of the coupled oscillator is given by $\mbox{Tr}\{\hat{H}_{s}(0)\,\hat{\rho}_{\beta}\}=\mbox{Tr}_{s}\{\hat{H}_{s}(0)\,\hat{R}_{s}(0)\}$, where the initial state of the oscillator $\hat{R}_{s}(0)=\mbox{Tr}_{b}(\hat{\rho}_{\beta})$. Here the symbol $\mbox{Tr}_{b}$ denotes the partial trace for the bath degrees of freedom only; it is explicitly given by GRA88 ; WEI08

expressed in terms of the equilibrium fluctuations, $\langle\hat{q}^{2}\rangle_{\beta}=\mbox{Tr}(\hat{q}^{2}\hat{\rho}_{\beta})$ and $\langle\hat{p}^{2}\rangle_{\beta}=\mbox{Tr}(\hat{p}^{2}\hat{\rho}_{\beta})=M^{2}\langle\dot{\hat{q}}^{2}\rangle_{\beta}$. Beyond the weak-coupling limit, this reduced density matrix is not any longer in form of a canonical thermal state $\propto e^{-\beta\hat{H}_{s}(0)}$, immediately leading to the fact that the coupled oscillator $\hat{H}_{s}(0)$ is not with its well-defined local equilibrium temperature KIM10 .

For the below discussions, we will need the fluctuation-dissipation theorem in the initial (equilibrium) state FOR88

Here the dynamic susceptibility is given by

where the symbol $\tilde{\gamma}(\omega)$ denotes the Fourier-Laplace transformed damping kernel. This can be rewritten as LEV88

in terms of the normal-mode frequencies $\{\bar{\omega}_{k}\}$ of the total system $\hat{H}(0)$. It is straightforward to verify that $\mbox{Im}\{\tilde{\chi}(\omega+i0^{+})\}\to\pi/(2My_{0})\cdot\delta(\omega-y_{0})$ for an uncoupled (or isolated) oscillator (i.e., all system-bath coupling strengths $c_{j}\equiv 0$).

From (2), it follows that FOR85

as well-known, both of which give the initial internal energy of the coupled oscillator

In the absence of the system-bath coupling, this reduces to the well-known expression of internal energy CAL85

where the average quantum number $\langle\hat{n}\rangle_{\beta}=1/(e^{\beta\hbar y_{0}}-1)$. With $\beta\hbar\to 0$, its classical value also appears as $e_{\text{\scriptsize cl}}(\beta)=1/\beta$.

In comparison, there is a widely used alternative approach to the thermodynamic energy of the coupled oscillator in equilibrium, given by ${\mathcal{U}}_{s}^{*}(0):=-(\partial/\partial\beta)\,\ln{\mathcal{Z}}_{\beta}^{*}(y_{0})$ FOR05 ; HAE05 ; HAE06 ; FOR06 ; KIM06 ; KIM07 ; the symbol ${\mathcal{Z}}_{\beta}^{*}(y_{0})$ denotes the reduced partition function associated with the Hamiltonian of mean force CAM09 ; JAR04

and thus ${\mathcal{Z}}_{\beta}^{*}(y_{0})=\mbox{Tr}_{s}\{e^{-\beta\hat{{\mathcal{H}}}_{s}(0)}\}=Z_{\beta}(y_{0})/(Z_{b})_{\beta}$ with the partition function $(Z_{b})_{\beta}$ associated with the isolated bath $\hat{H}_{b}$. With the vanishing coupling strengths ($c_{j}=0$), this partition function precisely reduces, as required, to its standard form of $z_{\beta}(y_{0})=\{\mbox{csch}(\beta\hbar y_{0}/2)\}/2$ for an isolated oscillator. Then it can be shown that HAE06

It is seen that ${\mathcal{U}}_{s}^{*}(0)\neq U_{s}(0)$ unless the damping model is Ohmic. In fact, all thermodynamic quantities resulting from the partition function ${\mathcal{Z}}_{\beta}^{*}(y_{0})$ cannot exactly describe the well-defined thermodynamics of the reduced system $\hat{H}_{s}(0)$ in its state (2) beyond the weak-coupling limit KIM10 ; see also ING12 for interesting discussions of the different behaviors between $\hat{{\mathcal{H}}}_{s}^{*}(0)$ and $\hat{H}_{s}(0)$ in terms of the specific heat, within a free damped quantum particle given by $\hat{H}(t)$ with $y(t)\equiv 0$ in (4).

It is also instructive to consider a quasi-static (or reversible) process briefly, which the system of interest undergoes change infinitely slowly throughout. Then the coupled oscillator remains in equilibrium exactly in form of Eq. (2) in every single step such that

Here the time $t$ is understood merely as a parameter specifying the frequency value $y(t)$. Accordingly, the thermodynamic energy ${\mathcal{U}}_{s}^{*}(t)$ turns out to be in form of (11a) with $y(t)\leftarrow y_{0}$. For later discussions, we rewrite it as FOR85 ; KIM07

where the second factor of the integrand

Here the susceptibility $\tilde{\chi}_{t}(\omega)$ is given by (7) with $y(t)\leftarrow y_{0}$, as well as $\{\bar{\omega}_{k,t}\}=\{\bar{\omega}_{k}\}_{y_{0}\to y(t)}$. In the absence of the system-bath coupling, Eq. (13a) reduces to $\pi\,\delta\{\omega-y(t)\}$, as required. Likewise, the free energy defined as ${\mathcal{F}}_{s}^{*}(t):=-(1/\beta)\,\ln{\mathcal{Z}}_{\beta}^{*}\{y(t)\}$, being the total system free energy minus the bare bath free energy, can also be expressed as FOR85

where the free energy of an isolated oscillator

with $f_{\text{\scriptsize cl}}(\beta,\omega)=\{\ln(\beta\hbar\omega)\}/\beta$ in the classical limit.

Further, to see explicitly the different behaviors of (10) from its classical value (1), we now apply to the exponentiated negative total Hamiltonian $\hat{H}(t)$ in (10) the Zassenhaus formula with $\hat{X}:=\hat{H}_{s}(t)$ and $\hat{Y}:=\hat{H}_{b}+\hat{H}_{sb}$ WIL67 ,

where $\hat{C}_{2}=-(1/2)\,[\hat{X},\hat{Y}]$, and $\hat{C}_{3}=-(2/3)\,[\hat{Y},\hat{C}_{2}]-(1/3)\,[\hat{X},\hat{C}_{2}]$; this enables Eq. (10) to be rewritten as

Here the operator $\hat{C}_{3}=\hat{C}_{3}(t)$, and so the second term on the right-hand side is time-dependent, which is not the case in its classical value (1), to be noted for our discussions below.

An extension of the Crooks fluctuation theorem (2) to arbitrary open quantum systems was introduced in CAM09 , which is valid regardless of the system-bath coupling strength; this reads as

where $\Delta{\mathcal{F}}_{s}^{*}(t_{\text{\scriptsize f}})={\mathcal{F}}_{s}^{*}(t_{\text{\scriptsize f}})-{\mathcal{F}}_{s}^{*}(0)$, explicitly given in (14) for a coupled oscillator. This fluctuation theorem leads to the Jarzynski equality

and it follows that the average work performed by the external perturbation satisfies $\langle W\rangle\geq\Delta{\mathcal{F}}_{s}^{*}$. However, it is normally non-trivial to determine the work distribution $P_{\text{\scriptsize f}}(W)$ explicitly, which can be obtained only from the spectrum information of the (isolated) total system $\hat{H}(t)$. Further, from the time-dependent behavior of (16) it is not apparent whether the free energy change $\Delta{\mathcal{F}}_{s}^{*}$ is precisely tantamount to the minimum work on the system-of-interest $\hat{H}_{s}(t)$ only, especially beyond the weak-coupling limit.

For later purposes, we also point out that an explicit expression of the work distribution is known for an isolated oscillator as TAL07 ; DEF08

with the microscopic work $W(t_{\text{\scriptsize f}},y_{0})$ in a single event of the variation $\{y_{0}\to y(t_{\text{\scriptsize f}})\}$. Here $E_{n}\{y(t)\}=\hbar y(t)\,(n+1/2)$ is the instantaneous energy eigenvalue, and the symbol $P_{n}(y_{0})=e^{-\beta\,E_{n}(y_{0})}/z_{\beta}(y_{0})$ denotes the probability of the occupation number in the initial preparation at temperature $T$, as well as $P_{m,n}\{y(t_{\text{\scriptsize f}})\}\;=\;|\langle\psi_{m}(t_{\text{\scriptsize f}})|\hat{U}(t_{\text{\scriptsize f}})|\psi_{n}(0)\rangle|^{2}$ the transition probability between initial and final states $n$ and $m$, expressed in terms of the instantaneous eigenstates $|\psi_{n}(t)\rangle$ and the time-evolution $\hat{U}(t_{\text{\scriptsize f}})$ of the time-dependent harmonic oscillator. Then the JE is explicitly given by $\langle e^{-\beta\,W}\rangle_{\scriptscriptstyle{P}}=e^{-\beta\,\Delta f\{y(t_{\text{\tiny f}}),y_{0}\}}$, where the free energy change $\Delta f\{y(t_{\text{\scriptsize f}}),y_{0}\}=f\{\beta,y(t_{\text{\scriptsize f}})\}-f(\beta,y_{0})$; and the average work turns out to be $\langle W\rangle_{\scriptscriptstyle{P}}=\int_{-\infty}^{\infty}dW\,W\,P(W)=(\hbar/2)\,\{Q^{*}\,y(t_{\text{\scriptsize f}})-y_{0}\}\,\coth(\beta\hbar y_{0}/2)\neq e\{\beta,y(t_{\text{\scriptsize f}})\}-e(\beta,y_{0})$ DEF08 , where $Q^{*}\{y_{0},y(t_{\text{\scriptsize f}})\}=$

with both $X$ and $Y$ expressed in terms of the Airy functions Ai and Bi. We stress that this average work evaluated with the distribution $P(W)$ is not a quantum-mechanical expectation value of an observable TAL07 .

Now we intend to discuss the second law within an oscillator coupled to a bath at an arbitrary strength. We do this in the Drude-Ullersma model for the damping kernel in (7) with $\tilde{\gamma}_{d}(\omega+i0^{+})=\gamma_{\mbox{\tiny o}}\,\omega_{d}/(\omega_{d}-i\omega)$, where a damping parameter $\gamma_{\mbox{\tiny o}}$, representing the system-bath coupling strength, and a cut-off frequency $\omega_{d}$ ULL66 . It is convenient to adopt, in place of $\{y(t),\omega_{d},\gamma_{\mbox{\tiny o}}\}$, the parameters $\{\bar{y}_{t},\Omega_{t},\bar{\gamma}_{t}\}$ through the relations FOR06

Then it can be shown that

where $(z_{t,1},z_{t,2})=(\bar{\gamma}_{t}/2+i\bar{{\mathbf{w}}}_{t},\bar{\gamma}_{t}/2-i\bar{{\mathbf{w}}}_{t})$ with $\bar{{\mathbf{w}}}_{t}=\{\bar{y}_{t}^{2}-(\bar{\gamma}_{t}/2)^{2}\}^{1/2}$. Assuming that $\gamma_{\mbox{\tiny o}}\neq 0$, the susceptibility (7) reduces to FOR06 ; KIM06 ; KIM07

In comparison, it turns out that in an isolated case ($\gamma_{\mbox{\tiny o}}=0$), we have $\bar{\gamma}_{t}=0$ and so $y(t)=\bar{y}_{t}=\bar{{\mathbf{w}}}_{t}$ and $\omega_{d}=\Omega_{t}$, as well as $z_{t,1}=iy(t)$ and $z_{t,2}=-iy(t)$, therefore Eq. (22) reduces to $(M\{y^{2}(t)-\omega^{2}\})^{-1}$, simply real-valued; in this case, the imaginary number, $\pi i/\{2My(t)\}\cdot\delta\{\omega-y(t)\}$ must be added to this real number for the actual susceptibility.

Now we substitute (22) into (13), which gives

where the distribution

with $(\underline{\omega_{1}},\underline{\omega_{2}},\underline{\omega_{3}}):=(\Omega_{t},z_{t,1},z_{t,2})$, as well as with the normalization $\int_{0}^{\infty}d\omega\,{\mathcal{P}}_{t}^{*}(\omega)=1$ for $\gamma_{\mbox{\tiny o}}\neq 0$ ILK13 . With the aid of (21), Eq. (23a) can be rewritten as a compact expression

where both factors

It is observed that the polynomial $g_{t}(\omega)$ is non-negative, while $g_{t}^{*}(\omega)$ can be negative and so can ${\mathcal{P}}_{t}^{*}(\omega)$; the behaviors of ${\mathcal{P}}_{t}^{*}(\omega)$ are plotted in Fig. 1. Then it is easy to see that in the classical case, Eq. (23) reduces to ${\mathcal{U}}_{s,\text{\scriptsize cl}}^{*}(t)=e_{\text{\scriptsize cl}}(\beta)$ regardless of the coupling strength $\gamma_{\mbox{\tiny o}}$. Likewise, we also have the free energy expressed as

In comparison, Eq. (23a) identically vanishes in an isolated case; however, for the aforementioned reason, we have ${\mathcal{P}}_{t}^{*}(\omega)=\delta\{\omega-y(t)\}$ indeed in this case. Therefore, we may say that the smoothness of the distribution (24) reflects the system-bath coupling.

To observe explicitly how the system-bath coupling strength affects both energies ${\mathcal{U}}_{s}^{*}(t)$ and ${\mathcal{F}}_{s}^{*}(t)$, we apply to (23a) the identity obtained from the interplay between generalized functions and the theory of moments KAN04 such that

Here the symbol $\delta^{(n)}\{\cdots\}$ denotes the $n$th derivative, and the $n$th moment $\mu_{n}^{*}=\int_{-\infty}^{\infty}d\omega\,\{\omega-y(t)\}^{n}\,{\mathcal{P}}_{t}^{*}(\omega)\,\Theta(\omega)$ with the Heaviside step function $\Theta(\omega)$. Then, with the aid of (21), the formal decomposition (26) can explicitly be evaluated as

where the moments are given by $\mu_{0}^{*}(t)=1$ and

It is easy to verify that in the weak-coupling limit $\gamma_{\mbox{\tiny o}}\to 0$, all moments $\mu_{n}^{*}\to 0$ where $n\geq 1$. The substitution of (3) into (23) then allows us to have

where the summation on the right-hand side reflects all system-bath coupling. An expression with the same structure immediately follows for the free energy ${\mathcal{F}}_{s}^{*}(t)$, too.

It is a noteworthy fact that as a simple case, we have $f(\infty,\omega)=\hbar\omega/2$ at zero temperature, and so Eq. (25) can be rewritten as a reduced expression $\int_{0}^{\infty}d\omega\,f(\infty,\omega)\cdot\delta\{\omega-y_{0}^{*}(t)\}$ with $y_{0}^{*}(t):=\sum_{k=0}\bar{\omega}_{k,t}-\sum_{j=1}\omega_{j}$ [cf. (13a)]. Therefore, it looks like a free energy of the Hamiltonian $\hat{{\mathcal{H}}}_{s}^{*}(t)$ which is in an isolated pure state; there is no heat flow between system and bath at zero temperature. On the other hand, the system-of-interest $\hat{H}_{s}(t)$ is in a mixed state due to the system-bath coupling even at zero temperature [cf. (2) and (12)]. This also suggests to us that the free energy ${\mathcal{F}}_{s}^{*}(t)$ cannot exactly be associated with the reduced system $\hat{H}_{s}(t)$ alone.

Similarly to (23), we next introduce another distribution which is useful for describing the the internal energy $U_{s}(t)$ beyond the weak-coupling. By using (9) with $y_{0}\to y(t)$, we can easily get

where the distribution

reducing to $\delta\{\omega-y(t)\}$ in the identically vanishing coupling. Within the Drude-Ullersma model, we have KIM07

directly obtained from (22). Here the coefficients

satisfy the relations

which will be useful below. In the weak-coupling limit $\gamma_{\mbox{\tiny o}}\to 0$, we easily get