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Jarzynski Equality for Conditional Stochastic Work

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Abstract

It has been established that the inclusive work for classical, Hamiltonian dynamics is equivalent to the two-time energy measurement paradigm in isolated quantum systems. However, a plethora of other notions of quantum work has emerged, and thus the natural question arises whether any other quantum notion can provide motivation for purely classical considerations. In the present analysis, we propose the conditional stochastic work for classical, Hamiltonian dynamics, which is inspired by the one-time measurement approach. This novel notion is built upon the change of expectation value of the energy conditioned on the initial energy surface. As main results, we obtain a generalized Jarzynski equality and a sharper maximum work theorem, which account for how non-adiabatic the process is. Our findings are illustrated with the parametric harmonic oscillator.

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Notes

  1. The two-time [36, 41, 42, 47, 51] as well as the one-time [50] measurement paradigm have been extended to open system dynamics. Thus, a generalization of our present discussion to open systems appears straight forward.

  2. Note that Eq. (43) is identical to a high-temperature expansion of the quantum expression \(\left\langle W\right\rangle = \hbar (Q^{*}\omega _{\tau } - \omega _{0})/2\, \coth \left( \beta \hbar \omega _0/2\right) \) [49].

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Acknowledgements

This work is supported by the U.S. Department of Energy, the Laboratory Directed Research and Development (LDRD) program and the Center for Nonlinear Studies at LANL. AS offers his gratitude to Yi-Xiang Liu and Paola Cappellaro for insightful discussions.

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Appendices

A: Proof of Eq. (26)

We show the detailed derivation of Eq. (26). The Kullblack–Leiber (KL) divergence of the conditional thermal distribution with respect to the canonical state of the final configuration is

$$\begin{aligned} D\left[ p_{B|A}\,||\,p^\mathrm {eq}_B\right] = \int dE_B\int dE_A p_{B|A}(E_A,E_B)\ln \left( \frac{p_{B|A}(E_A,E_B)}{p^{\mathrm {eq}}_B(E_B)}\right) \,. \end{aligned}$$
(51)

From Eq. (23), we have

$$\begin{aligned} \begin{aligned} \int dE_A\int dE_B&p_{B|A}(E_A,E_B)\ln p_{B|A}(E_A,E_B)\\ =&-\ln {\left( {\mathcal {Z}}(B|A)\right) }-\beta \int dE_A\int dE_B p_{B|A}(E_A,E_B)\varepsilon _B(E_A)\\&+\int dE_A\int dE_B p_{B|A}(E_A,E_B)\ln p(E_B|E_A)\,. \end{aligned} \end{aligned}$$
(52)

Here, we can explicitly write

$$\begin{aligned} \begin{aligned} \int dE_A\int dE_B&p_{B|A}(E_A,E_B)\varepsilon _B(E_A)\\&=\int dE_A\int dE_B\int dE_B'\, p(E_B|E_A)\frac{\exp {\left( -\beta \varepsilon _B(E_A)\right) }}{\mathcal {Z}(B|A)}E_B'p(E_B'|E_A)\\&=\int \!dE_A\!\int \!dE_B'\,p(E_B'|E_A) \frac{\exp {\left( -\beta \varepsilon _B(E_A)\right) }}{\mathcal {Z}(B|A)}E_B'\\&=\int dE_A \int dE_B'\,p_{B|A}(E_A,E_B')E_B'\,, \end{aligned} \end{aligned}$$
(53)

where we used

$$\begin{aligned} \int dE_B p(E_B|E_A)=1\, \end{aligned}$$
(54)

in the third line. Also, note that we have the vanishing conditional entropy

$$\begin{aligned} -\int dE_A\int dE_B p_{B|A}(E_A,E_B)\ln p(E_B|E_A) =0 \end{aligned}$$
(55)

because the time evolution of the system is deterministic due to Eq. (8) so that \(E_B\) is a function of \(E_A\) and vice versa. More precisely, by definition, we have \(E_A=H(\varGamma _0;\alpha _0)\) and \(E_B\equiv H(\varGamma _{\tau };\alpha _{\tau })\), where \(\varGamma _t\) obeys the Liouville equation, so that \(E_B\) is the function of \(E_A\). Therefore, we have

$$\begin{aligned} \begin{aligned} \int dE_A\int dE_B&p_{B|A}(E_A,E_B)\ln p_{B|A}(E_A,E_B)\\ =&-\ln {\left( {\mathcal {Z}}(B|A)\right) } -\beta \int dE_A\int dE_B p_{B|A}(E_A,E_B)E_B. \end{aligned} \end{aligned}$$
(56)

Furthermore, from \(p^{\mathrm {eq}}_B(E_B)=\exp {\left( -\beta E_B\right) }/Z_B\), we have

$$\begin{aligned}&\int dE_A\int dE_B p_{B|A}(E_A,E_B)\ln p^{\mathrm {eq}}_{B}(E_B)\nonumber \\&=-\ln {\left( Z_B\right) }-\beta \int dE_A \int dE_Bp_{B|A}(E_A,E_B)E_B. \end{aligned}$$
(57)

Therefore, from Eqs (56) and (57), we have

$$\begin{aligned} D\left[ p_{B|A}\,||\,p^\mathrm {eq}_B\right] = -\ln {\left( \frac{{\mathcal {Z}}(B|A)}{Z_B}\right) }, \end{aligned}$$
(58)

which proves Eq. (26).

B: Transition Probability Distribution in Energy and Phase Space Volume

In this appendix we prove the equivalence of the conditional transition probability distribution in energy [37] and phase space volume [63] representation. We start be considering,

$$\begin{aligned} p(\varOmega _\tau |\varOmega _0)=\frac{\int d\varGamma _0\,\delta \left( E(\varOmega _0; \alpha _0)-H(\varGamma _0; \alpha _0)\right) \,\delta \left( \varOmega _\tau -\varOmega (H(\varGamma _\tau (\varGamma _0); \alpha _\tau ); \alpha _\tau )\right) }{\int d\varGamma _0\,\delta \left( E(\varOmega _0; \alpha _0)-H(\varGamma _0; \alpha _0)\right) },\qquad \end{aligned}$$
(59)

and we recognize the denominator as density of states [37] of the initial energy surface. In general, we have

$$\begin{aligned} \chi (E; \alpha ) = \int d\varGamma \,\delta \left( E(\varOmega ; \alpha )-H(\varGamma ; \alpha )\right) =\frac{\partial \varOmega }{\partial E}\,. \end{aligned}$$
(60)

Thus, we can also write [63],

$$\begin{aligned} p(\varOmega _\tau |\varOmega _0)=\int d\varGamma _0\,\delta \left( \varOmega _\tau -\varOmega (H(\varGamma _\tau (\varGamma _0); \alpha _\tau ); \alpha _\tau )\right) \,\frac{\delta \left( E(\varOmega _0; \alpha _0)-H(\varGamma _0; \alpha _0)\right) }{\chi (E(\varOmega _0; \alpha _0); \alpha _0)}. \end{aligned}$$
(61)

Now, we note that from the elementary properties of the Dirac-\(\delta \) function we have

$$\begin{aligned} \delta \left( \varOmega _\tau -\varOmega (H(\varGamma _\tau (\varGamma _0); \alpha _\tau ); \alpha _\tau )\right) \chi (E_B; \alpha _\tau ) =\delta \left( E_B-H(\varGamma ; \alpha _\tau )\right) \,, \end{aligned}$$
(62)

where we used that \(E_B=E(\varOmega _\tau ; \alpha _\tau )\). Hence, we also have with \(E_A=E(\varOmega _0; \alpha _0)\) that

$$\begin{aligned} p(\varOmega _\tau |\varOmega _0)\, \chi (E_B; \alpha _\tau ) = p(E_B|E_A)\,. \end{aligned}$$
(63)

Equation (63) is then equivalent to

$$\begin{aligned} \int dE_{B}\,p(E_B|E_A)f(E_B)=\int _{0}^{\infty }d\varOmega _\tau \,p(\varOmega _\tau |\varOmega _0)f(E_B(\varOmega _\tau ; \alpha _\tau ))\,. \end{aligned}$$
(64)

which holds for any arbitrary test function \(f(E_B)\). In conclusion we have shown that the transition probability distributions in energy and phase space representation are, indeed, equivalent.

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Sone, A., Deffner, S. Jarzynski Equality for Conditional Stochastic Work. J Stat Phys 183, 11 (2021). https://doi.org/10.1007/s10955-021-02720-6

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