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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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
From Eq. (23), we have
Here, we can explicitly write
where we used
in the third line. Also, note that we have the vanishing conditional entropy
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
Furthermore, from \(p^{\mathrm {eq}}_B(E_B)=\exp {\left( -\beta E_B\right) }/Z_B\), we have
Therefore, from Eqs (56) and (57), we have
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,
and we recognize the denominator as density of states [37] of the initial energy surface. In general, we have
Thus, we can also write [63],
Now, we note that from the elementary properties of the Dirac-\(\delta \) function we have
where we used that \(E_B=E(\varOmega _\tau ; \alpha _\tau )\). Hence, we also have with \(E_A=E(\varOmega _0; \alpha _0)\) that
Equation (63) is then equivalent to
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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DOI: https://doi.org/10.1007/s10955-021-02720-6