Fluctuations in single-shot $\epsilon$ -deterministic work extraction

There has been an increasing interest in the quantification of nearly deterministic work extraction from a finite number of copies of microscopic particles in finite time. This paradigm, the so-called single-shot $\epsilon$-deterministic work extraction, considers processes with small failure probabilities. However, the resulting fluctuations in the extracted work entailed by this failure probability have yet to be studied. In the standard thermodynamics, paradigm fluctuation theorems are powerful tools to study fluctuating quantities. Given that standard fluctuation theorems are inadequate for a single-shot scenario, here we formulate and prove a fluctuation relation specific to the single-shot $\epsilon$-deterministic work extraction to bridge this gap. Our results are general in the sense that we allow the system to be in contact with the heat bath at all times. As a corollary of our theorem, we derive the known bounds on the $\epsilon$-deterministic work.

Figure 1

In the forward process, the system is initially prepared in the state ${\rho }_s$. Evolving the system and the bath together according to some global unitary operator correlates the state of the two. The transformation is such that a work $W^{\delta }$ is extracted and the final state of the system ${\tau }_s^{\delta }$ is close to the Gibbs state. The backward process is defined by the dual operator to the unitary in the forward process, and the initial state of the bath and the system are correlated, such that tracing out the state of the bath leaves the system in the Gibbs state. The final state of the backward process is in the ball of $\epsilon$-close states to the initial state ${\rho }_s$.

Figure 2

A weight $\epsilon$ is being taken out after $\beta$ ordering of the initial state. $\beta$ ordering refers to rearranging the eigenstates' energies in decreasing order of their weight $P\left(E_i\right)e^{\beta E_i}$. Since we are interested in removing as many eigenstates as possible while keeping a total weight of at least $1-\epsilon$, we cut from the far right end of the $\beta$-ordered spectrum. The white area shows the state that is $\epsilon$-close to the initial state and in our case maximizes the Renyi relative entropy of order 0 in Eq. (1).