Finite-bath corrections to the second law of thermodynamics

The second law of thermodynamics states that a system in contact with a heat bath can undergo a transformation if and only if its free energy decreases. However, the “if” part of this statement is only true when the effective heat bath is infinite. In this article we remove this idealization and derive corrections to the second law in the case where the bath has a finite size, or equivalently finite heat capacity. This can also be translated to processes lasting a finite time, and we show that thermodynamical reversibility is lost in this regime. We do so in full generality, without assuming any particular model for the bath; the only parameters defining the bath are its temperature and heat capacity. We find connections with second order Shannon information theory, in particular, in the case of Landauer erasure. We also consider the case of nonfluctuating work and derive finite-bath corrections to the min and max free energies employed in single-shot thermodynamics.

Figure 1

Majorization diagram of a particular subspace of total energy $E$ of the joint system-bath product state, where the system is in state $\rho ={\sum }_{s}p\left(s\right)|s\rangle \langle s|$ diagonal in energy, and with the energy levels $\beta$ ordered such that $p\left(k\right)e^{\beta E_k}\ge p(k+1)e^{\beta E_{k+1}}$.

Figure 2

Example of a thermomajorization diagram for three different states of a three-level system, $\rho , \sigma$, and the thermal state, or a total energy of $E$. The $\beta$ order in each is different, and at this particular energy the transition $\rho \to \sigma$ is possible, as the diagram of $\rho$ lies above that of $\sigma$.

Figure 3

Curves for the maximum work extracted and minimum work of formation for a given state $\rho$, given each particular total energy. The distribution of total energies (and hence the distribution of ${\beta }^{\prime }=\beta -\gamma E$) is the superposed Gaussian curve, for which a cut in the tails gives the values of effective temperatures ${\beta }_{-}$ and ${\beta }_{+}$ that determine the work of formation and the extractable work given some error probability, whose value is equal to the orange region.