Additional energy-information relations in thermodynamics of small systems

The Clausius inequality form of the second law of thermodynamics relates information changes (entropy) to changes in the first moment of the energy (heat and indirectly also work). Are there similar relations between other moments of the energy distribution, and other information measures, or is the Clausius inequality a one of a kind instance of the energy-information paradigm? If there are additional relations, can they be used to make predictions on measurable quantities? Changes in the energy distribution beyond the first moment (average heat or work) are especially important in small systems which are often very far from thermal equilibrium. The additional energy-information relations (AEIR's), here derived, provide positive answers to the two questions above and add another layer to the fundamental connection between energy and information. To illustrate the utility of the new AEIR's, we find scenarios where the AEIR's yield tighter constraints on performance (e.g., in thermal machines) compared to the second law. To obtain the AEIR's we use the Bregman divergence—a mathematical tool found to be highly suitable for energy-information studies. The quantum version of the AEIR's provides a thermodynamic meaning to various quantum coherence measures. It is intriguing to fully map the regime of validity of the AEIR's and extend the present results to more general scenarios including continuous systems and particles exchange with the baths.

Figure 1

Blue curve: high temperature limit of the $\tilde{\alpha }$ impurity-based AEIR, applied to a three-level thermalization process (isochore). For values of $\tilde{\alpha }$ that correspond to the shaded area, the AEIR (30) imposes a tighter constraint on the standard ${\mathcal{Q}}_1$ heat (red curve) compared to the standard second law $\tilde{\alpha }=1$ (dashed-horizontal line).

Figure 2

In systems with three or more levels it is possible to have very different distributions with the same Shannon entropy $S$. Thus, it is possible to have transformations for which the Shannon entropy does not change, $S\left({\vec{p}}_f\right)-S\left({\vec{p}}_i\right)=0$, while other information measures, such as ${\mathcal{S}}_2$ shown in the figure, do change, ${\mathcal{S}}_2\left({\vec{p}}_f\right)-{\mathcal{S}}_2\left({\vec{p}}_i\right)\ne 0$.

Figure 3

For a two-level four-stroke irreversible engine (a), the $\alpha <1$ AIER's provide a tighter bound on the efficiency [blue curve in (b)], compared to the $\alpha =1$ standard Carnot efficiency bound (red line). The green line shows the exact efficiency of this engine. Remarkably, when both temperatures are decreased by the same factor (3 in this example), the new $\alpha \mathrm{EIR}$ bound (magenta) converges to the exact efficiency even though the machine is irreversible.

Figure 4

This “half zero” heat machine reduces the energy variance of the cold bath ($-{\mathcal{Q}}_{2,c}\le 0$) without changing the average energy of the bath ${\mathcal{Q}}_{1,c}=0$. The standard Clausius inequality ($\alpha =1$) only predicts that the energy goes into the hot bath ${\mathcal{Q}}_{1,h}\le 0$ but gives no information on the changes in the cold bath (which concerns the main functionality of the device). In contrast, the $\alpha =2$ AEIR (12) predicts $-{\mathcal{Q}}_{2,c}/{\mathcal{Q}}_{2,h}\le {\left(T_c/T_h\right)}^2$. In this example $-{\mathcal{Q}}_{2,c}/{\mathcal{Q}}_{2,h}\simeq 0.234$ while the AEIR prediction is ${\left(T_c/T_h\right)}^2=0.25$.

Figure 5

(a) The (concave) Bregman divergence has a simple geometric interpretation. It is the difference between the linear extrapolation of a concave function (blue) and the actual function (green). (b) Three-state Bregman identity and its geometric interpretation. The right hand side determines the regime of validity of the AEIR's. Geometrically, it is the length of the blue line (initial divergence with respect to the reference) minus the length of the green line (final divergence with respect to the reference). If the final state is closer to the reference state (but does not overshoot to the other side) then the divergence difference is guaranteed to be positive as needed for the AEIR's. (c) By using concatenation of isochores and adiabats it is possible to construct various thermodynamic protocols that involve both heat and work such as isotherms. See text for the analysis.

Figure 6

A three-stage reversible state-preparation protocol $\{{\vec{p}}_i,H_i\}\to \{{\vec{p}}_f,H_f\}$. Stage A changes the populations but the end point Hamiltonian is the same as the initial Hamiltonian. Stage B is an isotherm in which $H_i$ is changed to $H_f$. Finally, stage C changes the thermal state to the desired final state. This protocol is identical to state preparation with $\alpha =1$ but we monitor other quantities in the process (${\mathcal{W}}_{\alpha }$ and ${\mathcal{Q}}_{\alpha }$). In this protocol the AEIR's become equalities and they give the exact amount of extracted ${\mathcal{W}}_{\alpha }$ and ${\mathcal{Q}}_{\alpha }$.

Figure 7

(a) Various system-bath configurations, and the interaction between elements (b). Resonant interaction: each energy quanta taken from the bath particles is given to the system and vice versa. This type of interaction enables to easily relate changes in energy moments of the system, to changes in the bath.