Third Law of Thermodynamics as a Single Inequality

The third law of thermodynamics in the form of the unattainability principle states that exact ground-state cooling requires infinite resources. Here, we investigate the amount of nonequilibrium resources needed for approximate cooling. We consider as a resource any system out of equilibrium, allowing for resources beyond the independent and identically distributed assumption and including the input of work as a particular case. We establish in full generality a sufficient and a necessary condition for cooling and show that, for a vast class of nonequilibrium resources, these two conditions coincide, providing a single necessary and sufficient criterion. Such conditions are expressed in terms of a single function playing a role for the third law similar to the one of the free energy for the second law. From a technical point of view, we provide new results about the concavity or convexity of certain Renyi divergences, which might be of independent interest.

Popular Summary

The laws of thermodynamics were originally qualitative expressions about the impossibility of certain tasks. The famous second law, for example, dictates (among other possible formulations) that perpetual motion machines are impossible. Nowadays, we understand the second law as a precise quantitative limitation on how much work one can extract given any initial condition or, more generally, which state transitions are possible via interaction with a source of heat. Those limitations are expressed with functions such as the free energy, which tells us precisely what is possible under the second law. The third law of thermodynamics, which says that it is impossible to cool any object to exactly absolute zero, has no such formulation. We have constructed a quantitative version of the third law that predicts exactly how close to zero one can get given some amount of fuel to be spent.

We formulate this limitation with a function that we call the vacancy, which is analogous to the free energy but here is applied to cooling. This function measures the cooling power of a given substance and allows one to formulate the third law compactly and in symmetry with the second law. Using the vacancy, we also quantify the amount of resources needed to cool a system close to its ground state.

Our formulation could open several pathways for future work, such as revealing the optimal size of the heat bath, identifying the particular catalyst needed for the cooling process, and quantifying the time or complexity needed to achieve our fundamental bounds. We expect that our findings will also be useful for estimating the fundamental resources required to operate quantum computers, which need a large amount of very cool qubits to operate.

Figure 1

The figure shows the behavior of $S_{\alpha }\mathbf{(}{\omega }_{{\beta }_S}(H_S)\parallel {\omega }_{\beta }(H_S)\mathbf{)}$ (orange line) and $S_{\alpha }\mathbf{(}{\rho }_R\parallel {\omega }_{\beta }(H_R)\mathbf{)}$ (blue line). The left plot shows a target state that is not very cold, together with an insufficient resource. The transition is not possible since the blue line is below the orange line for $\alpha \lesssim 1.25$. The right plot shows the behavior when ${\rho }_S$ becomes very cold. The function becomes more similar to a step function. The fact that $S_{\alpha }\mathbf{(}{\rho }_R\parallel {\omega }_{\beta }(H_R)\mathbf{)}$ (blue curve) is larger than the orange curve, which implies that the transition is possible, is determined by the behavior at very small values of $\alpha$, and up to a small error, by the fact that the derivate of the blue curve is larger than that of the orange curve at $\alpha =0$.

Figure 2

The left-hand side of Eq. (b11) is represented by the red striped area under the curve. The right-hand side corresponds to the blue striped region. This can be seen by noting that the blue striped region can be decomposed into a rectangle of sides $\tilde{\beta }-{\beta }_R$ and $E_{\tilde{\beta }}$ (light blue), which corresponds to the first term on the right-hand side of Eq. (b11), and a triangle that corresponds to the second term. If the function is $E_x$ convex, the red region is always larger than the blue region.