Beyond heat baths: Generalized resource theories for small-scale thermodynamics

Thermodynamics has recently been extended to small scales with resource theories that model heat exchanges. Real physical systems exchange diverse quantities: heat, particles, angular momentum, etc. We generalize thermodynamic resource theories to exchanges of observables other than heat, to baths other than heat baths, and to free energies other than the Helmholtz free energy. These generalizations are illustrated with “grand-potential” theories that model movements of heat and particles. Free operations include unitaries that conserve energy and particle number. From this conservation law and from resource-theory principles, the grand-canonical form of the free states is derived. States are shown to form a quasiorder characterized by free operations, $d$ majorization, the hypothesis-testing entropy, and rescaled Lorenz curves. We calculate the work distillable from—and we bound the work cost of creating—a state. These work quantities can differ but converge to the grand potential in the thermodynamic limit. Extending thermodynamic resource theories beyond heat baths, we open diverse realistic systems to modeling with one-shot statistical mechanics. Prospective applications such as electrochemical batteries are hoped to bridge one-shot theory to experiments.

Figure 1

Rescaled Lorenz curves for three resources ($R_1$, $R_2$, $R_3$) and an equilibrium state ($G$). The Lorenz curve encodes the quasiorder on states, as equilibrating operations can transform $R$ into $R^{\prime }$ if and only if $L_R\left(t\right)\ge L_{R^{\prime }}\left(t\right)t\in [0,1]$. Here, $R_1\stackrel{\beta ,\mu }{\mapsto }{R}_3$, and $R_2\stackrel{\beta ,\mu }{\mapsto }{R}_3$, but $R_1$ and $R_2$ are incomparable. The equilibrium state, having the linear Lorenz curve $L_G\left(t\right)=t$, is at the bottom of the quasiorder.