Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

Abstract

Thermodynamics is traditionally constrained to the study of macroscopic systems whose energy fluctuations are negligible compared to their average energy. Here, we push beyond this thermodynamic limit by developing a mathematical framework to rigorously address the problem of thermodynamic transformations of finite-size systems. More formally, we analyse state interconversion under thermal operations and between arbitrary energy-incoherent states. We find precise relations between the optimal rate at which interconversion can take place and the desired infidelity of the final state when the system size is sufficiently large. These so-called second-order asymptotics provide a bridge between the extreme cases of single-shot thermodynamics and the asymptotic limit of infinitely large systems. We illustrate the utility of our results with several examples. We first show how thermodynamic cycles are affected by irreversibility due to finite-size effects. We then provide a precise expression for the gap between the distillable work and work of formation that opens away from the thermodynamic limit. Finally, we explain how the performance of a heat engine gets affected when the working body is small. We find that while perfect work cannot generally be extracted at Carnot efficiency, there are conditions under which these finite-size effects vanish. In deriving our results we also clarify relations between different notions of approximate majorisation.

Collective operations can extremely reduce work fluctuations

“…With this choice, we recover the same behaviour found within single-shot thermodynamics [5,6,18,24,39], where the usual mindset is to allow a small finite error ò in order to extract w exactly. Summarising, by appropriately choosing γ as a function of N, one can interpolate between two regimes of work extraction from ρ ⊗ N : (i)the one presented here, where one fixes the extracted work, as…”

“…We also study the decay of work fluctuations in the presence of a thermal bath. In this case, fluctuation-free protocols have been extensively studied in the last years within the field of single-shot thermodynamics [5,6,[17][18][19][20][21][22][23][24][25], and our results provide new insights on the mesoscopic regime [23,24,[26][27][28][29], where N is finite and possibly small. Our considerations concern states that are diagonal in the energy basis, as the definition of work fluctuations in coherent systems is subtle and an active area of research [30].…”

“…, and observes an exponential decay of the failure probability, and (ii)the single-shot regime, where one fixes a (small) failure probability, and maximises the extracted work finding [5,6,18,24,39]. Other choices of γ lead to interesting interplays, such as (23), between the success probability and the extracted work.…”

Thermodynamic Capacity of Quantum Processes

“…Introduction.-In the quest of extending the laws of thermodynamics beyond the macroscopic regime, the resource theory of thermal operations was introduced to characterize possible transformations which could be carried out at the nano scale [1][2][3][4][5]. By imposing a set of physically motivated rules, an agent can only perform a restricted set of operations on a system, which we refer to generically as thermodynamic operations.…”

Extremal elements of a sublattice of the majorization lattice and approximate majorization

“…and k, s k (x − ν ( 1 + 2 ) ) d k=0 , respectively; and let L x−ν ( 2 ) (ω), L x ( 2 ) −ν ( 1 ) (ω) and L x−ν ( 1 + 2 ) (ω) their corresponding upper envelopes. Notice that (16) is equivalent to L x−ν ( 1 + 2 ) (ω) = L x ( 2 ) −ν ( 1 ) (ω) for all ω ∈ [0, d].…”