Exact analysis of work fluctuations in two-level systems

Abstract

This paper presents an exact probabilistic description of the work done by an external agent on a two-level system. We first develop a general scheme which is suitable for the treatment of functionals of the time-inhomogeneous Markov processes. Subsequently, we apply the procedure to the analysis of the isothermal-work probability density and we obtain its exact analytical forms in two specific settings. In both models, the state energies change with a constant velocity. On the other hand, the two models differ in their interstate transition rates. The explicit forms of the probability density allow a detailed discussion of the mean work. Moreover, we discuss the weight of the trajectories which display a smaller value of work than the corresponding equilibrium work. The results are controlled by a single dimensionless parameter which expresses the ratio of two underlying timescales: the velocity of the energy changes and the relaxation time in the case of frozen energies. If this parameter is large, the process is a strongly irreversible one and the work probability density differs substantially from a Gaussian curve.

Work Fluctuations in Slow Processes: Quantum Signatures and Optimal Control

“…The work FDR (1) is one of the pillars of classical stochastic thermodynamics: it shows that near equilibrium work fluctuations are responsible for dissipation, and conversely that any optimal slow process that minimises dissipation will subsequently minimise the fluctuations [10,11]. Furthermore, the work distribution P(w) is known to become a Gaussian distribution in slow processes, so that all higher cumulants beyond σ 2 w disappear [12][13][14][15][16]. In fact, combining the gaussianity of P(w) with Jarzynski's equality leads to Eq.…”

Unfolding kinetics of periodic DNA hairpins

“…While for a two-level system (m = 1) an explicit solution is given in [52,53], analytical treatments for larger m become increasingly more difficult. Under neglect of backward transitions, a solution for arbitrary m is given in [54].…”

Energetics and performance of a microscopic heat engine based on exact calculations of work and heat distributions

“…Secondly, some paths can enter (leave) the described family because they jump out of (into) the specified state. These two contributions correspond to the two terms on the right hand side of equation (21). Another derivation [21] is based on an explicit probabilistic construction of all possible paths and their respective probabilities.…”

“…Thereupon, in section 2.2, we particularize the generic solution to individual branches and, using the Chapman-Kolmogorov condition, we derive the solution for the limit cycle. In section 3 we employ the recently derived [21] analytical result for the work probability density under linear driving. Again, we first give the result for the generic linear driving and then we combine two such particular solutions into the final work distribution valid for the limit cycle.…”