Work and heat fluctuations in two-state systems: a trajectory thermodynamics formalism

Abstract

Two-state models provide phenomenological descriptions of many different systems, ranging from physics to chemistry and biology. We investigate work fluctuations in an ensemble of two-state systems driven out of equilibrium under the action of an external perturbation. We calculate the probability density P(W) that a work equal to W is exerted upon the system along a given non-equilibrium trajectory and introduce a trajectory thermodynamics formalism to quantify work fluctuations in the large-size limit. We then define a trajectory entropy S(W) that counts the number of non-equilibrium trajectories P(W)=exp(S(W)/kT) with work equal to W. A trajectory free-energy F(W) can also be defined, which has a minimum at a value of the work that has to be efficiently sampled to quantitatively test the Jarzynski equality. Within this formalism a Lagrange multiplier is also introduced, the inverse of which plays the role of a trajectory temperature. Our solution for P(W) exactly satisfies the fluctuation theorem by Crooks and allows us to investigate heat-fluctuations for a protocol that is invariant under time reversal. The heat distribution is then characterized by a Gaussian component (describing small and frequent heat exchange events) and exponential tails (describing the statistics of large deviations and rare events). For the latter, the width of the exponential tails is related to the aforementioned trajectory temperature. Finite-size effects to the large-N theory and the recovery of work distributions for finite N are also discussed. Finally, we pay particular attention to the case of magnetic nanoparticle systems under the action of a magnetic field H where work and heat fluctuations are predicted to be observable in ramping experiments in micro-SQUIDs.