Phase transition in the Jarzynski estimator of free energy differences

The transition between a regime in which thermodynamic relations apply only to ensembles of small systems coupled to a large environment and a regime in which they can be used to characterize individual macroscopic systems is analyzed in terms of the change in behavior of the Jarzynski estimator of equilibrium free energy differences from nonequilibrium work measurements. Given a fixed number of measurements, the Jarzynski estimator is unbiased for sufficiently small systems. In these systems, the directionality of time is poorly defined and configurations that dominate the empirical average, but which are in fact typical of the reverse process, are sufficiently well sampled. As the system size increases the arrow of time becomes better defined. The dominant atypical fluctuations become rare and eventually cannot be sampled with the limited resources that are available. Asymptotically, only typical work values are measured. The Jarzynski estimator becomes maximally biased and approaches the exponential of minus the average work, which is the result that is expected from standard macroscopic thermodynamics. In the proper scaling limit, this regime change can be described in terms of a phase transition in variants of the random energy model (REM). This correspondence is explicitly demonstrated in several examples of physical interest: near-equilibrium processes in which the work distribution is Gaussian, the sudden compression of an ideal gas and adiabatic quasi-static volume changes in a dilute real gas.Comment: 29 pages, 5 figures, accepted for publication in Physical Review E (2012

Rare Events and Scale--Invariant Dynamics of Perturbations in Delayed Dynamical Systems

We study the dynamics of perturbations in time delayed dynamical systems. Using a suitable space-time coordinate transformation, we find that the time evolution of the linearized perturbations (Lyapunov vector) can be mapped to the linear Zhang surface growth model [Y.-C. Zhang, J. Phys. France {\bf 51}, 2129 (1990)], which is known to describe surface roughening driven by power-law distributed noise. As a consequence, Lyapunov vector dynamics is dominated by rare random events that lead to non-Gaussian fluctuations and multiscaling properties.Comment: Final version to appear in Phys. Rev. Lett., 4 pages, 3 eps fig

On the Kinematics of GRB980425 and its association with SN1998bw

In this paper I put forward a model in which GRB980425 is both associated with SN1998bw and is also a standard canonical (long; ~seconds) gamma-ray burst. Herein it is argued that if gamma-ray bursts are relativistic jets with the fastest moving material at the core, then the range of observed jet inclinations to the line-of-sight produces a range in the observed properties of GRBs, i.e. the lag-luminosity relationship. In particular, if the jet inclination is high enough, the observed emitter will move slowly enough to render relativistic beaming ineffective, thus distinguishing the jet from apparent isotropic emission. Thus we expect a break in the lag-luminosity relationship. I propose that GRB980425 defines that break. The position of this break gives important physical parameters such as the Lorentz factor (\gamma_max ~ 1000), the jet opening angle (~1 degree), and thus the beaming fraction (~10^-4). Estimates of burst rates are consistent with observation. If correct, this model is evidence in favor of the collapsar mode as the progenitor of cosmological, long gamma-ray bursts.Comment: 7 pages, including 1 figure. Submitted to ApJ Letter

Extreme Quantum Advantage for Rare-Event Sampling

We introduce a quantum algorithm for efficient biased sampling of the rare events generated by classical memoryful stochastic processes. We show that this quantum algorithm gives an extreme advantage over known classical biased sampling algorithms in terms of the memory resources required. The quantum memory advantage ranges from polynomial to exponential and when sampling the rare equilibrium configurations of spin systems the quantum advantage diverges.Comment: 11 pages, 9 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/eqafbs.ht

Deformed Jarzynski Equality

The well-known Jarzynski equality, often written in the form $e^{-\beta\Delta F}=\langle e^{-\beta W}\rangle$, provides a non-equilibrium means to measure the free energy difference $\Delta F$ of a system at the same inverse temperature $\beta$ based on an ensemble average of non-equilibrium work $W$. The accuracy of Jarzynski's measurement scheme was known to be determined by the variance of exponential work, denoted as ${\rm var}\left(e^{-\beta W}\right)$. However, it was recently found that ${\rm var}\left(e^{-\beta W}\right)$ can systematically diverge in both classical and quantum cases. Such divergence will necessarily pose a challenge in the applications of Jarzynski equality because it may dramatically reduce the efficiency in determining $\Delta F$. In this work, we present a deformed Jarzynski equality for both classical and quantum non-equilibrium statistics, in efforts to reuse experimental data that already suffers from a diverging ${\rm var}\left(e^{-\beta W}\right)$. The main feature of our deformed Jarzynski equality is that it connects free energies at different temperatures and it may still work efficiently subject to a diverging ${\rm var}\left(e^{-\beta W}\right)$. The conditions for applying our deformed Jarzynski equality may be met in experimental and computational situations. If so, then there is no need to redesign experimental or simulation methods. Furthermore, using the deformed Jarzynski equality, we exemplify the distinct behaviors of classical and quantum work fluctuations for the case of a time-dependent driven harmonic oscillator dynamics and provide insights into the essential performance differences between classical and quantum Jarzynski equalities.Comment: 24 pages, 1 figure, accepted version to appear in Entropy (Special Issue on "Quantum Thermodynamics"