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On the Jarzynski relation for dissipative quantumdynamics

Description: In this note, we will discuss how to compactly express the Jarzynski identity for an open quantum system with dissipative dynamics. In quantum dynamics we must avoid explicitly measuring the work directly, which is tantamount to continuously monitoring the state of the system, and instead measure the heat ?ow from the environment. These measurements can be concisely represented with Hermitian map superoperators, which provide a convenient and compact representations of correlation functions and sequential measurements of quantum systems.
Date: October 30, 2008
Creator: Crooks, Gavin E
Partner: UNT Libraries Government Documents Department

On thermodynamic and microscopic reversibility

Description: The word 'reversible' has two (apparently) distinct applications in statistical thermodynamics. A thermodynamically reversible process indicates an experimental protocol for which the entropy change is zero, whereas the principle of microscopic reversibility asserts that the probability of any trajectory of a system through phase space equals that of the time reversed trajectory. However, these two terms are actually synonymous: a thermodynamically reversible process is microscopically reversible, and vice versa.
Date: July 12, 2011
Creator: Crooks, Gavin E.
Partner: UNT Libraries Government Documents Department

Quantum Operation Time Reversal

Description: The dynamics of an open quantum system can be described by a quantum operation: A linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes toward equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.
Date: March 25, 2008
Creator: Crooks, Gavin E.
Partner: UNT Libraries Government Documents Department

Measuring Thermodynamic Length

Description: Thermodynamic length is a metric distance between equilibrium thermodynamic states. Among other interesting properties, this metric asymptotically bounds the dissipation induced by a finite time transformation of a thermodynamic system. It is also connected to the Jensen-Shannon divergence, Fisher information, and Rao's entropy differential metric. Therefore, thermodynamic length is of central interestin understanding matter out of equilibrium. In this Letter, we will consider how to denethermodynamic length for a small system described by equilibrium statistical mechanics and how to measure thermodynamic length within a computer simulation. Surprisingly, Bennett's classic acceptance ratio method for measuring free energy differences also measures thermodynamic length.
Date: September 7, 2007
Creator: Crooks, Gavin E
Partner: UNT Libraries Government Documents Department

Comment regarding"On the Crooks fluctuation theorem and the Jarzynski equality" [J. Chem. Phys. 129, 091101 (2008)]and"Nonequilibrium fluctuation-dissipation theorem of Brownian dynamics" [J. Chem. Phys. 129, 144113 (2008)]

Description: The incongruous"unexpected inapplicability of the [Crook's fluctuation theorem]" is due to an inexplicable, inappropriate use of inconsistent expressions. The girding is secure.
Date: March 4, 2009
Creator: Crooks, Gavin E
Partner: UNT Libraries Government Documents Department

Far-from-equilibrium measurements of thermodynamic length

Description: Thermodynamic length is a path function that generalizes the notion of length to the surface of thermodynamic states. Here, we show how to measure thermodynamic length in far-from-equilibrium experiments using the work fluctuation relations. For these microscopic systems, it proves necessary to define the thermodynamic length in terms of the Fisher information. Consequently, the thermodynamic length can be directly related to the magnitude of fluctuations about equilibrium. The work fluctuation relations link the work and the free energy change during an external perturbation on a system. We use this result to determine equilibrium averages at intermediate points of the protocol in which the system is out-of-equilibrium. This allows us to extend Bennett's method to determine the potential of mean force, as well as the thermodynamic length, in single molecule experiments.
Date: November 5, 2008
Creator: Feng, Edward H. & Crooks, Gavin E.
Partner: UNT Libraries Government Documents Department

The Length of Time's Arrow

Description: An unresolved problem in physics is how the thermodynamic arrow of time arises from an underlying time reversible dynamics. We contribute to this issue by developing a measure of time-symmetry breaking, and by using the work fluctuation relations, we determine the time asymmetry of recent single molecule RNA unfolding experiments. We define time asymmetry as the Jensen-Shannon divergencebetween trajectory probability distributions of an experiment and its time-reversed conjugate. Among other interesting properties, the length of time's arrow bounds the average dissipation and determines the difficulty of accurately estimating free energy differences in nonequilibrium experiments.
Date: August 21, 2008
Creator: Feng, Edward H. & Crooks, Gavin E.
Partner: UNT Libraries Government Documents Department

Beyond Boltzmann-Gibbs statistics: Maximum entropy hyperensemblesout-of-equilibrium

Description: What is the best description that we can construct of athermodynamic system that is not in equilibrium, given only one, or afew, extra parameters over and above those needed for a description ofthe same system at equilibrium? Here, we argue the most appropriateadditional parameter is the non-equilibrium entropy of the system, andthat we should not attempt to estimate the probability distribution ofthe system, but rather the metaprobability (or hyperensemble) that thesystem is described by a particular probability distribution. The resultis an entropic distribution with two parameters, one a non-equilibriumtemperature, and the other a measure of distance from equilibrium. Thisdispersion parameter smoothly interpolates between certainty of acanonical distribution at equilibrium and great uncertainty as to theprobability distribution as we move away from equilibrium. We deducethat, in general, large, rare fluctuations become far more common as wemove away from equilibrium.
Date: February 23, 2006
Creator: Crooks, Gavin E.
Partner: UNT Libraries Government Documents Department

Nonequilibrium candidate Monte Carlo: A new tool for efficient equilibrium simulation

Description: Metropolis Monte Carlo simulation is a powerful tool for studying the equilibrium properties of matter. In complex condensed-phase systems, however, it is difficult to design Monte Carlo moves with high acceptance probabilities that also rapidly sample uncorrelated configurations. Here, we introduce a new class of moves based on nonequilibrium dynamics: candidate configurations are generated through a finite-time process in which a system is actively driven out of equilibrium, and accepted with criteria that preserve the equilibrium distribution. The acceptance rule is similar to the Metropolis acceptance probability, but related to the nonequilibrium work rather than the instantaneous energy difference. Our method is applicable to sampling from both a single thermodynamic state or a mixture of thermodynamic states, and allows both coordinates and thermodynamic parameters to be driven in nonequilibrium proposals. While generating finite-time switching trajectories incurs an additional cost, driving some degrees of freedom while allowing others to evolve naturally can lead to large enhancements in acceptance probabilities, greatly reducing structural correlation times. Using nonequilibrium driven processes vastly expands the repertoire of useful Monte Carlo proposals in simulations of dense solvated systems.
Date: November 8, 2011
Creator: Nilmeier, Jerome P.; Crooks, Gavin E.; Minh, David D. L. & Chodera, John D.
Partner: UNT Libraries Government Documents Department

Bayesian Estimates of Free Energies from Nonequilibrium Work Data in the Presence of Instrument Noise

Description: The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to nonequilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not been considered previously. Here, we present a Bayesian formalism for estimating free energy changes from nonequilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise and, therefore, provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with atomic force microscopy, optical, and magnetic tweezers
Date: July 8, 2008
Creator: Maragakis, Paul; Ritort, Felix; Bustamante, Carlos; Karplus, Martin & Crooks, Gavin E.
Partner: UNT Libraries Government Documents Department

On the work distribution for the adiabatic compression of a diluteclassical gas

Description: We consider the adiabatic and quasi-static compression of adilute classical gas, confined in a piston and initially equilibratedwith a heat bath. We find that the work performed during this process isdescribed statistically by a gamma distribution. We use this result toshow that the model satisfies the non-equilibrium work and fluctuationtheorems, but not the fluctation-dissipation relation. We discuss therare but dominant realizations that contribute most to the exponentialaverage of the work, and relate our results to potentially universal workdistributions.
Date: February 23, 2006
Creator: Crooks, Gavin E. & Jarzynski, Christopher
Partner: UNT Libraries Government Documents Department