Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages

Minh, David D. L.; Chodera, John D.

doi:10.1063/1.3242285

Cited by 53 publications

(88 citation statements)

References 62 publications

(132 reference statements)

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“…21 Though presented in the context of nonequilibrium experiments (in which the consequences of the Crooks fluctuation theorem 22,23 were explored), the estimator is sufficiently general that it applies to equilibrium trajectories sampled from different equilibrium thermodynamic states within the same thermodynamic ensemble, producing an asymptotically optimal estimate of properties within the thermodynamic state of interest. The generalized path ensemble estimator 21 is in turn based on the statistical inference framework of extended bridge sampling, [24][25][26] which provides a solid statistical foundation for earlier estimation and reweighting schemes found in statistical physics and chemistry. 12-14, 16, 27 Here, we briefly review the general estimator formalism and examine its application to common schemes used to model dynamics at constant temperature.…”

Section: B Dynamical Reweightingmentioning

confidence: 99%

“…This iterative procedure is continued until these estimates converge to within some specified tolerance. 17,21 For numerical stability, it is convenient to work with ln Z (n) i instead of Z (n) i directly. 17 In the canonical ensemble, the probability distributions are parameterized by a temperature T , or equivalently, the inverse temperature β ≡ (k B T ) −1 .…”

Section: B Dynamical Reweightingmentioning

confidence: 99%

“…17,21 Briefly, the estimating equations are linearized about the optimal estimator and the variance in the estimating equations is propagated into the corresponding variance in the estimator (see, for example, the Appendix in Ref. 26).…”

Section: B Dynamical Reweightingmentioning

confidence: 99%

“…(13) may run into numerical issues; in this case, it is recommended that the relative uncertainty varÂ(β)/[Â(β)] 2 be computed instead. Similarly, the covariance between two estimatesÂ(β) andB(β ) at respective temperatures β and β (which may be the same or different) can be computed by augmenting W with additional columns, 17,21 …”

Section: B Dynamical Reweightingmentioning

confidence: 99%

“…17,21 For purposes of numerical stability, it is convenient to work with the logarithms of quantities like q[…”

Section: B Dynamical Reweightingmentioning

confidence: 99%

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Dynamical reweighting: Improved estimates of dynamical properties from simulations at multiple temperatures

Chodera

Swope²,

Noé³

et al. 2011

The Journal of Chemical Physics

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Dynamical averages based on functionals of dynamical trajectories, such as time-correlation functions, play an important role in determining kinetic or transport properties of matter. At temperatures of interest, the expectations of these quantities are often dominated by contributions from rare events, making the precise calculation of these quantities by molecular dynamics simulation difficult. Here, we present a reweighting method for combining simulations from multiple temperatures (or from simulated or parallel tempering simulations) to compute an optimal estimate of the dynamical properties at the temperature of interest without the need to invoke an approximate kinetic model (such as the Arrhenius law). Continuous and differentiable estimates of these expectations at any temperature in the sampled range can also be computed, along with an assessment of the associated statistical uncertainty. For rare events, aggregating data from multiple temperatures can produce an estimate with the desired precision at greatly reduced computational cost compared with simulations conducted at a single temperature. Here, we describe use of the method for the canonical (NVT) ensemble using four common models of dynamics (canonical distribution of Hamiltonian trajectories, Andersen thermostatting, Langevin, and overdamped Langevin or Brownian dynamics), but it can be applied to any thermodynamic ensemble provided the ratio of path probabilities at different temperatures can be computed. To illustrate the method, we compute a time-correlation function for solvated terminallyblocked alanine peptide across a range of temperatures using trajectories harvested using a modified parallel tempering protocol.

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Section: B Dynamical Reweightingmentioning

confidence: 99%

Section: B Dynamical Reweightingmentioning

confidence: 99%

Section: B Dynamical Reweightingmentioning

confidence: 99%

Section: B Dynamical Reweightingmentioning

confidence: 99%

“…17,21 For purposes of numerical stability, it is convenient to work with the logarithms of quantities like q[…”

Section: B Dynamical Reweightingmentioning

confidence: 99%

See 3 more Smart Citations

Dynamical reweighting: Improved estimates of dynamical properties from simulations at multiple temperatures

Chodera

Swope²,

Noé³

et al. 2011

The Journal of Chemical Physics

Self Cite

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On the accurate estimation of free energies using the jarzynski equality

et al. 2018

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The Jarzynski equality is one of the most widely celebrated and scrutinized nonequilibrium work theorems, relating free energy to the external work performed in nonequilibrium transitions. In practice, the required ensemble average of the Boltzmann weights of infinite nonequilibrium transitions is estimated as a finite sample average, resulting in the so‐called Jarzynski estimator, normalΔFfalse^J. Alternatively, the second‐order approximation of the Jarzynski equality, though seldom invoked, is exact for Gaussian distributions and gives rise to the Fluctuation‐Dissipation estimator normalΔFfalse^FD. Here we derive the parametric maximum‐likelihood estimator (MLE) of the free energy normalΔFfalse^ML considering unidirectional work distributions belonging to Gaussian or Gamma families, and compare this estimator to normalΔFfalse^J. We further consider bidirectional work distributions belonging to the same families, and compare the corresponding bidirectional normalΔFfalse^italicML∗ to the Bennett acceptance ratio (normalΔFfalse^BAR) estimator. We show that, for Gaussian unidirectional work distributions, normalΔFfalse^FD is in fact the parametric MLE of the free energy, and as such, the most efficient estimator for this statistical family. We observe that normalΔFfalse^ML and normalΔFfalse^italicML∗ perform better than normalΔFfalse^J and normalΔFfalse^BAR, for unidirectional and bidirectional distributions, respectively. These results illustrate that the characterization of the underlying work distribution permits an optimal use of the Jarzynski equality. © 2018 Wiley Periodicals, Inc.

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An Information Theory Approach to Nonlinear, Nonequilibrium Thermodynamics

2011

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Using the problem of ion channel thermodynamics as an example, we illustrate the idea of building up complex thermodynamic models by successively adding physical information. We present a new formulation of information algebra that generalizes methods of both information theory and statistical mechanics. From this foundation we derive a theory for ion channel kinetics, identifying a nonequilibrium ‘process’ free energy functional in addition to the well-known integrated work functionals. The Gibbs-Maxwell relation for the free energy functional is a Green-Kubo relation, applicable arbitrarily far from equilibrium, that captures the effect of non-local and time-dependent behavior from transient thermal and mechanical driving forces. Comparing the physical significance of the Lagrange multipliers to the canonical ensemble suggests definitions of nonequilibrium ensembles at constant capacitance or inductance in addition to constant resistance. Our result is that statistical mechanical descriptions derived from a few primitive algebraic operations on information can be used to create experimentally-relevant and computable models. By construction, these models may use information from more detailed atomistic simulations. Two surprising consequences to be explored in further work are that (in)distinguishability factors are automatically predicted from the problem formulation and that a direct analogue of the second law for thermodynamic entropy production is found by considering information loss in stochastic processes. The information loss identifies a novel contribution from the instantaneous information entropy that ensures non-negative loss.

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Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages

Cited by 53 publications

References 62 publications

Dynamical reweighting: Improved estimates of dynamical properties from simulations at multiple temperatures

Dynamical reweighting: Improved estimates of dynamical properties from simulations at multiple temperatures

On the accurate estimation of free energies using the jarzynski equality

An Information Theory Approach to Nonlinear, Nonequilibrium Thermodynamics

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