On the accurate estimation of free energies using the jarzynski equality

Abstract

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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>J</mml:mi></mml:msub></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>FD</mml:mi></mml:msub></mml:math> . Here we derive the parametric maximum-likelihood estimator (MLE) of the free energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>ML</mml:mi></mml:msub></mml:math> considering unidirectional work distributions belonging to Gaussian or Gamma families, and compare this estimator to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>J</mml:mi></mml:msub></mml:math> . We further consider bidirectional work distributions belonging to the same families, and compare the corresponding bidirectional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mrow><mml:mi>ML</mml:mi><mml:mo>∗</mml:mo></mml:mrow></mml:msub></mml:math> to the Bennett acceptance ratio ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>BAR</mml:mi></mml:msub></mml:math> ) estimator. We show that, for Gaussian unidirectional work distributions, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>FD</mml:mi></mml:msub></mml:math> is in fact the parametric MLE of the free energy, and as such, the most efficient estimator for this statistical family. We observe that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>ML</mml:mi></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mrow><mml:mi>ML</mml:mi><mml:mo>∗</mml:mo></mml:mrow></mml:msub></mml:math> perform better than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>J</mml:mi></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Δ</mml:mi><mml:msub><mml:mover><mml:mi>F</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>BAR</mml:mi></mml:msub></mml:math> , 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.