Nonequilibrium work relations for systems subject to mechanical and thermal changes.

Generalized forms of the Crooks fluctuation theorem are derived for non-Hamiltonian and Hamiltonian systems subject to both mechanical and thermal changes. Almost identical derivations are provided for the two cases under rather general assumptions. The basic one is that the probability distribution is a stationary solution of the Liouville equation for fixed values of mechanical control parameters applied to collective variables of the system and for fixed temperature. Generalized expressions for several nonequilibrium work relations derivable from the Crooks fluctuation theorem, such as the Jarzynski equality, path-ensemble averages for systems driven far from equilibrium, Bennett acceptance ratio, and two work-based potential of mean force estimators, are also derived. Although this list is not complete, the extension to other related work theorems is straightforward. The application of the methodology is illustrated for two representative cases, namely, for systems evolving with isochoric-isokinetic and isothermal-isobaric equations of motion.

[1]  Debra J. Searles,et al.  The Fluctuation Theorem , 2002 .

[2]  Riccardo Chelli,et al.  A potential of mean force estimator based on nonequilibrium work exponential averages. , 2009, Physical chemistry chemical physics : PCCP.

[3]  Simone Marsili,et al.  Generalization of the Jarzynski and Crooks nonequilibrium work theorems in molecular dynamics simulations. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Hoover,et al.  Constant-pressure equations of motion. , 1986, Physical review. A, General physics.

[5]  Charles H. Bennett,et al.  Efficient estimation of free energy differences from Monte Carlo data , 1976 .

[6]  Gary P. Morriss,et al.  Statistical Mechanics of Nonequilibrium Liquids , 2008 .

[7]  G. Ciccotti,et al.  Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems , 2001 .

[8]  G. Crooks.  Path-ensemble averages in systems driven far from equilibrium , 1999, cond-mat/9908420.

[9]  Hoover,et al.  Canonical dynamics: Equilibrium phase-space distributions. , 1985, Physical review. A, General physics.

[10]  David D L Minh,et al.  Optimized free energies from bidirectional single-molecule force spectroscopy. , 2008, Physical review letters.

[11]  M. Klein,et al.  Nosé-Hoover chains : the canonical ensemble via continuous dynamics , 1992 .

[12]  Denis J. Evans,et al.  Ensemble dependence of the transient fluctuation theorem , 1999, cond-mat/9906002.

[13]  G. Ciccotti,et al.  Constant pressure-constant temperature molecular dynamics: a correct constrained NPT ensemble using the molecular virial , 2003 .

[14]  D. Chandler,et al.  Introduction To Modern Statistical Mechanics , 1987 .

[15]  G. Ciccotti,et al.  Hoover NPT dynamics for systems varying in shape and size , 1993 .

[16]  C. Dellago,et al.  A proof of Jarzynski's nonequilibrium work theorem for dynamical systems that conserve the canonical distribution. , 2006, The Journal of chemical physics.

[17]  C. Jarzynski.  Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach , 1997, cond-mat/9707325.

[18]  Christophe Chatelain.  A temperature-extended Jarzynski relation: application to the numerical calculation of surface tension , 2007, cond-mat/0702044.

[19]  G. Hummer,et al.  Free energy reconstruction from nonequilibrium single-molecule pulling experiments , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[20]  R. Astumian.  The unreasonable effectiveness of equilibrium theory for interpreting nonequilibrium experiments , 2005, cond-mat/0512024.

[21]  M. Cuendet.  The Jarzynski identity derived from general Hamiltonian or non-Hamiltonian dynamics reproducing NVT or NPT ensembles. , 2006, The Journal of chemical physics.

[22]  Simone Marsili,et al.  Calculation of the potential of mean force from nonequilibrium measurements via maximum likelihood estimators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  S. Nosé.  A unified formulation of the constant temperature molecular dynamics methods , 1984 .

[24]  J. Schofield,et al.  Extended state-space Monte Carlo methods. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Michael R. Shirts,et al.  Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods. , 2003, Physical review letters.

[26]  M. Klein,et al.  Constant pressure molecular dynamics algorithms , 1994 .

[27]  A. Barducci,et al.  Recovering the Crooks equation for dynamical systems in the isothermal-isobaric ensemble: a strategy based on the equations of motion. , 2007, The Journal of chemical physics.

[28]  Evans,et al.  Equilibrium microstates which generate second law violating steady states. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[30]  Simone Marsili,et al.  Numerical verification of the generalized Crooks nonequilibrium work theorem for non-Hamiltonian molecular dynamics simulations. , 2007, The Journal of chemical physics.

[31]  C. Dellago,et al.  Equilibrium free energies from fast-switching trajectories with large time steps. , 2005, The Journal of chemical physics.

[32]  G. Crooks.  Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems , 1998 .

[33]  C. Jarzynski.  Nonequilibrium Equality for Free Energy Differences , 1996, cond-mat/9610209.

[34]  Alan M. Ferrenberg,et al.  Optimized Monte Carlo data analysis. , 1989, Physical Review Letters.

[35]  D. Keffer,et al.  A generalized Hamiltonian-based algorithm for rigorous equilibrium molecular dynamics simulation in the isobaric–isothermal ensemble , 2006 .

[36]  C. Jarzynski.  Hamiltonian Derivation of a Detailed Fluctuation Theorem , 1999, cond-mat/9908286.

[37]  Stephen R. Williams,et al.  Nonequilibrium free-energy relations for thermal changes. , 2008, Physical review letters.

[38]  Simone Marsili,et al.  Crooks equation for steered molecular dynamics using a Nosé-Hoover thermostat. , 2006, The Journal of chemical physics.