Aspects of nonautonomous molecular dynamics IMA, University of Minnesota, Minneapolis January 28, 2007 Michel Cuendet Swiss Institute of Bioinformatics, Lausanne, Switzerland
Introduction to the Jarzynski identity Forced unfolding of 8-alanine peptide Usual thermodynamics : Jarzynski identity Xiong et al., Theor. Chem. Acc. 116 : 338 (2006) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). 2
Some quantities of interest Free energy difference between states and : Work performed to bring the system from to in time : Dissipation function : roughly such that is the entropy production. Usually : according to second law. 3
Nonequilibrium relations One way Equilibrium Zwanzig (1954) Work from state A to B Jarzynski (1997) Dissipation Kawazaki identity (1995) Two ways Bennett (1976) Crooks (1999) Fluctuation theo. (1993) R. W. Zwanzig, J. Chem. Phys 22, 1420 (1954). C. H. Bennett, J. Comp. Phys 22, 245 (1976). C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). G. E. Crooks, Phys. Rev. E 60, 2721 (1999). D. J. Evans and D. J. Searles, Phys. Rev. E 52, 5839 (1995). D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993). 4
Experimental Crooks 5
Nonautonomous thermostated MD Exploring slow reactions : drive the system system W ext integrator Heat bath : ΔF = W ext + W MD cutoffs Absorb excess heat + Q - TΔS W MD SHAKE Enforce right distribution twin-range - Q Symmetry breaking between time-reversible mechanics and irreversible thermodynamics* Thermostats as artificial heat baths : NOT the thermodynamic picture! Abstract heat bath replaced by few degrees of freedom Strongly coupled dynamics thermostat T 0 steady state : - Q = W ext + W MD Phase space compression / entropy production * W. G. Hoover, «Time Reversibility, Computer Simulation and Chaos», World Sientific, 1999 6
The Nosé-Hoover Thermostat Extended phase space EQUILIBRIUM physical variables NON-HAMILTONIAN Reproduces the canonical distribution : Conserved quantity : This is valid only with a correct discretization scheme (integrator). Time reversible Conserves pseudo-energy Preserves phase space volume Hoover, Phys. Rev. A 31 (1985), 1695 7
Is the Jarzynski identity valid for thermostated MD? NVT ensemble F MD N particles Try to use Jarzynski Jarzynski, Phys. Rev. Lett. 14 (1997), 2490 8
Free energy protocol Switch such that system Thermostat T 0 System energy change : NH dynamics The work is path-dependent : No variable is coupled to Jarzynski, J. Stat. Mech. : Theor. Exp. (2004) P09005 9
Formalism of Tuckerman et al. Invariant phase space measure : NH dynamics Metric factor : Invariant : Partition function for an isolated system : In the nonequilibrium case, also an invariant : * Tuckerman et al., JCP 115 (2001), 1678 10
Little proof of the Jarzynski identity Variable change Use invariant and metric factor Integrate on The Jarzynski identity!!! Cuendet, Phys. Rev. Lett. 12 : 120602 (2006) 11
Generalizations Nosé-Hoover chain thermostat Constant pressure ensemble, via volume coupling Generalized Nosé-Hoover with generic coupling terms define : - conserved quantity - Metric factor : Bulgac and Kusnezov, Phys. Rev. A 8 (1990), 5045 12
Generalization 2 : Hamiltonian thermostats Generalized time-dependent Nosé thermostat : Hamiltonian equations of motion integrated with respect to Laird and Leimkuhler, Phys. Rev. E 68 : 016704 (2003) Poincaré mapping : Any nonautonomous( ) Hamiltonian can be mapped to an autonomous( ) extended Hamiltonian Zare and Szebehely, Cel. Mech. 11 : 469 (1975) Struckmeier, J. Phys. A 38 : 1257 (2005) integrated with respect to Generalized Nosé-Poincaré thermostat! Bond et al., Comp. Phys. 151 : 114 (1999) Dettmann, Morriss, Phys. Rev. E 55 : 3693 (1997) Similar Jarzynski derivation with : Cuendet, J. Chem. Phys. 125 : 144109 (2006) 13
The JI as a property of the dynamics No need of hypotheses such as : Infinite number of particles Equivalence of microcanonical and canonical ensembles Infinite heat bath Weak or idealized coupling A priori canonical ensemble BUT : Generality loss : specific to the dynamics considered Relies only on properties of the thermostat: Inspired by this, Procacci et al. recently proved the Crooks theorem for thermostated dynamics: Procacci et al, J. Chem. Phys 125 : 164101 (2006) 14
Requirements for the dynamics Non-Ham. Ham. - Conserved quantity with a term linear in - Metric factor. - Representation where other variables are independent of NH dynamics Generalized Nosé-Poincaré 15
Unexpected Robustness Drift in Pseudo-energy : 320 kj/mol in 140 ps (!) Model the energy drift : Redo the Jarzynski proof : 16
Unexpected Robustness Steering potential = probe on only one degree of freedom. Drift per degree of freedom : 0.05 kj/mol (140ps) The free energy characterizes the whole system Convergence for "far" degrees of freedom out of reach 17
Outlook Improving MD sampling accuracy (calorimetric) Experiments time-resolved single molecule Nonequilibrium theory Jarzynski Kawazaki identity Crooks Fluctuation theo. 18
Usual Leap-frog : Leap-frog and Temperature Which squared velocity at step n? Squared velocity up to order 2 : 19
Leap-frog and Temperature 20
Thanks for your attention! Acknowledgments : Olivier Michielin Wilfred van Gunsteren Giovanni Ciccotti Chris Jarzynski Ben Leimkuhler 21