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Mark E Tuckerman - One of the best experts on this subject based on the ideXlab platform.
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generalized gaussian moment thermostatting a new continuous dynamical approach to the Canonical ensemble
Journal of Chemical Physics, 2000Co-Authors: Yi Liu, Mark E TuckermanAbstract:A new method for generating the Canonical ensemble via continuous dynamics is presented. The new method is based on controlling the fluctuations of an arbitrary number of moments of the multidimensional Gaussian momentum Distribution function. The equations of motion are non-Hamiltonian, and hence have a nonvanishing phase space compressibility. By applying the statistical mechanical theory of non-Hamiltonian systems recently introduced by the authors [M. E. Tuckerman, C. J. Mundy, and G. J. Martyna, Europhys. Lett. 45, 149 (1999)], the equations are shown to produce the correct Canonical phase space Distribution function. Reversible integrators for the new equations of motion are derived based on a Trotter-type factorization of the classical Liouville propagator. The new method is applied to a variety of simple one-dimensional example problems and is shown to generate ergodic trajectories and correct Canonical Distribution functions of both position and momentum. The new method is further shown to lead to rapid convergence in molecular dynamics based calculations of path integrals. The performance of the new method in these examples is compared to that of another Canonical dynamics method, the Nose–Hoover chain method [G. J. Martyna, M. L. Klein, and M. E. Tuckerman, J. Chem. Phys. 97, 2635 (1992)]. The comparison demonstrates the improvements afforded by the new method as a molecular dynamics tool. Finally, when employed in molecular dynamics simulations of biological macromolecules, the new method is shown to provide better energy equipartitioning and temperature control and to lead to improved spatial sampling over the Nose–Hoover chain method in a realistic application.
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nose hoover chains the Canonical ensemble via continuous dynamics
Journal of Chemical Physics, 1992Co-Authors: Glenn J Martyna, Michael L Klein, Mark E TuckermanAbstract:Nose has derived a set of dynamical equations that can be shown to give Canonically distributed positions and momenta provided the phase space average can be taken into the trajectory average, i.e., the system is ergodic [S. Nose, J. Chem. Phys. 81, 511 (1984), W. G. Hoover, Phys. Rev. A 31, 1695 (1985)]. Unfortunately, the Nose–Hoover dynamics is not ergodic for small or stiff systems. Here a modification of the dynamics is proposed which includes not a single thermostat variable but a chain of variables, Nose–Hoover chains. The ‘‘new’’ dynamics gives the Canonical Distribution where the simple formalism fails. In addition, the new method is easier to use than an extension [D. Kusnezov, A. Bulgac, and W. Bauer, Ann. Phys. 204, 155 (1990)] which also gives the Canonical Distribution for stiff cases.
Jian Liu - One of the best experts on this subject based on the ideXlab platform.
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two more approaches for generating trajectory based dynamics which conserves the Canonical Distribution in the phase space formulation of quantum mechanics
Journal of Chemical Physics, 2011Co-Authors: Jian LiuAbstract:We show two more approaches for generating trajectory-based dynamics in the phase space formulation of quantum mechanics: "equilibrium continuity dynamics" (ECD) in the spirit of the phase space continuity equation in classical mechanics, and "equilibrium Hamiltonian dynamics" (EHD) in the spirit of the Hamilton equations of motion in classical mechanics. Both ECD and EHD can recover exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Both ECD and EHD conserve the quasi-probability within the infinitesimal volume dx(t)dp(t) around the phase point (x(t), p(t)) along the trajectory. Numerical tests of both approaches in the Wigner phase space have been made for two strongly anharmonic model problems and a double well system, for each potential auto-correlation functions of both linear and nonlinear operators have been calculated. The results suggest EHD and ECD are two additional potential useful approaches for describing quantum effects for complex systems in condense phase.
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an approach for generating trajectory based dynamics which conserves the Canonical Distribution in the phase space formulation of quantum mechanics i theories
Journal of Chemical Physics, 2011Co-Authors: Jian Liu, William H MillerAbstract:We have reformulated and generalized our recent work [J. Liu and W. H. Miller, J. Chem. Phys. 126, 234110 (2007)] into an approach for generating a family of trajectory-based dynamics methods in the phase space formulation of quantum mechanics. The approach (equilibrium Liouville dynamics) is in the spirit of Liouville's theorem in classical mechanics. The trajectory-based dynamics is able to conserve the quantum Canonical Distribution for the thermal equilibrium system and approaches classical dynamics in the classical (ℏ → 0), high temperature (β → 0), and harmonic limits. Equilibrium Liouville dynamics provides the framework for the development of novel theoretical/computational tools for studying quantum dynamical effects in large/complex molecular systems.
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an approach for generating trajectory based dynamics which conserves the Canonical Distribution in the phase space formulation of quantum mechanics ii thermal correlation functions
Journal of Chemical Physics, 2011Co-Authors: Jian Liu, William H MillerAbstract:We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum Canonical Distribution–equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.
William H Miller - One of the best experts on this subject based on the ideXlab platform.
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an approach for generating trajectory based dynamics which conserves the Canonical Distribution in the phase space formulation of quantum mechanics i theories
Journal of Chemical Physics, 2011Co-Authors: Jian Liu, William H MillerAbstract:We have reformulated and generalized our recent work [J. Liu and W. H. Miller, J. Chem. Phys. 126, 234110 (2007)] into an approach for generating a family of trajectory-based dynamics methods in the phase space formulation of quantum mechanics. The approach (equilibrium Liouville dynamics) is in the spirit of Liouville's theorem in classical mechanics. The trajectory-based dynamics is able to conserve the quantum Canonical Distribution for the thermal equilibrium system and approaches classical dynamics in the classical (ℏ → 0), high temperature (β → 0), and harmonic limits. Equilibrium Liouville dynamics provides the framework for the development of novel theoretical/computational tools for studying quantum dynamical effects in large/complex molecular systems.
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an approach for generating trajectory based dynamics which conserves the Canonical Distribution in the phase space formulation of quantum mechanics ii thermal correlation functions
Journal of Chemical Physics, 2011Co-Authors: Jian Liu, William H MillerAbstract:We show the exact expression of the quantum mechanical time correlation function in the phase space formulation of quantum mechanics. The trajectory-based dynamics that conserves the quantum Canonical Distribution–equilibrium Liouville dynamics (ELD) proposed in Paper I is then used to approximately evaluate the exact expression. It gives exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits. Various methods have been presented for the implementation of ELD. Numerical tests of the ELD approach in the Wigner or Husimi phase space have been made for a harmonic oscillator and two strongly anharmonic model problems, for each potential autocorrelation functions of both linear and nonlinear operators have been calculated. It suggests ELD can be a potentially useful approach for describing quantum effects for complex systems in condense phase.
A.r. Plastino - One of the best experts on this subject based on the ideXlab platform.
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classical typicality of the Canonical Distribution
EPL, 2008Co-Authors: A.r. Plastino, Andreas DaffertshoferAbstract:We consider the typicality of the Canonical ensemble's probability Distribution from a classical perspective, resuming recent discussions on quantum-mechanical aspects of Canonical typicality. In the conventional derivation of the classical Canonical Distribution for a system S that is weakly coupled to a heat bath B, it is assumed that the composite S+B is represented by the microCanonical ensemble i.e., by a uniform probability Distribution on an energy shell of the composite S+B. Here we show that for a very large heat bath almost all probability Distributions defined on this energy shell behave according to the microCanonical ensemble, yielding a marginal probability Distribution for S of the Canonical form. Consequently, the classical Canonical Distribution can be regarded as much more "typical" than suggested by the standard derivation.
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from gibbs microCanonical ensemble to tsallis generalized Canonical Distribution
Physics Letters A, 1994Co-Authors: A.r. Plastino, Alexandre PlastinoAbstract:Abstract We derive the Tsallis generalized Canonical Distribution by recourse to considerations that originate in the traditional Gibbs microCanonical ensemble. The route to be followed has straightforward nature, and only invokes concepts that can be found in any undergraduate textbook on statistical mechanics. It is reasonable to claim, as a consequence, that the Tsallis generalized Distribution admits a clear physical interpretation.
Shuxin Guo - One of the best experts on this subject based on the ideXlab platform.
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Canonical Distribution implied binomial tree and the pricing of american options
Journal of Futures Markets, 2013Co-Authors: Qiang Liu, Shuxin GuoAbstract:In this study, a new approach to pricing American options is proposed and termed the Canonical implied binomial (CIB) tree method. CIB takes advantage of both Canonical valuation (Stutzer, 1996) and the implied binomial tree method (Rubinstein, 1994). Using simulated returns from geometric Brownian motions (GBM), CIB produced very similar prices for calls and European puts as those of Black–Scholes (BS). Applied to a set of over 15,000 American-style SP in addition, it outperformed the Canonical least-squares Monte Carlo (Liu, 2010) in the dynamic hedging of in-the-money options. Furthermore, CIB suggests that regular GBM-based Monte Carlo can be extended to American options pricing by also utilizing the implied binomial tree. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark
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Canonical Distribution implied binomial tree and the pricing of american options
Social Science Research Network, 2011Co-Authors: Qiang Liu, Shuxin GuoAbstract:In this paper a new approach to pricing American options is proposed and termed the Canonical implied binomial (CIB) tree method. CIB takes advantage of both Canonical valuation (Stutzer, 1996) and the implied binomial tree method (Rubinstein, 1994). Using simulated returns from geometric Brownian motions (GBM), CIB produced very similar prices for calls and European puts as those of Black-Scholes (BS). Applied to a set of over 15,000 American-style SP in addition, it outperformed the Canonical least-squares Monte Carlo (Liu, 2010) in the dynamic hedging of in-the-money options. Furthermore, CIB suggests that regular GBM-based Monte Carlo can be extended to American options pricing by also utilizing the implied binomial tree.