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William H Miller - One of the best experts on this subject based on the ideXlab platform.
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Using the thermal Gaussian approximation for the Boltzmann Operator in semiclassical initial value time correlation functions.
The Journal of chemical physics, 2006Co-Authors: Jian Liu, William H MillerAbstract:The thermal Gaussian approximation (TGA) recently developed by Mandelshtam et al has been demonstrated to be a practical way for approximating the Boltzmann Operator exp(-{beta}H) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the 'forward-backward semiclassical dynamics' (FBSD) approximation developed by Makri et al. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability.
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Quantum-instanton evaluation of the kinetic isotope effects
The Journal of chemical physics, 2005Co-Authors: Jiri Vanicek, William H Miller, Jesus F. Castillo, F. Javier AoizAbstract:A general quantum-mechanical method for computing kinetic isotope effects is presented. The method is based on the quantum-instanton approximation for the rate constant and on the path-integral Metropolis–Monte Carlo evaluation of the Boltzmann Operator matrix elements. It computes the kinetic isotope effect directly, using a thermodynamic integration with respect to the mass of the isotope, thus avoiding the more computationally expensive process of computing the individual rate constants. The method should be more accurate than variational transition-state theories or the semiclassical instanton method since it does not assume a single tunneling path and does not use a semiclassical approximation of the Boltzmann Operator. While the general Monte Carlo implementation makes the method accessible to systems with a large number of atoms, we present numerical results for the Eckart barrier and for the collinear and full three-dimensional isotope variants of the hydrogen exchange reaction H + H2 → H2 + H. In all seven test cases, for temperatures between 250 and 600 K, the error of the quantum instanton approximation for the kinetic isotope effects is less than 10%. © 2005 American Institute of Physics. DOI: 10.1063/1.1946740
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Different time slices for different degrees of freedom in Feynman path integration
Molecular Physics, 2005Co-Authors: William H MillerAbstract:A general scheme is presented for using different numbers of ‘time slices’ for different degrees of freedom in a path integral evaluation of the Boltzmann Operator for a large molecular system. This will be particularly useful, for example, in evaluating the ‘quantum instanton’ rate constant [cf. W.H. Miller, Y. Zhao, M. Ceotto, S. Yang. J. Chem. Phys., 119, 1329 (2003)] for H atom transfer reactions, or any applications involving atoms with largely differing masses.
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Quantum instanton approximation for thermal rate constants of chemical reactions
The Journal of Chemical Physics, 2003Co-Authors: William H Miller, Yi Zhao, Michele Ceotto, Sandy YangAbstract:A quantum mechanical theory for chemical reaction rates is presented which is modeled after the [semiclassical (SC)] instanton approximation. It incorporates the desirable aspects of the instanton picture, which involves only properties of the (SC approximation to the) Boltzmann Operator, but corrects its quantitative deficiencies by replacing the SC approximation for the Boltzmann Operator by the quantum Boltzmann Operator, exp(−βĤ). Since a calculation of the quantum Boltzmann Operator is feasible for quite complex molecular systems (by Monte Carlo path integral methods), having an accurate rate theory that involves only the Boltzmann Operator could be quite useful. The application of this quantum instanton approximation to several one- and two-dimensional model problems illustrates its potential; e.g., it is able to describe thermal rate constants accurately (∼10–20% error) from high to low temperatures deep in the tunneling regime, and applies equally well to asymmetric and symmetric potentials.
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Coherent state semiclassical initial value representation for the Boltzmann Operator in thermal correlation functions
The Journal of Chemical Physics, 2002Co-Authors: Nancy Makri, William H MillerAbstract:A semiclassical methodology for evaluating the Boltzmann Operator entering semiclassical approximations for finite temperature correlation functions is described. Specifically, Miller’s imaginary time semiclassical approach is applied to the Herman–Kluk coherent state initial value representation (IVR) for the time evolution Operator in order to obtain a coherent state IVR for the Boltzmann Operator. The phase-space representation gives rise to exponentially decaying factors for the coordinates and momenta of the real time trajectories employed in the dynamical part of the calculation. A Monte Carlo procedure is developed for evaluating dynamical observables, in which the absolute value of the entire exponential part of the integrand serves as the sampling function. Numerical tests presented show that the methodology is accurate as well as stable over the temperature range relevant to chemical applications.
Marc Briant - One of the best experts on this subject based on the ideXlab platform.
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Stability of the spectral gap for the Boltzmann multi-species Operator linearized around non-equilibrium Maxwell distributions
Communications on Pure and Applied Analysis, 2020Co-Authors: Andrea Bondesan, Marc Briant, Laurent Boudin, Bérénice GrecAbstract:We consider the Boltzmann Operator for mixtures with cutoff Maxwellian, hard potentials, or hard spheres collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species Operator L is known to possess an explicit spectral gap λ , in the global equilibrium weighted L^2 space. We study a new Operator L_ε obtained by linearizing the Boltzmann Operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order ε, ε > 0. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of L_ε in the global equilibrium weighted L^2 space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to L and we show that the same estimate holds for the entropy production functional associated to L_ε , up to a correction of order ε.
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Stability of the spectral gap for the Boltzmann multi-species Operator linearized around non-equilibrium Maxwell distributions
Communications on Pure & Applied Analysis, 2020Co-Authors: Andrea Bondesan, Marc Briant, Laurent Boudin, Bérénice GrecAbstract:We consider the Boltzmann Operator for mixtures with cutoff Maxwellian, hard potential, or hard-sphere collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species Operator \begin{document}$ \mathbf{L} $\end{document} is known to possess an explicit spectral gap \begin{document}$ \lambda_{ \mathbf{L}} $\end{document} , in the global equilibrium weighted \begin{document}$ L^2 $\end{document} space. We study a new Operator \begin{document}$ \mathbf{ L^{\varepsilon}} $\end{document} obtained by linearizing the Boltzmann Operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order \begin{document}$ \varepsilon $\end{document} , \begin{document}$ \varepsilon >0 $\end{document} . This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of \begin{document}$ \mathbf{ L^{\varepsilon}} $\end{document} in the global equilibrium weighted \begin{document}$ L^2 $\end{document} space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to \begin{document}$ \mathbf{L} $\end{document} and we show that the same estimate holds for the entropy production functional associated to \begin{document}$ \mathbf{ L^{\varepsilon}} $\end{document} , up to a correction of order \begin{document}$ \varepsilon $\end{document} .
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Stability of the spectral gap for the Boltzmann multi-species Operator linearized around non-equilibrium Maxwell distributions
arXiv: Mathematical Physics, 2018Co-Authors: Andrea Bondesan, Marc Briant, Laurent Boudin, Bérénice GrecAbstract:We consider the Boltzmann Operator for mixtures with cutoff Maxwellian, hard potentials, or hard spheres collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species Operator $\mathbf{L}$ is known to possess an explicit spectral gap $\lambda_{\mathbf{L}}$, in the global equilibrium weighted $L^2$ space. We study a new Operator $\mathbf{L^\varepsilon}$ obtained by linearizing the Boltzmann Operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order $\varepsilon$, $\varepsilon >0$. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of $\mathbf{L^\varepsilon}$ in the global equilibrium weighted $L^2$ space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to $\mathbf{L}$ and we show that the same estimate holds for the entropy production functional associated to $\mathbf{L^\varepsilon}$, up to a correction of order $\varepsilon$.
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Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings
Discrete and Continuous Dynamical Systems, 2016Co-Authors: Marc BriantAbstract:We prove the stability of global equilibrium in a multi-species mixture, where the different species can have different masses, on the $3$-dimensional torus. We establish stability estimates in $L^\infty_{x,v}(w)$ where $w=w(v)$ is either polynomial or exponential, with explicit threshold. Along the way we extend recent estimates and stability results for the mono-species Boltzmann Operator not only to the multi-species case but also to more general hard potential and Maxwellian kernels.
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the Boltzmann equation for a multi species mixture close to global equilibrium
Archive for Rational Mechanics and Analysis, 2016Co-Authors: Marc Briant, Esther S DausAbstract:We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in \({L^1_vL^\infty_x(m)}\), where \({m\sim (1+ |v|^k)}\) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann Operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann Operator. Then we develop an \({L^2-L^\infty}\) theory a la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann Operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the \({L^1_vL^\infty_x}\) framework is dealt with for any \({k > k_0}\), recovering the optimal physical threshold of finite energy \({k_0=2}\) in the particular case of a multi-species hard spheres mixture with the same masses.
Esther S Daus - One of the best experts on this subject based on the ideXlab platform.
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the Boltzmann equation for a multi species mixture close to global equilibrium
Archive for Rational Mechanics and Analysis, 2016Co-Authors: Marc Briant, Esther S DausAbstract:We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in \({L^1_vL^\infty_x(m)}\), where \({m\sim (1+ |v|^k)}\) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann Operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann Operator. Then we develop an \({L^2-L^\infty}\) theory a la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann Operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the \({L^1_vL^\infty_x}\) framework is dealt with for any \({k > k_0}\), recovering the optimal physical threshold of finite energy \({k_0=2}\) in the particular case of a multi-species hard spheres mixture with the same masses.
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The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
Archive for Rational Mechanics and Analysis, 2016Co-Authors: Marc Briant, Esther S DausAbstract:We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the 3-dimensional torus. The ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in L 1 v L ∞ x (m), where m ∼ (1 + |v| k) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann Operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann Operator. Then we develop an L 2 − L ∞ theoryàtheoryà la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann Operator which prevents the direct adaptation of standard mono-species methods (e.g. Carleman representation , Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the L 1 v L ∞ x framework is dealt with for any k > k 0 , recovering the optimal physical threshold of finite energy k 0 = 2 in the particular case of a multi-species hard spheres mixture with same masses.
Yi Zhao - One of the best experts on this subject based on the ideXlab platform.
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Quantum instanton approximation for thermal rate constants of chemical reactions
The Journal of Chemical Physics, 2003Co-Authors: William H Miller, Yi Zhao, Michele Ceotto, Sandy YangAbstract:A quantum mechanical theory for chemical reaction rates is presented which is modeled after the [semiclassical (SC)] instanton approximation. It incorporates the desirable aspects of the instanton picture, which involves only properties of the (SC approximation to the) Boltzmann Operator, but corrects its quantitative deficiencies by replacing the SC approximation for the Boltzmann Operator by the quantum Boltzmann Operator, exp(−βĤ). Since a calculation of the quantum Boltzmann Operator is feasible for quite complex molecular systems (by Monte Carlo path integral methods), having an accurate rate theory that involves only the Boltzmann Operator could be quite useful. The application of this quantum instanton approximation to several one- and two-dimensional model problems illustrates its potential; e.g., it is able to describe thermal rate constants accurately (∼10–20% error) from high to low temperatures deep in the tunneling regime, and applies equally well to asymmetric and symmetric potentials.
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Semiclassical initial value representation for the Boltzmann Operator in thermal rate constants
The Journal of Chemical Physics, 2002Co-Authors: Yi Zhao, William H MillerAbstract:The thermal rate constant for a chemical reaction, k(T), can be expressed as the long time limit of the flux-side correlation Cfs(t)=tr[e−βĤ/2Fe−βĤ/2eiĤt/ℏĥe−iĤt/ℏ]. Previous work has focused on semiclassical (SC) approximations [implemented via an initial value representation (IVR)] for the time evolution Operators exp(±iĤt/ℏ) in the correlation function, and this paper shows how an SC-IVR can also be used to approximate the Boltzmann Operators exp(−βĤ/2). Test calculations show that over a wide temperature range little error is introduced in the rate constant by this SC approximation for the Boltzmann Operator.
Jiushu Shao - One of the best experts on this subject based on the ideXlab platform.
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Quantum Phase Transition in the Spin-Boson Model: A Multilayer Multiconfiguration Time-Dependent Hartree Study.
The journal of physical chemistry. A, 2019Co-Authors: Haobin Wang, Jiushu ShaoAbstract:The multilayer improved relaxation is applied to study the delocalization–localization transition in the spin-boson model at zero temperature—a well-known example of quantum phase transition. Calculations of energy eigenstates are obtained by iteratively diagonalizing the matrix of the Boltzmann Operator in the top layer representation, using a Lanczos/Arnoldi method while relaxing the single particle functions of all layers using the multilayer multiconfiguration time-dependent Hartree imaginary time propagation. Two properties are used to examine the quantum phase transition: the energy splitting for the lowest pair of eigenstates and the magnetic susceptibility. Consistent findings are obtained with appropriate scaling parameters.
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A semiclassical initial-value representation for quantum propagator and Boltzmann Operator.
Journal of computational chemistry, 2018Co-Authors: Yun-an Yan, Jian Liu, Jiushu ShaoAbstract:Starting from the position-momentum integral representation, we apply the correction Operator method to the derivation of a uniform semiclassical approximation for the quantum propagator and then extend it to approximate the Boltzmann Operator. In this approach, the involved classical dynamics is determined by the method itself instead of given beforehand. For the approximate Boltzmann Operator, the corresponding classical dynamics is governed by a complex Hamiltonian, which can be described as a pair of real Hamiltonian systems. It is demonstrated that the semiclassical Boltzmann Operator is exact for linear systems. A quantum propagator in the complex time is thus proposed and preliminary numerical results show that it is a reasonable approximation for calculating thermal correlation functions of general systems. © 2018 Wiley Periodicals, Inc.