Boltzmann-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynamics”

Abstract

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or 'classical Wigner approximation') results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their 'Matsubara' values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of 2 at 0 (which gives back the standard LSC-IVR approximation), but in the normalmode derivatives corresponding to the lowest Matsubara frequencies. The resulting 'Matsubara' dynamics is inherently classical (since all terms O( 2 ) disappear from the Matsubara Liouvillian in the limit N → ∞), and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum timecorrelation function converges with respect to the number of modes, and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.

Non-equilibrium dynamics from RPMD and CMD

“…Several efforts to derive RPMD and CMD have been undertaken. [69][70][71][72] In particular, it has been found that both RPMD and CMD are related to Matsubara dynamics, 69,70 which approximatesC AB (t) by a mixture of quantum statistics and classical dynamics. Matsubara dynamics is derived from first principles and has been shown to reproduce TCFs more accurately than RPMD, CMD, or the linearized semiclassical-initial value representation approach, although its slow numerical convergence makes it less broadly applicable than these other approximate methods.…”

“…Matsubara dynamics is derived from first principles and has been shown to reproduce TCFs more accurately than RPMD, CMD, or the linearized semiclassical-initial value representation approach, although its slow numerical convergence makes it less broadly applicable than these other approximate methods. 69,70 B. Application of RPMD and CMD to non-equilibrium time-correlation functions Here, we apply RPMD and CMD to non-equilibrium TCFs of the form…”

“…To obtain RPMD from Matsubara dynamics, the Matsubara Liouvillian L M is written in terms of complex phase space as 69,70 M into real space. 73 This procedure can be done independently of the initial conditions, as the Matsubara Liouvillian L M only involves the Hamiltonian governing the time-evolution of the system (i.e., H (1) for the present case).…”

“…73 This procedure can be done independently of the initial conditions, as the Matsubara Liouvillian L M only involves the Hamiltonian governing the time-evolution of the system (i.e., H (1) for the present case). To obtain CMD from Matsubara dynamics, one invokes a centroid mean-field approximation of the exact force on the centroid 69,70 …”

“…It is found that nearly all important properties of RPMD and CMD are preserved when calculating non-equilibrium TCFs, as are the relationships of the two a) rwelsch@caltech.edu b) tfm@caltech.edu methods to Matsubara dynamics. 69,70 Furthermore, the numerical performance of the two methods for the calculation of non-equilibrium quantum TCFs is tested for a range of model systems.…”