Classical Langevin dynamics derived from quantum mechanics

Hoel, Håkon; Szepessy, Anders

doi:10.3934/dcdsb.2020135

Cited by 5 publications

(5 citation statements)

References 24 publications

(57 reference statements)

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“…Observations on quantum dynamics for a system in interaction with a heat bath in thermal equilibrium can be approximated by correlations (1.2) in the canonical ensemble, cf. [6], [16], [7], [10]. For instance, the classical observable for the diffusion constant…”

Section: The Main Results and Background From Weyl Calculusmentioning

confidence: 99%

Canonical mean-field molecular dynamics derived from quantum mechanics

Huang¹,

Plecháč²,

Sandberg³

et al. 2021

Preprint

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Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M −1 ), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M −1 ) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a meanfield molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr (He −βH )/Tr (e −βH ) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian H. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M −1 + tǫ 2 ), for correlation time t where ǫ 2 is related to the variance of mean value approximation h. The proof of this estimate does not rely on diagonalizing the electron operator and consequently coinciding electron eigenvalues are allowed. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation.

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Section: The Main Results and Background From Weyl Calculusmentioning

confidence: 99%

Canonical mean-field molecular dynamics derived from quantum mechanics

Huang¹,

Plecháč²,

Sandberg³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…{ ( )} = plays the role of an equilibrium configuration of the respective heat bath particle for any fixed X 3  Î . In fact, one can demonstrate [33] that such equilibrium map can be always derived once that a generic interaction potential is given.…”

Section: General Setting: Study Of the Equilibrium Mapmentioning

confidence: 99%

“…We now show that adopting definitions (44), one naturally recover the FDR. Hence, let us consider the heat bath Hamiltonian from equation (33), namely:…”

Section: Data Availability Statementmentioning

confidence: 99%

On the derivation of a Nonlinear Generalized Langevin Equation

Cairano¹

2022

J. Phys. Commun.

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We recast the Zwanzig's derivation of a non linear generalized Langevin equation (GLE) for a heavy particle interacting with a heat bath in a more general framework showing that it is necessary to readjust the Zwanzig's definitions of the kernel matrix and noise vector in the GLE in order to be able performing consistently the continuum limit. As shown by Zwanzig, the non linear feature of the resulting GLE is due to the non linear dependence of the equilibrium map by the heavy particle variables. Such an equilibrium map represents the global equilibrium configuration of the heat bath particles for a fixed (instantaneous) configuration of the system. Following the same derivation of the GLE, we show that a deeper investigation of the equilibrium map, considered in the Zwanzig's Hamiltonian, is necessary. Moreover, we discuss how to get an equilibrium map given a general interaction potential. Finally, we provide a renormalization procedure which allows to divide the dependence of the equilibrium map by coupling coefficient from the dependence by the system variables yielding a more rigorous mathematical structure of the non linear GLE.

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“…8 clearly contradicts the hypothesis that the mere presence of Langevin terms is detrimental for Davydov's model. A mathematically rigorous derivation of Langevin molecular dynamics, for the case of a composite quantum system that is weakly coupled to a heat bath of many fast particles, has been recently developed by Hoel and Szepessy [54]. Thus, we maintain the view that the system of equations ( 27) and ( 28) represents a quantum Langevin model [34][35][36] of protein α-helix immersed in water-based solvent at physiological temperature.…”

Section: Three-spine Model With Lateral Couplingmentioning

confidence: 99%

Thermal stability of solitons in protein $α$-helices

Georgiev,

Glazebrook

2022

Preprint

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Protein α-helices provide an ordered biological environment that is conducive to soliton-assisted energy transport. The nonlinear interaction between amide I excitons and phonon deformations induced in the hydrogen-bonded lattice of peptide groups leads to self-trapping of the amide I energy, thereby creating a localized quasiparticle (soliton) that persists at zero temperature. The presence of thermal noise, however, could destabilize the protein soliton and dissipate its energy within a finite lifetime. In this work, we have computationally solved the system of stochastic differential equations that govern the quantum dynamics of protein solitons at physiological temperature, T = 310 K, for either a single isolated α-helix spine of hydrogen bonded peptide groups or the full protein α-helix comprised of three parallel α-helix spines. The simulated stochastic dynamics revealed that although the thermal noise is detrimental for the single isolated α-helix spine, the cooperative action of three amide I exciton quanta in the full protein α-helix ensures soliton lifetime of over 30 ps, during which the amide I energy could be transported along the entire extent of an 18-nm-long α-helix. Thus, macromolecular protein complexes, which are built up of protein α-helices could harness soliton-assisted energy transport at physiological temperature. Because the hydrolysis of a single adenosine triphosphate molecule is able to initiate three amide I exciton quanta, it is feasible that multiquantal protein solitons subserve a variety of specialized physiological functions in living systems.

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Classical Langevin dynamics derived from quantum mechanics

Cited by 5 publications

References 24 publications

Canonical mean-field molecular dynamics derived from quantum mechanics

Canonical mean-field molecular dynamics derived from quantum mechanics

On the derivation of a Nonlinear Generalized Langevin Equation

Thermal stability of solitons in protein $α$-helices

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