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October  2020, 25(10): 4001-4038. doi: 10.3934/dcdsb.2020135

Classical Langevin dynamics derived from quantum mechanics

1. 

Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University, Germany

2. 

Institutionen för Matematik, Kungl. Tekniska Högskolan, 100 44 Stockholm, Sweden

* Corresponding author: Anders Szepessy (szepessy@kth.se)

Received  June 2019 Revised  January 2020 Published  October 2020 Early access  April 2020

Fund Project: The second author is supported by the Swedish Research Council grants 2014-04776 and 2019-03725

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

Citation: Håkon Hoel, Anders Szepessy. Classical Langevin dynamics derived from quantum mechanics. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4001-4038. doi: 10.3934/dcdsb.2020135
References:
[1]

A. AbdulleG. Vilmart and K. C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics, SIAM Journal on Numerical Analysis, 53 (2015), 1-16.  doi: 10.1137/140962644.

[2]

A. D. Baczewski and S. D. Bond, Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel, The Journal of Chemical Physics, 139 (2013), 044107. doi: 10.1063/1.4815917.

[3]

N. Bou-Rabee and H. Owhadi, Long-run accuracy of variational integrators in the stochastic context, SIAM Journal on Numerical Analysis, 48 (2010), 278-297.  doi: 10.1137/090758842.

[4]

A. BrüngerC. L. Brooks III and M. Karplus, Stochastic boundary conditions for molecular dynamics simulations of ST2 water, Chemical Physics Letters, 105 (1984), 495-500. 

[5]

G. M. Dall'ara, Discreteness of the spectrum of Schrödinger operators with non-negative matrix valued potentials, Journal of Functional Analysis, 268 (2015), 3649-3679.  doi: 10.1016/j.jfa.2014.10.007.

[6]

G. W. Ford and M. Kac, On the quantum Langevin equation, J. Statist. Phys., 46 (1987), 803-810.  doi: 10.1007/BF01011142.

[7]

G. W. FordM. Kac and P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Mathematical Phys., 6 (1965), 504-515.  doi: 10.1063/1.1704304.

[8]

M. HairerM. Hutzenthaler and A. Jentzen, Loss of regularity for Kolmogorov equations, The Annals of Probability, 43 (2015), 468-527.  doi: 10.1214/13-AOP838.

[9]

E. J. Hall, M. A. Katsoulakis and L. Rey-Bellet, Uncertainty quantification for generalized Langevin dynamics, The Journal of Chemical Physics, 145 (2016), 224108. doi: 10.1063/1.4971433.

[10]

A. Kammonen, P. Plecháč, M. Sandberg and A. Szepessy, Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics, Ann. Henri Poincaré, 19 (2018), 2727–2781. doi: 10.1007/s00023-018-0699-x.

[11]

D. P. Kroese, T. Taimre and Z. I. Botev, Handbook of monte carlo methods, John Wiley & Sons, 706 (2011). doi: 10.1002/9781118014967.

[12]

N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, Journal of Functional Analysis, 250 (2007), 521-558.  doi: 10.1016/j.jfa.2007.04.003.

[13]

R. Kupferman, Fractional kinetics in Kac-Zwanzig heat bath models, Journal of Statistical Physics, 114 (2004), 291-326.  doi: 10.1023/B:JOSS.0000003113.22621.f0.

[14]

P. Langevin, On the theory of Brownian movement, C. R. Acad. Sci., 146 (1908), 530-533. 

[15]

J. L. Lebowitz and E. Rubin, Dynamical study of Brownian motion, Phys. Rev., 131 (1963), 2381-2396.  doi: 10.1103/PhysRev.131.2381.

[16]

B. Leimkuhler and C. Matthews, Molecular Dynamics: With Deterministic and Stochastic Numerical Methods, Interdisciplinary Applied Mathematics, 39. Springer, Cham, 2015.

[17]

T. Leliévre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, 25 (2016), 681-880.  doi: 10.1017/S0962492916000039.

[18]

D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press, 2009. doi: 10.1017/CBO9780511609633.

[19]

J. C. MattinglyA. M. Stuart and D. J. Higham, Ergodicity for sdes and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Processes and Their Applications, 101 (2002), 185-232.  doi: 10.1016/S0304-4149(02)00150-3.

[20]

P. M. Mazur and I. Oppenheim, Molecular theory of Brownian motion, Physica, 50 (1970), 241-258.  doi: 10.1016/0031-8914(70)90005-4.

[21]

E. H. Müller, R. Scheichl and T. Shardlow, Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation, Proc. A., 471 (2015), 20140679, 20 pp. doi: 10.1098/rspa.2014.0679.

[22]

G. A. Pavliotis, Stochastic Processes and Applications. Diffusion Processes, the Fokker-Planck and Langevin Equations, Texts in Applied Mathematics, 60. Springer, New York, 2014. doi: 10.1007/978-1-4939-1323-7.

[23]

J.-E. Shea and I. Oppenheim, Fokker-Planck equation and Langevin equation for one Brownian particle in a nonequilibrium bath, J. Phys. Chem., 100 (1996), 19035-19042.  doi: 10.1021/jp961605d.

[24]

R. D. Skeel and J. A. Izaguirre, An impulse integrator for Langevin dynamics, Molecular Physics, 100 (2002), 3885-3891.  doi: 10.1080/0026897021000018321.

[25]

H.-M. Stiepan and S. Teufel, Semiclassical approximations for Hamiltonians with operator-valued symbols, Comm. Math. Phys., 320 (2013), 821-849.  doi: 10.1007/s00220-012-1650-5.

[26]

L. Verlet, Computer "Experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev., 159 (1967), 98-103.  doi: 10.1103/PhysRev.159.98.

[27] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford Univ. Press, New York, 2001. 
[28]

R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat. Phys., 9 (1973), 215-220.  doi: 10.1007/BF01008729.

[29]

M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, 138. American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/138.

show all references

References:
[1]

A. AbdulleG. Vilmart and K. C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics, SIAM Journal on Numerical Analysis, 53 (2015), 1-16.  doi: 10.1137/140962644.

[2]

A. D. Baczewski and S. D. Bond, Numerical integration of the extended variable generalized Langevin equation with a positive Prony representable memory kernel, The Journal of Chemical Physics, 139 (2013), 044107. doi: 10.1063/1.4815917.

[3]

N. Bou-Rabee and H. Owhadi, Long-run accuracy of variational integrators in the stochastic context, SIAM Journal on Numerical Analysis, 48 (2010), 278-297.  doi: 10.1137/090758842.

[4]

A. BrüngerC. L. Brooks III and M. Karplus, Stochastic boundary conditions for molecular dynamics simulations of ST2 water, Chemical Physics Letters, 105 (1984), 495-500. 

[5]

G. M. Dall'ara, Discreteness of the spectrum of Schrödinger operators with non-negative matrix valued potentials, Journal of Functional Analysis, 268 (2015), 3649-3679.  doi: 10.1016/j.jfa.2014.10.007.

[6]

G. W. Ford and M. Kac, On the quantum Langevin equation, J. Statist. Phys., 46 (1987), 803-810.  doi: 10.1007/BF01011142.

[7]

G. W. FordM. Kac and P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Mathematical Phys., 6 (1965), 504-515.  doi: 10.1063/1.1704304.

[8]

M. HairerM. Hutzenthaler and A. Jentzen, Loss of regularity for Kolmogorov equations, The Annals of Probability, 43 (2015), 468-527.  doi: 10.1214/13-AOP838.

[9]

E. J. Hall, M. A. Katsoulakis and L. Rey-Bellet, Uncertainty quantification for generalized Langevin dynamics, The Journal of Chemical Physics, 145 (2016), 224108. doi: 10.1063/1.4971433.

[10]

A. Kammonen, P. Plecháč, M. Sandberg and A. Szepessy, Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics, Ann. Henri Poincaré, 19 (2018), 2727–2781. doi: 10.1007/s00023-018-0699-x.

[11]

D. P. Kroese, T. Taimre and Z. I. Botev, Handbook of monte carlo methods, John Wiley & Sons, 706 (2011). doi: 10.1002/9781118014967.

[12]

N. V. Krylov, Parabolic equations with VMO coefficients in Sobolev spaces with mixed norms, Journal of Functional Analysis, 250 (2007), 521-558.  doi: 10.1016/j.jfa.2007.04.003.

[13]

R. Kupferman, Fractional kinetics in Kac-Zwanzig heat bath models, Journal of Statistical Physics, 114 (2004), 291-326.  doi: 10.1023/B:JOSS.0000003113.22621.f0.

[14]

P. Langevin, On the theory of Brownian movement, C. R. Acad. Sci., 146 (1908), 530-533. 

[15]

J. L. Lebowitz and E. Rubin, Dynamical study of Brownian motion, Phys. Rev., 131 (1963), 2381-2396.  doi: 10.1103/PhysRev.131.2381.

[16]

B. Leimkuhler and C. Matthews, Molecular Dynamics: With Deterministic and Stochastic Numerical Methods, Interdisciplinary Applied Mathematics, 39. Springer, Cham, 2015.

[17]

T. Leliévre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numerica, 25 (2016), 681-880.  doi: 10.1017/S0962492916000039.

[18]

D. Marx and J. Hutter, Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods, Cambridge University Press, 2009. doi: 10.1017/CBO9780511609633.

[19]

J. C. MattinglyA. M. Stuart and D. J. Higham, Ergodicity for sdes and approximations: Locally Lipschitz vector fields and degenerate noise, Stochastic Processes and Their Applications, 101 (2002), 185-232.  doi: 10.1016/S0304-4149(02)00150-3.

[20]

P. M. Mazur and I. Oppenheim, Molecular theory of Brownian motion, Physica, 50 (1970), 241-258.  doi: 10.1016/0031-8914(70)90005-4.

[21]

E. H. Müller, R. Scheichl and T. Shardlow, Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation, Proc. A., 471 (2015), 20140679, 20 pp. doi: 10.1098/rspa.2014.0679.

[22]

G. A. Pavliotis, Stochastic Processes and Applications. Diffusion Processes, the Fokker-Planck and Langevin Equations, Texts in Applied Mathematics, 60. Springer, New York, 2014. doi: 10.1007/978-1-4939-1323-7.

[23]

J.-E. Shea and I. Oppenheim, Fokker-Planck equation and Langevin equation for one Brownian particle in a nonequilibrium bath, J. Phys. Chem., 100 (1996), 19035-19042.  doi: 10.1021/jp961605d.

[24]

R. D. Skeel and J. A. Izaguirre, An impulse integrator for Langevin dynamics, Molecular Physics, 100 (2002), 3885-3891.  doi: 10.1080/0026897021000018321.

[25]

H.-M. Stiepan and S. Teufel, Semiclassical approximations for Hamiltonians with operator-valued symbols, Comm. Math. Phys., 320 (2013), 821-849.  doi: 10.1007/s00220-012-1650-5.

[26]

L. Verlet, Computer "Experiments" on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev., 159 (1967), 98-103.  doi: 10.1103/PhysRev.159.98.

[27] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford Univ. Press, New York, 2001. 
[28]

R. Zwanzig, Nonlinear generalized Langevin equations, J. Stat. Phys., 9 (1973), 215-220.  doi: 10.1007/BF01008729.

[29]

M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, 138. American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/138.

Figure 1.  (Left column) unit-volume scaled histogram for the final time position and momentum of the heat bath dynamics for a series of $ m $-values and (right column) corresponding histograms for the Langevin dynamics
Figure 2.  (Left column) heat bath dynamics autocorrelation function $ \mathbb E[X^1(t)X^1(10)] $ for a series of $ m $-values and (right column) corresponding Langevin dynamics autocorrelation functions $ \mathbb E[X^1_L(t)X^1_L(10)] $
Figure 3.  (Left column) heat bath dynamics autocorrelation function $ \mathbb E[P^1(t)P^1(10)] $ for a series of $ m $-values and (right column) corresponding Langevin dynamics autocorrelation functions $ \mathbb E[P^1_L(t)P^1_L(10)] $
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