Documentation and reviews
This Ph.D. thesis
contains a thorough introduction to the GLE thermostat,
which includes some background on stochastic differential
equations and a number of example applications.
A brief introduction to colored noise sampling for molecular dynamics is given in the
February 2014 highlight of the PsiK newsletter,
while the lecture notes for the
2015 Juelich school give
a discussion specific to the simulation of nuclear quantum effects.
The atomistic cookbook contains several recipes using GLE thermostats, especially for use in path integral molecular dynamics.
See also the FAQ
page.
References
 “Efficient firstprinciples calculation of the quantum kinetic energy and momentum distribution of nuclei”,
Michele Ceriotti and David E. Manolopoulos, Rev. Lett. 109, 100604 (2012)
doi: 10.1103/PhysRevLett.109.100604
The PI+GLE method is extended so as to guarantee that the quantum kinetic energy of nuclei
will converge as efficiently as the potential energy. This is done by simultaneously enforcing the correct
fluctuations of both individual beads and the centroid, in the harmonic limit. A transient anisotopic
Gaussian approximation to the particle momentum distribution is also developed, which makes it easy to
estimate the anisotropy of the PMD without using open path integrals.
 “The inefficiency of reweighted sampling and the curse of system size in highorder path integration”,
Michele Ceriotti, Guy A. R. Brain, Oliver Riordan and David E. Manolopoulos, Proc. R. Soc. A 468, 2 (2012)
doi: 10.1098/rspa.2011.0413
The problem of reweighted sampling is discussed in very general terms, demonstrating the pathological
behaviour of the sampling efficiency with increasing variance of the difference Hamiltonian. The implications with respect some recent
attempts to make highorder path integration practicable are discussed, and demonstrating that this is a
hopeless task and that PI+GLE obtains faster convergence without being affected by sampling problems.
 “Accelerating the convergence of path integral dynamics with a generalized Langevin equation”,
M. Ceriotti, D. E. Manolopoulos, and M. Parrinello, J. Chem. Phys. 134, 084104 (2011)
doi: 10.1063/1.3556661
The quantum thermostat can model nuclear quantum effects accurately in quasiharmonic
systems, but becomes less reliable as anharmonicities become more important. Here a nonMarkovian
GLE dynamics is combined with an imaginarytime path integral dynamics to recover systematic
convergence of nuclear quantum effects, at the same time greatly speeding up this convergence.
 “Efficient multiple time scale molecular dynamics: using colored noise thermostats to stabilize resonances”,
J. A. Morrone, T. E. Markland, M. Ceriotti, and B. J. Berne, J. Chem. Phys. 134, 014103 (2011)
doi: 10.1063/1.3518369
Multiple time step schemes can deliver consistent speed ups to classical molecular dynamics, but
can also lead to resonances and instabilities which may affect the reliability of the simulation. Stochastic terms
have been used in the past to control resonances, at the expense however of a slowingdown of diffusive motions
which ultimately cancels the benefits of MTS. Here a GLE is used to enforce tight control of resonances of high
frequency modes, while introducing minimal hindrance to slower, collective dynamics.
 “Efficient stochastic thermostatting of path integral molecular dynamics”, M. Ceriotti, M. Parrinello, T. E. Markland, D. E. Manolopoulos, J. Chem. Phys 133, 124104 (2010) doi: 10.1063/1.3489925
Thorough comparison of different thermostats applied to the challenging case of path integral molecular dynamics.
GLE turns out to be competitive with targeted techniques which exploit explicitly knowledge of the internal modes
of the necklace.
“The δthermostat: selective normalmodes excitation by colorednoise Langevin dynamics”, M. Ceriotti and M. Parrinello, Proc. Comp. Sci. 1, 1601 (2010) doi: 10.1016/j.procs.2010.04.180
Usage of a nonequilibrium GLE thermostat to freeze all but a narrow range of vibrational modes. Can be used to probe
the vibrational spectrum of a solid in a MD framework, or to approximately compute the eigenvalue spectrum of a
positivedefinite sparse matrix in linearscaling time.
 “Colorednoise thermostats à la carte”, M. Ceriotti, G. Bussi and M. Parrinello, J. Chem. Theory Comput. 6, 1170 (2010) doi: 10.1021/ct900563s
Long paper, describing in details the derivation and implementation of the GLE framework. Examples of optimal sampling LE and quantumthermostat dynamics.
 “Nuclear quantum effects in solids using a colorednoise thermostat”, M. Ceriotti, G. Bussi and M. Parrinello, Phys. Rev. Lett. 103, 030603 (2009) doi: 10.1103/PhysRevLett.103.030603
Description of the quantum thermostat, applications to the calculation of nuclear quantum effects in diamond and liquid water (proton momentum distribution).
 “Langevin equation with colored noise for constanttemperature molecular dynamics simulations”, M. Ceriotti, G. Bussi and M. Parrinello, Phys. Rev. Lett. 102, 020601 (2009) doi: 10.1103/PhysRevLett.102.020601
Introduction to the GLE framework for molecular dynamics. Applications to CarParrinello dynamics and to Drudeoscillator polarizable forcefields.
Slides and posters
A few (dated) posters and presentations introducing colored noise and
Generalized Langevin Equation thermostats.

"Nuclear quantum effects the colored way"
Poster presented in Lugano for the CECAM workshop "Quantum Monte Carlo meets Quantum Chemistry" (2010).
Example application of the "quantum thermostat" to estimate nuclear quantum effects in condensedphase
systems. Download PDF


"A novel framework for enhanced molecular dynamics based on the generalized Langevin equation"
Slides presented for M. Ceriotti's Ph.D. dissertation. General overview of both canonical sampling and
nonequilibrium GLE thermostats. Download PDF


"A tutorial on the Generalized Langevin Thermostat"
Slides for a group seminar. Introduction to the GLE, and thorough discussion of thermostat issues
in the simulation of liquid water (classical, flexible forcefield). Download PDF


"A tunable, allpurpose colorednoise thermostat"
Poster presented at Jülich, for the winter school "Multiscale simulation methods in molecular sciences" (2009).
Applications of GLE thermostat to efficient sampling and to CarParrinello dynamics. Download PDF
