Nose-Hoover-chain thermostat: Difference between revisions

Latest revision as of 12:58, 12 July 2023

The standard Nosé-Hoover thermostat suffers from well-known issues, such as the ergodicity violation in the case of simple harmonic oscillator^[1]. As proposed by Martyna and Klein^[1], these problems can be solved by using multiple Nose Hoover thermostats connected in a chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term:

{\displaystyle \mathcal{H'} = \mathcal{H}(\bold{r},\bold{p}) + \sum\limits_{j=1}^{M} \frac{p_{\eta_j}^2}{2Q_j} + (3N-N_c)k_{B} T \eta_1 + k_{B} T \sum\limits_{j=2}^{M} \eta_j, }

where $\mathcal{H}(\bold{r},\bold{p})$ is the Hamiltonian of the physical system, $M$ , $N$ and $N_c$ are the numbers of thermostats, atoms in the cell, and geometric constraints, respectively, and $\eta_{j}$ , $p_{\eta_j}$ , and $Q_{j}$ are the position, momentum, and mass-like parameter associated with the thermostat $j$ . Just like the total energy in the NVE ensemble, $\mathcal{H'}$ is valuable for diagnostics purposes. Indeed, a significant drift in $\mathcal{H'}$ indicates that the corresponding computational setting is suboptimal. Typical reasons for this behavior involve noisy forces (e.g., because of a poor SCF convergence) and/or a too large integration step (defined via POTIM).

The number of thermostats is controlled by the flag NHC_NCHAINS. Typically, this flag is set to a value between 1 and 5, the maximal allowed value is 20. In the special case of NHC_NCHAINS=0, the thermostat is switched off, leading to a MD in the microcanonical ensemble. Another special case of NHC_NCHAINS=1 corresponds to the standard Nose-Hoover thermostat.

The only parameter of this thermostat is the characteristic time scale ( $\tau$ ), defined via flag NHC_PERIOD. This parameter is used to setup the mass-like variables via the relations:

{\displaystyle Q_1 = 3 (N -N_c)k_{B} T \tau^2 }

{\displaystyle Q_j = k_{B} T \tau^2; \; \; \; j=2,\dots,M }

Furthermore, due to rapidly varying forces in thermostat variables propagators, the standard velocity Verlet algorithm with fixed integration step might be insufficiently accurate. As proposed by Tuckerman^[2], the RESPA^[3] methodology can be used to overcome this problem, in which the integration step used in thermostat variables propagation is split into NHC_NRESPA equal parts, each of which may be further divided into NHC_NS smaller parts treated by Suzuki-Yoshida scheme of fourth or sixth order.

References

↑ ^a ^b J. Martyna, M. L. Klein, and M. Tuckerman, J. Chem. Phys. 97, 2635 (1992).
↑ M. E. Tuckerman, Statistical mechanics: theory and molecular simulation, Oxford University Press Inc., New York, 2010; pp 194-199.
↑ M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97, 1900 (1992)

[martyna:jcp:92-1] J. Martyna, M. L. Klein, and M. Tuckerman, J. Chem. Phys. 97, 2635 (1992).

[2] M. E. Tuckerman, Statistical mechanics: theory and molecular simulation, Oxford University Press Inc., New York, 2010; pp 194-199.

[3] M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97, 1900 (1992)

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
-The standard Nose Hoover suffers from well known issues, such as the ergodicity violation in the case of simple harmonic oscillator. As proposed by Martyna and Klein<ref>J. Martyna, M. L. Klein, and M. Tuckerman, ''J. Chem. Phys.'' 97, 2635 (1992).[https://pubs.aip.org/aip/jcp/article/97/4/2635/927962/Nose-Hoover-chains-The-canonical-ensemble-via]</ref>, these problems can be solved by using multiple Nose Hoover thermostats connected in a chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term:
+The [[Nose-Hoover thermostat|standard Nosé-Hoover thermostat]] suffers from well-known issues, such as the ergodicity violation in the case of simple harmonic oscillator{{cite|martyna:jcp:92}}. As proposed by Martyna and Klein{{cite|martyna:jcp:92}}, these problems can be solved by using multiple Nose Hoover thermostats connected in a chain. Although the underlining dynamics is non-Hamiltonian, the corresponding equations of motion conserve the following energy term:
 ::<math>
-\mathcal{H'} = \mathcal{H}(\bold{r},\bold{p}) +  \sum\limits_{j=1}^{M} \frac{p_{\eta_j}^2}{2Q_j} + 3Nk_{B} T \eta_1 + k_{B} T \sum\limits_{j=2}^{M} \eta_j
+\mathcal{H'} = \mathcal{H}(\bold{r},\bold{p}) +  \sum\limits_{j=1}^{M} \frac{p_{\eta_j}^2}{2Q_j} + (3N-N_c)k_{B} T \eta_1 + k_{B} T \sum\limits_{j=2}^{M} \eta_j,
 </math>
-where <math>\mathcal{H}(\bold{r},\bold{p})</math> is the Hamiltonian of the physical system, <math>M</math> and <math>N</math> are the numbers of thermostats and atoms in the cell, respectively, and <math>\eta_{j}</math>, <math>p_{\eta_j}</math>, and <math>Q_{j}</math> are the position, momentum, and mass-like parameter associated with the thermostat <math>j</math>. Just like the total energy in NVE ensemble,<math>\mathcal{H'}</math> is valuable for diagnostics purposes. Indeed, a significant drift in <math>\mathcal{H'}</math> indicate that the corresponding computational setting is suboptimal. Typical reasons for this behavior involve noisy forces (e.g., because of a poor SCF convergence) and/or a too large integration step (defined via {{TAG|POTIM}}).
+where <math>\mathcal{H}(\bold{r},\bold{p})</math> is the Hamiltonian of the physical system, <math>M</math>, <math>N</math> and <math>N_c</math> are the numbers of thermostats, atoms in the cell, and geometric constraints, respectively, and <math>\eta_{j}</math>, <math>p_{\eta_j}</math>, and <math>Q_{j}</math> are the position, momentum, and mass-like parameter associated with the thermostat <math>j</math>. Just like the total energy in the NVE ensemble,<math>\mathcal{H'}</math> is valuable for diagnostics purposes. Indeed, a significant drift in <math>\mathcal{H'}</math> indicates that the corresponding computational setting is suboptimal. Typical reasons for this behavior involve noisy forces (e.g., because of a poor SCF convergence) and/or a too large integration step (defined via {{TAG|POTIM}}).
-The number of thermostats is controlled by the flag  {{TAG|NHC_NCHAINS}}. Typically, this flag is set to a value between 1 and 5, the maximal allowed value is 20. In the special case of {{TAG|NHC_NCHAINS}}=0, the thermostat is switched off, leading to a MD in microcanonical ensemble. Another special case of {{TAG|NHC_NCHAINS}}=1 corresponds to the  standard {{TAG|Nose-Hoover thermostat}}.
+The number of thermostats is controlled by the flag  {{TAG|NHC_NCHAINS}}. Typically, this flag is set to a value between 1 and 5, the maximal allowed value is 20. In the special case of {{TAG|NHC_NCHAINS}}=0, the thermostat is switched off, leading to a MD in the microcanonical ensemble. Another special case of {{TAG|NHC_NCHAINS}}=1 corresponds to the  standard {{TAG|Nose-Hoover thermostat}}.
-The only thermostat parameter is {{TAG|NHC_PERIOD}}, corresponding to a characteristic time scale (<math>\tau</math>) of the system expressed in time steps. This variable is used to setup the mass-like variables via the relations:
+The only parameter of this thermostat is the characteristic time scale (<math>\tau</math>), defined via flag {{TAG|NHC_PERIOD}}. This parameter is used to setup the mass-like variables via the relations:
 ::<math>
-Q_1 = 3 N k_{B} T \tau^2
+Q_1 = 3 (N -N_c)k_{B} T \tau^2
 </math>
 ::<math>
@@ Line 18: / Line 18: @@
 </math>
+Furthermore, due to rapidly varying forces in thermostat variables propagators, the standard velocity Verlet algorithm with fixed integration step might be insufficiently accurate. As proposed by Tuckerman<ref>M. E. Tuckerman, Statistical mechanics: theory and molecular simulation, Oxford University Press Inc., New York, 2010; pp 194-199.</ref>, the RESPA<ref>[https://pubs.aip.org/aip/jcp/article/97/3/1990/221848/Reversible-multiple-time-scale-molecular M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys. 97, 1900 (1992)]</ref> methodology can be used to overcome this problem, in which the integration step used in thermostat variables propagation is split into {{TAG|NHC_NRESPA}} equal parts, each of which may be further divided into {{TAG|NHC_NS}} smaller parts treated by Suzuki-Yoshida scheme of fourth or sixth order.
-{{TAG|NHC_NRESPA}}
+----
+==References==
+<references/>
-{{TAG|NHC_NS}}