Nose-Hoover-chain thermostat: Difference between revisions
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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 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}}). | 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 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: | 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: | ||
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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 | 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. | ||
---- | ---- | ||
==References== | ==References== | ||
<references/> | <references/> | ||
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:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \mathcal{H}(\bold{r},\bold{p})} is the Hamiltonian of the physical system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): M , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): N and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle N_c} are the numbers of thermostats, atoms in the cell, and geometric constraints, respectively, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \eta_{j}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle p_{\eta_j}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle Q_{j}} are the position, momentum, and mass-like parameter associated with the thermostat Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): j . Just like the total energy in the NVE ensemble,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \mathcal{H'}} is valuable for diagnostics purposes. Indeed, a significant drift in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \tau ), defined via flag NHC_PERIOD. This parameter is used to setup the mass-like variables via the relations:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle Q_1 = 3 (N -N_c)k_{B} T \tau^2 }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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)