Nose-Hoover thermostat: Difference between revisions
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<math> | <math> | ||
\mathcal{L} = \sum\limits_{i=1}^{N} \frac{m_{i}}{2} s^{2} \dot{\bold{r}}_{i}^{2} - U(\bold{r}) + \frac{Q}{2} \dot{s}^{2}- | \mathcal{L} = \sum\limits_{i=1}^{N} \frac{m_{i}}{2} s^{2} \dot{\bold{r}}_{i}^{2} - U(\bold{r}) + \frac{Q}{2} \dot{s}^{2}-g k_{B} T \mathrm{ln} \, s | ||
</math> | </math> | ||
where <math>m_{i}</math> and <math>k_{B}</math> are the mass of ion <math>i</math> and the Boltzmann constant, respectively. The first two terms are the kinetic and potential energy of the system. The third and fourth term represent the kinetic and potential energy of the fictitious coordinate <math>s</math>. The parameter <math>g</math> is usually equal to the number of degrees of freedom of the system <math>g=3N - N_{\mathrm{constraint}}</math>, where <math>N_{\mathrm{constraint}}</math> is equal to the number of constraint set (fixed coordinates in the {{TAG|POSCAR}} file). The parameter <math>Q</math> is an effective "mass" of <math>s</math>, which controls the coupling of the system to the heat bath. It is set by the {{TAG|INCAR}} tag {{TAG|SMASS}}. | |||
Revision as of 08:45, 31 May 2019
In the approach by Nosé and Hoover[1][2][3] an extra degree of freedom is introduced in the Hamiltonian. The heat bath is considered as an integral part of the system and has a fictious coordinate 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/":): s which is introduced into the Lagrangian of the system. This Lagrangian for an 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 is written as
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/":): {\mathcal {L}}=\sum \limits _{{i=1}}^{{N}}{\frac {m_{{i}}}{2}}s^{{2}}{\dot {{\mathbf {r}}}}_{{i}}^{{2}}-U({\mathbf {r}})+{\frac {Q}{2}}{\dot {s}}^{{2}}-gk_{{B}}T{\mathrm {ln}}\,s
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/":): m_{{i}} 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/":): k_{{B}} are the mass of ion 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/":): i and the Boltzmann constant, respectively. The first two terms are the kinetic and potential energy of the system. The third and fourth term represent the kinetic and potential energy of the fictitious coordinate 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/":): s . The parameter 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/":): g is usually equal to the number of degrees of freedom of the 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/":): g=3N-N_{{{\mathrm {constraint}}}} , 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/":): N_{{{\mathrm {constraint}}}} is equal to the number of constraint set (fixed coordinates in the POSCAR file). The parameter 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/":): Q is an effective "mass" of 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/":): s , which controls the coupling of the system to the heat bath. It is set by the INCAR tag SMASS.
References