# Difference between revisions of "Dynamical density-functional theory"

(Created page with "'''Dynamical Density functional theory''' is a set of theories in statistical mechanics that extends Density functional theory ...") |
Carl McBride (talk | contribs) m (Added some internal links) |
||

Line 1: | Line 1: | ||

'''Dynamical Density functional theory''' is a set of theories in [[statistical mechanics]] that | '''Dynamical Density functional theory''' is a set of theories in [[statistical mechanics]] that | ||

− | extends [[ | + | extends equilibrium [[density-functional theory]] to |

− | situations away from equilibrium. In the simplest case, only small deviations from equilibrium are | + | situations away from [[Non-equilibrium thermodynamics |equilibrium]]. In the simplest case, only small deviations from equilibrium are |

− | considered, so that linear response theory can be applied. | + | considered, so that [[linear response theory]] can be applied. |

A simple approach in this line is to consider this evolution of the density field: | A simple approach in this line is to consider this evolution of the density field: | ||

Line 8: | Line 8: | ||

:<math> \frac{\partial\rho}{\partial t}= - \mu \frac{\delta A}{\delta\rho}.</math> | :<math> \frac{\partial\rho}{\partial t}= - \mu \frac{\delta A}{\delta\rho}.</math> | ||

− | In equilibrium, the left hand side vanishes and | + | In equilibrium, the left hand side vanishes and one is left with the usual density functional |

− | expression. Away from it, an increase in the | + | expression. Away from it, an increase in the [[Helmholtz energy function]], <math>A</math>, causes |

− | a decrease in the density, mediated by the mobility coefficient <math>\mu</math>. | + | a decrease in the density, mediated by the [[Mobility |mobility coefficient]] <math>\mu</math>. |

This sort of evolution will not satisfy conservation of the number of particles | This sort of evolution will not satisfy conservation of the number of particles | ||

− | (that is, the space integral of the density field), and is therefore termed ''non-conserved dynamics''. This can be valid in cases in which this field is in fact not conserved, such as the | + | (that is, the space integral of the density field), and is therefore termed ''non-conserved dynamics''. This can be valid in cases in which this field is in fact not conserved, such as the magnetisation field in a model for magnets (such as the [[Ising model]]). |

In other cases, for example with actual particles, | In other cases, for example with actual particles, | ||

Line 22: | Line 22: | ||

The later sort of expressions are called ''non-conserved dynamics''. | The later sort of expressions are called ''non-conserved dynamics''. | ||

− | |||

− | |||

− | |||

− | |||

==See also== | ==See also== | ||

*[[Brownian dynamics]] | *[[Brownian dynamics]] | ||

− | + | ==References== | |

− | + | <references/> | |

==Interesting reading== | ==Interesting reading== | ||

− | *[http://dx.doi.org/10.1088/0953-8984/12/8A/356 Umberto Marini Bettolo Marconi and Pedro Tarazona "Dynamic density functional theory of fluids", | + | *[http://dx.doi.org/10.1088/0953-8984/12/8A/356 Umberto Marini Bettolo Marconi and Pedro Tarazona "Dynamic density functional theory of fluids", Journal of Physics: Condensed Matter '''12''' A413 (2000)] |

[[category: Density-functional theory]] | [[category: Density-functional theory]] |

## Latest revision as of 13:15, 28 February 2012

**Dynamical Density functional theory** is a set of theories in statistical mechanics that
extends equilibrium density-functional theory to
situations away from equilibrium. In the simplest case, only small deviations from equilibrium are
considered, so that linear response theory can be applied.

A simple approach in this line is to consider this evolution of the density field:

In equilibrium, the left hand side vanishes and one is left with the usual density functional expression. Away from it, an increase in the Helmholtz energy function, , causes a decrease in the density, mediated by the mobility coefficient .

This sort of evolution will not satisfy conservation of the number of particles
(that is, the space integral of the density field), and is therefore termed *non-conserved dynamics*. This can be valid in cases in which this field is in fact not conserved, such as the magnetisation field in a model for magnets (such as the Ising model).

In other cases, for example with actual particles, some evolution has to be postulated. For example, this evolution will conserve the number of particles:

The later sort of expressions are called *non-conserved dynamics*.