Difference between revisions of "Dynamical density-functional theory"

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'''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 [[equilibrium density functional theory | Density functional theory]] to
+
extends equilibrium  [[density-functional theory]] to
situations away from equilibrium. In the simplest case, only small deviations from equilibrium are
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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.
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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:
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:<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 we are left with the usual density functional
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In equilibrium, the left hand side vanishes and one is left with the usual density functional
expression. Away from it, an increase in the free energy ''A'' causes
+
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 magnetization field in a model for magnets (such as the [[Ising model]]).
+
(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,
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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]]
 
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==References==
 
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<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", J. Phys.: Condens. Matter '''12''' A413(2000)]
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*[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:

 \frac{\partial\rho}{\partial t}=  - \mu \frac{\delta A}{\delta\rho}.

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, A, causes a decrease in the density, mediated by the mobility coefficient \mu.

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:

 \frac{\partial\rho}{\partial t}=  - \nabla \mu \nabla \frac{\delta A}{\delta\rho}.

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

See also[edit]

References[edit]

Interesting reading[edit]