Dynamical density-functional theory
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:
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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 Helmholtz energy function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , causes a decrease in the density, mediated by the mobility coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:
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The later sort of expressions are called non-conserved dynamics.