Density functional theory (statistical physics)

The classical density functional theory ( DFT , also classical density functional theory ) is a method in statistical physics to describe the behavior of a many-particle system (e.g. a gas in a container). DFT is a standard technique in fluid theory today. In contrast to the older quantum mechanical density functional theory , it is applied to many-particle systems that are described using classical physics.

Classical DFT enables the location-dependent density of this system, correlation functions (including radial distribution function ) and thermodynamic properties ( free energy , equation of state , phase transitions ) to be calculated for given parameters (including temperature and externally specified interactions ) . The strength of DFT lies in the fact that it can be used for inhomogeneous systems (location-dependent particle density).

The theory was founded by Robert Evans (University of Bristol) in 1979, who proved the underlying principle of variation, using a corresponding principle for quantum mechanical many-body systems at finite temperature by N. David Mermin (1965, a generalization of the Hohenberg-Kohn theorem ). The theory has historical forerunners in classical studies by Johannes Diderik van der Waals on the liquid-gas interface (1893) and by Lars Onsager on phase transitions in liquid crystals (1949).

application areas

Classical DFT enables thermodynamic properties and correlation functions to be calculated for systems whose translation invariance and / or rotation invariance is broken. In inhomogeneous situations, effects can occur that do not exist in the homogeneous phase. Examples for this are:

• Restricting geometries create an external potential and change the structure of the fluid and its phase behavior: Due to packing effects, the density oscillates near a wall, while the effects averaged out further away. On solid substrates and walls, adsorption can lead to orderly structures near the wall. Macroscopically measurable effects such as wetting, sedimentation or surface phase transitions can occur. In the case of microchannels or porous media, effects such as capillary condensation or capillary filling can occur.

• Interfaces between different coexisting phases (liquid-gaseous, crystalline-liquid, liquid-liquid- segregation in liquid mixtures ...). The particle density varies greatly in the area of ​​the phase interface. This variation leads to the macroscopic effect of surface tension.

• Outer fields, e.g. B. Sedimentation of suspended colloids in a gravitational field, alignment of dipoles in an electric field (molecules of a liquid crystal ).

In principle, the crystalline phase can also be treated in the context of DFT, since it can be represented as a periodic density. So you can calculate the phase transition of freezing.

Compared to simulations ( MC or MD ), the DFT solution can usually be calculated much faster.

The classic DFT can only be used in the classic limit, i.e. not where quantum mechanical effects dominate. One criterion for this is that the thermal wavelength must be much smaller than the mean nearest neighbor distance.

Variation principle

The principle of density functional theory is based on the fact that the thermodynamic potential (e.g. the free energy or the grand canonical potential ) of an ensemble can be written as a functional of the microscopic density , i.e. or . This functional becomes minimal with the physically realized density , the equilibrium density . The following applies to the grand canonical potential ${\displaystyle F}$ ${\displaystyle \Omega }$ ${\displaystyle \rho }$ ${\displaystyle F[\rho ]}$ ${\displaystyle \Omega [\rho ]}$ ${\displaystyle \rho _{0}}$

${\displaystyle \Omega [\rho ]>\Omega [\rho _{0}]\ {\text{für}}\ \rho \neq \rho _{0}}$

whereby . If the density functional for a given system is known, the equilibrium density can thus be found by minimizing the functional. The functional derivative is zero for the equilibrium density: ${\displaystyle \Omega [\rho _{0}]=\Omega _{0}}$

${\displaystyle \left.{\frac {\delta \Omega [\rho ]}{\delta \rho ({\boldsymbol {r}})}}\right|_{\rho _{0}}=0}$

Derivation

Symbols used in the following: the Hamilton function , the chemical potential , the number of particles , the Boltzmann constant , the temperature, the inverse thermal energy , the grand canonical partition function , the classical trace formation in the grand canonical ensemble. ${\displaystyle H}$ ${\displaystyle \mu }$ ${\displaystyle N}$ ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle T}$ ${\displaystyle \beta =(k_{\mathrm {B} }T)^{-1}}$ ${\displaystyle Z_{\text{gk}}}$ ${\displaystyle \operatorname {Tr} }$

The equilibrium density and the grand canonical equilibrium potential are known from the statistics of the grand canonical ensemble.

${\displaystyle f_{0}=Z_{\mathrm {gk} }^{-1}\exp \!\left[-\beta (H-\mu N)\right]\quad \implies \quad \Omega _{0}=-k_{\mathrm {B} }T\ \ln Z_{\text{gk}}}$

The grand-canonical potential can be written in general (also for non-equilibrium states) as a functional of any probability density in phase space: ${\displaystyle f}$

${\displaystyle \Omega _{\text{gk}}[f]=\operatorname {Tr} \left\{f\left(H-\mu N+k_{\mathrm {B} }T\ln f\right)\right\}}$

One can use a Gibbs inequality to prove that the equilibrium density minimizes the functional: ${\displaystyle f_{0}}$

${\displaystyle \Omega _{0}=\min _{f}\Omega _{\text{gk}}[f]}$

The decisive step is the transition from a functional of to a functional of . It depends on coordinates (for particles, pulse coordinates and position coordinates ), on the other hand, only on position coordinates . The relationship is given by: . ${\displaystyle f}$ ${\displaystyle \rho }$ ${\displaystyle f}$ ${\displaystyle 6N}$ ${\displaystyle N}$ ${\displaystyle 3}$ ${\displaystyle {\boldsymbol {p}}_{i}}$ ${\displaystyle 3}$ ${\displaystyle {\boldsymbol {r}}_{i}}$ ${\displaystyle \rho }$ ${\displaystyle 3}$ ${\displaystyle {\boldsymbol {r}}}$ ${\displaystyle \textstyle \rho ({\boldsymbol {r}})=\operatorname {Tr} \{f\sum _{i=1}^{N}\delta ({\boldsymbol {r}}-{\boldsymbol {r}}_{i})\}}$

The minimization according to can be rewritten into a double minimization (Levy method): ${\displaystyle f}$

${\displaystyle \Omega _{0}=\min _{\rho }\Omega [\rho ]\quad {\text{wobei}}\quad \Omega [\rho ]=\min _{f\to \rho }\Omega _{\text{gk}}[f]}$

Inner minimization means that it is minimized under the condition that a certain one is generated by. In general, however, no analytical expression for (the excess part of) can be derived from this minimization . ${\displaystyle \Omega _{\text{gk}}[f]}$ ${\displaystyle \rho }$ ${\displaystyle f}$ ${\displaystyle \Omega [\rho ]}$

Intrinsic free energy

The Hamilton function can be split into kinetic energy, internal interaction and external interaction. These are: ${\displaystyle H=K_{N}+\Phi _{N}+V_{N}}$

${\displaystyle \textstyle K_{N}=\sum _{i=1}^{N}{\frac {{\boldsymbol {p}}_{i}^{2}}{2m}}\,,\quad \Phi _{N}=\sum _{i<j}^{N}\phi ({\boldsymbol {r}}_{i},{\boldsymbol {r}}_{j})\,,\quad V_{N}=\sum _{i=1}^{N}V({\boldsymbol {r}}_{i})}$

Here is the effective internal potential between the liquid particles and the external potential. ${\displaystyle \phi ({\boldsymbol {r}}_{i},{\boldsymbol {r}}_{j})}$ ${\displaystyle V({\boldsymbol {r}}_{i})}$

Inserting and splitting:

${\displaystyle \Omega [\rho ]=\underbrace {\underbrace {\overbrace {\min _{f\to \rho }\operatorname {Tr} \left\{f\left(K_{N}+k_{\mathrm {B} }T\ln f\right)\right\}} ^{{\mathcal {F}}_{\text{id}}[\rho ]}+\overbrace {\min _{f\to \rho }\operatorname {Tr} \left\{f\Phi _{N}\right\}} ^{{\mathcal {F}}_{\text{ex}}[\rho ]}} _{{\mathcal {F}}[\rho ]}+\overbrace {\min _{f\to \rho }\operatorname {Tr} \left\{fV_{N}\right\}} ^{\int \mathrm {d} {\boldsymbol {r}}\,\rho ({\boldsymbol {r}})\,V({\boldsymbol {r}})}} _{F[\rho ]}+\overbrace {\min _{f\to \rho }\operatorname {Tr} \left\{f\left(-\mu N\right)\right\}} ^{-\int \mathrm {d} {\boldsymbol {r}}\,\rho ({\boldsymbol {r}})\,\mu }}$

The functional free energy and intrinsic free energy were defined. The intrinsic free energy functional is introduced in such a way that it only depends on internal interactions , but not on external interactions . The analytical form of can therefore be used for all inhomogeneities, since these are caused by. ${\displaystyle F[\rho ]}$ ${\displaystyle {\mathcal {F}}[\rho ]}$ ${\displaystyle \phi }$ ${\displaystyle V}$ ${\displaystyle {\mathcal {F}}[\rho ]}$ ${\displaystyle V}$

The connection between and is given by ${\displaystyle \Omega [\rho ]}$ ${\displaystyle {\mathcal {F}}[\rho ]}$

${\displaystyle \Omega [\rho ]={\mathcal {F}}[\rho ]-\int \mathrm {d} {\boldsymbol {r}}\,\rho ({\boldsymbol {r}})\mu _{\text{in}}({\boldsymbol {r}})}$

where defines the intrinsic chemical potential. This corresponds to a Legendre transformation between the thermodynamic potentials. ${\displaystyle \mu _{\text{in}}({\boldsymbol {r}}):=\mu -V({\boldsymbol {r}})}$

Ideal and excessively functional

The intrinsic free energy is divided into an ideal and an excess part. The former describes the non-interacting part (see ideal gas ), the latter describes the interactions within the liquid.

${\displaystyle {\mathcal {F}}[\rho ]={\mathcal {F}}_{\text{id}}[\rho ]+{\mathcal {F}}_{\text{ex}}[\rho ]}$

The ideal part can be calculated exactly analytically ( is the thermal wavelength ): ${\displaystyle \Lambda }$

${\displaystyle {\mathcal {F}}_{\text{id}}[\rho ]=k_{\mathrm {B} }T\int \mathrm {d} {\boldsymbol {r}}\,\rho ({\boldsymbol {r}})\left\{\ln \!\left[\Lambda ^{3}\rho ({\boldsymbol {r}})\right]-1\right\}}$

The excess functional depends on the respective internal interaction potential and is generally unknown. The direct correlation functions are defined as functional derivatives: ${\displaystyle \phi }$

${\displaystyle c^{(n)}({\boldsymbol {r}}_{1},\ldots ,{\boldsymbol {r}}_{n},[\rho ]):=-\beta \left.{\frac {\delta ^{n}{\mathcal {F}}_{\text{ex}}[\rho ]}{\delta \rho ({\boldsymbol {r}}_{1})\ldots \delta \rho ({\boldsymbol {r}}_{n})}}\right|_{\rho }}$

For the ideal gas is and thus and ${\displaystyle \Phi =0}$ ${\displaystyle {\mathcal {F}}_{\text{ex}}=0}$ ${\displaystyle c^{(n)}=0}$

Application of the principle of variation

The following applies to the above variation principle:

${\displaystyle 0=\left.{\frac {\delta \Omega [\rho ]}{\delta \rho ({\boldsymbol {r}})}}\right|_{\rho =\rho _{0}}=k_{\mathrm {B} }T\ \ln \!\left[\Lambda ^{3}\rho _{0}({\boldsymbol {r}})\right]-k_{\mathrm {B} }T\ c^{(1)}({\boldsymbol {r}},[\rho _{0}])-\mu _{\text{in}}({\boldsymbol {r}})}$

Solving for the density gives the generalized barometric altitude formula

${\displaystyle \rho _{0}({\boldsymbol {r}})=\Lambda ^{-3}\exp \!\left[\beta \mu _{\text{in}}({\boldsymbol {r}})+c^{(1)}({\boldsymbol {r}},[\rho _{0}])\right]}$

In addition to the barometric altitude formula of the ideal gas, there is also the function , which includes the influence of the particle interaction on the density profile. If you write the exponent as , you can see that the external potential is modulated. ${\displaystyle c^{(1)}({\boldsymbol {r}})}$ ${\displaystyle \beta \mu -\beta \left\{V({\boldsymbol {r}})-k_{\mathrm {B} }T\,c^{(1)}({\boldsymbol {r}})\right\}}$ ${\displaystyle -k_{\mathrm {B} }T\,c^{(1)}({\boldsymbol {r}})}$

Since density is a functional, the generalized barometric height formula can be used to solve the equation self-consistently using fixed point iteration . ${\displaystyle c^{(1)}({\boldsymbol {r}})}$

Approximations for the excess functional

Different interactions between the observed particles (e.g. Lennard-Jones potential , hard spheres, soft repulsion between polymer coils) require different excess functionals. However, if a functional is known for a certain (internal) interaction, all inhomogeneous situations (for all external potentials) can be calculated with it.

The excess functional can only be constructed exactly for hard rods in one dimension; suitable approximations must be used for all other interacting systems. Thus, the central problem of DFT lies in obtaining a suitable approximation for this functional. Functional development based on microscopic properties (from an effective Hamilton function) requires a lot of experience. However, there are some standard functionalities that are very versatile.

Frequently used approximations are:

• Local density (Local Density Approximation, LDA)
• Molecular field approximation (MFA or Random Phase Approximation, RPA)
• Weighted density approximation (Weighted Density Approximation, WDA)
• Rosenfeld functionals (Fundamental Measure Theory, FMT)

Dynamic density functional theory

In addition to DFT, which considers states of equilibrium, there is also DDFT (dynamic DFT) for non-equilibrium states, which can be used to calculate the development of a system over time (e.g. colloidal suspensions that are subject to Brownian motion ).

literature

• J.-P. Hansen, IR McDonald: Theory of Simple Liquids . Academic Press / Elsevier, 4th edition, 2013, ISBN 978-0-12-387032-2
• D. Henderson: Fundamentals of Inhomogeneous Fluids . Dekker, 1992, ISBN 978-0-82-478711-0 , Chap. 3 by R. Evans.
• R. Evans: The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform, classical fluids . Adv. Phys. , 28 : 143-200 (1979). doi : 10.1080 / 00018737900101365
• Y. Rosenfeld: Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing . Phys. Rev. Lett. 63 : 980-983 (1989). doi : 10.1103 / PhysRevLett.63.980
• M. Schmidt, M. Burgis, WSB Dwandaru, G. Leithall, P. Hopkins: Recent developments in classical density functional theory: Internal energy functional and diagrammatic structure of fundamental measure theory, Condensed Matter Physics, Volume 15, 2002, 43603, 1 -15, pdf

See also

• Density functional theory (quantum physics) and Hohenberg-Kohn theorem .

Web links

• R. Evans, Density Functional Theory for Inhomogeneous Liquids I, 2009, pdf