**List of Symbols**


42 Will-be-set-by-IN-TECH

toys seems enough to evolve to as tools. Covering many topics in the renormalization group and quantum field theory is not so practical to learn in depth for typical material scientists. Such study will be done by the experts. The pedagogy for understanding the quantum field

To date there is no further fundamental theoretical tool in describing our physical world rather than the quantum field theory. It is hardly necessary to go beyond the quantum field theory for materials scientists. I would like to emphasize that we are standing at the starting point for a long journey to utilize the unified method with the reinterpreted terminology, *ab initio*

The author deeply appreciates Profs. Jae Il Lee and Se-Kyeun Kwon for their decades-long discussions and critics on this subject. A special acknowledgement is dedicated to Prof. Byeong-Joo Lee for his mentoring advises on the CALPHAD methods. Prof. H. K. D. H. Bhadeshia should receive thanks of the author for his comments from a metallurgist's view

This work was supported in part by POSCO Steel Innovation Program and in part by the Basic Science Research Program (Grant No. 2009-0088216) and the World Class University (WCU) Program (Project No. R32-2008-000-10147-0) through the National Research Foundation

• *G*: Gibbs free energy; full group in the order-parameter space *R* (depending on context)

funded by the Ministry of Education, Science and Technology of Republic of Korea.

This article is dedicated for the memory of Late Prof. Duk-Joo Kim.

• *E*xc [*n*]: exchange-correlation energy functional of density *n*

: external perturbation Hamiltonian operator

• *C* (*R*): correlation function of interaction range *R*

theory is also equally important.

thermodynamics authentically.

**9. Acknowledgements**

point.

**List of Symbols**

• *A*: spectral weight function

• *D*: long time interval • D: density of states • *E*: internal energy

• *F*: Helmholtz free energy

• *G*¯: real-time Green's function • *G*¯ *<sup>A</sup>*: advanced Green's function • *G*¯ *<sup>R</sup>*: retarded Green's function • G: temperature Green's function

• *G*/*H*: coset space

• *H*: subgroup of *G*

• *H*ˆ �

• *H*ˆ : Hamiltonian operator

• *CV*: volume constant heat capacity

• *F* (*t*): time dependent driving force

• *F* [*n* (**x**)]: energy functional of density *n* (**x**)


• *v*ext: static external potential

• Δ*p*Δ*q*: small phase interval

• Π: generalized polarization

• ΘD: Debye temperature

• |Φ0�: ground state of the system • Ξ: grand partition function

• *β*: inverse temperature 1/*k*B*T*

• *�*c: correlation energy density

• *κ*: generalized dielectric function • *λ*: interaction control parameter

• *μc*: correlation energy potential • *μ*xc: exchange-correlation potential

• *μi*: chemical potential of *i*th component

• *μ*: chemical potential

• *ψ*: wavefunction

• *�*xc: exchange-correlation energy density

• *δ*: Dirac delta function • *�*: single-particle energy

• *�*F: Fermi energy

• *�*<sup>0</sup>

• Ω: grand (thermodynamic) potential

• Γ: gamma function

• Σ: self-energy

• *x*: base point of order parameter space *R*

• *z*: coordinate of the order parameter space *R*

• Σ*λ*: self-energy for the interaction strength of *λ*

• Ω0: grand potential of the noninteracting system

• *γ***k**: damping of the quasiparticle of wavevector **k**

• *�***k**: energy spectrum of the particle at the wavevector **k** • *�***p**: energy spectrum of the particle at the momentum **p**

• *εR*: fractional error of the Landau theory as function of *R*

• *η*: infinitesimal real positive convergence factor

**<sup>k</sup>**: energy spectrum of the noninteracting single-particle at wavevector **k**

• *εij*: fractional error of the Landau theory as function of the *i*th and *j*th sites

• Γ¯: generic real-time Green's function in Lehmann representation

**k**

• **x**: position

• **v***n* (**k**): nonvanishing velocity of a Bloch wavefunction of the band *n* and the wavevector

Towards the Authentic *Ab Intio* Thermodynamics 587


44 Will-be-set-by-IN-TECH

• *c*.*c*.: complex conjutate

*f* : average of *f* (*p*, *q*) • *f* (*z*): order parameter map • *g*: spin degeneracy factor

• ¯*h*: Planck's constant

• **k**: wavevector

• *n*: band index

• **q**: wavevector

• *v*: static potential

• *n*<sup>0</sup>

• *t*

• *i*: index for the microstates

• *k*B: Boltzmann's constant

• *k*F: norm of the Fermi wavevector • *k* (*t*): solution of the equation of motion

• **l**: dislocation core axis vector

• *m*: mass of the particle • *m*∗: effective mass

• **m**: magnetization per spin

• *p*: norm of the momentum **p** • **p**: single-particle momentum

• *q*: norm of wavevector **q** • *qL*: liquifying wavevector

∗: critical temperature

• *n* (**x**): number density as function of **x**

• ¯

• *d*: dimensionality of the interaction • d1: differential operator for **x**1, *t*<sup>1</sup>

• *fAB*: generalized response function

• d*w*: propability of states represented in the phase volume

• *ms*: magnetization of the Landau function at the global minimum

*<sup>i</sup>* : occupation number of the state *i* of the noninteracting system

• *s*: degree of freedom of the system; number of spins (depending on context)

• *pi*: generalized momentum of the *i*th degree of freedom

• *qi*: generalized coordinate of the *i*th degree of freedom

• **u** (**x**): vector displacement field at position **x**

• *t*: time; reduced temperature *T* − *T*<sup>C</sup> (depending on context)

• *un***<sup>k</sup>** (**k**): Bloch function of band *n* and wavevector **k** at position **x**

• *f* (*p*, *q*): any physical quantity as a function of *p* and *q*

• *h*: reduced external magnetic field (*H* − *Hc*)


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