Entropic Density Functional Theory Entropic Inference and the Equilibrium
State of Inhomogeneous Fluids Doctor of Philosophy College of Arts and
Sciences Department of Physics Fall 2021 To my better half Parisa

ABSTRACT

> And your Lord revealed to the bees: Make your homes in the mountains, the
> trees, and in what they construct, and then feed from any fruit and follow
> the ways your Lord has made easy for you. From their bellies comes forth
> liquid of varying colours, in which there is healing for people. Surely in
> this is a sign for those who reflect.
>
> The Bees
> Quran

A unified formulation of the density functional theory is constructed on the
foundations of entropic inference in both the classical and the quantum
regimes. The theory is introduced as an application of entropic inference for
inhomogeneous fluids in thermal equilibrium. It is shown that entropic
inference reproduces the variational principle of DFT when information about
expected density of particles is imposed.
In the classical regime, this process introduces a family of trial density-
parametrized probability distributions, and consequently a trial entropy, from
which the preferred one is found using the method of Maximum Entropy (MaxEnt).
In the quantum regime, similarly, the process involves introduction of a
family of trial density-parametrized density matrices, and consequently a
trial entropy, from which the preferred density matrix is found using the
method of quantum MaxEnt. As illustrations some known approximation schemes of
the theory are discussed.

ACKNOWLEDGMENT

First and foremost, I’d like to thank my advisor Ariel Caticha for his
incredible teachings on foundations of physics, intriguing philosophical
discussions, his brilliant improvements to my lengthy calculations, and most
importantly the freedom he gave me throughout the grad school.
I am sincerely thankful to Oleg Lunin, not only for his amazing QM and QFT
classes, but also for all the hallway chats which clicked some of my deepest
understandings.
I would like to express my appreciation to Pedram Safari for his insightful
math lessons.
I am grateful to the faculty and the alumni of the University at Albany
Physics Department, specially Selman Ipek, Herbert Fotso, Daniel Robins, Kevin
Knuth, Anna Sharikova, Carlo Cafaro, and Professor Carolyn McDonald for
providing a pleasant work environment.
I must express my gratitude to my parents and my brother Ali, Shahnaz, and
Mohammad for their love, nurture, trust, and never ending supports.
Finally I am truly grateful to my best friend, my sharpest co-thinker, and my
beloved wife Parisa for standing by me throughout this journey, and
particularly for teaching me computational chemistry.

###### Contents

  1. 1 Background
    1. 1.1 Inductive Inference
      1. 1.1.1 Design of Entropic Inference
      2. 1.1.2 Quantum Entropic Inference
    2. 1.2 An Overview of this Work
  2. 2 Introduction to Density Functional Theory
    1. 2.1 DFT Theorem(s)
      1. 2.1.1 Classical DFT Theorem
      2. 2.1.2 Finite Temperature DFT Theorem
  3. 3 Entropic Classical DFT
    1. 3.1 Introduction
    2. 3.2 Entropic inference
    3. 3.3 Formalisms of Statistical Mechanics
      1. 3.3.1 Canonical Formalism
      2. 3.3.2 Grand-canonical Formalism
    4. 3.4 Maxent and Optimal Approximation of Probabilities
    5. 3.5 Density Functional Formalism
    6. 3.6 Slowly Varying Density and Gradient Expansion
    7. 3.7 Final Remarks
  4. 4 Entropic Quantum DFT
    1. 4.1 Introduction
    2. 4.2 Quantum Statistics
      1. 4.2.1 Density Operator
      2. 4.2.2 Quantum Liouville Equation
      3. 4.2.3 Many-body Density Matrix in Position and Momentum Representations
    3. 4.3 Quantum MaxEnt
      1. 4.3.1 Contact Structure of MaxEnt
      2. 4.3.2 Formalisms of Statistical Mechanics
      3. 4.3.3 Using Entropy to Find Optimal Approximations: the Bogolyubov Inequality
    4. 4.4 Density Functional Formalism
      1. 4.4.1 Constructing the Trial States
      2. 4.4.2 The Equilibrium Density Matrix
      3. 4.4.3 The DFT Theorem
    5. 4.5 Approximation Schemes of DFT
      1. 4.5.1 Free Energy Approximation for Almost Constant Density
      2. 4.5.2 Kohn-Sham Model
    6. 4.6 Final Remarks
  5. 5 Conclusion

##  Chapter 1 Background

###  1.1 Inductive Inference

The process of drawing conclusion from available information is called
inference [1]. In general, the available information may be insufficient to
make a unique assessment of truth, this process of reasoning with incomplete
information is called inductive inference. The methods deployed for such
reasoning are those of probability theory and entropic inference.
The method of entropic inference has its roots in works of Boltzmann, Maxwell
and Gibbs. From works of the three branch out the field of Statistical
Mechanics which aims for explaining, predicting, and/or understanding the
macroscopic properties of thermodynamic systems, using the dynamics of the
constituent particles of the system of interest, and probability theory. About
a half century later, Shannon [2] introduced a notion of entropy
$S[p]=\sum_{i}p_{i}logp_{i}$ which measures the amount of missing information
in communication theory, but it was doubted that the Shannon’s entropy had
anything to do with the thermodynamic entropy, heat, pressure, etc. until
Jaynes [3, 4] made it clear that the similarity is not accidental; Jaynes
showed that Thermodynamics, as fundamental as it is, should be interpreted as
a scheme for inference about nature. From Jaynes’ point of view -following
Shannon’s axioms entropy was the amount of information that is missing in a
probability distribution; with no concerns about updating probabilities.
In 1980, Shore and Johnson [5], proposed a new interpretation of the MaxEnt,
the idea was to axiomatize the updating method of probabilities instead of
entropy. They also proposed a variational principle for that matter.
In 1998, Skilling [6] justified the proposed variational principle by a simple
idea: in order to find the preferred posterior, one must rank all the
probability distributions in the order of preference and then pick the
highest.
Decades later, in the $21^{st}$ century, Caticha took the information theory
to the next level [7], by deriving the Kullback-Leibler divergence as the
unique relative entropy $S_{r}[p|q]$ which allows an ideal agent to update
their degree of rational belief from a prior probability $q$ to the posterior
probability $p$. In this approach, information is the constraint on the
probabilities in the updating process rather than anything stored in the
system or in the probability distribution in hand. Caticha’s framework has
evolved over the past two decades in a series of articles [7, 8, 9, 10, 11,
14]. Along the way of these publications, not only the number of the design
criteria of MaxEnt has been reduced from 5 to 2, but also it has been shown
that the information can come in form of any type of constraints on the
probability function, not just the expected values. This particular
flexibility of the theory opened room for applications in cases where
information is acquired from sources other than data and experiment. As an
important example, Caticha’s theory of Entropic Dynamics [12, 13] is
formulated on the basis of such a MaxEnt.
From a non-personalistic Bayesian point of view, probability is the rational
degree of belief of an ideal agent. From the entropic inference point of view,
the information about a system of interest, is what would change the agent’s
degree of belief about the system. In other words, information is a
constraint.
The entropic inference is a framework designed to update probabilities. Within
the framework, entropy is the unique mathematical construct which conforms to
certain design criteria described bellow. Therefore entropy is a tool that
transcends any of its applications with heat, temperature, data, signals, time
etc.. It is worth to emphasize again that our relative entropy is unique, in a
sense that the design criteria are constraining to such an extent that only
one functional is singled out, therefore inference with any other suggested
functional would contradict at least one of the design criteria.

####  1.1.1 Design of Entropic Inference

The general strategy of entropic inference is motivated by an irreducibly
simple idea primarily proposed by Skilling [6]: in order to find the posterior
probability distribution, one must rank all the possible probability
distributions in the order of preference and then pick the highest. Such a
strategy immediately suggests a variational principle: in order to rank a
family of probability distributions $P$ against a single prior $Q$, one
constructs a real valued functional $S_{r}[P|Q]$ which we call entropy of $P$
relative to $Q$.
The desired method of updating probability, ought to produce a posterior which
conforms to the information that changes the prior, but this constraint is not
sufficient to pick one preferred posterior, additional rules are required to
single out the one posterior probability, here are the suggested general
criteria:

###### GC 1

Universality: The method must be of universal applicability, meaning that we
should be able to apply the method to update any prior probability
distribution subject to any relevant information, regardless of the nature of
the proposition that the probability assigns a degree of belief to, or the
nature or the source of the information.

###### GC 2

Parsimony: The prior probability must be updated minimally as to conform to
the new information. In particular, in the absence of new information, an
ideal agent must not change their belief.

###### GC 3

Independence: If two systems are a priori believed to be independent, when we
receive information about one, then it shouldn’t matter if we include the
other in the updating process or not.

Skilling’s strategy and the above general criteria, are implemented through
the following specific Design Criteria (DC) of MaxEnt. Consider a lattice of
propositions generated by a set $\mathcal{X}$ of mutually exclusive
propositions  labeled discretely by $\\{i=1,2,\dots\\}$ for simplicity:

###### DC 1

For two non-overlapping subdomain
$\mathcal{D},\mathcal{D}^{\prime}\subset\mathcal{X}$, information about the
probabilities conditioned to one subdomain $\mathcal{D}$ should not change the
probabilities that are conditioned to the other subdomain
$\mathcal{D}^{\prime}$ .

It is proven in the reference [11] that this criterion implies that the non-
overlapping subdomains of $\mathcal{X}$ contribute additively to the entropy
$S_{r}[P|Q]$, in particular

$S_{r}[P|Q]=\sum_{i}F(p_{i}|q_{i})\;,$ (1.1)

where $p_{i}$ and $q_{i}$ are probabilities that distributions $P$ and $Q$
assign to the $i^{th}$ proposition respectively.

###### DC 2

If two systems are a priori believed to be independent, when we receive
information about one system with no reference to the other, the updating
process must yield the same result, either if we include the other system in
our analysis or not.

The DC 2 is such a constraining criterion that singles out only one particular
function from all possible candidates, so that

$S_{r}[P|Q]=-\sum_{i}p_{i}log\frac{p_{i}}{q_{i}}\;.$ (1.2)

The conclusion is that the posterior probability is found by maximization of
(1.2) subject to the information in hand.

####  1.1.2 Quantum Entropic Inference

In quantum mechanics, the state of knowledge about a system is represented by
a density matrix. Vanslette [14] showed that the Umegaki entropy [15] is the
unique tool to update density matrices in response to new information. The
idea is to rank density matrices according to design criteria similar to what
we reviewed above. The prescription is similar to the classical MaxEnt, given
a prior density matrix $\hat{\sigma}$, and information in form of expected
value of Hermitian operators $\langle\hat{A}_{i}\rangle=A_{i}$, the posterior
density matrix is found by maximizing the quantum entropy of $\hat{\rho}$
relative to $\hat{\sigma}$

$S_{r}[\hat{\rho}|\hat{\sigma}]=-Tr\hat{\rho}[log\hat{\rho}-\log\hat{\sigma}]$
(1.3)

subject to the information and the normalization of the posterior

$\delta\Bigg{[}[S_{r}[\hat{\rho}|\hat{\sigma}]]-\alpha_{0}[Tr\hat{\rho}-1]-\sum_{i}\alpha_{i}[Tr\hat{\rho}\hat{A}_{i}-A_{i}]\Bigg{]}_{\hat{\rho}=\hat{\rho}^{*}}=0\;.$
(1.4)

Such a maximization yields to posterior density matrix

$\hat{\rho}^{*}=\frac{1}{Z}e^{log\hat{\sigma}-\sum\alpha_{i}\hat{A}_{i}},$
(1.5)

which determines expected values.

###  1.2 An Overview of this Work

This thesis is not particularly concerned with any specific application of the
Density Functional Theory (DFT), nor is it concerned with DFT computer
programs, this work is about DFT itself as a computational method which has
been proved to be one of the most effective techniques to deal with quantum
and classical inhomogeneous fluids. We will show that the variational
principle of DFT is a special case of MaxEnt under appropriate constraints.
Such a study integrates DFT with statistical mechanics and quantum mechanics
as systematic applications of entropic inference and may offer a more general
framework for approximations from which other theories such as mean field
theory, dynamic DFT, Ginzburg-Landau theory, etc. can be recovered. Another
advantage of such a study is that it clarifies the abiguity of what is exact
and what is approximate in the DFT calculations, this clarification may help
computational physicists and chemists to make better decisions as to which
parts of the codes are modifiable and how they may want to compare their
results to experiments and to other computational methods.
In chapter 2 we briefly review the existing DFT in classical and quantum
regimes. In chapter 3 we first review the subject of entropic inference and
discuss how the well known formalisms of statistical mechanics are constructed
on the foundation of such an inference tool, then we introduce the density
functional formalism on the same basis and we recover the classical DFT
theorem. Then, as an illustration, we will recover the approximation for a
classical fluid with slowly varying density. In chapter 4, first we review
quantum statistics and quantum MaxEnt. We show that any quantum MaxEnt
produces a contact structure and therefore can be transformed by Legendre
Transformations. Then we construct the density functioanl formalism and show
that the DFT variatioanal equation is the Legendre Transform of the MaxEnt
where the inference method is used to pick the best density matrix from a
trial family of density parametrized density matrices. Next, we recover the
almost constant density approximation and the famous Kohn-Sham model as
illustrations.

##  Chapter 2 Introduction to Density Functional Theory

The Density Functional Theory (DFT) is an approach to the many-body particle
systems. The theory was first introduced in 1964 in the context of quantum
mechanics by Hohenberg and Kohn [16]. The article not only showed that for an
electron gas at zero temperature, the electronic density distribution uniquely
characterizes the ground state, but also proved that there exists a density
functional variational equation which yields the equilibrium density as its
solution, equal to the density calculated using the time independent
Schrödinger equation.
This equivalence of the Hohenberg-Kohn (HK) equation and the time independent
Schrödinger equation, extended to a new equivalence only a year later, when
Mermin [17] proved that for an electron gas at finite temperature in the
presence of an external potential, there exists a density functional
variational equation which gives the same equilibrium density as gives the
grand canonical ensemble formalism of statistical mechanics.
Unlike the usual historical order of theories in physics, the classical DFT
emerged years after the quantum DFT. Evans formulated the finite temperature
DFT in the classical regime [18] with motivation from Mermin’s work [17].
Before we proceed to a brief review of the three versions of the DFT theorem,
it is appropriate to look at the theory from a historical point of view.
In 1929, Paul Dirac announced that the non-relativistic quantum mechanics is
complete and approximation schemes are desired to simplify the sophisticated
quantum-mechanical calculations [19]. Physicists and chemists followed Dirac’s
advice and developed the mathematical frame work to solve the Schrödinger
equation of atoms, molecules, and solids over the years 1930’s, 40’s, 50’s,
and 60’s. Meanwhile the first digital computers were made during the 40’s and
they had rapidly grown to powerful machines capable of performing quantum
calculations of atoms and single molecules by 1965 [20]. At the time,
computers were able to give an approximate solution to the Schrödinger
equation of many-particle systems using the Hartree-Fock approximation method
for molecules and atoms, while solid state physicists preferred to borrow
analytical techniques from quantum field theory to study the electronic
correlation effects in materials, which evolved to the whole branch of
statistical physics of fields [21]. Given the above considerations, the
theoretical aspects of DFT have been neglected compared to the enormous
developments in its applications. Consequently the theory is mostly known as a
computational modeling method for investigation of electronic structure of
quantum mechanical systems.
This thesis, introduces the density functional formalism, as a straightforward
application of the maximum entropy principle, in both classical and quantum
regimes. In the upcoming chapters, we will show that the DFT variational
principle is nothing but the principle of maximum entropy subject to some
appropriate constraints, rewritten under an appropriate Legendre
Transformation. Before that, let us see what the three theorems are without
digging into the details of the proofs.

###  2.1 DFT Theorem(s)

We are going to review the DFT theorems in an order opposite to the historical
one. We start with the classical DFT, because it is conceptually the simplest,
then we proceed to the thermal DFT and finally we conclude the HK theorem as a
especial case of Mermin’s theorem where $\beta\rightarrow\infty$ and
$|\psi_{0}\rangle$ is non degenerate. This is exactly the same order that we
have developed the entropic DFT in, therefore this review of the already
existing theorems not only foreshadows the formal shape of a desired entropic
DFT, but also will ease the grasp of the upcoming chapters.

####  2.1.1 Classical DFT Theorem

The original classical DFT theorem [18], following its quantum precursors,
writes down a theorem in a form that ensures existence of a particular
function, makes a guess for the function, and then proceeds to show that the
guessed function is indeed one of such a kind to complete the proof.

###### Theorem 1

For a classical fluid at equilibrium with fixed temperature $\frac{1}{\beta}$
and chemical potential $\mu$ in the presence of an external potential $v(x)$,
there exists a density functional $F(\beta;n]$ 111The notation $F(\beta;n]$
means that $F$ is a function of $\beta$ and a functional of $n(x)$ .
independent of the external potential, such that

$\Omega(\beta;n]=\int d^{3}xv(x)n(x)+F(\beta;n]-\mu\int d^{3}xn(x)$ (2.1)

assumes its minimum equal to the grand potential of the system at the
equilibrium density

$\bigg{[}\delta\Omega(\beta;n]\bigg{]}_{n(x)=n_{eq}(x)}=0\quad\text{for fixed
$\beta$}\;.$ (2.2)

Now consider the trial grand potential

$\Omega[P]=Tr_{c}P\bigg{[}H_{N}-\mu N+\frac{1}{\beta}logP\bigg{]}\;,$ (2.3)

where

$Tr_{c}\equiv\sum_{N=0}^{\infty}\frac{1}{N!}\int d^{3N}xd^{3N}p\;,$ (2.4)

and $P=P\\{N,x_{1},\dots,x_{N},p_{q},\dots,p_{N}\\}$ is the probability that
the system has $N$ particles with positions and momenta
$\\{x_{1},\dots,x_{N},p_{q},\dots,p_{N}\\}$, and the $N$ particle Hamiltonian
function $H_{N}=H_{N}(x_{1},\dots,x_{N},p_{q},\dots,p_{N})$ is

$H_{N}=T_{N}+U_{N}+V_{N}\;,$ (2.5)

where $T_{N}$, $U_{N}$, and $V_{N}$ are the $N$ particle kinetic energy,
interaction energy, and potential energy functions respectively.
Now we have two tasks to conclude the proof of the theorem, first we have to
show that the functional $\Omega[P]$ assumes its minimum for equilibrium
probability distribution $P_{eq}$ given by the grand canonical ensemble, and
second, we need to show that $\Omega[P]$ is a functional of $n(x)$ which can
be written as equation (2.1).  The former is assured by the fact that
$\Omega[P]$ is a Legendre Transform of the grand-canonical entropy

$S[P]=-Tr_{cl}PlogP$ (2.6)

and therefore is minimum at equilibrium. In order to see that split
$\Omega[P]$ into $\Omega[P_{eq}]$ and the rest

$\Omega[P]=\Omega[P_{eq}]+\frac{1}{\beta}Tr_{c}\big{[}PlogP-
PlogP_{eq}\big{]}\;,$ (2.7)

where $P_{eq}$ is the grand canonical probability distribution

$P_{eq}=\frac{e^{-\beta(H_{N}-\mu N)}}{\Xi}\;.$ (2.8)

The second term in the RHS of equation (2.7) is non-negative following the
Gibbs inequality, therefore

$\Omega[P]\geq\Omega[P_{eq}]\;.$ (2.9)

Now we shall prove that $\Omega[P]$ fits in the functional definition given in
(2.1).  Quoting word by word from [18]: ”Since $P_{eq}$ is a function of
$v(x)$ it follows that $n_{eq}(x)$ is also a functional of $v(x)$. We can also
prove the more useful result that $P_{eq}$ is a functional of $n_{eq}(x)$. The
proof (see Appendix 1 of [cite evans79]) proceeds by showing that, for a given
interaction potential $U_{N}$, $v(x)$ is uniquely determined by $n_{eq}(x)$,
i.e. only one v(x) can determine a given $n_{eq}(x)$. The resultant $v(x)$
then determines $P_{eq}$. Thus, it follows that $P_{eq}$ is a functional of
$n_{eq}(x)$. This result implies that, for a given $U_{N}$, the functional

$F(\beta;n]\Big{|}_{n(x)=n_{eq}(x)}=Tr_{cl}P_{eq}\Bigg{[}T_{N}+U_{N}+\frac{1}{\beta}logP_{eq}\Bigg{]}$
(2.10)

is a unique functional of the equilibrium density $n_{eq}(x)$.”
Combine (2.10) and (2.3) to have

$\Omega(\beta;n]=\int d^{3}xv(x)n(x)+F-\mu\int d^{3}xn(x);,$ (2.11)

which concludes the proof of the theorem.

####  2.1.2 Finite Temperature DFT Theorem

The DFT was first introduced for a degenerate ground state and only later
extended to finite temperature, however the pathway to recover ground state
DFT from the thermal one is straightforward, namely $\beta\rightarrow\infty$ ,
therefore we prescribe the thermal DFT for the sake of generality; and leave
the ground state case to the reader. Bellow is a brief review of the Mermin’s
DFT theorem [17] and its proof.

###### Theorem 2

Finite temperature DFT: For an electron gas in equilibrium with fixed
temperature and chemical potential, in the presence of an external potential
$v(x)$, there exists a density functional $F(\beta;n]$ independent of the
external potential, such that

$\Omega(\beta;n]=\int d^{3}xv(x)n(x)+F(\beta;n]-\mu\int d^{3}xn(x)$ (2.12)

assumes its minimum equal to the grand potential of the system at the
equilibrium density

$\bigg{[}\delta\Omega(\beta;n]\bigg{]}_{n(x)=n_{eq}(x)}=0\quad\text{for fixed
$\beta$}\;.$ (2.13)

The proof is identical to the proof of the classical DFT, except that we shall
work in the quantum grand-canonical formalism.  Consider the grand potential

$\Omega[\hat{\rho}]=Tr\hat{\rho}\bigg{[}\hat{H}-\mu\hat{N}+\frac{1}{\beta}log\hat{\rho}\bigg{]}\;,$
(2.14)

where $\hat{\rho}$ is the density matrix of the system and $\hat{H}$ and
$\hat{N}$ are respectively the Hamiltonian and the particle number operator
defined on the Fock space of the electron gas

$\hat{H}=\hat{T}+\hat{U}+\hat{V}\;,$ (2.15)

where $\hat{T}$, $\hat{U}$, and $\hat{V}$ are the kinetic energy, interaction
energy, and potential energy operators respectively.
Now we have two tasks to conclude the proof of the theorem, first we have to
show that the functional $\Omega[\hat{\rho}]$ assumes its minimum for
equilibrium density matrix $\hat{\rho}_{eq}$ given by the grand canonical
ensemble, and second, we need to show that $\Omega[\hat{\rho}]$ is a
functional of $n(x)$ which can be written as equation (2.12). The former is
proved by introduction of a curve $\lambda$ in the space of density matrices
that connects , assured by the fact that $\Omega[\hat{\rho}]$ is a Legendre
Transform of the grand-canonical entropy

$S[\hat{\rho}]=-Tr\hat{\rho}log\hat{\rho}$ (2.16)

and therefore is minimum at equilibrium. In order to see that split
$\Omega[\hat{\rho}]$ into $\Omega[\hat{\rho}_{eq}]$ and the rest

$\Omega[\hat{\rho}]=\Omega[\hat{\rho}_{eq}]+\frac{1}{\beta}Tr_{c}\big{[}\hat{\rho}log\hat{\rho}-\hat{\rho}log\hat{\rho}_{eq}\big{]}\;,$
(2.17)

where $\hat{\rho}_{eq}$ is the grand canonical density matrix

$\hat{\rho}_{eq}=\frac{e^{-\beta(\hat{H}-\mu\hat{N})}}{\Xi}\;.$ (2.18)

The rest of the proof is compatible with the classical proof given previously,
and therefore we do not repeat it here, although we recommend the readers new
to the subject to refer to references [16, 17].

##  Chapter 3 Entropic Classical DFT

###  3.1 Introduction

The Density Functional Theory was first developed in the context of quantum
mechanics and only later extended to the classical regime. The theory was
first introduced by Kohn and Hohenberg (1964)[16] as a computational tool to
calculate the spatial density of an electron gas in the presence of an
external potential at zero temperature. Soon afterwards, Mermin provided the
extension to finite temperatures [17]. Ebner, Saam, and Stroud (1976)[22]
applied the idea to simple classical fluids, and Evans (1979) provided a
systematic formulation in his classic paper [18] ”The nature of the liquid-
vapour interface and other topics in the statistical mechanics of non-uniform,
classical fluids.”.
The majority of physicists and chemists today are aware of the quantum DFT and
the Kohn-Sham model [23], while fewer are familiar with the classical DFT, a
historical review of quantum DFT and its vast variety of applications is found
in [20, 24]. The classical DFT, similarly, is a ”formalism designed to tackle
the statistical mechanics of inhomogeneous fluids”[25], which has been used to
investigate a wide variety of equilibrium phenomena, including surface
tension, adsorption, wetting, fluids in porous materials, and the chemical
physics of solvation.
Just like the Thomas-Fermi-Dirac theory is usually regarded as a precursor of
quantum DFT, the van der Waals’ thermodynamic theory of capillarity under the
hypothesis of a continuous variation of density [26] can be regarded as the
earliest work on classical DFT, without a fundamental proof of existence for
such a variational principle.
”The long-term legacy of DFT depends largely on the continued value of the DFT
computer programs that practitioners use daily.”[20] The algorithms behind the
computer programs, all starting from an original Hartree-Fock method to solve
the N-particle Schrödinger equation, have evolved by many approximations and
extensions implemented over time by a series of individuals, although the
algorithms produce accurate results, they do not mention the HK variational
principle. Without the variational principle the computer codes are suspect of
being ad hoc or intuitively motivated without a solid theoretical foundation,
therefore, the DFT variational principle not only scientifically justifies the
DFT algorithms, but also provides us with a basis to understand the repeatedly
modified algorithms behind the codes.
In this work we derive the classical DFT as an application of the method of
maximum entropy [1, 3, 4, 27, 28]. This integrates the classical DFT with
other formalisms of classical statistical mechanics (canonical, grand
canonical, etc.) as an application of information theory. Our approach not
only enables one to understand the theory from the Bayesian point of view, but
also provides a framework to construct equilibrium theories on the foundation
of MaxEnt. We emphasize that our goal is not to derive an alternative to DFT.
Our goal is purely conceptual. We wish to find a new justification or
derivation of DFT that makes it explicit how DFT fits within the MaxEnt
approach to statistical mechanics. The advantage of such an understanding is
the potential for future applications that are outside the reach of the
current version of DFT.
In section 3.2, we briefly review entropic inference as an inference tool
which updates probabilities as degrees of rational belief in response to new
information. Then we show that any entropy maximization, produces a contact
structure which is invariant under Legendre Transformations; this enables us
to take advantage of these transformations for maximized entropy functions
(functionals here) found from constraints other than those of thermal
equilibrium as well as thermodynamic potentials.
In section 3.4 we briefly review the method of relative entropy for optimal
approximation of probabilities, which allows us to derive and then generalize
the Bogolyubov variational principle. Then we apply it for the especial case
wherein the trial family of probabilities are parametrized by the density
function $n(x)$.
In section 3.5, the Density Functional formalism is introduced as an extension
of the existing ensemble formalisms of statistical mechanics (canonical, grand
canonical, etc.) and we show that the core DFT theorem is an immediate
consequence of MaxEnt: we prove that in the presence of an external potential
$v(x)$, there exists a trial density functional entropy $S_{v}(E;n]$ maximized
at the equilibrium density. We also prove that this entropy maximization is
equivalent to minimization of a density functional potential
$\Omega_{v}(\beta;n]$ given by

$\Omega_{v}(\beta;n]=\int d^{3}xv(x)n(x)+F(\beta;n]$ (3.1)

where $F(\beta;n]$ is independent of $v(x)$. This formulation achieves two
objectives. i) It shows that the density functional variational principle is
an application of MaxEnt for non-uniform fluids at equilibrium and therefore,
varying the density $n(x)$ in $S_{v}(E;n]$ does not imply that the functional
represents entropy of any non-equilibrium system, this trial entropy, although
very useful, is just a mathematical construct which allows us to incorporate
constraints which are related to one another by definition. ii) By this
approach we show that the Bayesian interpretation of probability liberates the
fundamental theorem of the DFT from an imaginary grand-canonical ensemble,
i.e. the thermodynamic chemical potential is appropriately defined without
need to define microstates for varying number of particles.
Finally, in section 3.6, as an illustration we discuss the already well-known
example of a slowly varying inhomogeneous fluid. We show that our entropic DFT
allows us to reproduce the gradient approximation results derived by Evans
[18]. There are two different approximations involved, i) rewriting the non-
uniform direct correlation function in terms of the uniform one; and ii) the
use of linear response theory to evaluate Fourier transform of direct
correlation functions. The former assumes that the density is uniform inside
each volume element, and the latter assumes that difference of densities for
neighboring volume elements is small compared to their average.

###  3.2 Entropic inference

A discussion of the method of maximum entropy as a tool for inference is found
in [1]. Given a prior Probability Distribution Function (PDF) $Q(X)$, we want
to find the posterior PDF $P(X)$ subject to constraints on expected values of
functions of $X$.
Formally, we need to maximize the relative entropy

$S_{r}[P|Q]\equiv-\sum_{X}(PlogP-PlogQ)\;,$ (3.2)

under constraints $A_{i}=\sum_{X}P(X)\hat{A}_{i}(X)$ and
$1=\sum_{X}P({\color[rgb]{0,0,0}X})$; where $A_{i}$’s are real numbers, and
$\hat{A}_{i}$’s are real-valued functions on the space of $X$, and $1\leq
i\leq m$ for $m$ number of constraints.
The maximization process yields the posterior probability

$P(X)=Q(X)\frac{1}{Z}e^{-\sum_{i}\alpha_{i}\hat{A}_{i}(X)}\;,\quad\quad\text{where
}Z=\sum_{X}Q(X)e^{-\sum_{i}\alpha_{i}\hat{A}_{i}(X)}\;,$ (3.3)

and $\alpha_{i}$’s are Lagrange multipliers associated with $A_{i}$’s.
Consequently the maximized entropy is

$S=\sum_{i}\alpha_{i}A_{i}+logZ\;.$ (3.4)

Now we can show that the above entropy readily produces a contact structure,
calculate the complete differential of equation (3.4) to define the vanishing
one-form $\omega_{cl}$ as

$\omega_{cl}\equiv dS-\sum_{i}\alpha_{i}dA_{i}=0\;.$ (3.5)

Therefore any classical entropy maximization with $m$ constraints produces a
contact structure $\\{{\color[rgb]{0,0,0}\mathbb{T}},\omega_{cl}\\}$ in which
manifold $\mathbb{T}$ has $2m+1$ coordinates $\\{q_{0},q_{1},\dots
q_{m},p_{1},\dots,p_{m}\\}$.
The physically relevant manifold $\mathbb{M}$ is an $m$-dimensional sub-
manifold of $\mathbb{T}$, on which $\omega_{cl}$ vanishes; i.e. $\mathbb{M}$
is determined by $1+m$ equations

$q_{0}\equiv S(\\{A_{i}\\})\;,\quad\quad p_{i}\equiv\alpha_{i}=\frac{\partial
S}{\partial A_{i}}\;.$ (3.6)

Legendre Transformations defined as,

$q_{0}\longrightarrow\;q_{0}-\sum_{j=1}^{l}p_{j}q_{j}\;,$ (3.7)
$q_{i}\longrightarrow\;p_{i}\;,\quad
p_{i}\longrightarrow\;-q_{i}\;,\quad\quad\text{for }1\leq i\leq l\;,$

are coordinate transformations on space $\mathbb{T}$ under which $\omega_{cl}$
is conserved. It has been shown [29, 30] that the laws of thermodynamics
produce a contact structure conforming to the above prescription, here we are
emphasizing that the contact structure is an immediate consequence of MaxEnt,
and therefore it can be utilized in applications of information theory, beyond
thermodynamics.

###  3.3 Formalisms of Statistical Mechanics

An equilibrium formalism of statistical mechanics is a relative entropy
maximization process consisting of three crucial parts: i) The microstates
that describe the system of interest. ii) The uniform prior probability
distribution. iii) The constraints that represent our information about the
system of interest.

######  Remark 3 (About Uniform Prior)

Note that the choice of the uniform prior is not a consequence of maximum
entropy in our formulation, it is rather chosen for the sake of ultimate
honesty in the theory, at the very beginning, we have no information about
available microstates expect that they all exist and are defined, in other
words, before imposing inferential constraints we are equally ignorant about
different microstates , so we have no reason to break this symmetry; thus
comes the choice of uniform prior probability distribution in classical
statistical mechanics, and the choice of identity prior density matrix in
quantum statistical mechanics.

####  3.3.1 Canonical Formalism

The canonical formalism, obviously, is the most important formalism of
statistical mechanics, it plays such an important role to -virtually- all
statistical phenomena in physics, that Feynman called the canonical
probability distribution the key principle of statistical mechanics[31].
Feynman describes canonical distribution as the summit of statistical
mechanics and then teaches several subjects of the field as different paths
down the hill. our approach is similar to Feynman’s, except that we take
MaxEnt as the summit of statistical mechanics, and different formalisms appear
as the proper pistes down the hill.
For the sake of simplicity, lets assume that the constituent particles of the
system of interest are of the same kind. In canonical formalism the system of
interest consists of N particles and is in thermal equilibrium with a heat
bath. We mentioned earlier that any equilibrium formalism is founded on a
trinity of elements, here are the three elements for canonical formalism:

  1. 1.

Microstates are the positions and the momenta of the N particles
$\\{x_{1},\dots,x_{N},p_{1},\dots,p_{N}\\}$.

  2. 2.

The prior in the classical regime is given by uniform prior probability
distribution on the $N$ particle phase space, therefore we have prior PDF,
$Q(\\{x_{i},p_{i}\\})=\textit{const.}$

  3. 3.

There are only two constraints in the canonical formalism, one for
normalization $\langle 1\rangle=1$ and another constraint for the total energy
$\langle H\rangle=E$. 111Note that both normalization and energy constraints
have nature of inference, the former is imposed by laws of probability and the
latter is imposed by the experimental assumption that we can measure change of
total energy of any physical system of interest.

Now, following our argument and calculations in part 3.2, we must maximize the
relative entropy $S_{r}[P|Q]$ subject to above constraints to find the
posterior probability $P(\\{x_{1},\dots,x_{N},p_{1},\dots,p_{N}\\})$.
Analog to equations (3.3) and (3.4), the MaxEnt yields the posterior
probability distribution

$P(\\{x_{i},p_{i}\\})=\frac{1}{Z}exp{-\beta H(\\{x_{i},p_{i}\\})}$ (3.8)

and the maximized entropy

$S=\beta E+logZ\;,$ (3.9)

where $Z(\beta)=Tr_{cl}exp{-\beta H(\\{x_{i},p_{i}\\})}$.

####  3.3.2 Grand-canonical Formalism

In the grand-canonical formalism, we are interested in a system which is not
only in a thermal equilibrium with the bath, but also may exchange particles
with it, therefore the number of particles in the system of interest is not
fixed and can be any non-negative integer, here is the list of ingredient to
construct the grand canonical formalism:

  1. 1.

The microstates of the system are determined by the number of particles and
positions and momenta of them $\\{n,x_{1},\dots,x_{n},p_{1},\dots,p_{n}\\}$.

  2. 2.

The uniform prior probability distribution is given by,
$Q(n,\\{x_{i},p_{i}\\})=\textit{const.}$

  3. 3.

There are three constraints in grand-canonical formalism, one for
normalization $\langle 1\rangle=1$ one for total energy $\langle H\rangle=E$,
and another one for expected number of particles $\langle n\rangle=N$.

Again, analog to equations (3.3) and (3.4), the MaxEnt yields the posterior
probability distribution

$P(n,\\{x_{i},p_{i}\\})=\frac{1}{\Xi}e^{-\beta[H(\\{x_{i},p_{i}\\})-\mu
n]}\;,$ (3.10)

and the maximized entropy

$S=\beta[E-\mu N]+log\Xi\;,$ (3.11)

where the grand canonical partition function
$\Xi(\beta,\mu)=\sum_{n=0}^{\infty}Tr_{cl}e^{-\beta[H(\\{x_{i},p_{i}\\})-\mu
n]}$ determines the thermodynamic properties of the system.

###  3.4 Maxent and Optimal Approximation of Probabilities

The posterior PDF found from entropic inference, is usually too complicated to
be used for practical purposes. A common solution is to approximate the
posterior PDF by a more tractable family of PDFs $\\{p_{\theta}\\}$ [32].
Given the exact probability $p_{0}$, the preferred member of tractable family
$p_{\theta^{*}}$ is found by maximizing the entropy of $p_{\theta}$ relative
to $p_{0}$:

$\frac{\delta
S_{r}[p_{\theta}|p_{0}]}{\delta\theta}\Big{|}_{\theta=\theta^{*}}=0\;.$ (3.12)

The density functional formalism is a systematic method in which the family of
trial probabilities is parametrized by the density of particles; in section
3.5 we shall use the method of maximum entropy to determine the family of
trial distributions parametrized by $n(x)$, $p_{\theta}\equiv p_{n}$. So that
we can rewrite equation 3.12 as

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}S_{r}[p_{n}|p_{0}]+\alpha_{eq}[N-\int
d^{3}xn(x)]\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$ (3.13)

We will see that the canonical distribution itself is a member of the trial
family, therefore in this case, the exact solution to equation (3.13) is
$p_{0}$ itself:

$p_{n}\Big{|}_{n(x)=n_{eq}(x)}=p_{0}\;.$ (3.14)

###  3.5 Density Functional Formalism

An equilibrium formalism of statistical mechanics is a relative entropy
maximization process consisting of three crucial elements: i) One must choose
the microstates that describe the system of inference. ii) The prior is chosen
to be uniform. iii) One must select the constraints that represent the
information that is relevant to the problem in hand.
In the density ensemble, microstates of the system are given as positions and
momenta of all $N$ particles of the same kind, given the uniform prior
probability distribution

$Q(\\{\vec{x}_{1},\dots,\vec{x}_{N};\vec{p}_{1},\dots,\vec{p}_{N}\\})=constant.$
(3.15)

Having in mind that we are looking for thermal properties of inhomogeneous
fluids, it is natural to choose the density of particles $n(x)$ as
computational constraint, and the expected energy $E$ as thermodynamic
constraint, in which $n(x)$ represents the inhomogeneity and $E$ defines the
thermal equilibrium.
Note that all constraints (computational, thermal, etc.) in the framework can
be incorporated as inferential constraints and can be imposed as prescribed in
section 3.2.
The density constraint holds for every point in space, therefore we have
$1+1+\mathbb{R}^{3}$ constraints, one for normalization, one for total energy,
and one for density of particles at each point in space; so we have to
maximize the relative entropy

$S_{r}[P|Q]=-\frac{1}{N!}\int d^{3N}xd^{3N}p(PlogP-PlogQ)\equiv-Tr_{c}(PlogP-
PlogQ),$ (3.16)

subject to constraints

$\displaystyle 1=$ $\displaystyle\langle 1\rangle,\quad
E=\langle\hat{H}_{v}\rangle,$ (3.17a) $\displaystyle n(x)=$
$\displaystyle\langle\hat{n}_{x}\rangle\quad\text{where}\int d^{3}xn(x)=N,$
(3.17b)

where $\langle.\rangle\equiv\frac{1}{N!}\int(.)Pd^{3N}xd^{3N}p$. The classical
Hamiltonian operator $\hat{H}$ and the particle density operator $\hat{n}_{x}$
are given as

$\hat{H}_{v}\equiv\sum_{i=1}^{N}v(x_{i})+\hat{K}(p_{1},\dots,p_{N})+\hat{U}(x_{1},\dots,x_{N}),$
(3.18) $\hat{n}_{x}{\color[rgb]{0,0,0}\equiv}\sum_{i}^{N}\delta(x-x_{i}).$
(3.19)

The density $n(x)$ is not an arbitrary function; it is constrained by a fixed
total number of particles,

$\int d^{3}xn(x)=N.$ (3.20)

Maximizing (3.16) subject to (3.17) gives the posterior probability
$P(x_{1},\dots,x_{N};p_{1},\dots,p_{N})$ as

$P=\frac{1}{Z_{v}}e^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\quad\text{under condition}\int d^{3}xn(x)=N.$
(3.21)

where $\alpha(x)$ and $\beta$ are Lagrange multipliers.
The Lagrange multiplier function $\alpha(x)$ is implicitely determined by

$\frac{\delta logZ_{v}}{\delta\alpha(x)}=-n(x)$ (3.22)

and by equation (3.20)

$-\int d^{3}x\frac{\delta logZ_{v}}{\delta\alpha(x)}=N\;.$ (3.23)

Substituting the trial probabilities from (3.21) into (3.16) gives the trial
entropy $S_{v}(E{\color[rgb]{0,0,0};}n]$ as

$S_{v}(E;n]=\beta E+\int d^{3}x\alpha(x)n(x)+logZ_{v},$ (3.24)

where $Z_{v}(\beta;\alpha]$ is the trial partition function defined as

$Z_{v}(\beta;\alpha]=Tr_{c}e^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\;.$ (3.25)

The equilibrium density $n_{eq}(x)$ is that which maximizes
$S_{v}(E{\color[rgb]{0,0,0};}n]$ subject to $\int d^{3}xn(x)=N$:

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}S_{v}(E;n]+\alpha_{eq}[N-\int
d^{3}xn(x)]\Bigg{]}_{n(x)=n_{eq}(x)}=0\quad\textit{for fixed E}.$ (3.26)

Next, perform a Legendre transformation and define the Massieu functional
$\tilde{S}_{v}(\beta,n]$ as

$\tilde{S}_{v}\equiv S_{{\color[rgb]{0,0,0}v}}-\beta E,$ (3.27)

so that we can rewrite equation (3.26) as

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\tilde{S}_{v}(\beta;n]-\alpha_{eq}\int
d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\quad\textit{for fixed $\beta$}\;.$
(3.28)

Combine (3.24), (3.27), and (3.28), and use the variational derivative
identity $\frac{\delta n(x)}{\delta n(x^{\prime})}=\delta(x-x^{\prime})$ to
find

$\int d^{3}x^{\prime}\Bigg{[}\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x^{\prime})}+n(x^{\prime})\Bigg{]}\delta\alpha(x^{\prime})=\int
d^{3}x^{\prime\prime}\Bigg{[}\alpha_{eq}-\alpha(x^{\prime\prime})\Bigg{]}\delta
n(x^{\prime\prime})\;.$ (3.29)

The LHS of equation (3.29) vanishes by (3.20) and therefore the RHS must
vanish for an arbitrary $\delta n(x)$ which implies that

$\alpha(x)=\alpha_{eq}\;,\quad\text{and}\quad\frac{\delta
logZ_{v}}{\delta\alpha(x)}{\color[rgb]{0,0,0}\Big{|}_{\alpha(x)=\alpha_{eq}}}=-n_{{\color[rgb]{0,0,0}eq}}(x)\;.$
(3.30)

Substituting (3.30) into (3.21) yields the equilibrium probability
distribution

$P^{*}(x_{1},\dots,x_{N};p_{1},\dots,p_{N})=\frac{1}{Z_{v}^{*}}e^{-\beta\hat{H}_{v}-\alpha_{eq}\int
d^{3}x\hat{n}_{x}}=\frac{1}{Z_{v}^{*}}e^{-\beta\hat{H}_{v}-\alpha_{eq}N}$
(3.31)

where $Z_{v}^{*}(\beta,\alpha_{eq})=Tr_{c}e^{-\beta\hat{H}_{v}-\alpha_{eq}N}.$
From the inferential point of view, the variational principle for the grand
potential and the equilibrium density[18] is proved at this point; we showed
that for an arbitrary classical Hamiltonian $\hat{H}_{v}$, there exists a
trial entropy $S_{v}(E;n(x)]$ defined by equation (3.24), which assumes its
maximum at fixed energy and varying $n(x)$ under the condition $\int
d^{3}xn(x)=N$ at the equilibrium density, and gives the posterior PDF
(equation 3.31) equal to that of the canonical distribution.
The massieu function $\tilde{S}_{v}(\beta;n(x)]$ from equation (3.27) defines
the density functional potential $\Omega_{v}(\beta;n(x)]$ by

$\Omega_{v}(\beta;n]\equiv\frac{-\tilde{S}_{v}(\beta;n]}{\beta}=-\int
d^{3}x\frac{\alpha(x)}{\beta}n(x)-\frac{1}{\beta}logZ_{v}(\beta;\alpha]\;,$
(3.32)

so that the maximization of $S_{v}(E;n]$ (3.24) in the vicinity of
equilibrium, is equivalent to the minimization of $\Omega_{v}(\beta;n(x)]$
(3.32) around the same equilibrium point

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\Omega_{v}(\beta;n]+\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$ (3.33)

After we find $\Omega_{v}$, we just need to recall that
$\alpha(x)=-\beta\frac{\delta\Omega_{v}}{\delta n(x)}$ and substitute in
equation (3.30) to recover the ”core integro-differential equation”[18] of DFT
as

$\nabla\Big{(}\frac{\delta\Omega_{v}(\beta;n]}{\delta n(x)}\Big{)}_{eq}=0$
(3.34)

which implies that

$\Omega_{v;eq}\leq\Omega_{v}(\beta;n],$ (3.35)

where

$\Omega_{v,eq}(\beta;n]=-\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)-\frac{1}{\beta}logZ_{v}^{*}(\beta,\alpha_{eq}).$ (3.36)

From equation (3.32) it is clear that

$\Omega_{v}(\beta;n]=\int
d^{3}xv(x)n(x)+\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}(E;n]}{\beta}\;.$
(3.37)

It is convenient to define the intrinsic density functional potential $F_{v}$
as

$F_{v}(\beta;n]\equiv\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}}{\beta}$ (3.38)

to have

$\Omega_{v}(\beta;n]=\int v(x)n(x)+F_{v}(\beta;n]\;.$ (3.39)

Now we are ready to restate the fundamental theorem of the classical DFT:

###### Theorem 4

The intrinsic functional potential $F_{v}$ is a functional of density $n(x)$
and is independent of the external potential:

$\frac{\delta F_{v}(\beta;n]}{\delta v(x^{\prime})}=0\quad\text{for fixed
$\beta$ and $n(x)$.}$ (3.40)

######  Proof 3.5.5.

The crucial observation behind the DFT formalism is that $P$ and $Z_{v}$
depend on the external potential $v(x)$ and the Lagrange multipliers
$\alpha(x)$ only through the particular combination
$\bar{\alpha}(x)\equiv\beta v(x)+\alpha(x)$. Substitute equation (3.24) in
(3.38) to get

$\beta F_{v}(\beta;n]=logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x),$ (3.41)

where
$Z(\beta;\bar{\alpha}]=Z_{v}(\beta;\alpha]=Tr_{c}e^{-\beta(\hat{K}+\hat{U})-\int
d^{3}x\bar{\alpha}(x)\hat{n}_{x}}\;.$ The functional derivative of $\beta
F_{v}$ at fixed $n(x)$ and $\beta$ is

$\frac{\delta(\beta F_{v}(\beta;n])}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\int
d^{3}x^{\prime\prime}\frac{\delta}{\delta\bar{\alpha}(x^{\prime\prime})}\Big{[}logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x)\Big{]}\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}\;.$ (3.42)

Since $n(x^{\prime})=-\frac{\delta
logZ(\beta;\bar{\alpha}]}{\delta\bar{\alpha}(x^{\prime})}$, keeping $n(x)$
fixed is achieved by keeping $\bar{\alpha}(x)$ fixed:

$\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,\bar{\alpha}(x)}=0\;,$ (3.43)

so that

$\frac{\delta F_{v}(\beta,n]}{\delta v(x^{\prime})}\Big{|}_{\beta,n(x)}=0\;,$
(3.44)

which concludes the proof; thus, we can write down the intrinsic potential as

$F(\beta,n(x)]=F_{v}(\beta,n(x)]\;.$ (3.45)

######  Remark 3.5.6.

Note that since a change of the external potential $v(x)$ can be compensated
by a suitable change of the multiplier $\alpha(x)$ in such a way as to keep
$\bar{\alpha}(x)$ fixed, such changes in $v(x)$ will have no effect on $n(x)$.
Therefore keeping $n(x)$ fixed on the left hand side of (3.42) means that
$\bar{\alpha}(x)$ on the right side is fixed too.

Now we can substitute equation (3.39) in (3.33), and define the chemical
potential

$\mu\equiv\frac{-\alpha_{eq}}{\beta}\;,$ (3.46)

to have

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}\int
d^{3}xv(x)n(x)+F(\beta;n]-\mu\int d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$
(3.47)

We can also substitute (3.39) in (3.34) to find

$v(x)+\frac{\delta F}{\delta n(x)}\Big{|}_{eq}=\mu,$ (3.48)

which allows us to define and interpret $\mu_{in}(x;n]\equiv\frac{\delta
F}{\delta n(x)}$ as the intrinsic chemical potential of the system. To proceed
further we also split $F$ into that of ideal gas plus the interaction part as

$F(\beta;n]=F_{id}(\beta;n]-\phi(\beta;n].$ (3.49)

Differentiating with $\frac{\delta}{\delta n(x)}$ gives

$\beta\mu_{in}(x;n]=log(\lambda^{3}n(x))-c(x;n]\;,$ (3.50)

where for a monatomic gas
$\lambda=\Big{(}\frac{2\pi\hbar^{2}}{m}\Big{)}^{1/2}$. The additional one body
potential $c(x;n]=\frac{\delta\phi}{\delta n(x)}$ is related to the Ornstein-
Zernike direct correlation function of non-uniform fluids [33, 34, 35] by

$c^{(2)}(x,x^{\prime};n]\equiv\frac{\delta c(x;n]}{\delta
n(x^{\prime})}=\frac{\delta^{2}\phi(\beta;n]}{\delta n(x)\delta
n(x^{\prime})}\;.$ (3.51)

Up to here, we have proved the classical DFT theorem and clarified its
connection to thermodynamic correlations, now we shall procceed further and
recover the existing approximation for slowly varying densities [18] as an
illustration.

###  3.6 Slowly Varying Density and Gradient Expansion

We have proved that the solution to equation (3.47) is the equilibrium
density. But the functional $F(\beta;n]$ needs to be approximated because the
direct calculation of $F$ involves calculating the canonical partition
function, the task which we have been avoiding to begin with. Therefore
different models of DFT may vary in their approach for guessing $F(\beta;n]$.
Now assume that we are interested in a monatomic fluid with a slowly varying
external potential. In our language, it means that we use the approximation
$\int d^{3}x\equiv\sum(\Delta x)^{3}$ where $\Delta x$ is much longer than the
density correlation length and the change in density in each volume element is
small compared to its average density. This allows us to interpret each volume
element $(\Delta x)^{3}$ as a fluid at grand canonical equilibrium with the
rest of the fluid as its thermal and particle bath.
Similar to [18], we expand $F(\beta;n]$ as

$F(\beta;n]=\int d^{3}x\Big{[}f_{0}(n(x))+f_{2}(n(x))|\nabla
n(x)|^{2}+\mathcal{O}(\nabla^{4}n(x))\Big{]}\;.$ (3.52)

Differentiating with respect to $n(x)$ we have

$\mu_{in}(x;n]=\frac{\delta F}{\delta
n(x)}=f^{\prime}_{0}(n(x))-f^{\prime}_{2}(n(x))|\nabla
n(x)|^{2}-2f_{2}(n(x))\nabla^{2}n(x).$ (3.53)

In the absence of an external potential, $v(x)=0$, the second and the third
terms in the RHS of (3.53) vanish, and also from equation (3.48),
$\mu_{in}=\mu$ , therefore we have

$f_{0}^{\prime}(n)=\mu(n(x)),$ (3.54)

where $\mu(n(x))$ is the chemical potential of a uniform fluid with density
$n=n(x)$. On the other hand, with the assumption that each volume element
behaves as if it is in grand canonical ensemble for itself under influence of
both external potential and additional one body interaction $c(x;n]$ we know
that the second derivative of $F$ is related to Ornstein-Zernike theory by

$\beta\frac{\delta^{2}F}{\delta n(x)\delta
n(x^{\prime})}=\frac{\delta(x-x^{\prime})}{n(x)}-c^{(2)}(x,x^{\prime};n]\;.$
(3.55)

Therefore we have a Taylor expansion of $F$ around the uniform density as

$\displaystyle F[n(x)]=$ $\displaystyle F[n]+\int d^{3}x\Big{[}\frac{\delta
F}{\delta n(x)}\Big{]}_{n_{eq}(x)}\tilde{n}(x)$ (3.56)
$\displaystyle+\frac{1}{2\beta}\int\int
d^{3}xd^{3}x^{\prime}\Big{[}\frac{\delta(x-x^{\prime})}{n(x)}-c^{(2)}(|x-x^{\prime}|;n]\Big{]}_{n_{eq}(x)}\tilde{n}(x)\tilde{n}(x^{\prime})+\dots,$

where $\tilde{n}(x)\equiv n(x)-n$, and $c^{(2)}(|x-x^{\prime}|;n]$ is the
direct correlation function of a uniform fluid with density $n=n(x)$. The
Fourier transform of the second integral in (3.56) gives

$\displaystyle\frac{1}{2\beta}\int\int
d^{3}xd^{3}x^{\prime}\Big{[}\frac{\delta(x-x^{\prime})}{n(x)}-c^{(2)}(|x-x^{\prime}|;n]\Big{]}_{n_{eq}(x)}\tilde{n}(x)\tilde{n}(x^{\prime})$
(3.57) $\displaystyle=\frac{-1}{2\beta
V}\sum_{q}\Big{(}c^{(2)}(q;n]-\frac{1}{n(q)}\Big{)}\tilde{n}(q)\tilde{n}(-q)\;,$

and comparing with (3.52) yields

$f_{0}^{\prime\prime}(n)=\frac{-1}{\beta}(a(n)-\frac{1}{n})\;,\quad\quad
f_{2}(n)=\frac{-b(n)}{2\beta},$ (3.58)

where the functions $a(n)$ and $b(n)$ are defined as coefficients of Fourier
transform of the Ornstein-Zernike direct correlation function by
$c^{2}(q;n]=a(n(q))+b(n(q))q^{2}+\dots$.
$b(n)$ is evaluated by linear response theory to find that

$f_{2}(n(x))=\frac{1}{12\beta}\int
d^{3}x^{\prime}|x-x^{\prime}|^{2}c^{(2)}(|x-x^{\prime}|;n].$ (3.59)

We can substitute equations (3.59) and (3.54) in (3.53) and use equilibrium
identity $\nabla\mu=0$ to find the integro-differential equation

$\nabla\Bigg{[}v(x)+\mu(n(x))-f_{2}^{\prime}(n(x))|\nabla
n(x)|^{2}-2f_{2}(n(x))\nabla^{2}n(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0,$ (3.60)

that determines the equilibrium density $\hat{n}_{eq}(x)$ in the presence of
external potential $v(x)$ given Ornstein-Zernike direct correlation function
of uniform fluid $c^{(2)}[n(x),|x-x^{\prime}|]$.

###  3.7 Final Remarks

We showed that the variational principle of classical DFT is a special case of
applying the method of maximum entropy to construct optimal approximations in
terms of those variables that capture the relevant physical information
namely, the particle density $n(x)$. It is worth emphasizing once again: In
this work we have pursued the purely conceptual goal of finding how DFT fits
within the MaxEnt approach to statistical mechanics. The advantage of
achieving such an insight is the potential for future applications that lie
outside the reach of the current versions of DFT. As an illustration we have
discussed the already well-known example of a slowly varying inhomogeneous
fluid. Future research can be pursued in three different directions: i) To
show that the method of maximum entropy can also be used to derive the quantum
version of DFT (which is the subject of the next chapter). ii) To approach the
Dynamic DFT [36], generalizing the idea to non-equilibrium systems following
the theory of maximum caliber [37]. iii) To revisit the objective of section
3.6 and construct weighted DFTs [38, 39] using the method of maximum entropy.

##  Chapter 4 Entropic Quantum DFT

###  4.1 Introduction

In this chapter, we are going to formulate the density functional formalism as
an application of the quantum MaxEnt. In addition to opening the door to new
applications of DFT, we seek at least two main objectives by the approach: i)
to make it explicit how DFT fits within the MaxEnt framework ii) to clarify
the already existing and well-known approximation schemes of DFT. The former
objective is motivated by the fact that Jaynes [3, 4] showed that
Thermodynamics is an inference scheme about nature, and the original DFT
articles [16, 17, 23, 18] have used thermodynamic arguments repeatedly,
although the method may be applied to systems much smaller than the
macroscopic systems discussed in thermodynamics, therefore an inferential
interpretation of DFT is desired to eliminate the thermodynamic arguments from
the framework. The second objective is of both historical and conceptual
importance as well, the theory in the original form does not refer to a system
of fixed number of particles, while the electron gas of interest is not
necessarily free to exchange particle with the rest of the universe. Therefore
the choice of grand canonical ensemble is an approximation of a possible real
system to begin with, if DFT theorem is only correct for grand canonical
ensemble, then the equivalence between the DFT variational principle and the
time independent Schrödinger equation would be proven on a Fock space of
electrons not a single Hilbert space of $N$ electrons, this ambiguity about
what is exact and what is approximate in the DFT made a long term battle
between Pople and Kohn [40] which finally led to a ceasefire in 1998 when the
two shared the Nobel Prize in chemistry. In our entropic DFT, we prove the
variational principle for the case of a Hilbert space of $N$ electrons, and
then it becomes clear that the grand canonical formalism does not necessarily
appear in the theorem of DFT, the use of grand-canonical formalism is needed
when approximations are required. Indeed, our approach makes the theoretical
aspect of DFT independent of the second quantization, although it is useful
for practical purposes.

###  4.2 Quantum Statistics

”When we solve a quantum-mechanical problem, what we really do is divide the
universe into two parts–the system of interest and the rest of the
universe.”[31]
Let $|\phi_{i}\rangle$ be a complete set of vectors in the Hilbert space
describing the system, and $|\theta_{j}\rangle$ for the rest of the universe.
Let $|x\rangle$ form an orthonormal basis in the Hilbert space of the system,
and $|y\rangle$ form an orthonormal basis in the Hilbert space of the rest of
the universe:

$\phi_{i}(x)=\langle
x|\phi_{i}\rangle\quad\text{and}\quad\theta_{j}(y)=\langle
y|\theta_{j}\rangle\;.$ (4.2.1)

The most general vector in the total Hilbert space is given by $|\psi\rangle$
defined as

$|\psi\rangle\equiv\sum_{ij}C_{ij}|\phi_{i}\rangle|\theta_{j}\rangle\;,$
(4.2.2)

and consequently the wave function of the universe plus the system given as

$\psi(x,y)\equiv\langle x|\langle y|\psi\rangle=\sum_{ij}C_{ij}\langle
x|\phi_{i}\rangle\langle y|\theta_{j}\rangle\;.$ (4.2.3)

Let $\hat{A}$ be an operator which acts only on the system of interest, we
have

$\hat{A}=\sum_{ii^{\prime}j}A_{ii^{\prime}}|\phi_{i}\rangle|\theta_{j}\rangle\langle\theta_{j}|\langle\phi_{i^{\prime}}|\;,$
(4.2.4)

where $A_{ii^{\prime}}\equiv\langle\phi_{i}|\hat{A}|\phi_{i}^{\prime}\rangle$.
Therefore the expected value of $\hat{A}$ is given as

$\langle\psi|\hat{A}|\psi\rangle=\sum_{ii^{\prime}jj^{\prime}}C_{ij}^{*}C_{i^{\prime}j^{\prime}}\langle\theta_{j}|\langle\phi_{i}|\hat{A}|\phi_{i^{\prime}}\rangle|\theta_{j^{\prime}}\rangle=\sum_{ii^{\prime}j}C_{ij}^{*}C_{i^{\prime}j}\langle\phi_{i}|\hat{A}|\phi_{i^{\prime}}\rangle\;$
(4.2.5)

Define the density matrix $\rho_{i^{\prime}i}$ with elements

$\rho_{i^{\prime}i}=\sum_{j}C_{ij}^{*}C_{i^{\prime}j}$ (4.2.6)

so that we can rewrite (4.2.5) as

$\langle\hat{A}\rangle=\sum_{ii^{\prime}}A_{ii^{\prime}}\rho_{i^{\prime}i}=Tr\rho
A\;.$ (4.2.7)

Next, define the density operator $\hat{\rho}$ as

$\langle\phi_{i^{\prime}}|\hat{\rho}|\phi_{i}\rangle\equiv\rho_{i^{\prime}i}\;,$
(4.2.8)

equation (4.2.6) implies that $\hat{\rho}$ is Hermitian, therefore it can be
diagonalized by an orthonormal basis set $|i\rangle$ with real eigenvalues
$p_{i}$ as

$\hat{\rho}=\sum_{i}p_{i}|i\rangle\langle i|\;.$ (4.2.9)

Let $\hat{A}=1$ to rewrite equation (4.2.7) as

$Tr\rho=\sum_{i}p_{i}=1\;,$ (4.2.10)

and let $\hat{A}=|i^{\prime}\rangle\langle i^{\prime}|$ to rewrite equation
(4.2.7) as

$p_{i^{\prime}}=Tr\rho A=\sum_{j}|(\langle
i^{\prime}|\langle\theta_{j}|)|\psi\rangle|^{2}\geq 0\;.$ (4.2.11)

####  4.2.1 Density Operator

Given the motivation in the previous section, now we are interested in a
formulation of quantum mechanics focused on the system of interest and
ignorant about the rest of the universe.
In quantum mechanics, the most informative state of knowledge about a system
is represented by a state vector $|\psi\rangle$ . When we have incomplete
information about the system, we call it a mixed state. Mixed states are
treated with a density operator represented by the density matrix $\hat{\rho}$
instead of a single wave function. The density matrix assigns probabilities
$p_{i}$ to each member of an orthonormal basis set $|\psi_{i}\rangle$:

$\hat{\rho}\equiv\sum_{i,j}p_{i}\delta_{ij}|\psi_{i}\rangle\langle\psi_{j}|\;.$
(4.2.12)

The expected value of observable $A$ is given by

$\langle\hat{A}\rangle\equiv\sum_{i}p_{i}\langle\psi_{i}|\hat{A}|\psi_{i}\rangle\;.$
(4.2.13)

We can insert the identity operator $\mathbb{I}=\sum_{n}|n\rangle\langle n|$,
where $|n\rangle$ form an orthonormal basis

$\hat{A}=\mathbb{I}\hat{A}\mathbb{I}=\sum_{n}\sum_{m}|n\rangle\langle
n|\hat{A}|m\rangle\langle m|\;,$ (4.2.14) $\hat{A}=\sum_{n}\sum_{m}|n\rangle
A_{nm}\langle m|\;,\quad A_{nm}\equiv\langle n|\hat{A}|m\rangle\;.$ (4.2.15)

Therefore the expected value of operator $\hat{A}$ is

$\langle\hat{A}\rangle=\sum_{i}p_{i}\sum_{n}\sum_{m}\langle\psi_{i}|n\rangle\langle
m|\psi_{i}\rangle A_{nm}$ (4.2.16)
$\langle\hat{A}\rangle=\sum_{n}\sum_{m}\langle
m|\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|n\rangle A_{nm}\;.$ (4.2.17)

Substitute equation (4.2.12) in (4.2.17)

$\langle\hat{A}\rangle=\sum_{n}\sum_{m}\langle m|\hat{\rho}|n\rangle
A_{nm}=\sum_{n}\sum_{m}\rho_{mn}A_{nm}=Tr\hat{\rho}\hat{A}\;.$ (4.2.18)

Note that the result $\langle\hat{A}\rangle=Tr\hat{\rho}\hat{A}$ does not
depend on our choice of basis. The $|n\rangle$’s could be the eigenstates of
any complete set of commuting Hermitian Operators.

####  4.2.2 Quantum Liouville Equation

Given the definition of the density matrix in equation (4.2.12), if there is
no reason to update the probability distribution over time $\frac{\partial
p_{i}}{\partial t}=0$, then the time derivative of the density matrix is given
as

$\frac{\partial\hat{\rho}}{\partial
t}=\sum_{i}p_{i}\bigg{\\{}(\frac{\partial}{\partial
t}|\psi_{i}\rangle)\langle\psi_{i}|+|\psi_{i}\rangle(\frac{\partial}{\partial
t}\langle\psi_{i}|)\bigg{\\}}\;,$ (4.2.19)

substitute the Schrödinger equation
$\hat{H}(t)|\psi(t)\rangle=i\frac{\partial}{\partial t}|\psi(t)\rangle$ to
have

$\frac{\partial\hat{\rho}}{\partial
t}=\frac{1}{i}\sum_{i}p_{i}\bigg{\\{}(\hat{H}|\psi_{i}\rangle)\langle\psi_{i}|-|\psi_{i}\rangle(\langle\psi_{i}|\hat{H})\bigg{\\}}=\frac{1}{i}[\hat{H},\hat{\rho}]\;.$
(4.2.20)

####  4.2.3 Many-body Density Matrix in Position and Momentum Representations

In position representation, the eigenstates $|n\rangle$ are picked to be
eigenstates of operators $\\{\hat{x}_{1},\hat{x}_{2},\dots,\hat{x}_{N}\\}$ ;
for $N$ distinguishable particles

$\langle x|\psi_{i}\rangle\equiv\langle
x_{1},x_{2},\dots,x_{N}|\psi_{i}\rangle\equiv\psi_{i}(x_{1},x_{2},\dots,x_{N})$
(4.2.21)

therefore we can rewrite equation (4.2.17) for $N$ distinguishable particles

$\langle\hat{A}\rangle=\int d^{3N}x\int d^{3N}x^{\prime}\langle
x^{\prime}|\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|x\rangle
A_{xx^{\prime}}$ (4.2.22) $\langle\hat{A}\rangle=\int d^{3N}x\int
d^{3N}x^{\prime}\rho_{x^{\prime}x}A_{xx^{\prime}}=Tr\hat{\rho}\hat{A}\;,\quad\rho_{x^{\prime}x}=\langle
x^{\prime}|\hat{\rho}|x\rangle\;.$ (4.2.23)

and for $N$ identical particles as

$\langle\hat{A}\rangle=\frac{1}{N!}\sum_{\text{particle
permutations}}\hat{P}\int d^{3N}x\int d^{3N}x^{\prime}\langle
x^{\prime}|\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|x\rangle
A_{xx^{\prime}}\;,$ (4.2.24) $\langle\hat{A}\rangle=\int d^{3N}x\int
d^{3N}x^{\prime}\rho_{x^{\prime}x}A_{xx^{\prime}}=Tr\hat{\rho}\hat{A}\;,\quad\rho_{x^{\prime}x}=\frac{1}{N!}\sum_{\text{particle
permutations}}\hat{P}\langle x^{\prime}|\hat{\rho}|x\rangle\;,$ (4.2.25)

where $\sum_{P}$ stands for summation over all particle permutations, and the
permutation operator $\hat{P}$ is defined by

$\hat{P}\langle
x^{\prime}_{1},x^{\prime}_{2},\dots,x^{\prime}_{N}|\hat{\rho}|x_{1},x_{2},\dots,x_{N}\rangle\equiv\langle
x^{\prime}_{1},x^{\prime}_{2},\dots,x^{\prime}_{N}|\hat{\rho}|\hat{P}x_{1},\hat{P}x_{2},\dots,\hat{P}x_{N}\rangle\;.$
(4.2.26)

Note that $\psi_{i}(x_{1},\dots,x_{N})$ is symmetric for bosons and
antisymmetric for fermions, therefore they admit different sets of particle
permutations:

$\rho_{x^{\prime}x}=\frac{1}{N!}\sum_{P}\hat{P}\langle
x^{\prime}|\hat{\rho}|x\rangle\quad\quad\text{bosons}$ (4.2.27)
$\rho_{x^{\prime}x}=\frac{1}{N!}\sum_{P}(-1)^{P}\hat{P}\langle
x^{\prime}|\hat{\rho}|x\rangle\quad\quad\text{fermions}$ (4.2.28)

and $(-1)^{P}=\pm 1$ for even/odd permutations.
Similarly, in momentum representation, the expected value of operator
$\hat{A}$, is given by

$\langle\hat{A}\rangle=\int d^{3N}q\int
d^{3N}q^{\prime}\rho_{q^{\prime}q}A_{qq^{\prime}}=Tr\hat{\rho}\hat{A}\;,\quad\rho_{q^{\prime}q}=\frac{1}{N!}\sum_{\text{particle
permutations}}\hat{P}\langle q^{\prime}|\hat{\rho}|q\rangle\;,$ (4.2.29)

where

$\hat{P}\langle
q^{\prime}_{1},q^{\prime}_{2},\dots,q^{\prime}_{N}|\hat{\rho}|q_{1},q_{2},\dots,q_{N}\rangle\equiv\langle
q^{\prime}_{1},q^{\prime}_{2},\dots,q^{\prime}_{N}|\hat{\rho}|\hat{P}q_{1},\hat{P}q_{2},\dots,\hat{P}q_{N}\rangle\;.$
(4.2.30)

###  4.3 Quantum MaxEnt

The formal difference between classical and quantum regimes reduces
significantly when it comes to statistical mechanics, just like we would
update a prior probability distribution to the posterior probability
distribution given new information, we can update prior density matrix of a
quantum system in response to new information in form of expected value of
observables using the quantum method of maximum entropy.
Given prior density matrix 111The prior density matrix is usually picked to be
the identity matrix up to multiplication by a constant, but here we do not
limit the general discussion and proceed with arbitrary prior density matrix
and we postpone choosing of prior to later. $\hat{\sigma}$, and new
information in form of expected values of hermitian operators $\hat{A}_{i}$’s

$\langle\hat{A}_{i}\rangle=A_{i}$ (4.3.1)

the posterior density matrix $\hat{\rho}$ is found by maximizing quantum
relative entropy

$S_{r}[\hat{\rho}|\hat{\sigma}]=-Tr\hat{\rho}(log\hat{\rho}-log\hat{\sigma}),$
(4.3.2)

subject to constraints

$Tr\hat{A}_{i}\hat{\rho}-A_{i}=0$ (4.3.3a) $Tr\hat{\rho}-1=0.$ (4.3.3b)

So we need to solve the variational equation

$\delta\Big{[}S_{r}+\sum_{i}\alpha_{i}[A_{i}-Tr(\hat{A}_{i}\hat{\rho})]+\alpha_{0}[1-Tr\hat{\rho}]\Big{]}=0$
(4.3.4)

Where operators $\hat{A}_{i}$’s, expected values $A_{i}$’s, and Lagrange
multipliers $\alpha_{0}$, and $\alpha_{i}$’s are kept fixed, while density
matrix $\hat{\rho}$ varies.
Equation (4.3.4) can be solved to give the maximizing density operator

$\hat{\rho}=e^{log\hat{\sigma}-\alpha_{0}\hat{I}-\sum\alpha_{i}\hat{A}_{i}}$
(4.3.5)

in which, $\hat{I}$ is the identity matrix.
Using Baker–Campbell–Hausdorff formula, since the identity matrix commutes
with any matrix, we can rewrite the posterior density matrix as

$\hat{\rho}=\frac{1}{Z}e^{log\hat{\sigma}-\sum\alpha_{i}\hat{A}_{i}},$ (4.3.6)

where the partition function $Z$, is the normalization factor defined as

$Z\equiv e^{\alpha_{0}}=Tre^{log\hat{\sigma}-\sum\alpha_{i}\hat{A}_{i}}.$
(4.3.7)

Therefore one can substitute equation (4.3.6) in equation (4.3.2) to find
maximized entropy S as

$S=\sum\alpha_{i}A_{i}+logZ,$ (4.3.8)

And consequently

$dS=\sum\alpha_{i}dA_{i}+\sum d\alpha_{i}A_{i}+\frac{dZ}{Z}.$ (4.3.9)

Next, calculate $dZ(\alpha_{i})$,

$dZ=\sum\frac{\partial Z}{\partial\alpha_{i}}d\alpha_{i}.$ (4.3.10)

We can use matrix exponential differential identity [41] 222 Differential of a
matrix exponential is given by
$\frac{d}{dt}e^{X(t)}=e^{X}\frac{1-e^{-ad_{X}}}{ad_{X}}\frac{dX}{dt}$ (4.3.11)
where $ad_{X}Y$ is the adjoint action of X on Y, and  $ad_{X}Y=[X,Y]$ (4.3.12)
if $X$ and $Y$ belong to the same Hilbert space.  to calculate $dZ$ :

$dZ=Tr[\sum_{i}\frac{\partial
e^{log\hat{\sigma}-\sum_{j}\alpha_{j}\hat{A}_{j}}}{\partial\alpha_{i}}d\alpha_{i}]$
(4.3.13)

$dZ=Tr[\sum_{i}exp(log\hat{\sigma}-\sum\alpha_{j}\hat{A}_{j})\frac{1-e^{-ad_{exp(log\hat{\sigma}-\sum\alpha_{j}\hat{A}_{j})}}}{ad_{exp(log\hat{\sigma}-\sum\alpha_{j}\hat{A}^{j})}}(-\hat{A}_{i})d\alpha_{i}].$
(4.3.14)

Divide equation (4.3.14) by $Z$ to find

$\frac{dZ}{Z}=-Tr[\sum_{i}\hat{\rho}\frac{1-e^{-ad_{Z\hat{\rho}}}}{ad_{Z\hat{\rho}}}\hat{A}_{i}d\alpha_{i}]$
(4.3.15)

We can expand the exponential in above equation to find that

$\frac{dZ}{Z}=-\sum_{i}A_{i}d\alpha_{i}-\sum_{i}\sum_{n=1}^{\infty}\frac{1}{(n+1)!}Tr[\hat{\rho}(-Zad_{\hat{\rho}})^{n}\hat{A}_{i}]d\alpha_{i},$
(4.3.16)

and by substitution in equation (4.3.9) we find differential of entropy

$dS=\sum_{i}\alpha_{i}dA_{i}-\sum_{j}\sum_{n=1}^{\infty}\frac{(-Z)^{n}}{(n+1)!}\langle
ad^{n}_{\hat{\rho}}\hat{A}_{j}\rangle d\alpha_{j}.$ (4.3.17)

The second term in the RHS of equation (4.3.17) vanishes by the cyclic
property of trace, for any Hermitian operator $\hat{A}$

$\langle\hat{a}d_{\hat{\rho}}\hat{A}\rangle=Tr\hat{\rho}[\hat{\rho},\hat{A}]=Tr\hat{\rho}\hat{\rho}\hat{A}-Tr\hat{\rho}\hat{A}\hat{\rho}=0$
(4.3.18)

and expected value of higher order adjoint actions of $\hat{\rho}$ vanish by
the same argument

$\langle ad_{\hat{\rho}}^{n}\hat{A}\rangle=0\;,$ (4.3.19)

and we end up with

$dS=\sum_{i}\alpha_{i}dA_{i}\;.$ (4.3.20)

####  4.3.1 Contact Structure of MaxEnt

Equation (4.3.20) allows us to define the vanishing contact 1-form
$\omega_{q}$ for quantum MaxEnt as

$\omega_{q}\equiv dS-\sum_{i}\alpha_{i}A_{i}=0\;.$ (4.3.21)

Similar to the classical case [42], for any quantum MaxEnt with $m$
constraints, we define contact structure $\\{\mathbb{T},\omega_{q}\\}$, in
which manifold $\mathbb{T}$ is $2m+1$-dimensional with coordinates
$\\{q_{0},q_{1},\dots,q_{m},p_{0},\dots,p_{N}\\}$. This means that the
physically relevant manifold $\mathbb{M}$ is an $m$-dimensional submanifold on
$\mathbb{T}$ locally represented by vanishing one form $\omega_{q}$; or
equivalently, $\mathbb{M}$ is determined by $m+1$ equations

$q_{0}\equiv S(\\{A_{i}\\})$ (4.3.22) $p_{i}\equiv\alpha_{i}=\frac{\partial
S}{\partial A_{i}}\;.$ (4.3.23)

Legendre Transformations are coordinate transformations on $\mathbb{T}$ under
which contact form $\omega_{q}$ is invariant:

$q_{0}\rightarrow q_{0}-\sum_{j=1}^{l}p_{j}q_{j}\;,$ (4.3.24)
$q_{i}\rightarrow p_{i},\quad p_{i}\rightarrow-q_{i}\quad\text{for }1\leq
i\leq l\;.$ (4.3.25)

The existence of the prescribed contact structure makes it legitimate to
utilize Legendre Transformations in MaxEnt, regardless of thermal equilibrium
and the physical significance of the so called ”free energy” functions.

####  4.3.2 Formalisms of Statistical Mechanics

Any formalism of statistical mechanics -in the quantum context- is a quantum
MaxEnt which consists of a trinity of elements: i) One must choose the Hilbert
space that describes the system of inference. ii) The prior is chosen to be
uniform. iii) One must select the constraints that represent the information
that is relevant to the problem in hand.

##### Canonical

The canonical formalism is designed to treat many-body systems consisting of
$N$ fixed number of particles in thermal equilibrium with a heat bath, the
formalism is constructed similar to its classical counterpart:

  1. 1.

The Hilbert space of the system is a symmetrized/antisymmetrized product of
$N$ single particle Hilbert spaces for bosons/fermions.

  2. 2.

The prior density matrix is uniform: $\hat{\sigma}\propto\mathbb{I}$ .

  3. 3.

The relevant information is that the system is in a thermal equilibrium,
therefore the expected value of energy is fixed: $Tr\hat{\rho}\hat{H}=E$ .

The prescription above, implies that one must maximize
$S_{r}[\hat{\rho}|\hat{\sigma}]$ subject to $Tr\hat{\rho}\hat{H}=E$ and
$Tr\hat{\rho}=1$ to find the posterior density matrix, analog to (4.3.6) as

$\hat{\rho}^{*}=\frac{1}{Z}e^{-\beta\hat{H}}$ (4.3.26)

and the maximized entropy

$S_{r}[\hat{\rho}^{*}|\hat{\sigma}]=S(E)=\beta E+logZ(\beta)\;.$ (4.3.27)

##### Grand Canonical

The grand canonical formalism, is a quantum MaxEnt for inference about a
system of many particles in both thermal and particle equilibrium with a large
bath.

  1. 1.

The Hilbert space of the system is the direct sum of all $n$ particle Hilbert
spaces for $0\leq n\leq\infty$ (Fock space) .

  2. 2.

The prior density matrix is uniform: $\hat{\sigma}\propto\mathbb{I}$ .

  3. 3.

The relevant information is that system is in a thermal and particle
equilibrium, therefore the expected value of the Hamiltonian operator and the
expected value of the particle number operator are fixed:
$Tr\hat{\rho}\hat{H}=E$ , $Tr\hat{\rho}\hat{N}=N$ .

The MaxEnt process with the prescription above yields the posterior density
matrix analog to the equation (4.3.6) as

$\hat{\rho}^{*}=\frac{1}{\Xi}e^{-\beta(\hat{H}-\mu\hat{N})}$ (4.3.28)

and the maximized entropy analog to (4.3.8)

$S_{r}[\hat{\rho}^{*}|\hat{\sigma}]=S(E,N)=\beta E-\beta\mu N+log\Xi\;.$
(4.3.29)

where $\Xi=Tre^{-\beta(\hat{H}-\mu\hat{N})}$ is the grand canonical partition
function.

####  4.3.3 Using Entropy to Find Optimal Approximations: the Bogolyubov
Inequality

It has been shown [43, 14] that the relative entropy
$S_{r}[\hat{\rho}|\hat{\sigma}]$ defined in equation (4.3.2) provides a
ranking of all $\hat{\rho}$’s relative to a single $\hat{\sigma}$, on the
other hand, the relative entropy is non-positive and assumes zero only for
$\hat{\rho}=\hat{\sigma}$ [44] so that we have 333 Nielsen and Chuang proved
positivity of quantum relative entropy at page 513,  $S(\rho||\sigma)\geq
0\quad\text{with equality iff }\rho=\sigma\;,$ (4.3.30) but it is just because
their definition of relative entropy differs from ours by a multiplication by
minus one; apart from that, the proof concludes our claim.

$S_{r}[\hat{\rho}|\hat{\sigma}]\leq 0\quad\text{with equality iff
}\hat{\rho}=\hat{\sigma}.$ (4.3.31)

Similar to the classical framework developed in [32], given an exact density
matrix $\hat{\sigma}$, and a family of trial density matrices

$\hat{\rho}=\hat{\rho}_{\theta}$ (4.3.32)

parametrized by $\theta=\\{\theta_{1},\theta_{2},\dots\\}$, the member of the
trial family $\hat{\rho}_{\theta^{*}}$ that best approximates $\hat{\sigma}$
is the one which maximizes the relative entropy
$S_{r}[\hat{\rho}_{\theta}|\hat{\sigma}]$

$\frac{\partial}{\partial\theta}\Bigg{[}S_{r}[\hat{\rho}_{\theta}|\hat{\sigma}]\Bigg{]}_{\theta=\theta^{*}}=0\;.$
(4.3.33)

In a special case, when $\hat{\sigma}$ and $\hat{\rho}_{\theta}$ are
exponential density matrices defined as

$\hat{\sigma}=\frac{1}{Z}e^{-\beta\hat{H}}\;,$ (4.3.34)
$\hat{\rho}_{\theta}=\frac{1}{Z_{\theta}}e^{-\beta\hat{H}_{\theta}}\;,$
(4.3.35)

substitute equations (4.3.34) and (4.3.35) in equation (4.3.2) to have

$S[\hat{\rho}_{\theta}|\hat{\sigma}]=\beta(\langle\hat{H}_{\theta}-\hat{H}\rangle_{\theta}-F_{\theta}+F)$
(4.3.36)

where $\langle.\rangle_{\theta}=Tr(\hat{\rho}_{\theta}(.))$, $Z=e^{-\beta F}$,
$Z_{\theta}=e^{-\beta F_{\theta}}$. Substitute (4.3.36) in (4.3.31) to find
that

$F\leq F_{\theta}+\langle\hat{H}-\hat{H}_{\theta}\rangle_{\theta}\;,$ (4.3.37)

which is known as Bogolyubov inequality; with equality if and only if
$\hat{\sigma}$ itself is a member of the trial family $\hat{\rho}_{\theta}$’s.

###  4.4 Density Functional Formalism

Again, any equilibrium ensemble formalism includes three crucial ingredients

  * •

The Hilbert space associated with the time independent Schrödinger equation of
the system

  * •

Uniform prior density matrix

  * •

Inferential constraints

For non-relativistic electron gas at equilibrium, the wave function $\psi(x)$
is an antisymmetrized product of $N$ two-spinor orbitals, therefore the time
independent Schrödinger equation is given as

$\hat{H}|\psi\rangle=E|\psi\rangle\;,$ (4.4.1)

where (in absence of magnetic field) the Hamiltonian is given by

$\hat{H}_{v}=\hat{H}^{(0)}+\hat{V}=\hat{T}+\hat{U}+\hat{V}\equiv\sum_{i=1}^{N}\hat{p}_{i}^{2}+\frac{1}{2}\sum_{j\neq
k}^{N}\frac{1}{|\hat{x}_{j}-\hat{x}_{k}|}+\sum_{l=1}^{N}v(\hat{x}_{l})\;.$
(4.4.2)

####  4.4.1 Constructing the Trial States

Since we are interested in thermal properties of an inhomogeneous gas, we need
trial states that describe both thermal equilibrium and inhomogeneity, these
states will be constructed using the method of maximum entropy. Thermal
equilibrium is imposed by a constraint on the expected value of energy and the
inhomogeneity of the system is incorporated by a constraint on the expected
value of density at each point of space. Therefore we maximize the entropy

$S_{r}[\hat{\rho}|\hat{\mathbb{I}}]=Tr\hat{\rho}log\hat{\rho},$ (4.4.3)

subject to constraints

$Tr\hat{\rho}=1\;,\quad Tr\hat{\rho}\hat{H}=E,$ (4.4.4a)
$Tr\hat{\rho}\hat{n}_{x}=n(x)\quad\text{where}\int d^{3}xn(x)=N\;,$ (4.4.4b)

where $\hat{n}_{x}\equiv\sum_{i=1}^{N}\delta(\hat{x}_{i}-x)$. The density
$n(x)$ plays the role of the $\theta_{i}$ parameters in equation (4.3.35).
There is one constraint for each point in $\mathbb{R}^{3}$ in line (4.4.4b),
but the expected density function $n(x)$ is not arbitrary, it must conform to
the fact that the number of particles $N$ is fixed by definition of the wave
function. Here we deal with a MaxEnt under some constraints, the constraints
themselves are limited together under a condition imposed by the definition of
the Hilbert space. Proceed with the MaxEnt analog to equation (4.3.4) to find
the trial density matrix

$\hat{\rho}=\hat{\rho}_{n}=\frac{1}{Z}e^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\;.$ (4.4.5)

where $Z_{v}(\beta;\alpha]$ is the trial partition function, analog to
equation (4.3.7), defined as

$Z_{v}(\beta;\alpha]=Tre^{-\beta\hat{H}_{v}-\int
d^{3}x\alpha(x)\hat{n}_{x}}\;.$ (4.4.6)

The Lagrange multipliers $\alpha(x)$ are determined implicitly by

$\frac{\delta logZ}{\delta\alpha(x)}=-n(x)$ (4.4.7)

and the corresponding trial entropy is

$S_{v}(E;n]\equiv S_{r}[\hat{\rho}_{n}|\mathbb{I}]=\beta E+\int
d^{3}x\alpha(x)n(x)+logZ_{v}(\beta;\alpha]\;,$ (4.4.8)

where the Lagrange multiplier function $\alpha(x)$ is bound to the condition

$\int d^{3}xn(x)=-\int d^{3}x\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x)}=N\;.$ (4.4.9)

######  Remark 4.4.7.

Note that in all three equations (4.4.5), (4.4.6), and (4.4.8), $\alpha(x)$ is
not an arbitrary function, the only permissible $\alpha(x)$’s are those which
keep number of particles conserved as is shown in equation (4.4.9).

######  Remark 4.4.8.

An important symmetry in the DFT formalism is that $\hat{\rho}_{n}$ and
$Z_{v}(\beta;\alpha]$ depend on the external potential $v(x)$ and the Lagrange
multipliers $\alpha(x)$ only through the particular combination

$\bar{\alpha}(x)\equiv\beta v(x)+\alpha(x)\;.$ (4.4.10)

Substitute (4.4.10) in (4.4.6) to find that

$Z_{v}(\beta,\alpha]=Tre^{-\beta\hat{H}^{(0)}-\int
d^{3}x\bar{\alpha}(x)\hat{n}_{x}}\equiv Z(\beta,\bar{\alpha}]\;,$ (4.4.11)

and

$n(x)=\frac{\delta Z(\beta;\bar{\alpha}]}{\delta\bar{\alpha}(x)}\;.$ (4.4.12)

The symmetry implies that changing $\alpha(x)$ and $v(x)$ will not affect
expected density $n(x)$, as long as $\bar{\alpha}(x)$ remains unchanged; as we
shall see later, the core argument behind the DFT theorem is based on this
symmetry.

######  Remark 4.4.9.

A shift in $\alpha(x)$ can be compensated by shifting the external potential
to keep $\bar{\alpha}(x)$ unchanged, therefore if $\alpha(x)$ is permissible
in equation (4.4.9), so is $\alpha(x)+\textit{const.}$ .

####  4.4.2 The Equilibrium Density Matrix

Now that we have constructed our desired family of density matrices
$\hat{\rho}_{n}$, it is time to approximate the canonical density matrix of
the system by the best fitting member of $\hat{\rho}_{n}$’s. The canonical
density matrix is defined as

$\hat{c}=\frac{1}{Z_{v}(\beta)}e^{-\beta\hat{H}_{v}}\;,$ (4.4.13)

where $Z_{v}(\beta)=Tre^{-\beta\hat{H}_{v}}$ .
Substitute equations (4.4.13), and (4.4.5) in (4.3.2) to find that

$S_{r}[\hat{\rho}_{n}|\hat{c}]=-Tr\hat{\rho}_{n}log\hat{\rho}_{n}-Tr\hat{\rho}_{n}log\hat{c}=S_{v}(E;n]-\beta
E-logZ(\beta)\;,$ (4.4.14)

where $E$ is the expected energy calculated by the trial density matrix
$\hat{\rho}_{n}$.
Maximization of the relative entropy

$\delta S_{r}[\hat{\rho}_{n}|\hat{c}]=0\quad\text{for fixed $\beta$}$ (4.4.15)

gives the closest $\hat{\rho}_{n}$ to $\hat{c}$ . Note that the temperature
$\beta$ is fixed in (4.4.15) because it determines $\hat{c}$; therefore we can
perform Legendre transformation of the form $S_{r}\longrightarrow S_{r}+\beta
E$ to have

$\delta\Big{(}S_{r}+\beta E\Big{)}=0\quad\text{for fixed $E$}\;.$ (4.4.16)

In analogy to equation (4.3.33), the equilibrium density $n_{eq}(x)$ is found
by maximizing the relative entropy $S_{r}[\hat{\rho}_{n}|\hat{c}]$ defined in
(4.4.14)

$\frac{\delta}{\delta n(x^{\prime})}\bigg{[}S_{v}(E;n]-\beta
E+\alpha_{eq}[N-\int d^{3}xn(x)]\bigg{]}_{n(x)=n_{eq}(x)}=0\quad\text{for
fixed $\beta$}\;.$ (4.4.17)

Let us point out that the first two terms in equation (4.4.17) make a Legendre
transform of the trial entropy, which is a function of $\beta$ and a
functional of $n(x)$, the Massieu functional $\tilde{S}_{v}(\beta,n]$ defined
as

$\tilde{S}_{v}(\beta;n]\equiv S_{v}-\beta E\;.$ (4.4.18)

So we can rewrite equation (4.4.15) as

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\tilde{S}_{v}(\beta;n]-\alpha_{eq}\int
d^{3}xn(x)\Bigg{]}=0\quad\text{for fixed $\beta$}\;,$ (4.4.19)

and also (4.4.16)

$\frac{\delta}{\delta n(x^{\prime})}\bigg{[}S_{v}(E;n]+\alpha_{eq}[N-\int
d^{3}xn(x)]\bigg{]}_{n(x)=n_{eq}(x)}=0\quad\text{for fixed $E$}\;.$ (4.4.20)

Note that the term $\alpha_{eq}[N-\int d^{3}xn(x)]$ takes care of the fact
that $n(x)$ is not an arbitrary function in $S_{v}(E;n]$.

Up to here, we have shown that there exists a trial density functional entropy
$S_{v}(E;n]$ which assumes its maximum value at equilibrium density function
$n_{eq}(x)$, but so far it is not guaranteed that this variational principle
is equivalent to the DFT principle which was derived in 1965 [17].

####  4.4.3 The DFT Theorem

Next, substitute (4.4.18) and (4.4.8) in (4.4.19) to find

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}logZ_{v}(\beta;\alpha]+\int
d^{3}x[\alpha(x)-\alpha_{eq}]n(x)\Bigg{]}=0\quad\text{for fixed $\beta$}.$
(4.4.21)

The first term of equation (4.4.21) is given by

$\frac{\delta}{\delta n(x^{\prime})}logZ_{v}(\beta;\alpha]\Big{|}_{\beta}=\int
d^{3}x\frac{\delta logZ_{v}}{\delta\alpha(x)}\frac{\delta\alpha(x)}{\delta
n(x^{\prime})},$ (4.4.22)

therefore we can rewrite (4.4.21) as

$\int d^{3}x\Bigg{[}\frac{\delta
logZ_{v}(\beta;\alpha]}{\delta\alpha(x)}+n(x)\Bigg{]}\frac{\delta\alpha(x)}{\delta
n(x^{\prime})}=\int d^{3}x\Bigg{[}\alpha_{eq}-\alpha(x)\Bigg{]}\frac{\delta
n(x)}{\delta n(x^{\prime})}.$ (4.4.23)

The LHS of (4.4.23) vanishes by (4.4.7), and $\frac{\delta n(x)}{\delta
n(x^{\prime})}=\delta(x-x^{\prime})$, therefore we end up with

$\alpha(x^{\prime})=\alpha_{eq}\;,$ (4.4.24)

and

$\frac{\delta
logZ_{v}}{\delta\alpha(x)}\Big{|}_{\alpha(x)=\alpha_{eq}}=-n_{eq}(x)\;.$
(4.4.25)

Substituting (4.4.24) into (4.4.5) yields the equilibrium density matrix equal
to that of canonical formalism

$\hat{\rho}_{eq}=\frac{1}{Z_{v,eq}}e^{-\beta\hat{H}_{v}-\alpha_{eq}\int
d^{3}x\hat{n}_{x}}=\frac{1}{Z_{v,eq}}e^{-\beta\hat{H}_{v}-\alpha_{eq}N}=\hat{c}\;,$
(4.4.26)

where $Z_{v,eq}(\beta,\alpha_{eq})=Tre^{-\beta\hat{H}_{v}-\alpha_{eq}N}$ .

Although we could proceed further with the Massieu function
$\tilde{S}_{v}(\beta;n(x)]$, in order to have a notation more similar to the
original DFT articles [16, 17, 18], it is convenient to define density
functional potential $\Omega_{v}(\beta;n]$ as

$\Omega_{v}(\beta;n]\equiv\frac{-\tilde{S}_{v}(\beta;n]}{\beta}=-\int
d^{3}x\frac{\alpha(x)}{\beta}n(x)-\frac{1}{\beta}logZ_{v}(\beta;\alpha]\;,$
(4.4.27)

so that the maximization of $S_{v}(E;n]$ (4.4.8) in the vicinity of
equilibrium, is equivalent to the minimization of $\Omega_{v}(\beta;n(x)]$
(4.4.27) around the same equilibrium point

$\frac{\delta}{\delta
n(x^{\prime})}\Bigg{[}\Omega_{v}(\beta;n]+\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$ (4.4.28)

After we find $\Omega_{v}$, we just need to note that
$\alpha(x)=-\beta\frac{\delta\Omega_{v}}{\delta n(x)}$ and substitute in
equation (4.4.24) to find the ”core integro-differential equation of DFT”[18]
as

$\nabla\Big{(}\frac{\delta\Omega(\beta;n]}{\delta
n(x)}\Big{)}_{n_{eq}(x)}=0\;,$ (4.4.29)

which implies that

$\Omega_{v;eq}\leq\Omega_{v}(\beta;n]\;,$ (4.4.30)

where

$\Omega_{v,eq}(\beta;n]=-\frac{\alpha_{eq}}{\beta}\int
d^{3}xn(x)-\frac{1}{\beta}logZ_{v}^{*}(\beta,\alpha_{eq}).$ (4.4.31)

Combine equations (4.4.27), (4.4.18), and (4.4.8) to find that

$\Omega_{v}(\beta;n]=\int
d^{3}xv(x)n(x)+\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}(E;n]}{\beta}\;.$
(4.4.32)

We are now ready to state the DFT theorem, let us define the so called
intrinsic density functional potential $F_{v}$ as

$F_{v}(\beta;n]\equiv\langle\hat{K}+\hat{U}\rangle-\frac{S_{v}}{\beta}\;,$
(4.4.33)

to have

$\Omega_{v}(\beta;n]=\int v(x)n(x)+F_{v}(\beta;n]\;.$ (4.4.34)

######  Theorem 4.4.10 (Density Functional).

The intrinsic functional potential $F_{v}$ is a functional of density $n(x)$
and is independent of the external potential $v(x)$:

$\frac{\delta F_{v}(\beta;n]}{\delta v(x^{\prime})}=0\quad\text{for fixed
$\beta$ and $n(x)$.}$ (4.4.35)

######  Proof 4.4.11.

The crucial observation behind the DFT formalism is that $\hat{\rho}_{n}$ and
$Z_{v}$ depend on the external potential $v(x)$ and the Lagrange multipliers
$\alpha(x)$ only through the particular combination $\bar{\alpha}(x)$ defined
in (4.4.10). Substitute (4.4.8), and (4.4.10) in equation (4.4.33) to get

$\beta F_{v}(\beta;n]=logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x),$ (4.4.36)

where
$Z(\beta;\bar{\alpha}]=Z_{v}(\beta;\alpha]=Tr_{c}e^{-\beta(\hat{K}+\hat{U})-\int
d^{3}x\bar{\alpha}(x)\hat{n}_{x}}\;.$ The functional derivative of $\beta
F_{v}$ at fixed $n(x)$ and $\beta$ is

$\frac{\delta(\beta F_{v}(\beta;n])}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\int
d^{3}x^{\prime\prime}\frac{\delta}{\delta\bar{\alpha}(x^{\prime\prime})}\Big{[}logZ(\beta;\bar{\alpha}]+\int
d^{3}x\bar{\alpha}(x)n(x)\Big{]}\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}\;.$ (4.4.37)

Since $n(x^{\prime})=-\frac{\delta
logZ(\beta;\bar{\alpha}]}{\delta\bar{\alpha}(x^{\prime})}$, keeping $n(x)$
fixed is achieved by keeping $\bar{\alpha}(x)$ fixed:

$\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,n(x)}=\frac{\delta\bar{\alpha}(x^{\prime\prime})}{\delta
v(x^{\prime})}\Big{|}_{\beta,\bar{\alpha}(x)}=0\;,$ (4.4.38)

so that

$\frac{\delta F_{v}(\beta;n]}{\delta v(x^{\prime})}\Big{|}_{\beta,n(x)}=0\;,$
(4.4.39)

which concludes the proof; thus, we can write down the intrinsic potential as

$F(\beta;n]=F_{v}(\beta;n]\;.$ (4.4.40)

######  Remark 4.4.12.

Note that a change of the external potential $v(x)$ can be compensated by a
suitable change of the multiplier $\alpha(x)$ in such a way as to keep
$\bar{\alpha}(x)$ fixed, such changes in $v(x)$ will have no effect on $n(x)$.
Therefore keeping $n(x)$ fixed on the left hand side of (4.4.36) means that
$\bar{\alpha}(x)$ on the right side is fixed too.

Now we can substitute equation (4.4.34) in (4.4.28), and define the chemical
potential

$\mu\equiv\frac{-\alpha_{eq}}{\beta}\;,$ (4.4.41)

to have

$\frac{\delta}{\delta n(x^{\prime})}\Bigg{[}\int
d^{3}xv(x)n(x)+F(\beta;n]-\mu\int d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$
(4.4.42)

We can also substitute (4.4.34) in (4.4.29) to find

$v(x)+\frac{\delta F}{\delta n(x)}\Big{|}_{eq}=\mu,$ (4.4.43)

which allows us to interpret $\mu_{in}(x;n]\equiv\frac{\delta F}{\delta n(x)}$
as intrinsic chemical potential of the system.

###  4.5 Approximation Schemes of DFT

In section 4.4 we proved that there exists a density functional
$\Omega_{v}(\beta,n]$ defined as

$\Omega_{v}(\beta;n]=\int d^{3}xv(x)n(x)+F(\beta;n]$ (4.5.1)

which assumes its minimum at equilibrium density, exact calculation of
$F(\beta;n]$ requires calculation of the canonical partition function of the
system, this is what an approximation scheme of statistical mechanics must
avoid to begin with, therefore all different DFT models are different
approximations of $F(\beta;n]$; in this section, as an illustration of the
method, we reformulate some well-known approximations.

####  4.5.1 Free Energy Approximation for Almost Constant Density

Similar to [16] and [17], consider an electron gas 444 Note that our approach
is different from [16, 17], in the sense that the number of particles is
fixed, but our argument follows the similar pathway.  with mean density
$n_{0}$, imposed to a small electric field produced by an external positive
charge distribution $n_{ext}(x)$

$n_{ext}(x)=\lambda\int\frac{d^{3}q}{(2\pi)^{3}}a(q)e^{-ix.q}\;,$ (4.5.2)

for an almost uniform density, $n_{0}$ with a small deformation $\tilde{n}(x)$
write the electron density to the first order in $\lambda$

$n(x)=n_{0}+\tilde{n}(x)=n_{0}+\lambda\int\frac{d^{3}q}{(2\pi)^{3}}b_{1}(q)e^{-ix.q}\;.$
(4.5.3)

Let us write down all the relevant perturbations in the same form as functions
of $\lambda$:

$\displaystyle n_{ext}(x)=$ $\displaystyle 0+\lambda n_{ext}^{(1)}(x)+\dots$
(4.5.4) $\displaystyle n(x)=$ $\displaystyle n^{(0)}+\lambda n^{(1)}(x)+\dots$
(4.5.5) $\displaystyle\alpha(x)=$ $\displaystyle
0+\lambda\alpha^{(1)}(x)+\dots$ (4.5.6) $\displaystyle\Omega_{v}=$
$\displaystyle\Omega_{v}^{(0)}+\lambda\Omega_{v}^{(1)}+\lambda^{2}\Omega_{v}^{(2)}+\dots$
(4.5.7)

Note that we discussed in remark 4.4.9 that shifting $\alpha(x)$ by a constant
doesn’t change the physics, therefore we have chosen $\alpha^{(0)}=0$.
Compare (4.5.2) and (4.5.4) to have

$n_{ext}^{(1)}(x)=\int\frac{d^{3}q}{(2\pi)^{3}}a(q)e^{-ix.q}\;.$ (4.5.8)

and compare (4.5.3) and (4.5.5) to find that

$\displaystyle n^{(0)}=$ $\displaystyle n_{0}$ (4.5.9) $\displaystyle
n^{(1)}(x)=$ $\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}b_{1}(q)e^{-ix.q}\;.$

To calculate $\Omega_{v}(\beta;n]$ Combine (4.5.5) and (4.5.6) with equation
(4.4.27) to find

$\Omega_{v}(\beta;n]\approx-\int
d^{3}x\frac{\lambda\alpha^{(1)}(x)}{\beta}\Bigg{[}n_{0}+\lambda
n^{(1)}(x)\Bigg{]}-\frac{1}{\beta}logZ_{v}(\beta;\alpha]\;.$ (4.5.10)

In order to expand $Z_{v}(\beta;\alpha]$ in terms of $\lambda$, write down the
Taylor expansion of the functional around $\alpha(x)=0$

$\displaystyle logZ_{v}(\beta;\alpha]=$ $\displaystyle
logZ_{v}(\beta,0]\Big{|}_{v(x)=0}+\int d^{3}x\frac{\delta
logZ_{v}}{\delta\alpha(x)}\Bigg{|}_{v(x)=0}\delta\alpha(x)$ (4.5.11)
$\displaystyle+\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\frac{\delta^{2}logZ_{v}}{\delta\alpha(x)\delta\alpha(x^{\prime})}\Bigg{|}_{v(x)=0}\delta\alpha(x)\delta\alpha(x^{\prime})\;.$

Since we have set $n(x)=n_{0}$ in the absence of external potential, taking
the functional derivative of $Z_{v}$ given in (4.4.6) yields

$\displaystyle\frac{\delta logZ_{v}}{\delta{\alpha(x)}}\Big{|}_{v(x)}$
$\displaystyle=-n_{0}$ (4.5.12)
$\displaystyle\frac{\delta^{2}logZ_{v}}{\delta\alpha(x)\delta\alpha(x^{\prime})}\Big{|}_{v(x)=0}$
$\displaystyle=n_{0}^{(2)}(x,x^{\prime})=n_{0}^{(2)}(|x-x^{\prime}|)\;,$
(4.5.13)

where $n_{0}^{(2)}(|x-x^{\prime}|)$ is the density correlation function of
uniform electron gas at temperature $\beta$ and mean density $n_{0}$. the
zeroth order term

$Z_{v}(\beta,0]\Big{|}_{v(x)=0}=Tre^{-\beta\hat{H}^{(}0)}=Z_{N}(\beta)$
(4.5.14)

is the canonical partition function of a uniform electron gas in absence of
external potential.
Equation (4.5.6) tells that

$\delta\alpha(x)\equiv\lambda\alpha^{(1)}(x)$ (4.5.15)

Substitute (4.5.12), (4.5.13), and (4.5.14), in equation (4.5.11) to find that

$\displaystyle logZ_{v}(\beta;\alpha]\approx$ $\displaystyle
logZ_{N}(\beta)-\lambda\int d^{3}xn_{0}\alpha^{(1)}(x)$ (4.5.16)
$\displaystyle+\frac{\lambda^{2}}{2}\int
d^{3}xd^{3}x^{\prime}n_{0}^{(2)}(|x-x^{\prime}|)\alpha^{(1)}(x)\alpha^{(1)}(x^{\prime})+\dots\;.$

Substitute (4.5.16) in (4.5.10) to get

$\displaystyle\Omega_{v}(\beta;n]\approx$
$\displaystyle\frac{1}{\beta}logZ_{N}(\beta)$ (4.5.17)
$\displaystyle+\lambda^{2}\Bigg{[}-\frac{1}{\beta}\int
d^{3}x\alpha^{(1)}(x)n^{(1)}(x)\Bigg{]}+\lambda^{2}\Bigg{[}\frac{1}{2\beta}\int
d^{3}x^{\prime}d^{3}x^{\prime}\alpha^{(1)}(x)\alpha^{(1)}(x^{\prime})n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}\;.$

Now that we are done with perturbative calculations, we can take back
$\lambda\rightarrow 1$ and consequently use $\alpha^{(1)}(x)=\alpha(x)$ and
$n^{(1)}(x)\approx n(x)-n_{0}$, to have

$\displaystyle\Omega_{v}(\beta;n]\approx$
$\displaystyle\frac{1}{\beta}logZ_{N}(\beta)$ (4.5.18)
$\displaystyle+\Bigg{[}-\frac{1}{\beta}\int
d^{3}x\alpha(x)[n(x)-n_{0}]\Bigg{]}+\Bigg{[}\frac{1}{2\beta}\int
d^{3}xd^{3}x^{\prime}\alpha(x)\alpha(x^{\prime})n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}\;.$

The last approximation is to replace the local chemical potential
$\alpha(x)=\alpha(x;n]$ with the chemical potential of a uniform electron gas
with density $n(x)$

$\alpha(x;n]\approx-\beta\mu(n(x))\;,$ (4.5.19)

so that

$\displaystyle\Omega_{v}(\beta;n]\approx$
$\displaystyle\frac{1}{\beta}logZ_{N}(\beta)$ (4.5.20)
$\displaystyle+\Bigg{[}\int
d^{3}x\mu(n(x))[n(x)-n_{0}]\Bigg{]}+\Bigg{[}\frac{\beta}{2}\int
d^{3}xd^{3}x^{\prime}\mu(n(x))\mu(n(x^{\prime}))n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}\;.$

And we can substitute (4.5.20) in (4.4.28) and define the chemical potential
$\alpha_{eq}\equiv-\beta\mu$ to find an approximation for the DFT variational
principle which is written solely in terms of thermodynamic properties of a
uniform electron gas

$\displaystyle\frac{\delta}{\delta n(x^{\prime\prime})}\Bigg{[}$
$\displaystyle\int d^{3}x\mu(n(x))[n(x)-n_{0}]$ (4.5.21)
$\displaystyle+\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\mu(n(x))\mu(n(x^{\prime}))n_{0}^{(2)}(|x-x^{\prime}|)-\mu\int
d^{3}xn(x)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$

Recall that

$v(x)+\mu_{in}(x)=\mu$ (4.5.22)

and once again replace $\mu_{in}(x)$ with $\mu(n(x))$ to find

$\displaystyle\frac{\delta}{\delta n(x^{\prime\prime})}\Bigg{[}$
$\displaystyle\int d^{3}xv(x)n(x)+\int d^{3}x\mu(n(x))n_{0}$ (4.5.23)
$\displaystyle-\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\mu(n(x))\mu(n(x^{\prime}))n_{0}^{(2)}(|x-x^{\prime}|)\Bigg{]}_{n(x)=n_{eq}(x)}=0\;.$

This means that we reduced analysis of an inhomogeneous gas to that of a
uniform gas for the case of almost constant density.

####  4.5.2 Kohn-Sham Model

Another approach to approximate $\Omega_{v}$ is the Kohn-Sham Model for slowly
varying potential, the condition of slowly varying potential is of course one
degree loser than the almost constant density, in a sense that although the
density does not change dramatically from one volume element to the next, but
the differences may accumulate to make a considerable difference of density
between far away coordinates.
The novelty of the Kohn-Sham model is that instead of calculating approximate
value of $\Omega_{v}$, they showed that for a slowly varying density, the DFT
variational principle gives an equilibrium density equal to that of a non-
interacting electron gas in the presence of an effective potential. This means
that the Kohn-Sham model simplifies an interacting many-body Schrödinger
equation, to that of a single particle in the presence of an effective
potential.
Let us see how the Kohn-Sham model fits in our framework. First split
$F(\beta;n]$ into the Coulomb interaction and the rest: regarding the long
range nature of the Coulomb interaction we can approximate it with the
classical one:

$F(\beta;n]=\int
d^{3}xd^{3}x^{\prime}\frac{n(x)n(x^{\prime})}{|x-x^{\prime}|}+G(\beta;n]\;,$
(4.5.24)

where $G(\beta;n]$ is independent of $v(x)$ just like $F(\beta;n]$. Compare
equation (4.4.33) with (4.5.24) to find

$G(\beta;n]=\langle\hat{K}\rangle-S_{v}(E;n]\;.$ (4.5.25)

Again, split $G(\beta;n]$ into that of a non-interacting electron gas and the
rest

$G(\beta;n]=G_{s}(\beta;n]+F_{xc}(\beta;n]\;,$ (4.5.26)

where

$G_{s}(\beta;n]=\langle\hat{K}\rangle_{0}-S_{0,v}(E;n]\;,$ (4.5.27)

and the non-interacting trial kinetic energy $\langle\hat{K}\rangle_{0}$, and
non-interacting trial entropy $S_{0,v}$ are found by

$\displaystyle\langle.\rangle_{0}=Tr\hat{\rho}_{0,n}(.)$ (4.5.28)
$\displaystyle\hat{\rho}_{0,n}=\frac{e^{-\beta\hat{H}_{0}-\int\alpha(x)\hat{n}(x)}}{Tre^{-\beta\hat{H}_{0}-\int\alpha(x)\hat{n}(x)}}$
(4.5.29) $\displaystyle\hat{H}_{0}=\hat{K}+\hat{V}$ (4.5.30) $\displaystyle
S_{0,v}=-Tr\hat{\rho}_{0,n}log\hat{\rho}_{0,n}$ (4.5.31)

where $\hat{\rho}_{0,n}$ is the non-interacting trial density matrix and
$\hat{H}_{0}$ the non-interacting Hamiltonian operator. Substitute equations
(4–4–25) and (4–4–23) in (4–4–1) to have an almost exact approximation of the
functional $\Omega_{v}$

$\Omega_{v}(\beta;n]=\int d^{3}xv(x)n(x)+\frac{1}{2}\int
d^{3}xd^{3}x^{\prime}\frac{n(x)n(x^{\prime})}{|x-x^{\prime}|}+F_{xc}(\beta;n]\;.$
(4.5.32)

The expression (4.5.32) is almost exact and not exact in a sense that the
Coulomb potential is approximated by its classical counterpart, otherwise the
expression $F_{xc}(\beta;n]$ is exactly the difference between the intrinsic
free energy $F(\beta;n]$ and the intrinsic free energy of a non-interacting
electron gas with the same density $F_{0}(\beta;n]$

$F_{xc}(\beta;n]=F(\beta;n]-F_{0}(\beta;n]\;.$ (4.5.33)

The non-interacting intrinsic free energy is

$F_{0}(\beta;n]=\int d^{3}xf_{0}(n(x))n(x)\;,$ (4.5.34)

where $f_{0}(n(x))$ is the intrinsic free energy per particle for a uniform
non-interacting electron gas with density $n=n(x)$. Although, it is not
guaranteed that one can write a $F(\beta;n]$ in such a form as well, indeed
one can not, because exchange correlation energy is non-local by definition,
but it is a reasonable approximation for a thermal system to assume that
entanglement of particles only appear in short distances, so that we can
approximate the total exchange correlation energy with sum of that of volume
elements

$F_{xc}(\beta;n]=\int d^{3}xf_{xc}(n(x))n(x)\;,$ (4.5.35)

where $f_{xc}(n(x))$ is the exchange correlation free energy per particle for
a uniform electron gas with density $n=n(x)$.
Combine equations (4.5.35), (4.5.32), (4.5.26), and (4.4.42) to get

$0=\phi(x)+\frac{\delta G_{s}(\beta;n]}{\delta n(x)}+\mu_{xc}(n(x))-\mu\;,$
(4.5.36)

where

$\phi(x)\equiv v(x)+\int
d^{3}x^{\prime}\frac{n(x^{\prime})}{|x-x^{\prime}|}\;,$ (4.5.37)

and

$\mu_{xc}(n(x))=\frac{d[n(x)f_{xc}(x)]}{d[n(x)]}\;.$ (4.5.38)

Equation (4.5.36) is identical to equation (3.6) of reference [23], it is
”identical to the corresponding equation for a system of non-interacting
electrons in the effective potential $\phi(x)+\mu_{xc}(x)$. Its solution is
therefore determined by the following system of equations:

$\big{[}-\frac{1}{2}\nabla^{2}+\phi(x)+\mu_{xc}(n(x))\big{]}\psi_{i}=\epsilon_{i}\psi_{i}\;,$
(4.5.39)
$n(r)=\sum_{i=1}^{N}\frac{|\psi_{i}(x)|^{2}}{e^{\beta(\epsilon_{i}-\mu)}+1}\;.$
(4.5.40)

###  4.6 Final Remarks

In this chapter, we reformulated the quantum DFT from the MaxEnt point of
view. We showed that the variational principle of DFT is a special case of
quantum entropic inference, where the method is incorporated to treat an
inhomogeneous electron gas. we clarified that the practical advantage of DFT
is that it enables one to analyze an inhomogeneous electron gas by just
knowing characteristics of the homogeneous gas. As a matter of fact we did not
even mention the second quantization in the theoretical framework of DFT. The
use of the grand canonical formalism for approximation of $F_{xc}$ (or
equivalently of $\mu_{xc}$) is intuitively justifiable.
There are three different approximations in the theory which finally lead to
the famous Kohn-Sham equation (4.5.39): i) the Coulomb interaction is treated
classically ii) the volume elements of the system are assumed to contribute
additively to the exchange correlation part of the intrinsic free energy iii)
the expected number of particles in a volume element is negligible compared to
the total number of particles.

##  Chapter 5 Conclusion

In this research we reconstructed the concept of the density functional theory
within the framework of entropic inference. We mainly explored the theoretical
aspects of the theory, starting with a non-personalistic and objective
Bayesian interpretation of the probability theory and Caticha’s theory of
entropic inference, we showed that the variational principle of DFT is
equivalent to the maximum entropy principle. Furthermore, we showed that this
can be formulated within a canonical ensemble of fixed number of particles.
Along our way, we showed that both the classical and the quantum maximum
entropy principles can be recognized as contact structures which are invariant
under Legendre transformations. Apart from the possible practical advantages
of our approach, not only we integrated the density functional formalism as an
straightforward implementation of MaxEnt which can inspire new equilibrium
theories to emerge from the framework, but also we reopened a line of research
which is primarily concerned with the theoretical aspects of the DFT. The main
achievement of this research is that DFT has been derived according to the
same principles of entropic inference that are the foundations on which
statistical mechanics and quantum mechanics are built.

## References

  * [1] Caticha, A. Entropic physics, probability, entropy, and The foundations of physics. Available online accessed on 12 April 2021: https://www.albany.edu/physics/faculty/ariel-caticha.
  * [2] Shannon, C.E. A mathematical theory of communication The Bell System Technical Journal 1948 27, 3.
  * [3] Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev., 1957, 106, 620.
  * [4] Jaynes, E.T. Information theory and statistical mechanics II. Phys. Rev. 1957, 108, 171.
  * [5] Shore, J., Johnson, R. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Transactions on Information Theory, 1980, 26.1, 26-37.
  * [6] Skilling, J. The Axioms of Maximum Entropy. Maximum-Entropy and Bayesian Methods in Science and Engineering, 1988, 173–187.
  * [7] Caticha, A. Relative Entropy and Inductive Inference. AIP Conf. Proc. 2004, 707, 75.
  * [8] Caticha, A.; Giffin, A. Updating Probabilities. AIP Conf. Proc. 2006, 872, 31.
  * [9] Caticha, A. Information and Entropy. AIP Conf. Proc., 2007, 954, 11.
  * [10] Caticha, A. Towards an Informational Pragmatic Realism. Minds and Machines, 2014, 24, 37.
  * [11] Caticha, A. Entropy, Information, and the Updating of Probabilities. Entropy, 2021, 23, 895.
  * [12] Caticha, A. The Entropic Dynamics Approach to Quantum Mechanics. Entropy, 2019, 21, 943.
  * [13] Caticha, A. Quantum Mechanics as Hamilton-Killing Flows on a Statistical Manifold arxiv.org, 2021, avilable at https://arxiv.org/abs/2107.08502.
  * [14] Vanslette, K., Entropic Updating of Probabilities and Density Matrices.  Entropy, 2017, 19-2,664; and at https://arxiv.org/abs/1710.09373.
  * [15] Umegaki, H., Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math Sem. Rep., 1962 14-2, 59-85.
  * [16] Kohn, W., Hohenberg, P., Inhomogeneous electron gas. Phys. Rev., 1964, 136, B864.
  * [17] Mermin, D.,Thermal properties of inhomogeneous electron gas. Phys. Rev. 1965, 137, A1441.
  * [18] Evans, R.,The nature of the liquid-vapor interface and other topic in statistical mechanics of non uniform, classical fluids. Advances in Physics, 1979, 28-2, 143-200.
  * [19] Dirac, P.A.M. Quantum mechanics of many-electron systems. Proc. R. Soc. Lond. 1929, A123, 714-733.
  * [20] Zangwill, A. A half century of density functional theory. Physics Today, 2015, 68, 7.
  * [21] Kardar, M. Statistical physics of fields. Cambridge University Press, 2007.
  * [22] Ebner, C., Saam, W.F., Stroud, D., Density-functional theory of simple classical fluids. I. surfaces. Phys. Rev. A, 1976 14, 2264.
  * [23] Kohn, W., Sham, L.J., Self-consistent equations including exchange and correlation effects. Phys. Rev., 1965, 140, A1133.
  * [24] Jones, R.O., Density functional theory: Its origins, rise to prominence, and future. Rev. Mod. Phys., 2015, 87, 897.
  * [25] Evans, R., Oettel, M., Roth, R., Kahl, G., New developments in classical density functional theory. J. Phys.: Condens. Matter, 2016, 28, 240401.
  * [26] van der Waals, J.D.,Z. Phys. Chem., 1894, 13, 657.
Rowlinson, J.S., The thermodynamik theory of capillarity under the hypothesis
of a continuous variation of density. 1979 J. Stat. Phys., 1979, 20, 197.
(translation)

  * [27] Jaynes, E.T., Where do we stand on maximum entropy?. The Maximum Entropy Principle ed. by R. D. Levine and M. Tribus (MITPress); 1979; and at http://bayes.wustl.edu.
  * [28] Jaynes, E.T. Predictive statistical mechanics. in Frontiers of Nonequilibrium Statistical Physics, G.T. Moore and M.O. Scully (eds.)(Plenum Press, New York), 1986; and at http://bayes.wustl.edu.
  * [29] Rajeev, S.G., A Hamilton-Jacobi formalism for thermodynamics. Annals of Physics, 323-9, 2265-2285; and at https://arxiv.org/pdf/0711.4319.pdf.
  * [30] Balian, R., Valentin, P., Hamiltonian structure of thermodynamics with gauge. Eur. Phys. J. B 2001 21, 269–282; and at https://arxiv.org/pdf/cond-mat/0007292.pdf.
  * [31] Feynman, R.P., Statistical mechanics: a set of lectures. Reading, Mass. : W. A. Benjamin, 1972.
  * [32] Caticha, A.,Tseng, C., Using relative entropy to find optimal approximations: an application to simple fluids. Physica A 2008, 387, 6759; and at http://arxiv.org/abs/0808.4160v1.
  * [33] Toxvaerd, S., Perturbation theory for nonuniform fluids: surface tension, J. Chem. Phys., 1971 55, 3116.
  * [34] Singh, Y., and Abraham, F. F., Structure and thermodynamics of the liquid–vapor interface. J. Chem. Phys., 1977 67,537.
  * [35] Hansen, J-P., McDonald, I.R., Theory of simple liquids. Academic Press, 1976.
  * [36] Marconi U.M.B., Tarazona, P., Dynamic density functional theory of fluids. J. Chem. Phys. 1999, 110, 8032.
  * [37] Jaynes, E.T., The minimum entropy production principle. Ann. Rev. Phys. Chem., 1980 31,579-601.
  * [38] Tarazona, P., Free-energy density functional for hard spheres. Phys. Rev. A, 1985, 31,2672.
  * [39] Rosenfeld, Y., Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing. Phys. Rev. Lett., 1989, 63,980.
  * [40] Kohn, W., Sherrill, CD. Editorial: Reflections on fifty years of density functional theory. The Journal of Chemical Physics, 2014, 140, 18A201.
  * [41] Wilcox, R. M., Exponential Operators and Parameter Differentiation in Quantum Physics. Journal of Mathematical Physics, 1967, 8, 962.
  * [42] Yousefi, A., Caticha, A., An entropic approach to classical density functional theory, submitted to MaxEnt 2021 Proceedings; and at https://arxiv.org/abs/2108.01594.
  * [43] Vanslette, K., The inferential design of entropy and its application to quantum measurements State University of New York at Albany. ProQuest Dissertations Publishing, 2018. 10791592. and at https://arxiv.org/abs/1804.09142.
  * [44] Nielsen, M.A., Chuang. I.L., Quantum computation and quantum information, 10th anniversary edition. Cambridge University Press 2011.