#  Numerical fluid dynamics for FRG flow equations:
Zero-dimensional QFTs as numerical test cases.
II. Entropy production and irreversibility of RG flows

Adrian Koenigstein<EMAIL_ADDRESS>Institut für
Theoretische Physik, Goethe University,
Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany  Martin J. Steil
<EMAIL_ADDRESS>Technische Universität Darmstadt,
Department of Physics, Institut für Kernphysik, Theoriezentrum,
Schlossgartenstraße 2, D-64289 Darmstadt, Germany  Nicolas Wink
<EMAIL_ADDRESS>Technische Universität Darmstadt, Department of
Physics, Institut für Kernphysik, Theoriezentrum,
Schlossgartenstraße 2, D-64289 Darmstadt, Germany  Institut für Theoretische
Physik, University Heidelberg,
Philosophenweg 16, D-69120 Heidelberg, Germany  Eduardo Grossi
<EMAIL_ADDRESS>Center for Nuclear Theory, Department of Physics
and Astronomy,
Stony Brook University, Stony Brook, NY 11794, U.S.A.  Jens Braun
<EMAIL_ADDRESS>Technische Universität Darmstadt, Department
of Physics, Institut für Kernphysik, Theoriezentrum,
Schlossgartenstraße 2, D-64289 Darmstadt, Germany  Helmholtz Research Academy
Hesse for FAIR, Campus Darmstadt,
D-64289 Darmstadt, Germany  ExtreMe Matter Institute EMMI, GSI,
Planckstraße 1, D-64291 Darmstadt, Germany

###### Abstract

We demonstrate that the reformulation of renormalization group (RG) flow
equations as non-linear heat equations has severe implications on the
understanding of RG flows in general. We demonstrate by explicitly
constructing an entropy function for a zero-dimensional
$\mathbb{Z}_{2}$-symmetric model that the dissipative character of generic
non-linear diffusion equations is also hard-coded in the functional RG
equation. This renders RG flows manifestly irreversible, revealing the semi-
group property of RG transformations on the level of the flow equation itself.
Additionally, we argue that the dissipative character of RG flows, its
irreversibility and the entropy production during the RG flow may be linked to
the existence of a so-called $\mathcal{C}$-/$\mathcal{A}$-function. In total,
this introduces an asymmetry in the so-called RG time – in complete analogy to
the thermodynamic arrow of time – and allows for an interpretation of infrared
actions as equilibrium solutions of dissipative RG flows equations. The
impossibility of resolving microphysics from macrophysics is evident in this
framework.

Furthermore, we directly link the irreversibility and the entropy production
in RG flows to an explicit numerical entropy production, which is manifest in
diffusive and non-linear partial differential equations (PDEs) and a standard
mathematical tool for the analysis of PDEs. Using exactly solvable zero-
dimensional $\mathbb{Z}_{2}$-symmetric models, we explicitly compute the
(numerical) entropy production related to the total variation non-increasing
property of the PDE during RG flows toward the infrared limit.

Finally, we discuss generalizations of our findings and relations to the
$\mathcal{C}$-/$\mathcal{A}$-theorem as well as how our work may help to
construct truncations of RG flow equations in the future, including
numerically stable schemes for solving the corresponding PDEs.

entropy, numerical entropy, total variation diminishing, Functional
Renormalization Group, conservation laws, numerical fluid dynamics,
irreversibility, $\mathbb{Z}_{2}$ model, zero-dimensional QFT,
$\mathcal{C}$-theorem

##  I Introduction

Our modern understanding of Quantum Field Theories (QTFs) and particularly
phase transitions is built upon the analysis of renormalization group (RG)
trajectories. In fact, RG theory facilitates our understanding by connecting
microscopic and macroscopic physics in a continuous manner. This is often
visualized at the example of block spin transformations [1, 2] which provides
an intuitive picture of so-called RG flows in position space. A modern,
functional approach to RG theory is provided by the Functional Renormalization
Group (FRG). It allows for non-perturbative studies of QTFs with applications
ranging from biophysics over condensed matter to high-energy physics and
quantum gravity, see Ref. [3] for a recent overview.

In Refs. [4, 5, 6, 7] it is shown that renormalization group flows can be seen
as flows in the literal sense. The RG time
$t=-\ln\big{(}\tfrac{k}{\Lambda}\big{)}$, where $k$ is the RG scale in units
of energy and $\Lambda$ is some ultraviolet (UV) reference scale, can be
identified with an abstract time and directions in field space correspond to
spatial directions111A specific example is the RG flow of (the field-space
derivative of) a local potential, which can involve advective and diffusive
contributions as well as source/sink terms [4, 5, 8, 9, 10, 6, 11, 7, 12,
13]., cf. Refs. [14, 15, 16, 17, 18] for similar identifications in related
flow equations. With this at hand, it becomes appealing to look for further
connections between the research fields of (numerical) fluid dynamics and RG
theory.

Such a connection is discussed in this paper. To be specific, we shall show
that the numerical entropy, which is of utmost importance in the theoretical
treatment of partial differential equations (PDEs), see, e.g., the textbooks
[19, 20, 21, 22, 23, 24, 25], has a very close connection to an entropy in the
RG flow222In this context we also have to mention the subsequent publication
[26] by J. Cotler and S. Rezchikov who were able to interpret the Polchinski
equation as an “optimal transport gradient flow of a field-theoretic relative
entropy” thus establishing a firm and explicit connection between an
information-theoretic entropy and (F)RG flows. and further possible
connections to the so-called $\mathcal{C}$-/$\mathcal{A}$-functions for RG
theories, cf. Refs. [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41] and Sec. V for more details on $\mathcal{C}$-/$\mathcal{A}$-functions.

One of the most important direct consequences of this is that the same
“(thermodynamic) arrow of time” or “thermodynamic time asymmetry” [42]
identified by the entropy of a PDE, is also present from an RG perspective.

In nature as well as in the PDEs that describe our physical world, entropy is
produced by diffusion (dissipation) as well as discontinuities of all kind.
Consequently the evolution of such systems and also their numerical solutions
are irreversible and usually only weak solutions are accessible numerically
[19, 20, 21, 22, 23, 24, 25]. As we will demonstrate in this paper, the total
variation non-increasing property and related numerical entropy, used to
guarantee the stability of numeric solution schemes, can be promoted to a
“physical” entropy function sharing some characteristics with a
$\mathcal{C}$-function and its properties transfer from the PDE to the QFT and
vice versa. Therefore, RG flows are also not reversible.333Note that similar
arguments, which link the dissipative character of RG flow equations to the
irreversibility of the RG flow, were already brought up in Refs. [16] and [27]
already before or parallel to the development of the functional RG framework
pioneered in Ref. [43]. This makes the semi-group character of the RG, see,
e.g., Ref. [44], explicit. This semi-group character also becomes manifest in
Kadanoffs block-spin picture [1, 45, 46, 2]. The irreversibility of RG flows
is not just an abstract concept but presents on a practical level in rather
simple truncations of the Functional Renormalization Group (FRG) equation.

These statements may have no severe practical implications for studies of,
e.g., QCD and condensed-matter systems, where the RG flow is in general
followed from small (UV limit) to large length scales (IR limit). In these
cases, the dynamics in the long-range limit is predicted from a given known UV
action by integrating out high momentum modes along the “natural” RG-time
direction. However, in situations where RG flows are followed from large to
small length scales, such as studies of the asymptotic safety scenario in QFTs
(see Refs. [47, 48, 49, 50, 28], Refs. [51, 52] for a recent review in the
context of (quantum) gravity, and Refs. [53, 54] for applications in
condensed-matter physics), the question of irreversibility of RG flows and the
associated production of entropy may indeed be very relevant.

Whereas RG flows are indeed reversible for certain classes of truncations (of
the underlying effective action), we shall demonstrate in the present work
(with the aid of simple models) that it becomes formally impossible to reverse
RG flows in cases where no truncations of the effective action are made. Even
more, already for often employed truncation schemes (e.g., local potential
approximations), we shall see that irreversibility associated with numerical
entropy production can already be a manifest feature of RG flows. Of course,
irreversibility of RG flows does not imply that it is not possible to
construct theories which are valid on all scales. It only implies that the
search for such theories may in general be more complicated. In any case,
generalizations of the arguments presented in our present work may help to
provide a fresh view on these aspects (and/or revive some already existing
arguments [27, 14, 15, 16, 28, 18, 17]).

As we shall discuss below, fixed points still play an important role within
the fluid dynamic interpretation of RG flows. In fact, fixed points can be
identified with steady-flow solutions and/or (thermal) equilibrium situations
on the level of the rescaled dimensionless flow equations, which have
advective and diffusive character.

One major benefit of the connection revealed in this paper is that a measure
for the irreversibility of the RG flow is explicitly provided via the
identification with numerical entropy and especially total variation [21, 22,
25, 55, 56]. Hence, the construction and analysis of such a measure, at least
in certain truncations, might be greatly simplified.444We note that
observations similar to ours have already been pointed out in the works of
Refs. [28, 14, 15, 16] for related (partially linearized) flow equations. In
future, this might also help to single out adequate truncation schemes for RG
flow equations as those truncations, which maintain the irreversible character
of the flow of the full untruncated system.

This paper is organized as follows: In Sec. II, we briefly discuss the
methodological framework of our present study. This includes both the
functional RG approach and its correspondence to fluid dynamics. Moreover, we
introduce the zero-dimensional $O(1)$ model which underlies our numerical
studies. Numerical entropy and the total variation non-increasing property is
then discussed in detail in Sec. III. Explicit computations and a detailed
analysis of numerical entropy production in a variety of test cases are
presented in Sec. IV. In Sec. V, we then give a discussion of the possibility
of a generalization of our present findings with respect to the
irreversibility of RG flows, entropy production and the $\mathcal{C}$-theorem
to higher-dimensional theories. Finally, our conclusions and a brief outlook
can be found in Sec. VI.

##  II Functional RG, fluid dynamics, and the zero-dimensional $O(1)$ model

###  II.1 FRG framework

This section is dedicated to a brief summary of the key aspects of the FRG and
the zero-dimensional $O(1)$ model within this framework. For a comprehensive
discussion, we refer to Part I of our series of publications on numerical
fluid dynamics and FRG flow equation [6] as well as to Refs. [4, 5, 13, 8, 7]
and upcoming publications [12, 11]. For more general reviews on the FRG
method, we refer to Refs. [57, 28, 3, 58, 59, 60, 61, 62].

The FRG framework is built on an exact RG equation [63, 64, 65, 43], which is
a functional partial integro-differential equation for the scale-dependent
effective average action $\bar{\Gamma}_{t}[\Phi]$:

$\displaystyle\partial_{t}\bar{\Gamma}_{t}[\Phi]=\mathrm{STr}\Big{[}\big{(}\tfrac{1}{2}\,\partial_{t}R_{t}\big{)}\,\big{(}\bar{\Gamma}_{t}^{(2)}[\Phi]+R_{t}\big{)}^{-1}\Big{]}\,.$
(1)

The equation holds for arbitrary dimensions and arbitrary field content, which
is summarized in the “super”-field $\Phi$, cf. Refs. [66, 67, 68, 69]. Here,

$\displaystyle t=-\ln\big{(}\tfrac{k}{\Lambda}\big{)}$ (2)

is the RG time (note our sign convention), while $k/\Lambda$ constitutes the
ratio of the RG scale $k$ and the UV cutoff scale $\Lambda$. The latter is the
scale, where the Exact Renormalization Group (ERG) equation is initialized
with the classical action ${\bar{\Gamma}_{t=0}[\Phi]=\mathcal{S}[\Phi]}$. The
“super”-trace stands for a trace in field space, momentum-/position-space, as
well as all internal spaces, e.g., color, flavor etc. and $R_{t}$ denotes an
monotonically decreasing scale-dependent IR regulator function, see, e.g.,
Refs. [28, 70, 71, 57, 72, 73] for details. Solving the ERG equation (1) by
integrating the full set of PDEs that can be generated from the ERG via
suitable projections, from $t=0$ to $t\rightarrow\infty$, thus calculating the
full quantum IR effective action
$\Gamma[\Phi]\equiv\bar{\Gamma}_{t\rightarrow\infty}[\Phi]$, is equivalent to
calculating all $1$PI-$n$-point-correlation (vertex) functions via a partition
function/functional integral [74, 75, 76, 77, 78, 79, 80, 61, 57]. The ERG
equation (1) is the direct mathematical implementation of Wilson’s idea of the
Renormalization Group [45, 46, 81]: Obtaining the macrophysics from the micro-
physics via gradually integrating out momentum-shells from the UV to the IR,
which corresponds to a coarse-graining process in position space, e.g.,
Kadanoff’s block-spin transformations [2, 1, 62]. Earlier formulations of the
RG in terms of similar flow equations can be found in Refs. [82, 17, 81, 83,
14, 15, 16].

###  II.2 The zero-dimensional $O(1)$-model

For what concerns this paper, we study one of the probably most simplistic
QFTs imaginable within this framework – a zero-dimensional $\mathbb{Z}_{2}$
symmetric model or $O(1)$ model555Although being technically imprecise, we
mainly refer to the model as the $O(1)$ model, thus the special case $N=1$ of
an $O(N)$ symmetry, where the symmetry is only realized in terms of discrete
transformations.. Still, as the interested reader will experience by studying
this and the parallel publications of our series [6, 7], as well as Refs. [84,
85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102,
103, 104, 105], this model (and its extension, the zero-dimensional $O(N)$
(vector) model) is non-trivial and can serve as a minimalistic tool to
highlight and test fundamental features of and basic methods for QFTs, not
only for pedagogical purposes, but also as real benchmark scenarios.

For the zero-dimensional $O(1)$ model, the most general “truncation” to solve
the ERG equation (1) is the local potential approximation

$\displaystyle\bar{\Gamma}_{t}[\varphi]=U(t,\varphi)\,,$ (3)

where $U(t,\varphi)$ is simply a function of $t$ and the real scalar (mean)
field $\varphi$ that is called the scale-dependent effective potential. Space-
time or momentum-space dependences of the field or integrations over the
former as well as derivatives of the field do not exist. The entire QFT
consists of a scalar $\phi$ (field), which can assume arbitrary real values
and can be thought of as a self-interacting “particle” in a single point. The
theory is therefore maximally coupled and ultra-local. The only additional
requirement, which we impose on ${\langle\phi\rangle=\varphi}$ and
$U(t,\varphi)$, is that the (mean) field transforms as follows

$\displaystyle\varphi\mapsto\varphi^{\prime}=-\varphi$ (4)

under $\mathbb{Z}_{2}$-transformations and that $U(t,\varphi)$ is in turn
invariant under these transformations

$\displaystyle U(t,\varphi)=U(t,-\varphi)\,.$ (5)

The RG flow from $t=0$ to $t\rightarrow\infty$ is initialized with the
classical action (here the classical potential),

$\displaystyle\bar{\Gamma}_{t=0}(\varphi)=U(t=0,\varphi)=\mathcal{S}(\varphi)=U(\varphi)\,,$
(6)

which also represents an ordinary function of $\varphi$ with
$\mathbb{Z}_{2}$-symmetry. Still, in order to render the corresponding
partition function as well as the expectation values (10) convergent,
$U(\varphi)$ has to be bounded from below and, for
$|\varphi|\rightarrow\infty$, it has to grow at least with $\varphi^{2}$.
Nevertheless, there is no need to claim smoothness or analyticity for
$U(\varphi)$, as discussed in detail in part I and III of this series of
publications [6, 7] and Sec. IV of this work.

The corresponding ERG equation (1) simplifies drastically for the zero-
dimensional $O(1)$ model,666Note that we evaluated $\varphi$ on a “background-
field configuration” $\sigma$, which is actually not needed for the zero-
dimensional $O(1)$ model because the field $\varphi$ is already space-time-
independent. Still, we adopt the conventions used in Ref. [6] and higher-
dimensional scenarios.

$\displaystyle\partial_{t}U(t,\sigma)=\frac{\tfrac{1}{2}\,\partial_{t}r(t)}{r(t)+\partial_{\sigma}^{2}U(t,\sigma)}=\begin{gathered}\includegraphics{diagrams/potential_02.pdf}\end{gathered}\,.$
(8)

However, it retains its fundamental one-loop structure and all its
characteristic properties as a non-linear parabolic PDE and initial value
problem in one “temporal” $t\in[0,\infty)$ and one “spatial”
$\sigma\in(-\infty,\infty)$ direction. Furthermore, it remains an exact
equation without any truncation.

A major difference to higher-dimensional $O(1)$ models (e.g., when using the
LPA-optimized regulator [70, 71]) is the absence of an additional
$t$-dependent factor on the right hand side of the equations, cf. Refs. [65,
43, 106], which, however, does not conceptually spoil any of our further
reasoning. For what follows, we always use a zero-dimensional version of the
monotonically decreasing exponential regulator

$\displaystyle r(t)=\Lambda\,\mathrm{e}^{-t}\,.$ (9)

Here, the UV cutoff must be chosen sufficiently large, cf. Refs. [6, 107,
108]. A peculiar feature of the zero-dimensional version of the ERG equation
(1) is that an integration to $t\rightarrow\infty$ is indeed possible (which
can be seen via reparametrization of the RG time and there is no need for a
numerical IR cutoff [92, 6]). Nevertheless, we will use non-vanishing IR
cutoffs for our numerical calculations in Sec. IV, to be as close as possible
to higher dimensional scenarios.

Having performed the $t$-integration down to the IR limit, we can extract the
vertex functions $\Gamma^{(2n)}$ at the physical point777There is no
(spontaneous) symmetry breaking in zero dimensions [91, 6]. This is a
consequence of a special version of the Coleman-Mermin-Wagner-Hohenberg
theorem [109, 110, 111] or on the level of Eq. (10) simply a consequence of
the discrete $\mathbb{Z}_{2}$-symmetry. $\langle\phi\rangle=\varphi=\sigma=0$
by taking (numerical) derivatives of $U(t=0,\sigma)$ w.r.t. $\sigma$. These
vertex functions are in direct relation to the expectation values, see Refs.
[92, 6],

$\displaystyle\langle\phi^{2n}\rangle=\frac{\int_{-\infty}^{\infty}\mathrm{d}\phi\,\phi^{2n}\,\mathrm{e}^{-\mathcal{S}(\phi)}}{\int_{-\infty}^{\infty}\mathrm{d}\phi\,\mathrm{e}^{-\mathcal{S}(\phi)}}\,,$
$\displaystyle n\in\mathbb{N}_{0}\,,$ (10)

which can be calculated numerically up to machine precision (or sometimes even
be evaluated analytically). (Here, $\phi$ denotes the “fluctuating quantum
field”.) This fact makes zero-dimensional QFTs an interesting test ground
because numerical FRG calculations can be compared against easily attainable
exact results from (numerical) integration of Eq. (10), cf. Refs. [84, 85, 86,
87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104,
105].

###  II.3 Fluid-dynamic reformulation of the RG flows

In this section, we briefly summarize the main findings of our parallel and
upcoming publications [6, 7, 11, 12] and Refs. [4, 5] on the reformulation of
the RG flow equation in terms of a fluid-dynamical conservation law.

Taking a derivative w.r.t. $\sigma$ of the flow equation (8), we obtain a
scalar one-dimensional (here parabolic) conservation law,

$\displaystyle\partial_{t}u(t,x)=\frac{\mathrm{d}}{\mathrm{d}x}\,\bigg{(}\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{1}{r(t)+\partial_{x}u(t,x)}\bigg{)}\,,$
(11)

where $\sigma=x$ is identified with a spatial dimension and
$u(t,x)\equiv\partial_{x}U(t,x)$ is the conserved quantity. In fact, Eq. (11)
is a non-linear diffusion/heat equation,888In part I of this series of
publications [6] we demonstrate that the conservative formulation generalizes
to zero-dimensional $O(N)$ models, which turn out to be a non-linear
advection-diffusion equations, and can even be generalized to higher
dimensional models involving fermions in terms of advection-diffusion-
source/sink equations [4, 9, 8, 13, 10, 5, 6, 7, 11, 12].,999Note that the
similarities between RG flow equations and the heat equation were already
observed before, cf. Refs. [83, 15, 28, 99, 112], but did – to the best of our
knowledge – never result in a comprehensive picture. which can actually be
generalized to arbitrary dimension. This conservative formulation and
interpretation in terms of (numerical) fluid dynamics has tremendous
consequences and benefits for understanding and solving the RG flow equation:

  1. 1.

Conservative formulations of RG flow equations provide direct access to the
highly developed toolbox of numerical fluid dynamics.

  2. 2.

An interpretation of the RG flow equations as flow equations in the narrow
sense of the word makes the dynamics during the flow intuitively
understandable. Advective contributions (pions in the $O(N)$-scenario)
transport the conserved quantity $u(t,x)$ along the field space direction
$\sigma=x$ (bulk motion) and can cause non-analyticities like shocks and
rarefaction waves in field space [4, 5, 113], as is well known for non-linear
hyperbolic conservation laws [114, 115, 19, 20, 21, 22, 23, 24, 25]. The non-
linear diffusive contribution (the radial sigma mode) smears out cusps and
jumps in $u(t,x)$ and corresponds to undirected movement of $u(t,x)$ depending
on the local “concentration differences”, the gradient $\partial_{x}u(t,x)$,
via a highly non-linear diffusion coefficient. On the level of the LPA
approximation, also fermionic contributions to the flow can be easily
understood in this framework in terms of source/sink terms, see ,e.g. Ref.
[12].

The reformulation of the flow equation (8) as a diffusion equation (11) has
direct implications for the goal of the present paper. As outlined in our
parallel discussion in Sec. IV of Ref. [6], diffusion is one specific
dissipative process. Dissipative processes go hand in hand with entropy
production and irreversibility.101010This argument also generalizes to the
$O(N)$ model involving advection and the large-$N$ limit [116]. The
corresponding flow equation in the limit $N\rightarrow\infty$ is a purely
hyperbolic advection equation. Interacting and arising non-analyticities like
shocks and rarefaction waves in non-linear advection equations are sources of
entropy and as such lead so irreversible flows, cf. Refs. [4, 5, 7, 117, 113]
for examples of non-analytical dynamics in RG flows. We can therefore conclude
that the irreversibility of the RG transformations during the RG flow is hard
coded in the diffusive character of the ERG equation (1), not only in zero
space-time dimensions, but for any dimension and any QFT, cf. Refs. [15, 15,
16, 14]. Hence, the rise of entropy during the RG flow might therefore be
directly linked to $\mathcal{C}$-/$\mathcal{A}$-theorems. This is explained in
the next sections in the context of our minimalistic toy model QFT.

##  III (Numerical) entropy and the total variation

The first part of this section deals with the explicit construction of a
(numerical) entropy for the conservation law (11). This entropy has to be a
functional of the conserved quantity $u(t,x)$ and/or its derivatives111111The
purely diffusive character of Eq. (11) is expected to smoothen $u$ during the
RG flow which renders $u(t,x)$ differentiable (but not necessarily analytic)
at least for $0<t<\infty$. This does not need to be the case for hyperbolic
conservation laws where taking derivatives of $u(t,x)$ has to be handled with
great care, e.g., around shocks. that is monotonically rising during the RG
flow. Monotonicity is explicitly proven for valid initial conditions
$U(t=0,x)$. Since $u(t,x)$ is by definition a function of all couplings of the
theory, the (numerical) entropy function might therefore be linked to a zero-
dimensional version of $\mathcal{C}$-/$\mathcal{A}$-function. In fact it might
have some practical advantages compared to some practical approaches toward
$\mathcal{C}$-/$\mathcal{A}$-functions studied in literature, since $u(t,x)$
does not even need to be expandable in explicit couplings at all, but still
contains all degrees of freedom.

In the second part of this section, we derive a discrete formulation of this
entropy functional and demonstrate that it can be directly related to the
total variation (TV) of $u(t,x)$ and the total variation diminishing/non-
increasing property (TVD/TVNI) of commonly used numeric schemes for
conservation laws [56, 19, 55], which is, why we denote it as a “numerical”
entropy.

###  III.1 Construction of the (numerical) entropy

The construction of our (numerical) entropy function is directly inspired by
the construction of entropy/energy functionals for the Bateman-Burgers
equation [118, 119] or the heat-equation [120].

Let $y\in\mathbb{R}$ and

$\displaystyle s:\mathbb{R}\to\mathbb{R}\,,$ $\displaystyle y\mapsto s(y)\,,$
(12)

be a continuously twice differentiable convex function on $\mathbb{R}$, hence

$\displaystyle s(y)\in C^{2}(\mathbb{R})\,,$ $\displaystyle
s^{\prime\prime}(y)\geq 0\,,$ (13)

for all $y\in\mathbb{R}$. Furthermore, we require that $s(y)$ shall not grow
faster than $y^{2}$ for $|y|\rightarrow\infty$, which is explained below.
Using $s(y)$ we define the functional

$\displaystyle S[f(x)]\equiv-\int_{-\infty}^{\infty}\mathrm{d}x\,s(f(x))\,,$
(14)

to which we shall refer as entropy functional. In general, the bounds of
integration are chosen according to the domain of our problem at hand. Next,
we prove that, choosing $f(x)=\partial_{x}u(t,x)$, Eq. (14) indeed plays the
role of an (numerical) entropy for the partial differential equation (11).
Hence, it measures similarly to $\mathcal{C}$-/$\mathcal{A}$-functions for the
RG flows the degrees of freedom and irreversibility. To this end, we
explicitly demonstrate that $S[\partial_{x}u(t,x)]$ is a monotonically
increasing during the RG flow, thus being a monotonic function on
$t\in[0,\infty)$:

$\displaystyle\tfrac{\mathrm{d}}{\mathrm{d}t}\,S[\partial_{x}u(t,x)]\geq 0\,.$
(15)

The only further ingredient, which is needed for the proof is the spatial
derivative of the flow-equation (11):

$\displaystyle\partial_{t}[\partial_{x}u(t,x)]=\vphantom{\bigg{(}\bigg{)}}$
(16) $\displaystyle=$
$\displaystyle-\frac{\mathrm{d}}{\mathrm{d}x}\,\bigg{(}\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{\partial_{x}^{2}u(t,x)}{[r(t)+\partial_{x}u(t,x)]^{2}}\bigg{)}\,.$

Taking spatial derivatives of $u(t,\sigma)$ should be allowed at any
$t\in(0,\infty)$ because of the smoothening character of the diffusion – at
least in zero space-time dimensions.121212The generalization of this argument
to higher-dimensional $O(N)$-type models might be delicate, because the non-
linear diffusion can also cause non-analyticities in potentials in the IR if
these end up in the symmetry broken phase. Here it might be unavoidable to
base and repeat the entire discussion using a rigorous weak/integral
formulation of the PDEs under consideration. Only for $t=0$ the initial
condition may violate smoothness, see part I of this series of publications
[6] for a detailed discussion of this subtle issue. Below, we normalize the
entropy and subtract the entropy of the initial condition
$S[\partial_{x}u(t=0,x)]$, such that this should not spoil any of our
subsequent arguments.

Let us now evaluate Eq. (15):

$\displaystyle\tfrac{\mathrm{d}}{\mathrm{d}t}\,S[\partial_{x}u(t,x)]=\vphantom{\Bigg{(}\Bigg{)}}$
(17) $\displaystyle=\,$
$\displaystyle-\tfrac{\mathrm{d}}{\mathrm{d}t}\,\int_{-\infty}^{\infty}\mathrm{d}x\,s(\partial_{x}u(t,x))=\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle=\,$
$\displaystyle-\int_{-\infty}^{\infty}\mathrm{d}x\,\big{(}\partial_{t}[\partial_{x}u(t,x)]\big{)}\,s^{\prime}(\partial_{x}u(t,x))=\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle=\,$
$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}x\,\bigg{[}\frac{\mathrm{d}}{\mathrm{d}x}\,\bigg{(}\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{\partial_{x}^{2}u(t,x)}{[r(t)+\partial_{x}u(t,x)]^{2}}\bigg{)}\bigg{]}\,s^{\prime}(\partial_{x}u(t,x))=\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle=\,$
$\displaystyle\int_{-\infty}^{\infty}\mathrm{d}x\,\big{[}-\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{[\partial_{x}^{2}u(t,x)]^{2}}{[r(t)+\partial_{x}u(t,x)]^{2}}\,s^{\prime\prime}(\partial_{x}u(t,x))+\bigg{[}\big{[}\tfrac{1}{2}\,\partial_{t}r(t)\big{]}\,\frac{\partial_{x}^{2}u(t,x)}{[r(t)+\partial_{x}u(t,x)]^{2}}\,s^{\prime}(\partial_{x}u(t,x))\bigg{]}_{-\infty}^{\infty}\vphantom{\Bigg{(}\Bigg{)}}$

Next, we analyze both terms in the last line separately.

  1. 1.

We note that all factors in the integrand of the first term are greater or
equal to zero: For the regulator insertion, we have

$\displaystyle-\tfrac{1}{2}\,\partial_{t}r(t)\geq 0\,,$ (18)

because $r(t)$ is a monotonically decreasing function. The numerator and the
denominator are obviously positive. In fact, for the denominator of the
fraction

$\displaystyle r(t)>\partial_{x}u(t,x)\,,$ (19)

for all $t$ anyhow, as long as the initial condition $u(t=0,x)$ and the UV
cutoff $\Lambda$ are chosen accordingly, cf. Ref. [6]. Finally,

$\displaystyle s^{\prime\prime}(\partial_{x}u(t,x))\geq 0\,,$ (20)

holds by construction according to Eq. (13).

In total, we find that the integrand of the first term is always greater or
equal to zero, which directly transfers to the integral itself.

  2. 2.

For the second term, we first use that, for large $|x|$, the potential
$U(t,x)$ and all its derivatives do not change during the RG flow, see also
part I of this series of publications [6]. Furthermore, we use that $s(y)$
maximally grows like $y^{2}$ for $|y|\rightarrow\infty$ by definition. This
implies that its derivative $s^{\prime}(y)$ increases asymptotically as $y$ at
most. Additionally, we use that $U(t,x)$ is at least proportional to $x^{2}$
for $|x|\rightarrow\infty$ in order to have well-defined expectation values
(10). Consequently, we have to distinguish two scenarios. If $U(t,x)\sim
x^{2}$ for large $|x|$, the second term vanishes identically, due to the third
spatial derivative of $U(t,x)$, namely $\partial_{x}^{2}u(t,x)$, in the
numerator. Otherwise, if $U(t,x)$ grows faster than $x^{2}$ for large $|x|$,
the denominator $[r(t)+\partial_{x}u(t,x)]^{2}$ will always grow faster than
the product $[\partial_{x}^{2}u(t,x)]\,s^{\prime}(\partial_{x}u(t,x))$ for
$|x|\rightarrow\infty$. We conclude that the second term always vanishes,
provided that the initial conditions come with the assumed
large-$|x|$-asymptotic behavior.

In total, we have shown the statement of Eq. (15), which promotes $S$ to an
entropy (functional) of our system that can only increase.

For what follows, we choose the twice differentiable convex function
$s(y)=y^{2}$. This implies

$\displaystyle
S[\partial_{x}u(t,x)]=-\int_{-\infty}^{\infty}\mathrm{d}x\,[\partial_{x}u(t,x)]^{2}\,.$
(21)

$S$ can be viewed as measure for the richness of structure of the potential –
the information encoded in the potential – by integrating the square of the
gradient of $u(t,x)$ over all positions $x$ in field space.

With the definition (21) the following practical problem arises: For practical
purposes $S[\partial_{x}u(t,x)]$ formally diverges at any time $t$ because
$\partial_{x}u(t,x)$ is at least constant for $|x|\rightarrow\infty$. This
problem can be cured, by subtracting the entropy $S[\partial_{x}u(t=0,x)]$ of
the initial condition. Since $u(t,x)$ does not change for large $|x|$ during
the entire RG flow, the infinite but constant contributions cancel and we can
observe the relative rise in entropy. This should be a valid approach, since
we are only interested in these relative changes anyhow. We therefore define
and consider the normalized entropy

$\displaystyle\mathcal{C}[\partial_{x}u(t,x)]=S[\partial_{x}u(t,x)]-S[\partial_{x}u(t=0,x)]\,,$
(22)

which is finite. The alphabetic character “$\mathcal{C}$” is chosen because
this function quantifies irreversibility similarly to
$\mathcal{C}$-/$\mathcal{A}$-functions. We are aware of the fact that a real
$\mathcal{C}$-/$\mathcal{A}$-function should be based on the dimensionless
rescaled flow equation. This issue is discussed in Sub.Sec. V.1.

Eq. (22) for the $\mathcal{C}$-function makes the loss of information/richness
of structure of the effective potential $u(t,x)$ during RG time evolution
explicit. $\mathcal{C}$ monotonically increases with RG time $t$ because the
richness of structure/information decreases with $t$. A loss of information
about a system (effective potential $u(t,x)$) during RG time evolution goes
hand in hand with the impossibility to reconstruct/recover earlier states of
the system (which had more information) and thus the RG time evolution is
irreversible. In the present setup the purely diffusive flow equation of the
zero-dimensional $O(1)$ model is responsible for this loss of information as
large gradients are smeared out by diffusion during RG time evolution, cf.
Sec. IV for explicit numerical examples.

###  III.2 Discrete formulation and relation to the total variation non-
increasing property

In this subsection we discuss a discretized version of Eq. (22), which is
suited for practical computations. In the following and without loss of
generality we consider a finite volume (FV) discretization [21, 22, 25, 55] of
$u(t,x)$ in $x$ with $n$ volume cells of constant width $\Delta x$, centered
at $x_{i}$, $i\in\\{0,1,\ldots,n-1\\}$, see our parallel discussion in Sec. IV
of Ref. [6] for details. The RG flow is described by the temporal evolution of
the $n$ volume averages $\bar{u}_{i}(t)$, which are formally defined as the
spatial averages of $u(t,x)$ over $[x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]$,
where $x_{i\pm\frac{1}{2}}\equiv x_{i}\pm\tfrac{\Delta x}{2}$. For the purpose
of calculating $\mathcal{C}[\partial_{x}u(t,x)]$, we reconstruct the first
derivatives from the set of volume averages $\\{\bar{u}_{i}(t)\\}$ by a first
order finite difference (FD) forward stencil,

$\displaystyle\partial_{x}u(t,x_{i})=\frac{\bar{u}_{i+1}(t)-\bar{u}_{i}(t)}{\Delta
x}+\mathcal{O}(\Delta x)\,.$ (23)

For the scope of this work this has proven sufficient since the purely
diffusive character of the PDE smoothens $u(t,x)$.131313However, for non-
smooth/non-differentiable initial conditions at $t=0$, such as Eqs. (28) and
(31) of our numeric examples, a naive finite difference stencil is of course
generically ill-conditioned at the discontinuities. As a direct consequence,
the absolute value of $S[\partial_{x}u(t=0,x)]$ strongly depends on the
explicit discretization points and the “capturing of the discontinuity” in the
respective volume cells. For $t\rightarrow\infty$ (as a direct consequence of
the Coleman-Mermin-Wagner-Hohenberg theorem [109, 110, 111]), $u(t,x)$ is
smooth and the finite difference approximation is well-behaved as long as
$\Delta x$ is not too small. We conclude that the absolute values of our
entropy function (22) will strongly depend on $\Delta x$ for non-
differentiable initial conditions in the IR because we use
$S[\partial_{x}u(t=0,x)]$ as normalization, while the qualitative behavior
(monotonic rise) is independent of the discretization, which is also true for
the discrete total variation (27). For the smooth initial conditions (29) and
(30), we observed little dependence of the absolute values of the
$\mathcal{C}[\partial_{x}u(t,x)]$ on $\Delta x$, as expected. Similar
discussions will arise for $O(N)$-type models in higher-spacetime dimensions,
when their RG flows end in the symmetry broken phase with a non-analytic IR-
potential.

We use a grid with the first volume cell of the computational domain centered
at zero, $x_{0}=0$, and the last centered at a finite $x_{\mathrm{max}}$,
hence $x_{n-1}=x_{\mathrm{max}}$. $x_{\mathrm{max}}$ is chosen large enough,
such that $u(t,x_{\mathrm{max}})=u(t=0,x_{\mathrm{max}})$ holds to a
sufficient level for all $t$, compare our discussion in Ref. [6] as well as
Refs. [4, 121, 122, 123]. This enables a computation of
$\mathcal{C}[\partial_{x}u(t,x)]$ considering only
$x\in[-x_{\mathrm{max}},+x_{\mathrm{max}}]$ since the difference
$S[\partial_{x}u(t,x)]-S[\partial_{x}u(t=0,x)]$ practically vanishes for
$|x|\geq x_{\mathrm{max}}$. We therefore study the following quantity:

$\displaystyle\mathcal{C}[\partial_{x}u(t,x)]=\,$
$\displaystyle-2\int_{0}^{x_{\mathrm{max}}}\mathrm{d}x\,\big{[}\partial_{x}u(t,x)\big{]}^{2}+\vphantom{\bigg{(}\bigg{)}}$
(24)
$\displaystyle+2\int_{0}^{x_{\mathrm{max}}}\mathrm{d}x\,\big{[}\partial_{x}u(t=0,x)\big{]}^{2}\vphantom{\bigg{(}\bigg{)}},$

leveraging the $\mathbb{Z}_{2}$-symmetry of the problem at hand. Inserting Eq.
(23) and performing the integrals over the constant segments in the volume
cells leads to our semi-discrete formulation

$\displaystyle\mathcal{C}[\\{\bar{u}_{i}(t)\\}]=\,$
$\displaystyle-\frac{2}{\Delta
x}\Bigg{(}\sum_{i=0}^{n-1}\frac{[\bar{u}_{i+1}(t)-\bar{u}_{i}(t)]^{2}}{(1+\delta_{i,0}+\delta_{i,n-1})}-$
(25)
$\displaystyle-\sum_{i=0}^{n-1}\frac{[\bar{u}_{i+1}(0)-\bar{u}_{i}(0)]^{2}}{(1+\delta_{i,0}+\delta_{i,n-1})}\Bigg{)}\,,$

where the factor $(1+\delta_{i,0}+\delta_{i,n-1})$ takes into account the fact
that we only integrate over the right half of the first and the left half of
the last volume cell.

Practical computations of solutions to the PDE (11) on the compact interval
$x\in[0,x_{\mathrm{max}}]$ require carefully chosen boundary conditions [6, 7]
to be consistent with solutions of the pure initial value problem posed by Eq.
(11) on the interval $x\in(-\infty,+\infty)$ [122, 124]. In the present finite
volume setup we implement boundary conditions with “ghost cells” at $x_{-2}$,
$x_{-1}$, $x_{n}$ and $x_{n+1}$, where the corresponding cell averages are
chosen due to the $\mathbb{Z}_{2}$-anti-symmetry of $u(t,x)$ in cases of
$\bar{u}_{-2}(t)$ and $\bar{u}_{-1}(t)$ and by means of linear extrapolation
in the cases of $\bar{u}_{n}(t)$ and $\bar{u}_{n+1}(t)$, see Sub.Sec. IV D of
part I of this series of publications [6] for details. For the computation of
$\mathcal{C}[\\{\bar{u}_{i}(t)\\}]$ we require only the ghost-cell average
$\bar{u}_{n}(t)=2\bar{u}_{n-1}(t)-\bar{u}_{n-2}(t)$ as well as the cell
averages at $x_{i}$ for $i\in\\{0,1,\ldots,n-1\\}$.

The entropy functional (21) introduced in Sub.Sec. III.1 is closely related to
the total variation [56] – which is simply the arc length – of the solution
$u(t,x)$,

$\displaystyle\mathrm{TV}[\partial_{x}u(t,x)]\equiv\int_{0}^{x_{\mathrm{max}}}\mathrm{d}x\,|\partial_{x}u(t,x)|\,,$
(26)

on the (computational) interval $[0,x_{\mathrm{max}}]$. The TV qualitatively
differs only by a global sign from the entropy functional $S$, where the sign
used for the TV is compatible with the mathematical convention for (numerical)
entropy. The use of the absolute value $|\partial_{x}u(t,x)|$ in Eq. (26)
instead of the square $[\partial_{x}u(t,x)]^{2}$ used for $S$ presents only as
quantitative difference, which is not of any practical relevance in this work.

On a FV grid, a typical discretized version of Eq. (26) is given by, cf. Refs.
[56, 21, 22, 25],

$\displaystyle\mathrm{TV}[\\{\bar{u}_{i}(t)\\}]\equiv\sum_{i=0}^{n-1}|\bar{u}_{i+1}(t)-\bar{u}_{i}(t)|\,,$
(27)

where a first order forward FD stencil is used to discretize the first
derivative. The differences
$\mathcal{C}[\\{\bar{u}_{i}(t^{m})\\}]-\mathcal{C}[\\{\bar{u}_{i}(t^{m+1})\\}]$
and
$\mathrm{TV}[\\{\bar{u}_{i}(t^{m+1})\\}]-\mathrm{TV}[\\{\bar{u}_{i}(t^{m})\\}]$
on a discrete trajectory $\bar{u}_{i}(t)$ of an admissible solution at
different times separated by one time step $\Delta t$, where
$t^{m+1}=t^{m}+\Delta t$, are both greater or equal to zero for all $t^{m}$.
Thus total variation – arc length – is non-increasing141414In literature total
variation diminishing (TVD) is often used as a less precise synonym for total
variation non-increasing (TVNI), cf. Sec. 9.2.2 of Ref. [25]. and the
corresponding entropy in the sign convention of this paper is non-decreasing –
monotonically increasing.

(Weak) solutions of broad classes of hyperbolic and parabolic conservation
laws are total variation non-increasing during time evolution when considered
on a finite interval, see, e.g., Refs. [56, 21, 24] and especially Ref. [125].
The flow Eq. (11) under consideration in this paper is a non-linear, parabolic
pure diffusion equation and the construction of the normalized entropy
functional $\mathcal{C}$ of Eq. (22) can be adapted to prove directly that
solutions of the flow Eq. (11) are TVNI. The notion of numerical entropy (and
TV as a possible candidate for it) is very important in the study,
construction and numerical computation of physical weak solutions of
conservative equations, see, e.g., the textbooks [19, 20, 21, 22, 23, 24, 25]
for further details.

##  IV Numerical entropy production in zero-dimensional models

In this section we present explicit numerical results for the RG flows of the
(numerical) entropy function (22) for some selected zero-dimensional $O(1)$
models (different UV initial conditions). As examples, we choose the test
cases which are introduced and discussed in great detail in Sec. V of part I
of this series of publications [6]. All information on the explicit numerical
treatment is presented in Sec. IV of part I of this series of publications
[6], where the Kurganov-Tadmor (KT) central scheme [55] is discussed and
applied to RG flow equations. The numerical parameters for the RG flows of
this paper are stated in the figures and their respective captions. An
elaborated discussion on the choice and tests of numerical and model
parameters can also be found in Chap. V of part I of this series of
publications [6]. For the sake of completeness and as proof of reliability of
our numerical scheme and the choice of our numerical parameters, we
nevertheless provide a comparison in Tab. 1 between numerical results for the
1PI-two-point-function $\Gamma^{(2)}$ calculated via the solution of the flow
equation (11) with the KT-scheme and “exact” results calculated via
expectation values (10) from the partition function.

Note that all plots of the entropy in this section are based on a direct
implementation of Eq. (25).

Table 1: The table lists the “exact” results for $\Gamma^{(2)}$ of the $O(1)$ model (second column) for the various UV initial potentials of our test cases (first column), which are calculated by a brute force high-precision one-dimensional numerical integration of the expectation values (10) using NIntegrate in Mathematica [126] with a PrecisionGoal and AccuracyGoal of $10$. Here, we shall present the first ten digits. The last column lists the relative errors of the numerical solution of the RG flow equation with diffusion obtained with the second order accurate KT-central scheme [55] using the parameters listed in the corresponding Figs. 1, 3, 4, 7 and 9, see also Ref. [6] for a detailed discussion of such errors.  UV potential | $\Gamma^{(2)}$ | $|\Gamma^{(2)}_{\mathrm{FRG}}/\Gamma^{(2)}-1|$
---|---|---
Eq. (28) | $0.1768130358$ | $6.0\cdot 10^{-6}$
Eq. (29) (neg. mass) | $0.1995098930$ | $1.1\cdot 10^{-5}$
Eq. (29) (pos. mass) | $1.3324252475$ | $1.4\cdot 10^{-5}$
Eq. (30) | $0.1740508127$ | $2.5\cdot 10^{-5}$
Eq. (31) | $0.2046977422$ | $5.8\cdot 10^{-6}$

###  IV.1 Test case I: Non-analytic initial condition

As our first test case, we choose a UV potential associated with a broken
$\mathbb{Z}_{2}$ symmetry. Moreover, it shall come with infinitely many
degenerate minima as well as non-analytic points at $|\sigma|=2$ and
$|\sigma|=3$,

$\displaystyle
U(\sigma)=\begin{cases}-\tfrac{1}{2}\,\sigma^{2}\,,&\text{if}\quad|\sigma|\leq
2\,,\vphantom{\bigg{(}\bigg{)}}\\\ -2\,,&\text{if}\quad 2<|\sigma|\leq
3\,,\vphantom{\bigg{(}\bigg{)}}\\\
+\tfrac{1}{2}\,(\sigma^{2}-13)\,,&\text{if}\quad
3<|\sigma|\,,\vphantom{\bigg{(}\bigg{)}}\end{cases}$ (28)

see Fig. 4 of Ref. [6] and also top panel of Fig. 1 for visualizations. This
UV potential amounts to a piecewise linear discontinuous initial condition
$u(t=0,x)$ for the RG flow equations (11). The corresponding RG flow of
$u(t,x)$ is presented in Fig. 1.

Figure 1: RG flow of the effective potential $U(t,\sigma)$ (upper panel) and
its derivative $u(t,\sigma)=\partial_{\sigma}U(t,\sigma)$ (lower panel) for
the zero-dimensional $O(N=1)$-model with initial condition Eq. (28) evaluated
at $t=0,\,2,\,4,\,\ldots,\,60$ (integer values for $t$ were only chosen for
convenience and readability). The blue curve corresponds to the UV while the
red curve to the IR. We used the exponential regulator Eq. (9) with UV cutoff
$\Lambda=10^{6}$. For convenience only, the plot does not show the region
$x=5$ to $x=10$ because the tiny differences between $u(t,\sigma)$ and
$u(t_{\mathrm{UV}},\sigma)$ are not visible in this region and vanish for
large $x=\sigma$ anyhow. The lower panel is identical to Fig. 8 (upper panel)
of Ref. [6].

The diffusive character of the $\sigma$-mode is clearly visible from the fact
that it smoothens the discontinuities at $x=2$ and $x=3$, without any directed
propagation (advection) of the conserved quantity $u(t,x)$. As discussed in
Refs. [6, 91], the system has to restore the $\mathbb{Z}_{2}$ symmetry in the
ground state as dictated the Coleman-Mermin-Wagner-Hohenberg theorem [109,
110, 111]. In particular, the potential has to become convex [80, 127]. This
can be directly observed in the plot of the RG flow and read off from Tab. 1 –
the two-point function is positive at $\sigma=0$.

In Fig. 2 we present the RG flow of the (discretized numerical) entropy
function for our first test case.

Figure 2: The plot shows the monotonic growth of the (numerical) entropy/the
$\mathcal{C}$-function $\mathcal{C}[\partial_{x}u(t,x)]$ during the RG flow of
the test case (28) and corresponds to Fig. 1.

As expected from our discussion in Sec. III, the entropy grows monotonically.
It increases by two orders of magnitude starting at zero in the UV until it
reaches (again) a plateau in the IR. We find that the entropy grows most when
the regulator (9) reaches the model scales. Loosely speaking, this is where
most of the dynamics takes place, see Fig. 1 (approximately between $t\approx
4$ and $t\approx 16$). This is the RG time frame in which the diffusion smears
out the discontinuities. From a fluid and thermodynamic perspective and
directly on the level of the PDE, the whole process is intuitively
understandable: Diffusion goes hand in hand with strong dissipation and a loss
of information about the initial state of the system – the UV, cf. Ref. [27,
16]. This is directly comparable to heat conduction, where the information
about the initial temperature distribution gets lost during the flow toward
“thermal” equilibrium [120, 21, 42]. In the RG framework, this translates to
integrating out degrees of freedom from the UV to the IR and a growth in the
number of coupling constants in $U(t,\sigma)$, which is directly related to
the growth of entropy. The entropy plateau in the IR is identified with the
interacting IR regime and an “thermal” equilibrium on the level of the
diffusive PDE, whereas a plateau in the UV is associated with a Gaussian UV
fixed point [128, 76]. As expected the entropy stops changing at these points.
IR solutions therefore correspond either to steady-flow solutions (in
advection dominated systems for a large number of “Goldstone” modes [129, 130,
131]) or to (thermal) equilibrium solutions (in diffusion dominated
$O(1)$-symmetric systems) in the fluid dynamical picture [6].

Note that $t\in[0,60]$ corresponds to a integration over $26$ orders of
magnitude in $r(t)$, starting $6$ orders of magnitude above the model scales
(which are of order one) and ending up $20$ orders of magnitude below the
model scales. In part I of this series of publications [6], we discussed that
large/low UV/IR cutoffs are needed to ensure cutoff independence of the IR
effective action, which is also known as RG consistency [107]. Interestingly,
we find that the almost total absence of a plateau in the entropy in the UV of
our first case implies that we almost violated RG consistency.151515The
absence of the zero-entropy plateau can also be seen by closer inspection of
Fig. 16 of Ref. [6], where $\Lambda=10^{6}$ is barely on the plateau of RG
consistent UV scales. For all other test cases this is avoided by choosing
larger UV cutoffs $\Lambda$, see below.

Before we continue with our next test case, we again note that the absolute
value of $\mathcal{C}[\partial_{x}u(t,x)]$ in the IR in Fig. 2 has no
quantitative meaning, due to the ill-conditioned behavior when applied to the
discontinuous initial condition (28) of the numerical derivative (23).
However, this does not spoil our qualitative arguments at all.

###  IV.2 Test case II: $\phi^{4}$-theory

The second test case is the zero-dimensional analogue of higher-dimensional
$\phi^{4}$-models. We consider two UV initial conditions, differing in the
sign of the mass-like $\phi^{2}$-contribution,

$\displaystyle
U(\sigma)=\mp\tfrac{1}{2}\,\sigma^{2}+\tfrac{1}{4!}\,\sigma^{4}\,.$ (29)

Hence, depending on the sign, we either start the RG flow with a broken
$\mathbb{Z}_{2}$-symmetry in the ground state or with a
$\mathbb{Z}_{2}$-symmetric ground state. For a visualization of the initial
condition with negative mass term, see Fig. 18 of Ref. [6]161616RG flows for
both choices of sign within a low-order FRG Taylor expansion are also provided
in, e.g., Refs. [95, 92, 91] for the zero-dimensional $O(N)$ model.. This
initial condition is chosen because of its relevance in higher dimensions,
e.g., for studies of spontaneous symmetry breaking and symmetry restoration
(ranging from applications in statistical mechanics and condensed-matter
theory to high-energy physics). Additionally, in contrast to our first test
case (28), due to the analyticity of Eq. (29), a generic expansion of the
potential in polynomials at any $\sigma$ is possible in the UV at $t=0$. This
property can be used to study the convergence of a common FRG truncation
scheme, the Taylor expansion of the effective action. In Sec. V of Ref. [6],
we find that only for positive mass-like terms, where the physical point does
not move during the RG flow, the FRG Taylor expansion about the IR minimum
$\sigma=0$ exhibits “apparent” convergence by increasing the expansion order.
For negative mass terms (also using a fixed expansion point at the IR minimum
$\sigma=0$), we do not find convergence while increasing the expansion order.
In Sec. V of Ref. [6], we argue that during the RG flow, while the physical
point moves from $\sigma=\pm\sqrt{6}$ to $\sigma=0$, presumably an excessively
large or even infinitely many new couplings are generated in $U(t,\sigma)$.
This renders FRG Taylor expansion at a finite order a potentially problematic
approximation scheme in such a scenario.

In this subsection, we reinforce our findings about the non-convergence of
expansions of the potential during the RG flow by studying the (numerical)
entropy production during the RG flows. The RG flows of $u(t,x)$ for both
initial conditions are depicted in Fig. 3 (for negative mass term) and in Fig.
4 (for positive mass term).171717Note that the plot ranges for the “positive
mass”-case are different from all other plots of RG flows of $u(t,x)$ in this
section. Otherwise, the tiny changes during the RG flow would not be visible
at all.

Figure 3: RG flow of the effective potential $U(t,\sigma)$ (upper panel) and
its derivative $u(t,\sigma)=\partial_{\sigma}U(t,\sigma)$ (lower panel) for
the zero-dimensional $O(N=1)$-model with initial condition Eq. (29) (with
negative mass term) evaluated at $t=0,\,2,\,4,\,\ldots,\,60$ (integer values
for $t$ were only chosen for convenience and readability). The blue/magenta
curve corresponds to the UV while the red curve to the IR. We used the
exponential regulator Eq. (9) with UV cutoff $\Lambda=10^{12}$. For
convenience only, the plot does not show the region $x=5$ to $x=10$ because
the tiny differences between $u(t,\sigma)$ and $u(t_{\mathrm{UV}},\sigma)$ are
not visible in this region and vanish for large $x=\sigma$ anyhow.  Figure 4:
RG flow of the effective potential $U(t,\sigma)$ (upper panel) and its
derivative $u(t,\sigma)=\partial_{\sigma}U(t,\sigma)$ (lower panel) for the
zero-dimensional $O(N=1)$-model with initial condition Eq. (29) (with positive
mass term) evaluated at $t=0,\,2,\,4,\,\ldots,\,60$ (integer values for $t$
were only chosen for convenience and readability). The blue/magenta curve
corresponds to the UV while the red curve to the IR. We used the exponential
regulator Eq. (9) with UV cutoff $\Lambda=10^{12}$. For convenience only, the
plot does not show the region $x=3$ to $x=10$ because otherwise the
differences between the UV and the IR curves would not be visible at all for
the entire domain.

Both RG flows are by visual inspection not really spectacular: For the
“negative mass”-case, we find that, according to the Coleman-Mermin-Wagner-
Hohenberg theorem [109, 110, 111], the diffusion via the $\sigma$-mode
restores the $\mathbb{Z}_{2}$-symmetry and drives the potential convex during
the RG flow before the system equilibrates in the IR. For the RG flow of the
“positive mass”-case we only find minimal changes in the shape of $u(t,x)$
also originating from the non-linear diffusion during the RG flow. Hence, the
equilibrated solution in the IR is relatively close to the UV initial
potential.

Figure 5: The plot shows the monotonic growth of the (numerical) entropy/the
$\mathcal{C}$-function $\mathcal{C}[\partial_{x}u(t,x)]$ during the RG flow of
the test case (29) (with negative mass term) and corresponds to Fig. 3.
Figure 6: The plot shows the monotonic growth of the (numerical) entropy/the
$\mathcal{C}$-function $\mathcal{C}[\partial_{x}u(t,x)]$ during the RG flow of
the test case (29) (with positive mass term) and corresponds to Fig. 4.

The plots of the corresponding entropies in Fig. 5 (for negative mass term)
and Fig. 6 (for positive mass term) are more instructive. In both cases we
find a clear monotonic rise of the (numerical) entropy exactly in the RG time
period, in which most of the dynamics takes place. Furthermore, we clearly
find plateaus in the UV and the IR, which correspond to the trivial UV regime
and the non-trivial interacting IR regime. This plateau-like behavior signals
RG consistency. In comparison with our first test case (28), where we used
exactly the same discretization points (volume cells), the monotonic growth of
entropy is less drastic and significantly smaller. This is expected because
the jumps in $u(t=0,x)$ at $x=2$ and $x=3$ in the first test case (28) lead to
greater changes in the discrete total variation [the arc length in $x$ of
$u(t,x)$] than the rather small changes of the profiles of $u(t,x)$ for the
$\phi^{4}$-models, compare Sub.Sec. III.2. Also from a fluid dynamic
perspective, this is intuitively understandable because the smoothening of
huge gradients (rarefaction waves) is a substantial source of entropy and
obviously an irreversible process, whereas only a small transport of a fluid
is not a source of excessive but rather small entropy production, even though
it is diffusion driven. Still, also for both $\phi^{4}$-cases the entropy
increases during the RG flow, which first signals an increasing number of
coupling constants generated during the RG flow, and second also renders the
RG flows irreversible.

The second observation has severe consequences: Any RG flow in a FRG Taylor
expansion employs a finite set of coupled ODEs for the couplings (vertices).
Since the system is finite, it seems to be theoretically possible integrate in
either RG time direction. In higher dimensions, one can formally integrate to
larger or lower (energy) scales (associated with resolutions in position
space), compare with, e.g., the perturbative $\beta$-functions of QCD, QED
etc. [132, 133, 134, 135]. However, this is in principle not compatible with
the irreversibility of RG flows as shown in our present work (as, e.g.,
signaled by the rise of entropy) and may only be reliable within small
subspaces of the theory space associated with a given theory. In fact, the
computation of fundamental couplings at small scales (high energies) from
effective couplings at large scales (low energies) is in general not possible,
cf. Ref. [2]. We conclude that the increase of entropy, which we also observe
during the RG flow of our analytic initial conditions (29) reveals potential
limitations of Taylor expansion of effective actions because most likely an
extremely large number (or even infinite number) of couplings is generated in
the RG flow and would be required to correctly describe the RG flow. 181818At
this point, one might be tempted to apply our definition of the normalized
(numerical) entropy directly to some $\partial_{x}u(t,x)$ that is
reconstructed from the flow of the coefficients of a Taylor expansion of the
potential to study the validity of the expansion. However, this is not
possible, because the FRG Taylor expansion in general provides only an
adequate local description of the potential, while our (numerical) entropy or
the TV requires knowledge about the global shape of the potential or its
derivatives, respectively. 191919This reasoning might also resolve some
issues, which are discussed in Ref. [34]. In Ref. [34], it is argued that the
“$\mathcal{C}$-theorem folklore” about the existence of a monotonically rising
$\mathcal{C}$-function prevents the RG flow from entering limiting cycles (or
even chaotic behavior) is wrong. However, their arguments are entirely based
on examples of $\beta$-functions with a finite number of couplings. As
explicitly shown in Ref. [34], such systems can indeed show limiting cycles
and still have monotonic flows etc. as shown by the authors. Similar to what
is explained at several occasions in our work, such systems do however not
show irreversibility in the sense of a true diffusive/dissipative process or
via the interaction/generation of discontinuities in field space – “the
theory-space of couplings” [4, 7]. The only irreversible character of these
systems might indeed be that they can enter limiting cycles, show chaotic
behavior or enter fixed points on a finite set of couplings.

In practical computations with the zero-dimensional $O(N)$ model a complete
inversion of the RG flow – integrating up from the IR to the UV – is possible
only under certain conditions. In Sub.Sub.Sec. V B 2 of part I of this series
of publications [6] we performed tests with the $\phi^{4}$ model discussed in
this subsection and found that a complete and accurate practical inversion is
only possible for the $\phi^{4}$ model with positive mass term when
considering a small set of running couplings. Numerical instabilities related
to massive oscillations in the higher-order couplings prevent a numerical
inversion of the RG flow for larger systems of couplings. In the $\phi^{4}$
model with negative mass term a numerical reconstruction of the non-convex UV
potential by integrating up from the convex IR potential seems to be
practically impossible with the employed Taylor expansions. While an inversion
of the RG flow seems theoretically possible on first sight when considering
the finite ODE systems of the FRG Taylor (vertex) expansion the
practical/numerical realization is not obvious.

When considering an expansion in vertices, it might be possible that higher-
order couplings/vertices are strongly suppressed (especially when considering
higher dimensional QFTs), such that an expansion of the ERG equation (1) in
vertices is applicable and meaningful in practice, see, e.g., Refs. [136, 137,
138, 108]. This should go hand in hand with only a small growth of an entropy
for the exact RG flow. Exactly this seems to be the case for our “positive
mass” case (29), which shows almost no dynamics at all and yields the smallest
increase in entropy of all our test cases. A reason, why here a rather small
number of couplings might be sufficient to describe the entire RG flow is that
the potential is convex during the entire flow and has a single unique non-
moving minimum. Hence, the UV regime of this model and the IR regime do not
differ much and, as long as the quartic coupling is extremely small, also
perturbation theory [93] leads to results which are consistent with the exact
values for the lowest $1$PI-$n$-point-correlation (vertex) functions [92].

###  IV.3 Test case III: $\phi^{6}$-potential

The third test case describes a potential that is analytic with a
$\mathbb{Z}_{2}$-symmetric ground state in the UV. However, the potential
exhibits two non-trivial local minima and behaves asymptotically
$\propto\phi^{6}$,

$\displaystyle
U(\sigma)=\tfrac{1}{2}\,\sigma^{2}-\tfrac{1}{20}\,\sigma^{4}+\tfrac{1}{6!}\,\sigma^{6}\,,$
(30)

such that it is not convex in the UV. A plot of the UV potential can be found
in Fig. 27 of Ref. [6]. This initial condition of the RG flow is chosen in
Ref. [6] to test whether the poor convergence of the FRG Taylor expansion of
the $\phi^{4}$-potential with negative mass term (29) is merely an artifact of
the moving scale-dependent global minimum in the RG flow and to figure out
whether the FRG Taylor expansion should have actually be performed at a moving
expansion point, i.e., about the moving global minimum instead of expanding
around the IR minimum $\sigma=0$ during the entire flow. However, although the
global minimum at $\sigma=0$ is not moving at all in the RG flow of the
$\phi^{6}$-case (30), the Taylor expansion completely fails here,202020We
thank J. Eser for discussions on this issue and a cross check which reproduced
our findings for this test case, using his FRG code for Taylor-expanded
effective actions [138, 108, 136, 137]. see Ref. [6]. In Ref. [6], we conclude
that there has to be a time interval during the RG flow, where $u(t,x)$
exhibits a highly non-local dynamics. The latter cannot be captured by a local
expansion with a finite number of couplings about a single point, even though
the expansion point is unique and does not move.

Figure 7: RG flow of the effective potential $U(t,\sigma)$ (upper panel) and
its derivative $u(t,\sigma)=\partial_{\sigma}U(t,\sigma)$ (lower panel) for
the zero-dimensional $O(N=1)$-model with initial condition Eq. (30) evaluated
at $t=0,\,2,\,4,\,\ldots,\,60$ (integer values for $t$ were only chosen for
convenience and readability). The blue curve corresponds to the UV while the
red curve to the IR. We used the exponential regulator Eq. (9) with UV cutoff
$\Lambda=10^{6}$. For convenience only, the plot does not show the region
$x=5$ to $x=10$ because the tiny differences between $u(t,\sigma)$ and
$u(t_{\mathrm{UV}},\sigma)$ are not visible in this region and vanish for
large $x=\sigma$ anyhow.

Actually, this can be seen directly from the RG flow of $u(t,x)$ in Fig. 7 at
approximately $t\approx 27$, which is also the time when the RG flow of the
Taylor-/vertex expansion collapses due to strongly oscillating and ultimately
diverging couplings at $t\approx 27$. At this RG time, the local minimum (the
second non-trivial zero-crossing) vaporizes via the diffusion and merges with
the global minimum at $x=0$. Already from the curves in Fig. 7 one can observe
that the $u(t,x)$ is hardly describable over the entire RG flow with only a
finite set of couplings. The breakdown of any expansion can also be directly
related to Wilbraham-Gibbs-oscillations [139, 140, 141, 142] in the flat
region of $u(t,x)$. This was already (indirectly) described before in the
context of FRG studies [143] and represents another direct interplay between
characteristic properties of the RG and the numeric treatment of PDEs.

Figure 8: The plot shows the monotonic growth of the (numerical) entropy/the
$\mathcal{C}$-function $\mathcal{C}[\partial_{x}u(t,x)]$ during the RG flow of
the test case (30) and corresponds to Fig. 7

Interestingly, also the (numerical) entropy function signals exactly the
discussed non-local behavior at $t\approx 27$. Exactly at that point in time
when the local minimum merges with the global minimum, we observe the
strongest increase of entropy, see Fig. 8. We also find that by absolute
measures, the entropy production for the $\phi^{6}$-initial potential (30) is
greater than the entropy production observed for both quartic initial
conditions (29). Nevertheless, the entropy production for the non-analytic
initial condition (28) is still greater than the one in the $\phi^{6}$-case.
This can be easily understood from the relation of the numerical entropy to
the total variation, i.e., the arc length of $u(t,x)$ which even formally
diverges for Eq. (28) in the UV.212121Absolute values of the numerical entropy
as well as their comparison should be considered with some care as explained
above.

We conclude from this section that the (numerical) entropy might be a tool to
detect if the RG flow “moves” far from the perturbative region, while going
from $t=0$ to $t\rightarrow\infty$. In other words, it is a tool to discuss
whether the RG flow is governed by strong (non-perturbative) dynamics and
cannot be captured within any kind of local or perturbative expansion. This is
analogous to a thermal system or fluid evolving through an out-of-equilibrium
state, before finally equilibrating or showing steady flow behavior, in
contrast to a temperature distribution or fluid that is already close to its
equilibrium state.

###  IV.4 Test case IV: the $\sigma=0$ boundary

Our final test case has originally been used in part I of this series of
publications [6] to test the correct implementation of (spatial) boundary
conditions for PDEs of the form (11):

$\displaystyle
U(\sigma)=\begin{cases}-(\sigma^{2})^{\tfrac{1}{3}}\,,&\text{if}\quad|\sigma|\leq\sqrt{8}\,,\vphantom{\bigg{(}\bigg{)}}\\\
\tfrac{1}{2}\,\sigma^{\,2}-6\,,&\text{if}\quad|\sigma|>\sqrt{8}\,.\vphantom{\bigg{(}\bigg{)}}\end{cases}$
(31)

The UV initial potential $U(\sigma)$ now exhibits a non-analyticity – a cusp –
at ${\sigma=0}$ which leads to a pole in the conserved quantity
${u(t=0,\sigma)=\partial_{\sigma}U(\sigma)}$.222222Potentials with cusps in
field space can be found in the context of, e.g., theories in 2+1 spacetime
dimensions, such as the Gross-Neveu model [53]. Additionally, to put the tests
of our numerical approach to the extremes, we incorporated two non-trivial
minima at $\sigma=\pm\sqrt{8}$, which are on top of that also non-analytic
points, causing again discontinuities in $u(t=0,\sigma)$. A visualization of
Eq. (31) is shown in Fig. 30 of Ref. [6].

Also in the context of this work, the test case (31) turns out to be a highly
interesting almost pathological example. The RG flow of $u(t,x)$ is shown in
Fig. 9, where one can see that the numerical scheme indeed perfectly copes
with the aforementioned somewhat artificial challenges and correctly
reproduces symmetry restoration, convexity and smoothness of the potential in
the IR regime.

Figure 9: The plot shows the RG flow of the effective potential $U(t,\sigma)$
(upper panel) and its derivative $u(t,\sigma)=\partial_{\sigma}U(t,\sigma)$
(lower panel) for the zero-dimensional $O(N=1)$-model with initial condition
Eq. (31) evaluated at $t=0,\,2,\,4,\,\ldots,\,60$ (integer values for $t$ were
only chosen for convenience and readability). The blue curve corresponds to
the UV while the red curve to the IR. We used the exponential regulator Eq.
(9) with UV cutoff $\Lambda=10^{6}$. For convenience only, the plot does not
show the region $x=5$ to $x=10$ because the tiny differences between
$u(t,\sigma)$ and $u(t_{\mathrm{UV}},\sigma)$ are not visible in this region
and vanish for large $x=\sigma$ anyhow.

Of specific interest regarding the (numeric) entropy is of course also the
pole of $u(t=0,x)$ at $x=0$. Formally, the arc length (the total variation) of
$u(t,x)$, which is directly related to our entropy function, diverges due to
the pole at $t=0$ for all $t>0$. This divergence is of different nature than
the divergence caused by integrating from $x=-\infty$ to $x=+\infty$ in Eq.
(21). Whereas the latter can be cured by normalizing the entropy w.r.t. the
entropy of $u(t=0,x)$, the present divergence also occurs on the level of the
“normalized” entropy function (22) similar to the other non-analytic jumps in
the UV. The reason for the infinite entropy production while going from $t=0$
to $t>0$ is exactly that the total variation between $-x_{\mathrm{max}}$ and
$+x_{\mathrm{max}}$ turns finite for $u(t,x)$ during the flow because the
potential becomes convex and smooth. Moreover, symmetry restoration in the
ground state sets in for $t\rightarrow\infty$. However, it is still normalized
against the infinite total variation of $u(t=0,x)$. Interestingly, this
problem can be traced back to the initialization of the RG flow equations at
$t=0$ with the classical UV action
$\bar{\Gamma}_{t=0}(\varphi)=\mathcal{S}(\varphi)$, which is actually not
totally exact but rather an almost perfect approximation for sufficiently
large $\Lambda$, see also our discussion in Ref. [6]. In Ref. [6], we argue
that the extremely tiny errors stemming from this approximation of the correct
initial condition are immediately “washed out” after the first RG steps (after
an infinitesimal RG time step $\varepsilon>0$) because of the diffusive
character of the ERG equation (as long as $\Lambda$ is chosen sufficiently
large). This implies that we can safely ignore the problem of the infinite arc
length at $t=0$, and formally start considering the entropy from
$t=\varepsilon$ onward.

Figure 10: The plot shows the monotonic growth of the (numerical) entropy/the
$\mathcal{C}$-function $\mathcal{C}[\partial_{x}u(t,x)]$ during the RG flow of
the test case (31) and corresponds to Fig. 9.

From a purely practical and numerical perspective, these details may appear
somewhat academic anyhow. Via the finite-volume discretization, a smallest
resolution $\Delta x$ in field space enters the problem which technically
renders the pole at $\sigma=0$ a huge but already finite jump captured in
three volume cells on the level of the cell averages $\bar{u}_{i}(t)$ at
$t=0$. Therefore, we can use $u(t=0,x)$ as our reference entropy for the
normalization of Eq. (22) as it is numerically finite right from the beginning
of the flow.

An explicit result for the RG flow of our (numerical) entropy is shown in Fig.
10. Irrespective of the subtleties of the preceding discussion, we find a
rather large entropy production at exactly those times when the pole vanishes
and the jumps at $x=\pm\sqrt{8}$ are smeared out via the diffusion.

Additionally, we find that the total entropy production is much larger for
this test case than for the previous ones. Again, this is of course directly
related to the huge gradients in the initial condition, which are tremendous
sources of entropy via dissipation, directly analogous to the heat equation.

In this section, we confronted our theoretical findings with direct numerical
computations. We verified the behavior of the function
$\mathcal{C}[\partial_{x}u(t,x)]$ from Eq. (22) by means of its discretized
version in Eq. (25) as a valid numerical entropy in four test cases. Using the
numerical entropy and the ERG equation in the form (11), we made several at
this point almost intuitive connections between phenomena known in fluid and
thermodynamic processes and directly related processes and aspects of RG
flows. Most notable, the diffusive character of the flow equation (11) results
directly in irreversible RG flows. This also establishes a connection between
steady-state/(thermal) equilibrium solutions and the UV and IR regime.
Moreover, the application of the numerical entropy and total variation appears
to be an attractive monitor for RG consistency and the origin of an
“thermodynamic” time asymmetry.

##  V Irreversibility of the RG flow, entropy and the $\mathcal{C}$-theorem –
generalizations

The (re)discoveries within this work unravel the connection between the
(numerical) entropy and total variation, employed in applied mathematics, and
the irreversibility inherent to RG flows. Furthermore, they might even provide
some connections to $\mathcal{C}$-/$\mathcal{A}$-theorems within the framework
of truncated RG flow equations. This section is dedicated to a discussion of
these aspects of the RG and first generalizations of our findings from our
zero-dimensional toy model to higher-dimensional theories.

###  V.1 The $\mathcal{C}$-theorem, fixed points and generalizations to
(higher-dimensional) $O(N)$ models

The original formulation of the $\mathcal{C}$-theorem [27] states that for a
two-dimensional field theory the following properties hold:

  1. 1.

There exists a positive function

$\displaystyle\mathcal{C}(\\{g_{i}\\},t)\geq 0\,,$ (32)

of all (possibly infinitely many) dimensionless couplings $\\{g_{i}\\}$ of the
theory and RG time $t$, with the additional property

$\displaystyle\tfrac{\mathrm{d}}{\mathrm{d}t}\,\mathcal{C}(\\{g_{i}\\},t)\geq
0\,.$ (33)

(The choice of sign is convention.)

  2. 2.

The $\mathcal{C}$-function takes a fixed value at (critical) fixed points of
the theory, these fixed values can be identified with the central charge $c$
(of the Virasoro algebra)

$\displaystyle\mathcal{C}(\\{g_{i}^{\ast}\\},t)=c\,.$ (34)

The central charge is different for different fixed points.

At first glance, it seems as if our numerical entropy (22) shares a lot of
these properties because all (infinitely many) coupling constants are by
definition dimensionless in ${d=0}$ and all included via $u(t,x)$ in
$\mathcal{C}[\partial_{x}u(t,x)]$. Furthermore, the function
$\mathcal{C}[\partial_{x}u(t,x)]$ from Eq. (22) monotonically rises during the
RG flow and clearly signals irreversibility – a central aspect of
$\mathcal{C}$-/$\mathcal{A}$-theorems [27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41]232323Generalizations of Zamolodchikov’s
$\mathcal{C}$-theorem [27], especially from two to four dimensions, are often
refereed to as $\mathcal{A}$-theorems, referring to Ref. [30] in which an
anomaly coefficient – hence $\mathcal{A}$-theorem – in four dimensions is
proposed to take the role of the central charge which has given the original
$\mathcal{C}$-theorem in two dimensions its name.. However, there is a crucial
difference between our numerical entropy $\mathcal{C}[\partial_{x}u(t,x)]$ and
the $\mathcal{C}$-function defined via Eqs. (32)-(34), which is more apparent
if the discussion is generalized to higher dimensions.

As already briefly discussed, our previous results should also apply to
higher-dimensional $O(1)$ models. This is directly understood by considering
the LPA-flow equation of the $d$-dimensional $O(1)$ model using the LPA-
optimized regulator [70, 71]. The corresponding flow equation for
$u(t,x)=\partial_{x}U(t,x)$ then reads

$\displaystyle\partial_{t}u(t,x)=\frac{\mathrm{d}}{\mathrm{d}x}\bigg{(}-\frac{A_{d}\,k^{d+2}(t)}{k^{2}(t)+\partial_{\sigma}u(t,\sigma)}\bigg{)}\,,$
(35)

where $A_{d}=\tfrac{\Omega_{d}}{d\,(2\pi)^{d}}$, and
$\Omega_{d}=\frac{2\,\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})}$ is the surface
of the $d$-dimensional sphere and the definition (2) was used for the RG scale
$k(t)$. Apart from numerical and $k$-dependent prefactors, which however do
not affect the discussion for the numerical entropy in Sub.Sec. III.1 aside
from the caveats already mentioned, the main differences between zero- and
higher dimensions are

  1. 1.

In the zero-dimensional case, the PDE (11) represents a complete untruncated
description of the QFT, while for higher-dimensional $O(1)$ models, the
corresponding equation constitutes a truncation of the ERG equation (1).
Still, on the level of the PDE – thus within the LPA truncation – the
definition of the numerical entropy function (22) can be used as an entropy
and a detector for irreversibility. Additionally, it is not expected that a
generic rise of the entropy during the RG flow gets lost, if more
sophisticated truncations (like the inclusion of field-dependent wave function
renormalizations) are studied because already the ERG equation (1) itself has
the form of a non-linear diffusion equation. The numerical entropy (22) might
even help to construct suitable truncation schemes as well as stable numerical
methods.

  2. 2.

In contrast to the zero-dimensional model, the couplings have in general non-
zero energy dimensions. Thus, our numerical entropy (22) cannot adequately
describe the second property of the $\mathcal{C}$-theorem (34) – namely
capturing the properties of fixed points, which are defined via the zeroes of
the $\beta$-functions of all dimensionless couplings and additionally a
constant $\mathcal{C}$-/$\mathcal{A}$-function. To capture the fixed-point
structure, one has to replace (35) by its rescaled version and derive a
corresponding numerical entropy functional.

Applying the transformations

$\displaystyle y=\,$ $\displaystyle
A_{d}^{\frac{1}{2}}\,k^{-\frac{d-2}{2}}(t)\,x\,,\vphantom{\bigg{(}\bigg{)}}$
(36) $\displaystyle v(t,y)=\,$ $\displaystyle
A_{d}^{\frac{1}{2}}\,k^{-d}(t)\,u(t,x)\,\vphantom{\bigg{(}\bigg{)}}$ (37)

to Eq. (35), one arrives at the rescaled dimensionless RG flow equation for
the derivative of the scale dependent effective potential,

$\displaystyle\partial_{t}v(t,y)+\tfrac{d}{\mathrm{d}y}\,F[y,v(t,y)]=\vphantom{\bigg{(}\bigg{)}}$
(38) $\displaystyle=\,$
$\displaystyle\tfrac{d}{\mathrm{d}y}\,Q[\partial_{y}v(t,y)]+S[v(t,y)]\,,\vphantom{\bigg{(}\bigg{)}}$

where

$\displaystyle F[y,v(t,y)]\equiv\,$
$\displaystyle\tfrac{d-2}{2}\,y\,v(t,y)\,,\vphantom{\bigg{(}\bigg{)}}$ (39)
$\displaystyle Q[\partial_{y}v(t,y)]\equiv\,$
$\displaystyle-\frac{1}{1+\partial_{y}v(t,y)}\,,\vphantom{\bigg{(}\bigg{)}}$
(40) $\displaystyle S[v(t,y)]\equiv\,$ $\displaystyle
d\,v(t,y)\,.\vphantom{\bigg{(}\bigg{)}}$ (41)

Note that also in its rescaled dimensionless version, the flow equation (38)
for the $d$-dimensional $O(1)$ model has the structure of a non-linear
advection-diffusion-source/sink equation, similar to related RG flow equations
in Refs. [18, 14, 15, 16, 17], where definition (39) corresponds to a
position-dependent advection flux,242424For the $O(N)$ model with $N>1$ the
advection flux gains an additional contribution $-(N-1)/(1+v/y)$. definition
(40) corresponds to a non-linear diffusion flux, and, finally, definition (41)
corresponds to a source/sink term. We can therefore completely stick to our
fluid-dynamic interpretation in terms of conservation laws. Most notably, the
diffusive character of the flow equation, thus irreversibility, is also
manifest in its rescaled form, independent of the spacetime dimension $d$, see
also Refs. [18, 14, 15, 16, 17] for related RG flow equations.

Using Eq. (38), the fixed-point solutions are defined as solutions of the
equation with $\partial_{t}v(t,x)=0$ because all other terms of the flow
equation do not explicitly depend on $t$ anymore. A corresponding (numerical)
entropy function for Eq. (38), analogously to Eq. (22) should therefore signal
irreversibility, associated with the change of the number of degrees of
freedom for increasing RG time. In particular, it should also assume fixed
values at fixed points of the RG flow. We would therefore expect a direct
relation between (numerical) entropy in RG flows and Zamolodchikov’s
formulation [27] or more recent [38, 39] formulations of the
$\mathcal{C}-$function. In this respect, we note that it was possible to
formulate $\mathcal{C}$-functions for the linearized version of the RG flow
equation of the LPA [14, 15, 16, 28, 36].

Unfortunately, due to the explicitly position-dependent advection flux (38) as
well as the source term (41), we were not able to formulate an appropriate
numerical entropy measure yet.252525The explicit position dependency of the
advection flux also prevented us from formulating a numerical entropy for the
zero-dimensional $O(N)$ model for finite $N>1$. The corresponding contribution
to the entropy function (22) allows for
$\tfrac{\mathrm{d}}{\mathrm{d}t}\,\mathcal{C}[\partial_{x}u(t,x)]<0$ during RG
flow for certain initial conditions and $N$ in the case of $N>1$. For the
zero-dimensional cases (29) and (30) discussed in this paper we find
$\tfrac{\mathrm{d}}{\mathrm{d}t}\,\mathcal{C}[\partial_{x}u(t,x)]<0$ during
the RG evolutions for $N\geq 8$. For the cases (28) and (31) with their
$\sigma^{2}$ asymptotics for large $\sigma$ the inequality
$\tfrac{\mathrm{d}}{\mathrm{d}t}\,\mathcal{C}[\partial_{x}u(t,x)]\geq 0$ seems
to hold for all $N$ and $t$. We further strongly believe that there are also
counterexamples for higher-dimensional $O(N)$ models. Nevertheless, our
numerical entropy function (22) for $N=1$ might be a good starting point for
generalizations to finite $N>1$, other models and other truncations. When
performing the $y$-derivative of the advection flux (39) in Eq. (38), thus
considering the advection term in its primitive form, we can distinguish
between a purely advective contribution $\propto\partial_{y}v(t,y)$ and an
internal source term $\propto v(t,y)$. Source terms like $S[v(t,y)]$ from Eq.
(41) and from position dependences in non-linear advection or also diffusion
fluxes make the construction of numerical entropy functions (like the TV)
generally difficult. It is intuitively clear that source and sink terms can
change and crucially increase the arc length – total variation – of solutions.
The construction of numerical entropy functions and related numerical schemes
for generic advection-diffusion-source/sink equations is still subject of
ongoing research in numerical mathematics [144, 145, 146, 147].

A promising next step might be studies of two-dimensional quantum field
theories, where the advection flux (39) vanishes for the $O(1)$ model also for
the rescaled flow equation (38). For this case, the system entirely lacks
advective contributions and the fields are not rescaled with $k$. Especially
the last property might be interesting w.r.t. the spatial integration
contained in the total variation (26), which consequently is $k$-independent
for both – the dimensionful and dimensionless – formulations of the RG flow
equation. Furthermore, for $d=2$ one operates closest to the original version
of Zamolodchikov’s $\mathcal{C}$-theorem [27], which should provide some
additional guidance to relate the $\mathcal{C}$-theorem to numeric aspects of
the PDEs that describe the RG flows.

Source terms arising from position-depended advection- (in a formulation in
$x$) and diffusion-terms (in a formulation in the invariant
$\tfrac{1}{2}\,x^{2}$) when considering the PDEs in primitive form prevent an
obvious generalization of the presented results to zero-dimensional $O(N)$
models with a finite amount $N>1$ of scalars. In App. E 2 b of part III of
this series of publications [7] we discuss numerical entropy and especially
total variation/arc length as a candidate for the flow equations of the zero-
dimensional $O(N)$ model at finite and notably infinite $N$. Reformulating the
RG flow equation in the rescaled invariant ${\tfrac{1}{2N}\,\sigma^{2}}$ leads
to a pure advection equation (with a non-linear but crucially position-
independent flux) which are known to be TVNI [56, 19, 125], thus we identify
the difference between initial and current TV as a numerical entropy
functional in Eq. (E18) of Ref. [7].

Formulating a numerical entropy for the flow equation of the zero-dimensional
$O(N)$ model at finite $N>1$ might be an important first step toward a
formulation in higher dimensions, since it would involve a treatment of source
terms $\propto u(t,x)$. In this sense such a study would be similar to the
study in two dimensions proposed in the previous paragraph. Here, a possible
starting point might be the observations by Refs. [87, 86] that the zero-
dimensional analogues of Dyson-Schwinger equations for the $O(N)$ model can be
recast into a Virasoro algebra. The Virasoro algebra plays a central role in
Zamolodchikov derivation of the $\mathcal{C}$-function for two-dimensional
conformal field theories [27], while the Dyson-Schwinger equations are in
direct relation to the RG equations. Also notable in this context is the fact
that the Virasoro algebra arising in the study of zero-dimensional $O(N)$
model appears with vanishing central charge – in this limit strictly speaking
as a Witt algebra – which considering Zamolodchikov definition of a
$\mathcal{C}$-function indicates the absence of fixed points in RG flows of
the zero-dimensional $O(N)$ model. This will be discussed elsewhere.

###  V.2 Pointwise monotonicity

Interestingly, the discussion of the previous paragraphs shows many
similarities with the findings of Refs. [40, 41]. References [40, 41] discuss
that for purely bosonic models, the effective average action itself is a
“pointwise monotonic function”, which can be directly seen from the signs of
the bosonic contributions in the flow equation (1)262626Related statements
about the general monotonicity of solutions of broad classes of
PDEs/conservations laws are directly linked to the TVNI property, see e.g.
Refs. [56, 21, 24] and especially Ref. [125].. This implies

$\displaystyle\partial_{t}\bar{\Gamma}_{k}[\Phi]\geq 0\,,\quad\forall\Phi\,.$
(42)

However, Refs. [40, 41] argue that $\bar{\Gamma}_{k}[\Phi]$ cannot be directly
used as a $\mathcal{C}$-function because – similar to our numerical entropy
(22) – the above statement is only true for dimensionful field arguments and
dimensionful couplings. However, as stated in [40, p. 3], “the $c$-theorem and
its generalizations apply to the RG flow on theory space, $\mathcal{T}$, a
manifold which is coordinatized by the dimensionless couplings. The latter
differ from the dimensionful ones by explicit powers of $k$ fixed by the
canonical scaling dimensions. As a consequence, when rewritten in terms of
dimensionless fields and couplings, the property [(42)] does not precisely
translate into a monotonicity statement about the de-dimensionalized theory
space analog of [$\bar{\Gamma}_{k}$], henceforth denoted $\mathcal{A}_{k}$.
Rather, when the derivative $\partial_{k}$ hits the explicit powers of $k$,
additional canonical scaling terms arise which prevent us from concluding
simply “by inspection” that $\mathcal{A}_{k}$ is monotonic along RG
trajectories.

In fact, we run into the same problems as Refs. [40, 41]. It is the trivial
rescaling of dimensionful couplings in terms of the source term (41) that
prevents us from immediately writing down an entropy functional or adapted
total variation for our flow equation (38) (at least for $d=2$) which could
then be directly interpreted as a $\mathcal{C}$-function. In Refs. [40, 41],
it is also the term related to the trivial rescaling which spoils the
“pointwise monotonicity”.

##  VI Conclusion and Outlook

In this article, we discussed several generic aspects of (F)RG flow equations.
We based the discussion of our main findings on the rudimentary example of a
zero-dimensional scalar QFT with $\mathbb{Z}_{2}$-symmetry.

We started off by repeating and deepening the discussion of similarities
between RG flow equations and (numerical) fluid dynamics, which was already
started in part I of our series of publications [6]. Based on the formulation
of RG flows as advection and diffusion driven dissipative flows in the field
space of the corresponding QFT along RG scale (time), we argued that RG flows
“produce” entropy. The RG scale (time) defines a rather natural
“thermodynamic” arrow of time in this respect. We concluded that this
dissipative character of the RG, which causes irreversibility of RG flows, is
hard coded in the ERG equation (1). This implies that the irreversibility of
Kadanoff’s block-spin picture is directly encoded in the PDEs (the field
dependent $\beta$-functions), which describe the RG flows. Hereby, the IR
solutions of RG flows represent equilibrium solutions of fluid dynamic
equations. The impossibility of an unambiguous resolution of the microphysics
(UV) from the macrophysics (IR) becomes apparent from this standpoint.

Furthermore, we explicitly demonstrated that the entropy production and the
irreversibility during the RG flow from the UV to the IR are not only of
abstract manner, but can – at least for flow equations in certain truncations
– be quantified. Thereby, we directly related the entropy production to the
numerical entropy production from the research field of PDEs and numerical
fluid dynamics as well as to the total variation non-increasing property. The
latter is used to ensure stability of numerical schemes for broad classes of
PDEs.

Using our zero-dimensional toy model, we explicitly demonstrated for various
test cases taken from our parallel publication [6] how numerical entropy is
produced by diffusion in RG flows and non-analyticities in the UV-initial
conditions.

Furthermore, we related certain aspects of the (numeric) entropy production in
RG flows to the concept of $\mathcal{C}$-/$\mathcal{A}$-theorems in RG theory
since both manifestly encode the irreversible character of RG flows. However,
our present study is not yet conclusive in this respect. Another interesting
aspect related to the introduction of a numeric entropy for RG flows was
pointed out by the referee: the present formulation based on the effective
potential shares some similarities with the macroscopic description of systems
in statistical mechanics. Instead of working with an infinite set of couplings
(microstates in statistical mechanics) we switch to a description in terms of
an effective potential (a macroscopic formulation in statistical mechanics).
The inability (of a macroscopic observer) to track the dynamics of an infinite
set of couplings (microstates in statistical mechanics) leads to a macroscopic
entropy production/information loss and irreversible processes. An approach to
formalize this notion in statistical mechanics was made by Boltzmann [148] and
later Gibbs [149] with the introduction of $H$-theorems, see e.g. Chap. VI and
XII of the textbook [150] for further details. Exploring this connection and
possible relations between $\mathcal{C}$-/$\mathcal{A}$\- and $H$-theorems
further could be a very interesting prospect for further research.

Although certain aspects of our discussion are still on an abstract level and
could not yet be formalized in terms of explicit equations, we believe that
our present work provides a fresh view on certain aspects of RG theory,
embellished with at at least a few new insights. In particular, our approach
may help in the future to construct approximation schemes for RG studies which
fully preserve the fundamentally dissipative character of RG flows.

###### Acknowledgements.

A.K., M.J.S., and J.B. acknowledge the support of the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the
collaborative research center trans-regio CRC-TR 211 “Strong-interaction
matter under extreme conditions”– project number 315477589 – TRR 211. A.K.
acknowledges the support of the Friedrich-Naumann-Foundation for Freedom. A.K.
and M.J.S. acknowledge the support of the Giersch Foundation and the Helmholtz
Graduate School for Hadron and Ion Research. J.B. acknowledges support by the
DFG under Grant No. BR 4005/4-1 and BR 4005/6-1 (Heisenberg program). E.G. is
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, grants Nos. DE-FG-02-08ER41450. N.W. is supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy EXC 2181/1 - 390900948 (the Heidelberg
STRUCTURES Excellence Cluster), the Collaborative Research Centre SFB 1225
(ISOQUANT), the BMBF grant 05P18VHFCA and from the Hessian collaborative
research cluster ELEMENTS.
We thank the co-authors of the first part of this series of publications [6],
M. Buballa and D. H. Rischke, for their collaboration, their support, and a
lot of exceptionally valuable discussions. We further thank F. Divotgey, J.
Eser, F. Giacosa, F. Ihssen, L. Kurth, J. M. Pawlowski, K. Otto, R. D.
Pisarski, S. Rechenberger, A. Sciarra, J. Stoll, and N. Zorbach for valuable
discussions.
We are grateful to the referee of the manuscript for his very insightful
comments and questions especially regarding the similarities to statistical
mechanics and the $H$-theorems mentioned in the outlook.
All numerical results as well as all figures in this work were obtained and
designed using Mathematica [126] including the following ResourceFunction(s)
from the Wolfram Function Repository: PlotGrid [151], PolygonMarker [152], and
MaTeXInstall [153]. The “Feynman” diagram in Eq. (8) was generated with
Axodraw Version 2 [154].

## References

  * Kadanoff [1966] L. P. Kadanoff, Scaling laws for Ising models near $T_{c}$, Physics Physique Fizika 2, 263 (1966).
  * Wilson [1979] K. G. Wilson, Problems in physics with many scales of length, Sci. Am. 241, 158 (1979).
  * Dupuis  _et al._ [2021] N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rept. 910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech] .
  * Grossi and Wink [2019] E. Grossi and N. Wink, Resolving phase transitions with Discontinuous Galerkin methods (2019), arXiv:1903.09503 [hep-th] .
  * Grossi  _et al._ [2021] E. Grossi, F. J. Ihssen, J. M. Pawlowski, and N. Wink, Shocks and quark-meson scatterings at large density, Phys. Rev. D 104, 016028 (2021), arXiv:2102.01602 [hep-ph] .
  * Koenigstein  _et al._ [2022] A. Koenigstein, M. J. Steil, N. Wink, E. Grossi, J. Braun, M. Buballa, and D. H. Rischke, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. I. The $O(N)$ model, Phys. Rev. D 106, 065012 (2022), arXiv:2108.02504 [cond-mat.stat-mech] .
  * Steil and Koenigstein [2022] M. J. Steil and A. Koenigstein, Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. III. Shock and rarefaction waves in RG flows reveal limitations of the $N\rightarrow\infty$ limit in $O(N)$-type models, Phys. Rev. D 106, 065014 (2022), arXiv:2108.04037 [cond-mat.stat-mech] .
  * Wink [2020] N. Wink,  _Towards the spectral properties and phase structure of QCD_ , Phd thesis, University of Heidelberg (2020).
  * Wink [2019] N. Wink, Resolving phase transitions with Discontinuous Galerkin methods, Talk at the EMMI Workshop Functional Methods in Strongly Correlated Systems (2019), [Online; accessed 2020.11.24].
  * Steil  _et al._ [2020] M. J. Steil, A. Koenigstein, J. Braun, M. Buballa, E. Grossi, D. H. Rischke, and N. Wink, Numerical fluid dynamics for FRG-flow equations: Zero-dimensional QFTs as numerical test cases, Slides for the 10th International Conference on Exact Renormalization Group 2020 (ERG2020) (2020), [Online; accessed 2021.02.01].
  * Braun  _et al._ [2021a] J. Braun, M. Buballa, E. Grossi, A. Koenigstein, D. H. Rischke, M. J. Steil, and N. Wink, Reanalysis of the phase diagram of the quark meson model in local potential approximation via the FRG, in preparation (2021a).
  * Stoll  _et al._ [2021] J. Stoll, N. Zorbach, A. Koenigstein, M. J. Steil, and S. Rechenberger, Bosonic fluctuations in the $(1+1)$-dimensional Gross-Neveu(-Yukawa) model at varying $\mu$ and $T$ and finite $N$ (2021), arXiv:2108.10616 [hep-ph] .
  * Ihssen [2020] F. J. Ihssen,  _Low-energy effective models of QCD at finite temperatures and densities_ , master’s thesis, University of Heidelberg (2020).
  * Zumbach [1993] G. Zumbach, Almost second order phase transitions, Phys. Rev. Lett. 71, 2421 (1993).
  * Zumbach [1994a] G. Zumbach, The Local Potential Approximation of the Renormalization Group and its applications, Phys. Lett. A 190, 225 (1994a).
  * Zumbach [1994b] G. Zumbach, The Renormalization Group in the Local Potential Approximation and its applications to the $O(N)$ model, Nucl. Phys. B 413, 754 (1994b).
  * Hasenfratz and Hasenfratz [1986] A. Hasenfratz and P. Hasenfratz, Renormalization Group study of scalar field theories, Nucl. Phys. B 270, 687 (1986).
  * Felder [1987] G. Felder, Renormalization group in the local potential approximation, Comm. Math. Phys. 111, 101 (1987).
  * Lax [1973] P. D. Lax, 1. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, in  _Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves_ (Society for Industrial and Applied Mathematics, 1973) pp. 1–48.
  * Ames [1992] W. F. Ames,  _Numerical Methods for Partial Differential Equations_ , 3rd ed., Computer science and scientific computing (Academic Press, Boston [u.a.], 1992).
  * LeVeque [1992] R. J. LeVeque,  _Numerical methods for conservation laws_, 2nd ed. (Birkhäuser, Basel, 1992).
  * LeVeque [2002] R. J. LeVeque,  _Finite-volume methods for hyperbolic problems_, Cambridge Texts in Applied Mathematics (Cambridge University Press, 2002).
  * Hesthaven and Warburton [2007] J. S. Hesthaven and T. Warburton,  _Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications_, 1st ed. (Springer Publishing Company, Incorporated, 2007).
  * Toro [2009] E. F. Toro,  _Riemann Solvers and Numerical Methods for Fluid Dynamics_, 3rd ed. (Springer-Verlag, Berlin, Germany, 2009).
  * Rezzolla and Zanotti [2018] L. Rezzolla and O. Zanotti,  _Relativistic hydrodynamics_ (Oxford University Press, Oxford, England, UK, 2018).
  * Cotler and Rezchikov [2022] J. Cotler and S. Rezchikov, Renormalization Group Flow as Optimal Transport (2022), arXiv:2202.11737 [hep-th] .
  * Zamolodchikov [1986] A. B. Zamolodchikov, Irreversibility of the flux of the Renormalization Group in a $2D$ field theory, JETP Lett. 43, 730 (1986).
  * Rosten [2012] O. J. Rosten, Fundamentals of the Exact Renormalization Group, Phys. Rept. 511, 177 (2012), arXiv:1003.1366 [hep-th] .
  * Banks and Martinec [1987] T. Banks and E. J. Martinec, The Renormalization Group and String Field Theory, Nucl. Phys. B 294, 733 (1987).
  * Cardy [1988] J. L. Cardy, Is there a $c$-theorem in four dimensions?, Phys. Lett. B 215, 749 (1988).
  * Osborn [1989] H. Osborn, Derivation of a four dimensional $c$-theorem for renormaliseable quantum field theories, Phys. Lett. B 222, 97 (1989).
  * Jack and Osborn [1990] I. Jack and H. Osborn, Analogs for the $c$-theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343, 647 (1990).
  * Komargodski and Schwimmer [2011] Z. Komargodski and A. Schwimmer, On Renormalization Group flows in four dimensions, JHEP 12, 099, arXiv:1107.3987 [hep-th] .
  * Curtright  _et al._ [2012] T. L. Curtright, X. Jin, and C. K. Zachos, RG flows, cycles, and $c$-theorem folklore, Phys. Rev. Lett. 108, 131601 (2012), arXiv:1111.2649 [hep-th] .
  * Haagensen  _et al._ [1994] P. E. Haagensen, Y. Kubyshin, J. I. Latorre, and E. Moreno, Gradient flows from an approximation to the Exact Renormalization Group, Phys. Lett. B 323, 330 (1994), arXiv:hep-th/9310032 .
  * Generowicz  _et al._ [1997] J. Generowicz, C. Harvey-Fros, and T. R. Morris, $C$ function representation of the Local Potential Approximation, Phys. Lett. B 407, 27 (1997), arXiv:hep-th/9705088 .
  * Forte and Latorre [1998] S. Forte and J. I. Latorre, A proof of the irreversibility of Renormalization Group flows in four-dimensions, Nucl. Phys. B 535, 709 (1998), arXiv:hep-th/9805015 .
  * Codello  _et al._ [2014] A. Codello, G. D’Odorico, and C. Pagani, A Functional RG equation for the $c$-function, JHEP 07, 040, arXiv:1312.7097 [hep-th] .
  * Codello  _et al._ [2015] A. Codello, G. D’Odorico, and C. Pagani, Functional and local Renormalization Groups, Phys. Rev. D 91, 125016 (2015), arXiv:1502.02439 [hep-th] .
  * Becker and Reuter [2015] D. Becker and M. Reuter, Towards a $C$-function in $4D$ quantum gravity, JHEP 03, 065, arXiv:1412.0468 [hep-th] .
  * Becker [2016] D. Becker,  _Asymptotically safe quantum gravity: Bimetric actions, boundary terms, $\&$ a C-function_, Phd thesis, Johannes Gutenberg University Mainz (2016).
  * Lebowitz [2008] J. L. Lebowitz, Time’s arrow and Boltzmann’s entropy, Scholarpedia 3, 3448 (2008), revision #137152, [Online; accessed 2021.02.01].
  * Wetterich [1993a] C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993a), arXiv:1710.05815 [hep-th] .
  * Deligne  _et al._ [1999] P. Deligne, P. I. Etingof, D. S. Freed, and A. M. Society,  _Quantum fields and strings: a course for mathematicians_ , Vol. 1 (American Mathematical Society Providence, 1999).
  * Wilson [1971a] K. G. Wilson, Renormalization group and critical phenomena. 1. Renormalization group and the Kadanoff scaling picture, Phys. Rev. B 4, 3174 (1971a).
  * Wilson [1971b] K. G. Wilson, Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior, Phys. Rev. B 4, 3184 (1971b).
  * Weinberg [1976] S. Weinberg, Critical Phenomena for Field Theorists, in  _14th International School of Subnuclear Physics: Understanding the Fundamental Constitutents of Matter_ (1976).
  * Weinberg [1996a] S. Weinberg, What is quantum field theory, and what did we think it is?, in  _Conference on Historical Examination and Philosophical Reflections on the Foundations of Quantum Field Theory_ (1996) arXiv:hep-th/9702027 .
  * Percacci [2007] R. Percacci, Asymptotic Safety, in  _Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter_ , edited by D. Oriti (Cambridge University Press, Cambridge, 2007) Chap. 8, pp. 111–128, arXiv:0709.3851 [hep-th] .
  * Weinberg [2009] S. Weinberg, Living with Infinities (2009) arXiv:0903.0568 [hep-th] .
  * Niedermaier and Reuter [2006] M. Niedermaier and M. Reuter, The Asymptotic Safety Scenario in Quantum Gravity, Living Rev. Rel. 9, 5 (2006).
  * Bonanno  _et al._ [2020] A. Bonanno, A. Eichhorn, H. Gies, J. M. Pawlowski, R. Percacci, M. Reuter, F. Saueressig, and G. P. Vacca, Critical reflections on asymptotically safe gravity, Front. in Phys. 8, 269 (2020), arXiv:2004.06810 [gr-qc] .
  * Braun  _et al._ [2011] J. Braun, H. Gies, and D. D. Scherer, Asymptotic safety: a simple example, Phys. Rev. D 83, 085012 (2011), arXiv:1011.1456 [hep-th] .
  * Jakovác  _et al._ [2015] A. Jakovác, A. Patkós, and P. Pósfay, Non-Gaussian fixed points in fermionic field theories without auxiliary Bose-fields, Eur. Phys. J. C 75, 2 (2015), arXiv:1406.3195 [hep-th] .
  * Kurganov and Tadmor [2000] A. Kurganov and E. Tadmor, New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection–Diffusion Equations, Journal of Computational Physics 160, 241 (2000).
  * Harten [1983] A. Harten, High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics 49, 357 (1983).
  * Pawlowski [2007] J. M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322, 2831 (2007), arXiv:hep-th/0512261 .
  * Berges  _et al._ [2002] J. Berges, N. Tetradis, and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363, 223 (2002), arXiv:hep-ph/0005122 .
  * Kopietz  _et al._ [2010] P. Kopietz, L. Bartosch, and F. Schütz,  _Introduction to the Functional Renormalization Group_, Lecture Notes in Physics, Vol. 798 (Springer-Verlag Berlin Heidelberg, 2010).
  * Gies [2012] H. Gies, Introduction to the Functional RG and applications to gauge theories, Lect. Notes Phys. 852, 287 (2012), arXiv:hep-ph/0611146 .
  * [61] J. M. Pawlowski, J. A. Bonnet, S. Rechenberger, M. Reichert, and N. Wink, The functional renormalization group - applications to gauge theories and gravity, unpublished lecture notes, in preparation.
  * Delamotte [2012] B. Delamotte, An introduction to the nonperturbative Renormalization Group, Lect. Notes Phys. 852, 49 (2012), arXiv:cond-mat/0702365 .
  * Ellwanger [1994] U. Ellwanger, Flow equations for $N$ point functions and bound states, Z. Phys. C 62, 503 (1994), arXiv:hep-ph/9308260 .
  * Morris [1994] T. R. Morris, The Exact Renormalization Group and approximate solutions, Int. J. Mod. Phys. A 09, 2411 (1994), arXiv:hep-ph/9308265 .
  * Wetterich [1993b] C. Wetterich, The average action for scalar fields near phase transitions, Z. Phys. C 57, 451 (1993b).
  * Reuter and Wetterich [1994] M. Reuter and C. Wetterich, Effective average action for gauge theories and exact evolution equations, Nucl. Phys. B 417, 181 (1994).
  * Reuter [1998] M. Reuter, Nonperturbative evolution equation for quantum gravity, Phys. Rev. D 57, 971 (1998), arXiv:hep-th/9605030 .
  * Reuter and Saueressig [2002] M. Reuter and F. Saueressig, Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation, Phys. Rev. D 65, 065016 (2002), arXiv:hep-th/0110054 .
  * Braun  _et al._ [2010] J. Braun, H. Gies, and J. M. Pawlowski, Quark confinement from color confinement, Phys. Lett. B 684, 262 (2010), arXiv:0708.2413 [hep-th] .
  * Litim [2000] D. F. Litim, Optimization of the exact renormalization group, Phys. Lett. B 486, 92 (2000), arXiv:hep-th/0005245 .
  * Pawlowski  _et al._ [2018] J. M. Pawlowski, N. Strodthoff, and N. Wink, Finite temperature spectral functions in the $O(N)$-model, Phys. Rev. D 98, 074008 (2018), arXiv:1711.07444 [hep-th] .
  * Braun  _et al._ [2021b] J. Braun, T. Dörnfeld, B. Schallmo, and S. Töpfel, Renormalization group studies of dense relativistic systems, Phys. Rev. D 104, 096002 (2021b), arXiv:2008.05978 [hep-ph] .
  * Osborn and Twigg [2012] H. Osborn and D. E. Twigg, Remarks on Exact RG equations, Annals Phys. 327, 29 (2012), arXiv:1108.5340 [hep-th] .
  * Weinberg [1996b] S. Weinberg,  _The quantum theory of fields: Modern applications_, Vol. 2 (Cambridge University Press, Cambridge, England, UK, 1996).
  * Peskin and Schroeder [1995] M. E. Peskin and D. V. Schroeder,  _An introduction to quantum field theory_ (Addison-Wesley, Reading, USA, 1995).
  * Zinn-Justin [2002] J. Zinn-Justin,  _Quantum field theory and critical phenomena_, 4th ed., Int. Ser. Monogr. Phys., Vol. 113 (Oxford University Press, 2002) pp. 1–1054, a Clarendon Press Publication.
  * Iliopoulos  _et al._ [1975] J. Iliopoulos, C. Itzykson, and A. Martin, Functional methods and perturbation theory, Rev. Mod. Phys. 47, 165 (1975).
  * DeWitt [1965] B. S. DeWitt,  _Dynamical theory of groups and fields_ (Gordon and Breach, New York, USA, 1965).
  * Greiner and Reinhardt [1996] W. Greiner and J. Reinhardt,  _Field quantization_ (Springer, Berlin Heidelberg, 1996).
  * Wipf [2013] A. Wipf,  _Statistical Approach to Quantum Field Theory_, Lect.Notes Phys. No. 864 (Springer-Verlag, Berlin, Germany, 2013).
  * Polchinski [1984] J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys. B 231, 269 (1984).
  * Wegner and Houghton [1973] F. J. Wegner and A. Houghton, Renormalization Group equation for critical phenomena, Phys. Rev. A 8, 401 (1973).
  * Brydges and Kennedy [1987] D. C. Brydges and T. Kennedy, Mayer expansions and the Hamilton-Jacobi equation, J. Stat. Phys. 48, 19 (1987).
  * Hikami and Brezin [1979] S. Hikami and E. Brezin, Large order behavior of the $1/N$ expansion in zero and one dimensions, J. Phys. A 12, 759 (1979).
  * Bessis  _et al._ [1980] D. Bessis, C. Itzykson, and J.-B. Zuber, Quantum field theory techniques in graphical enumeration, Adv. Appl. Math. 1, 109 (1980).
  * Di Vecchia  _et al._ [1991] P. Di Vecchia, M. Kato, and N. Ohta, Double scaling limit in $O(N)$ vector models, Nucl. Phys. B 357, 495 (1991).
  * Nishigaki and Yoneya [1991] S. Nishigaki and T. Yoneya, A nonperturbative theory of randomly branching chains, Nucl. Phys. B 348, 787 (1991).
  * Schelstraete and Verschelde [1994] S. Schelstraete and H. Verschelde, Large $N$ limit of $O(N)$ vector models, Phys. Lett. B 332, 36 (1994), arXiv:hep-th/9405158 .
  * Zinn-Justin [1998] J. Zinn-Justin, Vector models in the large $N$ limit: A Few applications, in  _11th Taiwan Spring School on Particles and Fields_ (1998) arXiv:hep-th/9810198 .
  * Flörchinger [2010] S. Flörchinger, Functional Renormalization and Ultracold Quantum Gases, Springer Theses 10.1007/978-3-642-14113-3 (2010).
  * Moroz [2011] S. Moroz,  _Few-body physics with functional renormalization_ , Phd thesis, University of Heidelberg (2011).
  * Keitel and Bartosch [2012] J. Keitel and L. Bartosch, The zero-dimensional $O(N)$ vector model as a benchmark for perturbation theory, the large-$N$ expansion and the Functional Renormalization Group, J. Phys. A45, 105401 (2012), arXiv:1109.3013 [cond-mat.stat-mech] .
  * Strocchi [2013] F. Strocchi,  _An introduction to non-perturbative foundations of quantum field theory_, Vol. 158 (Oxford University Press, Oxford, 2013).
  * Kemler and Braun [2013] S. Kemler and J. Braun, Towards a Renormalization Group approach to density functional theory – general formalism and case studies, J. Phys. G 40, 085105 (2013), arXiv:1304.1161 [nucl-th] .
  * Pawlowski [2013] J. M. Pawlowski, Solving integrals with flow equations, Slides for the lecture Non-perturbative aspects of gauge theories winter term 2012/2013 (2013), [Online; accessed 2020.10.29].
  * Rentrop  _et al._ [2015] J. F. Rentrop, S. G. Jakobs, and V. Meden, Two-particle irreducible Functional Renormalization Group schemes – a comparative study, Journal of Physics A: Mathematical and Theoretical 48, 145002 (2015).
  * Rosa  _et al._ [2016] D. S. Rosa, R. L. S. Farias, and R. O. Ramos, Reliability of the optimized perturbation theory in the 0-dimensional $O(N)$ scalar field model, Physica A 464, 11 (2016), arXiv:1604.00537 [hep-ph] .
  * Liang  _et al._ [2018] H. Liang, Y. Niu, and T. Hatsuda, Functional Renormalization Group and Kohn-Sham scheme in density functional theory, Phys. Lett. B 779, 436 (2018), arXiv:1710.00650 [cond-mat.str-el] .
  * Skinner [2018] D. Skinner, Lecture notes: Quantum Field Theory II (2018), [Online; accessed 2021.01.12].
  * Millington and Saffin [2019] P. Millington and P. M. Saffin, Visualising quantum effective action calculations in zero dimensions, J. Phys. A 52, 405401 (2019), arXiv:1905.09674 [hep-th] .
  * Alexander  _et al._ [2019] E. Alexander, P. Millington, J. Nursey, and P. M. Saffin, Alternative flow equation for the functional renormalization group, Phys. Rev. D 100, 101702 (2019), arXiv:1907.06503 [hep-th] .
  * Catalano [2019] A. G. Catalano,  _Application of renormalization group techniques to the solution of integrals and Schrödinger eigenvalue equations_ , master’s thesis, Politecnico di Torino (2019).
  * Millington [2020] P. Millington, An alternative flow equation from the regulator-sourced $2$PI effective action, Talk at the 10th International Conference on Exact Renormalization Group 2020 (ERG2020) (2020), [Online; accessed 2021.01.12].
  * Millington and Saffin [2021] P. Millington and P. M. Saffin, Benchmarking regulator-sourced 2PI and average 1PI flow equations in zero dimensions, J. Phys. A 54, 465401 (2021), arXiv:2107.12914 [hep-th] .
  * Kades  _et al._ [2021] L. Kades, M. Gärttner, T. Gasenzer, and J. M. Pawlowski, Towards sampling complex actions (2021), arXiv:2106.09367 [hep-lat] .
  * Schaefer  _et al._ [2002] B.-J. Schaefer, O. Bohr, and J. Wambach, Finite temperature gluon condensate with renormalization group flow equations, Phys. Rev. D 65, 105008 (2002), arXiv:hep-th/0112087 .
  * Braun  _et al._ [2019] J. Braun, M. Leonhardt, and J. M. Pawlowski, Renormalization group consistency and low-energy effective theories, SciPost Phys. 6, 056 (2019), arXiv:1806.04432 [hep-ph] .
  * Cichutek  _et al._ [2020] N. Cichutek, F. Divotgey, and J. Eser, Fluctuation-induced higher-derivative couplings and infrared dynamics of the quark-meson-diquark model, Phys. Rev. D 102, 034030 (2020), arXiv:2006.12473 [hep-ph] .
  * Mermin and Wagner [1966] N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966).
  * Hohenberg [1967] P. C. Hohenberg, Existence of long-range order in one and two dimensions, Phys. Rev. 158, 383 (1967).
  * Coleman [1973] S. R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31, 259 (1973).
  * Salmhofer [2020] M. Salmhofer, Rigorous Renormalization Group, Talk at the 10th International Conference on Exact Renormalization Group 2020 (ERG2020) (2020), [Online; accessed 2021.01.12].
  * Aoki  _et al._ [2018] K.-I. Aoki, S.-I. Kumamoto, and M. Yamada, Phase structure of NJL model with weak renormalization group, Nucl. Phys. B 931, 105 (2018), arXiv:1705.03273 [hep-th] .
  * Rankine [1870] W. J. M. Rankine, On the thermodynamic theory of waves of finite longitudinal disturbance, Phil. Trans. R. Soc. 160, 277 (1870).
  * Hugoniot [1998] P.-H. Hugoniot, On the propagation of motion in bodies and in perfect gases in particular – $\mathrm{I}$, in  _Classic papers in shock compression science_, edited by J. N. Johnson and R. Chéret (Springer, New York, 1998) pp. 161–243, translation of: Hugoniot, Pierre-Henri, Journal de l‘École Polytechnique, Vol. 57, pp. 3 – 97 (1887).
  * Tetradis and Litim [1996] N. Tetradis and D. F. Litim, Analytical solutions of exact renormalization group equations, Nucl. Phys. B 464, 492 (1996), arXiv:hep-th/9512073 .
  * Aoki  _et al._ [2014] K.-I. Aoki, S.-I. Kumamoto, and D. Sato, Weak solution of the non-perturbative renormalization group equation to describe dynamical chiral symmetry breaking, PTEP 2014, 043B05 (2014), arXiv:1403.0174 [hep-th] .
  * Bateman [1915] H. Bateman, Some recent researches on the motion of fluids, Monthly Weather Review 43, 163 (1915).
  * Burgers [1948] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics 1, 171 (1948).
  * Cannon [1984] J. R. Cannon,  _The one-dimensional heat equation_ (Cambridge University Press, 1984).
  * Pangon  _et al._ [2011] V. Pangon, S. Nagy, J. Polonyi, and K. G. Sailer, Onset of symmetry breaking by the Functional RG method, Int. J. Mod. Phys. A 26, 1327 (2011), arXiv:0907.0144 [hep-th] .
  * Borchardt and Knorr [2015] J. Borchardt and B. Knorr, Global solutions of functional fixed point equations via pseudospectral methods, Phys. Rev. D 91, 105011 (2015), [Erratum: Phys.Rev.D 93, 089904 (2016)], arXiv:1502.07511 [hep-th] .
  * Caillol [2012] J.-M. Caillol, The non-perturbative renormalization group in the ordered phase, Nucl. Phys. B 855, 854 (2012), arXiv:1109.4024 [cond-mat.stat-mech] .
  * Borchardt and Knorr [2016] J. Borchardt and B. Knorr, Solving functional flow equations with pseudo-spectral methods, Phys. Rev. D 94, 025027 (2016), arXiv:1603.06726 [hep-th] .
  * Redheffer and Walter [1974] R. M. Redheffer and W. Walter, The total variation of solutions of parabolic differential equations and a maximum principle in unbounded domains, Math. Ann. 209, 57 (1974).
  * Wolfram Research, Inc. [2020] Wolfram Research, Inc., Mathematica, Version 12.1 (2020), Champaign, IL.
  * Fujimoto  _et al._ [1983] Y. Fujimoto, L. O’Raifeartaigh, and G. Parravicini, Effective potential for non-convex potentials, Nucl. Phys. B 212, 268 (1983).
  * Zinn-Justin [2010] J. Zinn-Justin, Critical Phenomena: field theoretical approach, Scholarpedia 5, 8346 (2010), revision #148508.
  * Nambu [1960] Y. Nambu, Quasiparticles and gauge invariance in the theory of superconductivity, Phys. Rev. 117, 648 (1960).
  * Goldstone [1961] J. Goldstone, Field theories with superconductor solutions, Nuovo Cim. 19, 154 (1961).
  * Goldstone  _et al._ [1962] J. Goldstone, A. Salam, and S. Weinberg, Broken symmetries, Phys. Rev. 127, 965 (1962).
  * Politzer [1973] H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
  * Gross and Wilczek [1973a] D. J. Gross and F. A. Wilczek, Ultraviolet behavior of non-Abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973a).
  * Gross and Wilczek [1973b] D. J. Gross and F. A. Wilczek, Asymptotically free gauge theories I, Phys. Rev. D 8, 3633 (1973b).
  * Gross and Wilczek [1974] D. J. Gross and F. A. Wilczek, Asymptotically free gauge theories II, Phys. Rev. D 9, 980 (1974).
  * Eser  _et al._ [2018] J. Eser, F. Divotgey, M. Mitter, and D. H. Rischke, Low-energy limit of the $O(4)$ quark-meson model from the functional renormalization group approach, Phys. Rev. D 98, 014024 (2018), arXiv:1804.01787 [hep-ph] .
  * Eser  _et al._ [2019] J. Eser, F. Divotgey, and M. Mitter, Low-energy limit of the $O(4)$ quark-meson model, PoS CD2018, 060 (2019), arXiv:1902.04804 [hep-ph] .
  * Divotgey  _et al._ [2019] F. Divotgey, J. Eser, and M. Mitter, Dynamical generation of low-energy couplings from quark-meson fluctuations, Phys. Rev. D 99, 054023 (2019), arXiv:1901.02472 [hep-ph] .
  * Wilbraham [1848] H. Wilbraham, On a certain periodic function, Cambridge and Dublin Mathematical Journal 3, 198 (1848).
  * Gibbs [1898] J. W. Gibbs, Fourier’s series, Nature 59, 200 (1898).
  * Gibbs [1899] J. W. Gibbs, Fourier’s series, Nature 59, 606 (1899).
  * Boyd [2001] J. P. Boyd,  _Chebyshev and Fourier spectral methods_, 2nd ed. (Dover Publications, Mineola, New York, 2001) 1st ed. in 1969 by Springer-Verlag, Berlin Heidelberg, in the series Lecture Notes in Engeneering, Vol. 49.
  * Pangon [2012] V. Pangon, Structure of the broken phase of the sine-Gordon model using Functional Renormalization, Int. J. Mod. Phys. A 27, 1250014 (2012), arXiv:1008.0281 [hep-th] .
  * Monthé [2001] L. Monthé, A study of splitting scheme for hyperbolic conservation laws with source terms, Journal of Computational and Applied Mathematics 137, 1 (2001).
  * Beneito and Gavara [2008] R. D. Beneito and A. M. Gavara, A ‘TVD-like’ Scheme for Conservation Laws with Source Terms, in  _Numerical Mathematics and Advanced Applications_ (Springer, Berlin, Germany, 2008) pp. 265–272.
  * Chen and Mao [2011] R. Chen and D.-k. Mao, Entropy-TVD Scheme for Nonlinear Scalar Conservation Laws, J. Sci. Comput. 47, 150 (2011).
  * Bessemoulin-Chatard and Filbet [2012] M. Bessemoulin-Chatard and F. Filbet, A Finite Volume Scheme for Nonlinear Degenerate Parabolic Equations, SIAM Journal on Scientific Computing 34, B559–B583 (2012).
  * Boltzmann [2003] L. Boltzmann, Further studies on the thermal equilibrium of gas molecules, in  _The Kinetic Theory of Gases_ (2003) pp. 262–349, [Originally published under the title ”Weitere Studien über das Wörmegleichgewicht unter Gasmolekülen”, in Sitzungsberichte Akad. Wiss., Vienna, part II, 66, 275–370 (1872); reprinted in Boltzmann’s Wissenschaftliche Abhandlungen, Vol. I, Leipzig, J. A. Barth, 1909, pp. 316–402].
  * Gibbs [2010] J. W. Gibbs,  _Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics_ (Cambridge University Press, Cambridge, England, UK, 2010).
  * Tolman [1979] R. C. Tolman,  _The Principles of Statistical Mechanics (Dover Books on Physics)_ (Dover Publications Inc., Mineola, NY, USA, 1979).
  * Lang [2020] L. Lang, Wolfram Mathematica function repository: ResourceFunction PlotGrid (2020), [Online; accessed 2020.10.08].
  * Popkov [2020] A. Popkov, Wolfram Mathematica function repository: ResourceFunction PolygonMarker (2020), [Online; accessed 2020.10.08].
  * Horvát [2020] S. Horvát, MaTeX (2020), [Online; accessed 2020.10.08].
  * Collins and Vermaseren [2016] J. C. Collins and J. A. Vermaseren, Axodraw version 2 (2016), arXiv:1606.01177 [cs.OH] .