#  Uncertainty Relations in Thermodynamics of Irreversible Processes on a
Mesoscopic Scale

Giorgio SONNINO  Université Libre de Bruxelles (U.L.B.)
Department of Theoretical Physics and Mathematics
Campus de la Plaine C.P. 231 - Bvd du Triomphe
B-1050 Brussels - Belgium
Email<EMAIL_ADDRESS>
###### Abstract

Studies of mesoscopic structures have now become a leading and rapidly
evolving research field ranging from physics, chemistry, and mineralogy to
life sciences. The increasing miniaturization of devices with length scales of
a few nanometers is leading to radical changes not only in the realization of
new materials but also in shedding light on our understanding of the
fundamental laws of nature that govern the dynamics of systems at the
mesoscopic scale. On the basis of recent experimental results and previous
theoretical research, we investigate thermodynamic processes in small systems
in Onsager’s region. We show that fundamental quantities such as the total
entropy production, the thermodynamic variables conjugate to the thermodynamic
forces, and the Glansdorff-Prigogine’s dissipative variable may be discretized
at the mesoscopic scale. We establish the canonical commutation rules (ccr)
valid at the mesoscopic scale. The numerical value of the discretization
constant is estimated experimentally. The ultraviolet divergence problem is
solved by applying the correspondence principle with Einstein-Prigogine’s
fluctuations theory in the limit of macroscopic systems.
PACS numbers: 05.60.Cd; 03.70.+k; 05.70.Ln; 89.75.-k.

Key Words: Mesoscopic Scale; Canonical Commutation Rules; Thermodynamics of
Irreversible Processes; Onsager’s Theory, Complex Systems.

##  I Introduction

The physical world can exhibit different behaviors and be described by
different laws depending on the scale of observation. This is known as scale-
dependent behavior or scale-dependent laws. The laws that govern macroscopic
systems may not necessarily apply directly to mesoscopic or microscopic
systems, and vice-versa. Classical physics laws, such as Newtonian mechanics,
thermodynamics, and electromagnetism, are typically applicable at the
macroscopic scale. These laws have been extensively validated and are
effective in describing the behavior of objects and phenomena observable to
the naked eye. They provide a good approximation for many everyday situations.
However, as we move to smaller scales, such as the mesoscopic and microscopic
levels, quantum mechanics becomes increasingly relevant. Quantum mechanics
governs the behavior of particles at the atomic, molecular, and subatomic
scales. It introduces concepts like wave-particle duality, quantum
entanglement, and uncertainty. Quantum effects can also manifest in mesoscopic
systems, where collective behavior or quantum coherence comes into play. In
certain cases, there can be a smooth transition between classical and quantum
behavior as we move from the macroscopic to the microscopic scale. This is
known as the correspondence principle, where classical laws emerge from
quantum mechanics in the limit of large systems and high energies. However,
emergent phenomena, as seen at the mesoscopic scale, can arise due to the
collective behavior of a large number of particles. These phenomena may not
have direct counterparts in either classical or quantum mechanics alone, and
they often require to establish a new theoretical framework to describe them.
A classification, albeit generic, of these three scales can be the following.

i) Macroscopic Scale. The macroscopic scale refers to the largest scale of
observation, where objects or systems are observed and analyzed on a human
scale. It involves the study of phenomena that are visible to the naked eye
and can be measured directly. Examples of macroscopic scale observations
include the behavior of everyday objects, the motion of planets, or the flow
of fluids. At this scale, classical physics laws, such as Newtonian mechanics
and thermodynamics, are often applicable;

ii) Mesoscopic Scale. The mesoscopic scale is an intermediate scale between
the macroscopic and microscopic scales. It refers to the study of systems or
objects that are larger than individual atoms or molecules but smaller than
what can be considered macroscopic. The mesoscopic scale often deals with
phenomena that arise from the collective behavior of a large number of
particles or constituents, exhibiting properties that are not observed at
larger or smaller scales. Examples of mesoscopic phenomena include quantum
dots, nanoscale devices, and certain biological systems. At this scale, both
classical and quantum mechanics may be applicable, but not always, depending
on the specific system.

iii) Microscopic Scale. The microscopic scale involves the study of objects or
systems at the smallest level of observation, where individual atoms,
molecules, or subatomic particles are examined. It deals with phenomena
governed by quantum mechanics and involves understanding fundamental
particles’ behavior, interactions, and properties. Examples of microscopic
scale studies include molecular dynamics simulations, atomic structure
analysis, or particle physics experiments.

Currently several research activities are focused on seeking fundamental laws
and understanding the behavior of systems at the mesoscopic scale. Scientists
and researchers explore mesoscopic phenomena to gain insights into the
emergence of novel properties and to bridge the gap between macroscopic and
microscopic descriptions. One area of research is mesoscopic physics, which
investigates systems that are larger than individual atoms or molecules but
smaller than macroscopic objects bachtold , cleland1 , cleland2 , ekinci ,
lifshitz . This field aims to understand how collective behavior and quantum
effects manifest at the mesoscopic scale. Various experimental techniques,
such as nanofabrication and scanning probe microscopy, are employed to study
and manipulate mesoscopic systems. Another active research area is quantum
mesoscopics, which explores mesoscopic systems exhibiting quantum coherence
and interference effects poot , seifert , jarzynski , sagawa , berut , ribezzi
, dago . Researchers investigate phenomena like quantum transport, quantum
dots, and mesoscopic superconductivity to uncover the fundamental laws that
govern these systems. In this case, advanced experimental techniques and
theoretical models rooted in quantum mechanics are employed. Furthermore,
research in condensed matter physics often delves into mesoscopic systems to
investigate emergent phenomena dinis , julicher , sheth , salle , shilo . This
involves studying the behavior of materials with nanoscale dimensions or
complex structures, where mesoscopic effects can play a crucial role. For
example, the study of topological materials and quantum Hall systems explores
the emergence of novel states of matter at the mesoscopic scale. Additionally,
studies on nanomechanical vibrational systems have also opened a new direction
in the field. Nanomechanical vibrational systems are resonators for mechanical
vibrations, and by their nature they are mesoscopic aspelmeyer , schmid ,
steeneken . Overall, the research activities in these fields aim to uncover
fundamental principles and laws that govern the behavior of systems at the
mesoscopic scale, bridging the gap between the macroscopic and microscopic
worlds and enhancing our understanding of complex phenomena.

The main objective of this work is to investigate, and consequently try to
establish, the thermodynamic laws governing nonequilibrium mesoscopic
structures in Onsager’s region. We shall see that it is technically possible
to propose canonical commutation rules (ccr) for the pair of variables (time,
entropic production) and (thermodynamic variable, conjugate thermodynamic
force). This is motivated by the experimental results: At the mesoscopic
scale, it is not possible to measure the values of transport coefficients and
time with infinite precision simultaneously. Then by extension, it is also not
possible to measure the values of entropy production and time with infinite
precision simultaneously.

The manuscript is organized as follows. For easy reference, section II
summarizes the general formalism developed by I. Prigogine and L. Onsager for
treating transport processes in thermodynamic systems close to equilibrium.
The canonical commutation rules for thermodynamic systems in Onsager’s at the
mesoscopic scale are proposed in section III. The numerical value of the
discretization constant is estimated by using experimental findings in roldan
. Detailed calculations in the Onsager region, leading to the discretization
of the key thermodynamic quantities (such as the total entropy production, the
thermodynamic variables conjugate to its thermodynamic forces, and the
Glansdorff-Prigogine dissipative quantity) are reported in section IV. In
section V the correspondence limit to Einstein-Prigogine’s theory of
fluctuations is investigated. In section VI we show that the formalism of the
canonical commutation rules does not affect the validity of Onsager’s
reciprocity relation as long as the flux-force transport relations remain
linear. The effect of the volume of the system on the canonical commutation
rules is discussed in section VII. Concluding remarks with a brief
introduction on future works being currently finalized can be found in section
VIII.

##  II The Onsager-Prigogine Formalism for Systems out of Thermodynamic
Equilibrium

In order to establish vocabulary, for easy reference, it is useful to recall,
very briefly, the formalism used by L. Onsager and I. Prigogine to deal with
the thermodynamics of systems out-of-equilibrium.

###  II.1 Prigogine’s Theory

L. Boltzmann deals with isolated systems, which satisfy the following law: The
total entropy of an isolated system tends to increase over time and reaches
thermodynamic equilibrium when the total entropy reaches its maximum value. I.
Prigogine deals with open systems and his theory also includes isolated
systems as a special case prigogine1 , prigogine2 , prigogine3 , prigogine4 .
Exploiting the additivity property of entropy, Prigogine writes the entropy
balance as

$dS=d_{e}S+d_{I}S$ (1)

where $dS$ is the total entropy variation, $d_{e}S$ the entropic flow i.e.,
the entropy contribution transmitted reversibly, and $d_{I}S$ denotes the
entropic production. Entropy production provides an irreversible contribution
and refers to the generation or increase in entropy within a system. By the
second law of thermodynamics $d_{I}S\geq 0$. $dS$, $d_{e}S$ and $d_{I}S$ have
the dimensions of Energy/Temperature. Note that Prigogine writes $d_{I}S$ and
not $dS_{I}$ because $d_{I}S$ is not an exact differential. Let us now
consider a system characterized by $n$ degrees of advancement
$\xi_{1},\cdots,\xi_{n}$. The deviations of $\xi_{i}$ from the values
$\xi_{\mu}^{eq.}$, assumed by the degrees of advance when the system is in
local equilibrium, are denoted by ${\bar{\alpha}}_{\mu}$, that is
${\bar{\alpha}}_{\mu}\equiv\xi_{\mu}-\xi^{eq.}_{i}$. Note that the
${\bar{\alpha}}_{\mu}$ can also represent the fluctuations of the various
thermodynamic quantities (for example, temperature, pressure, etc.).The
entropy variation due to the fluctuations is

$\Delta S=\Delta_{e}S+\Delta_{I}S$ (2)

where

$\Delta S=\int_{\xi^{eq.}}^{\xi}dS=S(\xi)-S(\xi^{eq.})\ \ ;\ \
\Delta_{e}S=\int_{\xi^{eq.}}^{\xi}d_{e}S\ \ ;\ \
\Delta_{I}S=\int_{\xi^{eq.}}^{\xi}d_{I}S$ (3)

####  II.1.1 Homogeneous Systems

For a non-spatially extended system, the thermodynamic forces are defined as:

${\tilde{X}}^{\mu}=\frac{\partial\Delta_{I}S}{\partial{\bar{\alpha}}_{\mu}}\ $
(4)

Note that In the Boltzmann case, we get

${\tilde{X}}^{\mu}=\frac{\partial\Delta
S}{\partial{\bar{\alpha}}_{\mu}}=\frac{\partial
S}{\partial{\bar{\alpha}}_{\mu}}$ (5)

since for an isolated system, we have $\Delta_{e}S=0$.

####  II.1.2 Spatially Extended Systems

Let us introduce the thermodynamic forces

$x^{\mu}=\frac{\partial\Delta_{I}s}{\partial{\bar{\alpha}}_{\mu}}$ (6)

where $\Delta_{I}s$ is the the entropy production per unit volume having
dimension [$\Delta_{I}s$]=[Energy]/([Temperature] $\times$[Volume]). Let us
now rewrite Eq. (1) as

$\frac{dS}{dt}=\frac{d_{e}S}{dt}+\frac{d_{I}S}{dt}$ (7)

where

$\frac{dS}{dt}=\int_{V}\partial_{t}s\
dV\quad;\quad\frac{d_{e}S}{dt}=-\int_{V}\nabla\cdot{\bf j}_{s}\ dV\quad{\rm
and}\quad\frac{d_{I}S}{dt}=\int_{V}\sigma\ dV$ (8)

Here, ${\bf j}_{s}$ is the (reversible) entropy flux and $\sigma$ the entropy
production strength. $dV$ denotes an infinitesimal spatial volume element and
the integrals are performed on the whole volume $V$ occupied by the system.
From Eq. (8) we get

$\partial_{t}s=-\nabla\cdot{\bf j}_{s}+{\bar{\sigma}}$ (9)

Eq. (8) also gives

${\bar{\sigma}}=\frac{\delta}{\delta
V}\Bigl{(}\frac{d\Delta_{I}S}{dt}\Bigr{)}=\frac{d}{dt}\Bigl{(}\frac{\delta\Delta_{I}S}{\delta
V}\Bigr{)}=\frac{\partial\Delta_{I}s}{\partial{\bar{\alpha}}_{\mu}}\frac{d{\bar{\alpha}}_{\mu}}{dt}=x^{\mu}j_{\mu}\quad{\rm
with}\quad j_{\mu}\equiv\frac{d{\bar{\alpha}}_{\mu}}{dt}$ (10)

where Eq. (6) has been taken into account. $j_{\mu}$ denotes the thermodynamic
fluxes conjugate to the thermodynamic force $x^{\mu}$ fitts , gyarmati . In
Eq. (10) we have adopted the Einstein convention of repeated indices. Unless
stated otherwise, this convention will also be adopted in the sequel of this
manuscript. Note that at local equilibrium (i.e., when $\alpha_{\mu}=0$), we
get

$x^{\mu}=\frac{\partial\Delta_{I}s}{\partial\alpha_{\mu}}\mid_{\alpha_{\mu}=0}=0$
(11)

To work with entropy production strength, which has dimension
[Energy]/([Temperature] $\times$ [time]), we adopt the following definition
for the thermodynamic forces, the thermodynamic fluxes, and the thermodynamic
variables, respectively:

$X^{\mu}({\bf r},t)\equiv\sqrt{V}x^{\mu}\quad;\quad J_{\mu}({\bf
r},t)\equiv\sqrt{V}j_{\mu}\quad;\quad\alpha_{\mu}({\bf
r},t)\equiv\sqrt{V}{\bar{\alpha}}_{\mu}$ (12)

with (${\bf r},t$) denoting the space-time. Eq. (10) links the entropy
production strength with the thermodynamic forces and the conjugate fluxes. To
obtain the expression for the entropy production strength solely in terms of
the thermodynamic forces, it is necessary to relate the dissipative fluxes to
the thermodynamic forces that produce them. These closure relations are called
transport flux-force relations. The study of these relations is the object of
non-equilibrium thermodynamics. The most used transport relations are balescu1
, balescu2 , vidal , ottinger

$J_{\mu}=\tau_{\mu\nu}X^{\nu}$ (13)

with $\tau_{\mu\nu}$ denoting the transport coefficients. Note that to perform
calculations, the transport coefficients must be written in a dimensionless
form (see the forthcoming subsection and sonnino1 ). In terms of the transport
coefficients, the local entropy production strength can be brought into the
form

$\sigma=g_{\mu\nu}X^{\mu}X^{\nu}=Vg_{\mu\nu}x^{\mu}x^{\nu}=V{\bar{\sigma}}$
(14)

with $g_{\mu\nu}$ denoting the symmetric piece of the transport coefficients.
So, as we wish, $\sigma$ has dimension [Energy]/([Temperature]$\times$[time])
while ${\bar{\sigma}}$ has dimension
[Energy]/([Temperature]$\times$[time]$\times$[Volume]).

###  II.2 Onsager’s Theory

Close to equilibrium, the transport equations of a thermodynamic system are
provided by the well-known Onsager theory onsager1 , onsager2 . The Onsager
transport relations are

$J_{\mu}=\tau_{0\mu\nu}X^{\nu}$ (15)

where $\tau_{0\mu\nu}$ are the transport coefficients. In this equation, the
Einstein summation convention on the repeated indexes is adopted. Matrix
$\tau_{0\mu\nu}$ can be decomposed into a sum of two matrices, one symmetric
and the other skew-symmetric, denoted by $L_{\mu\nu}$ and $f_{0\mu\nu}$,
respectively i.e.,

$\tau_{0\mu\nu}=L_{\mu\nu}+f_{0\mu\nu}\qquad{\rm with}\quad
L_{\mu\nu}=L_{\nu\mu}\quad;\quad f_{0\mu\nu}=-f_{0\nu\mu}$ (16)

The second law of thermodynamics imposes that $L_{\mu\nu}$ be a positive
definite matrix. The most important property of Eqs (15) is that near
equilibrium, the coefficients $\tau_{\mu\nu}$ are independent of the
thermodynamic forces, i.e.,

$\frac{\partial\tau_{0\mu\nu}}{\partial X^{\lambda}}=\frac{\partial
L_{\mu\nu}}{\partial X^{\lambda}}=\frac{\partial f_{0\mu\nu}}{\partial
X^{\lambda}}=0$ (17)

The region where Eqs (15) and (17) hold, is called Onsager’s region or, the
linear region of thermodynamics. A well-founded microscopic explanation of the
validity of the linear phenomenological laws was developed by Onsager in 1931
onsager1 , onsager2 . Onsager’s theory is based on three assumptions: i) The
probability distribution function for the fluctuations of thermodynamic
quantities (Temperature, pressure, degree of advancement of a chemical
reaction, etc.) is a Maxwellian ii) Fluctuations decay according to a linear
law and iii) The principle of the detailed balance (or the microscopic
reversibility) is satisfied. Onsager showed the equivalence of Eqs (15) and
(17) with the assumptions i)-iii) (assumption iii) allows deriving the
reciprocity relations $L_{\mu\nu}=L_{\nu\mu}$). In Onsager’s region, the
expression for the local entropy production strength reads

$\sigma=L_{\mu\nu}X^{\mu}X^{\nu}=L^{\mu\nu}J_{\mu}J_{\nu}$ (18)

###  II.3 The Minimum Entropy Production Theorem

In 1947, I. Prigogine proved the Minimum Entropy Production Theorem prigogine1
, prigogine2 , concerning the relaxation of thermodynamic systems near
equilibrium. This theorem states that: Minimum Entropy Production Theorem

A thermodynamic system, near equilibrium, relaxes to a steady state in such a
way that the inequality

$\int_{V}\partial_{t}\sigma dV\leq 0$ (19)

is satisfied throughout the evolution. The inequality is saturated only at the
steady state.

It is worth noting that the term $\partial_{t}\sigma$ consists of two
contributions i.e.,

$\partial_{t}\sigma=J_{\mu}\partial_{t}X^{\mu}+X^{\mu}\partial J_{\mu}$ (20)

These two contributions are identical (only) in the Onsager region:

$J_{\mu}\partial_{t}X^{\mu}=X^{\mu}\partial
J_{\mu}=2L_{\mu\nu}X^{\mu}\partial_{t}X^{\nu}$ (21)

being $L_{\mu\nu}=L_{\nu\mu}$. Quantity

$P=J_{\mu}\partial_{t}X^{\mu}$ (22)

is called the Glansdorff-Prigogine dissipative quantity and plays a key role
in the thermodynamics of irreversible processes prigogine5 , prigogine6 ,
degroot , sonnino1 , sonnino2 , sonnino3 , sonnino4 , sonnino5 , sonnino6 .

###  II.4 The Space of the Thermodynamic Forces

To continue with the formalism it is necessary to define the space where we
can perform calculations. For this, we have to specify two quantities: the
metric tensor (denoted by $g_{\mu\nu}$ and the affine connection (denoted by
$\Gamma^{\lambda}_{\mu\nu}$) sonnino1 , sonnino2 , sonnino3 , sonnino4 ,
sonnino5 , sonnino6 .

a) A metric tensor is a central object in the theory; it describes the local
geometry of space. The metric tensor is a dimensionless symmetric tensor used
to raise and lower the indicative tensors and generate the connections used to
determine the field equations, that have to be satisfied by the metric tensor,
and to construct the Riemann curvature tensor.

b) The curvature of a space can be identified by taking a vector at some point
and transporting it parallel along a curve in space-time. An affine connection
is a rule that describes how to legitimately move a vector along a curve on
the variety without changing its direction. We adopt the following
definitions: the space of the thermodynamic forces (or, simply, the
thermodynamic space) is the space spanned by the thermodynamic forces. The
metric tensor and the affine connection are determined by physics. More
specifically, the metric tensor is identified with the symmetric piece
$g_{\mu\nu}$ of the transport coefficients, and the expression of the affine
connection $\Gamma^{\kappa}_{\mu\nu}$ is determined by imposing the validity
of the Glansdorff-Prigogine Universal Criterion of Evolution prigogine6 ,
sonnino3 , sonnino6 . Note that, for the second law of thermodynamics, the
square (infinitesimal) distance $ds^{2}=d{\bf s}\cdot d{\bf s}$ is always a
nonnegative quantity - see Fig. 1.

Figure 1: The Thermodynamic Space. The space is spanned by the thermodynamic
forces, the metric tensor is identified with the symmetric piece of the
transport coefficients, and the expression of the affine connection is
determined by the Universal Criterion of Evolution.

In the thermodynamic space, the total entropy production $\sigma_{T}$, and the
minimum entropy production theorem read, respectively

$\sigma_{T}(t)=\frac{1}{\Omega}\int_{\Omega}g_{\mu\nu}X^{\mu}X^{\nu}\sqrt{g}d^{n}X\qquad;\quad\frac{1}{\Omega}\int_{\Omega}\partial_{t}\sigma\sqrt{L}d^{n}X\leq
0$ (23)

with $\Omega$ denoting the (finite) volume of the thermodynamic space where
the thermodynamic forces act, and $g$ and $L$ are the determinant of the
symmetric piece of the transport coefficients and the determinant of Onsager’s
matrix, respectively.

##  III The Entropy Production at the Mesoscopic Scale

###  III.1 Preliminary Notions

In most cases, the study of wave phenomena deals with groups of waves or wave
packets which are phenomena limited in time and space. For example, the note
played by a piano lasts a finite time and the electromagnetic wave produced by
the strike of lightning lasts only a fraction of a second. In this case, the
function is called a wave packet because it is a packet of waves with
frequencies/wave numbers clustered around a single value, say $\omega_{0}$. In
these conditions it can be demonstrated that the wave emitted cannot be
monochromatic but, on the contrary, it is the sum of several monochromatic
waves whose frequency is included in an interval whose measure $\Delta\omega$
is connected to the duration $\Delta t$ of the perturbation. Briefly, a
wavepacket is a localized disturbance that results from the sum of many
different waveforms. If the packed is strongly localized, more frequencies are
needed to allow constructive superposition in the region of localization and
destructive superposition outside the region. Fig. 2 shows a typical
wavepacket.

Figure 2: Wavepacket. A wavepacket is an infinite set of waves with different
wave numbers that interfere constructively in a small region and destructively
in the rest of space. Fourier’s theorem establishes the general property of
the wavepackets: $\Delta t\Delta\omega\geq 1$ or ${\Delta\bf r}\Delta{\bf
k}\geq 1$.

In the $3+1$ dimensional space-time (${\bf r},t$), the general form of a real
wavepacket $\phi({\bf r},t)$ reads

$\phi({\bf r},t)=\frac{1}{(2\pi)^{3}}\int_{-\infty}^{+\infty}\left(\phi({\bf
k})e^{i({\bf r}\cdot{\bf k}-\omega({\bf k})t)}+\phi^{\star}({\bf k})e^{-i({\bf
r}\cdot{\bf k}-\omega({\bf k})t)}\right)d{\bf k}$ (24)

where $\phi^{\star}({\bf k})$ denotes the complex conjugate of $\phi({\bf
k})$. The physical meaning of the above expression is simple: by adding an
infinite number of monochromatic waves, chosen appropriately, a localization
of the wave perturbation is obtained; or vice-versa, one can imagine that each
packet of waves is made up of infinite suitable monochromatic waves. The
crucial aspect is that, according to Fourier’s theorem, for the amplitudes of
any wavepacket, the standard deviations $\Delta t=\sqrt{<t^{2}>-<t>^{2}}$
characterizing the width of the spectral distribution in the angular frequency
domain, $\Delta\omega$, and the width in time $\Delta t$ are related:

$\Delta t\Delta\omega\geq 1$ (25)

This product of the standard deviations equals unity only for the special case
of Gaussian-shaped spectral distributions and is greater than unity for all
other shaped spectral distributions. This states that the uncertainties with
which we can simultaneously measure the time and frequency for the intensity
of a given wavepacket are related. Eq. (25) expresses the celebrated Fourier’s
uncertainty principle for frequency-time intensities.

###  III.2 The Entropy Production in Small Systems

To avoid misunderstanding, we specify right now that in this approach, entropy
production is not considered a wave packet at all. However, as we shall see,
Fourier analysis will prove to be a very powerful investigation tool. Entropy
production is a macroscopic concept that emerges from the statistical behavior
of a large number of particles, and the probabilities and distributions of
their microstates. When performing measurements involving a large number of
particles and considering the statistical behavior of the system, the
resulting macroscopic phenomena, including entropy production, are generally
described by continuous variables and distributions. However, it is also true
that entropy production arises from collisions among particles in a system.
When particles collide and interact, they can lead to changes in the system’s
microstate configurations, resulting in entropy production. It should be
emphasized that all transport coefficients vanish in the absence of
collisions. If we perform an experiment at a mesoscopic scale we make
observations that reflect the discrete nature of particle interactions. Being
the entropic production directly dependent on the transport coefficients (and
therefore on the collisions of the particles’ system), at the mesoscopic
level, the observed entropy production may exhibit a more discrete behavior
due to the discrete nature of particle collisions. This is because the
discrete nature of individual particle interactions can become more apparent
on such a small scale. Fluctuations arise due to the probabilistic nature of
individual particle interactions and movements. In smaller systems, these
fluctuations have a greater impact on the system’s overall behavior. At a
macroscopic scale, the averaging over a large number of particles and the
statistical nature of the system smoothes out the microscopic discretization
or discrete behavior that may be observed on a smaller scale. Recent
experiments show the existence of lower bounds for the rate of entropy
production in active fluctuations of the hair-cell bundle by quantifying the
irreversibility of stochastic traces obtained from mesoscopic degrees of
freedom roldan . In particular, it has been observed that, on a mesoscopic
scale, the simultaneous measurement of entropy production and time becomes
more indeterminate due to the inherent randomness introduced by these
fluctuations. Experiments like roldan  lead to two different (and opposite)
interpretations: 1) The limited number of particles in the system leads to a
greater sensitivity to individual events, making it experimentally more
challenging to precisely determine entropy production at any specific moment.

2) On a mesoscopic scale, there is a fundamental limit to the accuracy with
which the values for time and entropy production can be predicted from initial
conditions. Here, we shall investigate the second hypothesis. A quasi-
localized disturbance of entropy production can be obtained using a linear
superposition of modes with close mode numbers. So, if the system is subject
to ”n” independent thermodynamic forces, in case of periodic boundary
conditions, it is convenient to choose as basis-functions of a local
disturbance of entropy production strength that of plane waves with the
generic wave number ${\bf K}$ compatible with periodicity conditions:

$\sigma({\bf
X},t)=\frac{1}{(2\pi)^{n}}\int_{-\infty}^{+\infty}\left(\sigma_{\bf
K}e^{i({\bf K}\cdot{\bf X}-\omega_{\bf K}t)}+\sigma^{\star}_{\bf K}e^{-i({\bf
K}\cdot{\bf X}-\omega_{\bf K}t)}\right)d{\bf K}$ (26)

Eq. (26) corresponds to a perturbation of the entropy production strength
localized in a small region of the thermodynamic space. Notice that the modes
are in the space of thermodynamic forces (and not in the ordinary space). Of
course, as in ordinary space, even in thermodynamic space it is possible to
introduce the phase velocity $v_{p}$ and the group velocity $v_{g}$, defined
in the usual way:

$v_{p}=\mid\frac{\omega}{{\bf K}}\mid\qquad;\qquad
v_{g}=\mid\frac{\partial\omega}{\partial{\bf K}}\mid$ (27)

According to the Fourier theorem, we have

$\Delta t\Delta\omega\geq 1\quad;\quad\Delta K_{\mu}\Delta X^{\mu}\geq
1\quad({\rm no\ summation\ convention\ on}\ \mu;\ \mu=1,\cdots,n)$ (28)

We may investigate what happens at a mesoscopic scale assuming that in the
space of the thermodynamic forces, the entropy production strength is
proportional to the frequency and the thermodynamic variable $\alpha_{\mu}$,
conjugate to the thermodynamic forces $X^{\mu}$, is proportional to the wave-
vector $K_{\mu}$ i.e.,

$\displaystyle\sigma={/\\!\\!\\!k}_{\\!{B}}\omega$ (29)
$\displaystyle\alpha_{\mu,{\bf K}}={/\\!\\!\\!k}_{\\!{B}}{\bf
K}_{\mu}\qquad{\rm with}\quad{/\\!\\!\\!k}_{\\!{B}}\equiv\beta k_{B}$

with ${/\\!\\!\\!k}_{\\!{B}}$ denoting Boltzmann’s constant $k_{B}$ times a
pure number, say $\beta$, undetermined at this stage:
${/\\!\\!\\!k}_{\\!{B}}\equiv\beta k_{B}$. Combining Eq. (28) with Eq. (29) we
get

$\displaystyle\Delta t\Delta\sigma\geq{/\\!\\!\\!k}_{\\!{B}}$ (30)
$\displaystyle\Delta\alpha_{\mu}\Delta
X^{\mu}\geq{/\\!\\!\\!k}_{\\!{B}}\quad\forall\mu\qquad({\rm no\ summation\
convention\ on}\ \mu;\ \mu=1,\cdots,n)$

These inequalities assert a fundamental limit to how accurately a system’s
values of certain pairs of physical quantities, such as entropy production
strength and time or thermodynamic force and its conjugate thermodynamic
variable, can be predicted from initial conditions. In other words, we cannot
know with perfect accuracy both the value of the entropy production strength
of a system and time; the more we fix the entropy of the system, the less we
know about time and vice versa. Similarly, we also have the uncertainty
relation between the value of a thermodynamic force and its conjugate
thermodynamic variable. The pairs of variables ($t,\sigma$) and
($\alpha_{\mu},X^{\mu}$) may be called complementary variables or canonically
conjugate (in analogy with the quantum mechanics’ terminology).

###  III.3 The Canonical Commutation Rules

At this point, to progress in the formalism, it is necessary to promote the
single variables $\sigma$ and $t$, and $X^{\mu}$ and $\alpha_{\mu}$ to
operators (which act on some state space of the system as yet unspecified),
separately, imposing a ”bind” between them so that their products behave ”as
we wish”. So, at the mesoscopic scale, we write 111This condition finds its
justification later. Precisely, thanks to this condition we shall obtain a
coherent theory.:

$\displaystyle[t,\sigma]=i\frac{{/\\!\\!\\!k}_{\\!{B}}}{2}$ (31)
$\displaystyle[\alpha_{\mu,{\bf K}},X_{\bf
K^{\prime}}^{\nu}]=i\frac{{/\\!\\!\\!k}_{\\!{B}}}{2}\delta_{\mu\nu}\delta_{{\bf
K}{\bf K^{\prime}}}\qquad{\rm with}\quad{/\\!\\!\\!k}_{\\!{B}}=\beta k_{B}$

with $[\cdots]$ denoting the commutator between two operators: $[A,B]=AB-BA$
and $\delta_{\mu\nu}$ the Kronecker delta, respectively 222Note that the
dimensions of the expressions (31) are correct. Indeed, the dimension of
Boltzmann’s constant is [$k_{B}$]=[Energy]/[Temperature]. The dimension of the
quantity $[t,\sigma]$ is [time]$\times$ [Energy]/([Temperature]×[time])=
[Energy]/[Temperature]=[$k_{B}$]. The dimension of the quantity
$[\alpha_{\mu},X^{\nu}$] is [time]$\times$[$\sigma^{1/2}$]$\times$
[$\sigma^{1/2}$]=[$k_{B}$].. In roldan  we find several examples of
experimental traces for the tip position of different mechano-sensory hair
bundles as a function of time. The authors estimated the local irreversibility
measure obtained from single $30\ sec.$ recordings of the oscillations shown
in these examples. The sampling rate was $\omega=2.5kHz$. From these
experiments, we can estimate (albeit approximately) the numerical value of
$\beta$. We find $\beta\sim 1.2\times 10^{-8}$ so, ${/\\!\\!\\!k}_{\\!{B}}\sim
1.6\times 10^{-31}\ J/K$.

##  IV Discretization of the Key Thermodynamic Quantities of a Small System in
Onsager’s Region

In this first work, we shall limit ourselves to discretizing the entropic
production of a small system in the Onsager region (i.e., in a region outside
the thermodynamic equilibrium but still sufficiently close to it). In this
situation, as mentioned in section II, the metric of the thermodynamic space
is flat (i.e., it does not depend on the thermodynamic forces) and the
metric’s coefficients are the elements of Onsager’s matrix $L_{\mu\nu}$.

###  IV.1 Discretization of the Total Entropy Production Strength

The total entropy production strength is

$\sigma_{T}(t)=\frac{1}{\Omega}\int_{\Omega}\sigma({\bf
X},t)\sqrt{L}d^{n}X=\frac{1}{\Omega}\int_{\Omega}L_{\mu\nu}X^{\mu}X^{\nu}\sqrt{L}d^{n}X$
(32)

The local entropy production strength can be split into two contributions

$\sigma({\bf
X},t)=L_{\mu\nu}X^{\mu}X^{\nu}=\frac{1}{2}L_{\mu\nu}X^{\mu}X^{\nu}+\frac{1}{2}L^{\mu\nu}J_{\mu}J_{\nu}$
(33)

Let us now perform the following linear coordinate transformation 333Note that
linear transformations of coordinates are allowed because this class of
transformations belongs to the TCT-group sonnino4 .:

${X^{\prime}}^{\lambda}=A^{\lambda}_{\kappa}X^{\kappa},\quad{\rm with}\ \
A^{\mu}_{\nu}\ {\rm such\ that}\ \
A^{\alpha}_{\lambda}L^{\lambda\kappa}A^{\beta}_{\kappa}={\rm I}^{\alpha\beta}$
(34)

with ${\rm I}^{\alpha\beta}$ denoting the Identity matrix. So, after
transformation, $\sqrt{L}\rightarrow 1$. Notice that, since the matrix
$L_{\mu\nu}$ is a positive definite matrix, there exists always a matrix
$A^{\mu}_{\nu}$, which satisfies condition (34). After transformation we get

$\displaystyle\sigma_{T}(t)=\frac{1}{{\Omega}^{\prime}}\int_{\Omega^{\prime}}\sigma({\bf
X}^{\prime},t)d^{n}X^{\prime}\qquad{\rm with}$ (35) $\displaystyle\sigma({\bf
X}^{\prime},t)=\frac{1}{2}\sum_{\mu=1}^{n}\left(\mid
X^{{}^{\prime}\mu}\mid^{2}+\mid J^{\prime}_{\mu}\mid^{2}\right)$

In the space of the thermodynamic forces, the Fourier expansion in a finite
box of volume $\Omega$ of the thermodynamic fluxes reads

$J^{\prime}_{\mu}({\bf X}^{\prime},t)=\sum_{\bf K}\left(J_{\mu,{\bf
K}}(t)e^{i({\bf K}\cdot{\bf X^{\prime}})}+J^{\star}_{\mu,{\bf K}}(t)e^{-i({\bf
K}\cdot{\bf X^{\prime}})}\right)$ (36)

where

$J_{\mu,{\bf K}}(t)=i\omega_{\bf K}\alpha_{\mu,{\bf K}}(t)\qquad{\rm
and}\qquad J^{\star}_{\mu,{\bf K}}(t)=-i\omega_{\bf K}\alpha^{\star}_{\mu,{\bf
K}}(t)$ (37)

Let us now evaluate the total entropy production strength $\sigma_{T}(t)$.
This task is easily accomplished by taking into account the orthogonality of
the plane waves, assumed normalized in a box of volume $\Omega$. We begin by
considering the second contribution:

$\displaystyle\frac{1}{2\Omega^{\prime}}\int_{\Omega^{\prime}}\sum_{\mu=1}^{n}\mid
J_{\mu}^{\prime}\mid^{2}d^{n}X^{\prime}=$ (38)
$\displaystyle\frac{1}{2\Omega^{\prime}}\sum_{\mu=1}^{n}\sum_{{\bf K},{\bf
K}^{\prime}}\int_{\Omega^{\prime}}\\!\\!\left(J_{\mu,{\bf K}}(t)e^{i({\bf
K}\cdot{\bf X}^{\prime})}\\!+\\!J^{\star}_{\mu,{\bf K}}(t)e^{-i({\bf
K}\cdot{\bf X}^{\prime})}\right)\\!\\!\left(J_{\mu,{\bf
K}^{\prime}}(t)e^{i({\bf K^{\prime}}\cdot{\bf
X}^{\prime})}\\!+\\!J^{\star}_{\mu,{\bf K}^{\prime}}(t)e^{-i({\bf
K^{\prime}}\cdot{\bf X}^{\prime})}\right)d^{n}X^{\prime}$

The orthogonality relation reads

$\frac{1}{\Omega}\int_{\Omega}e^{i{\bf K}\cdot{\bf X}}e^{-i{\bf
K^{\prime}}\cdot{\bf X}}d^{n}X=\delta_{{\bf K}{\bf K}^{\prime}}$ (39)

Additionally, by looking at Eq. (37), we have to take into account that
$J_{\mu,{\bf K}^{\prime}}=J^{\star}_{\mu,{-\bf K}^{\prime}}$. So, upon
integration over the volume, we get two terms with ${\bf K}={\bf K}^{\prime}$
and two terms with ${\bf K}=-{\bf K}^{\prime}$, finally giving four equivalent
terms in all. Similar calculations apply to the first contribution
$\int_{\Omega^{\prime}}\mid X^{{}^{\prime}\mu}\mid^{2}d^{n}X^{\prime}$ (in
this case, $X^{\mu}_{\bf K}={X^{\mu}}^{\star}_{\\!\\!\\!\\!-{\bf
K}^{\prime}}$). The final result is

$\sigma_{T}(t)=2\sum_{\mu=1}^{n}\sum_{\bf K}\left(\mid X^{\mu}_{\bf
K}(t)\mid^{2}+\mid J_{\mu,{\bf
K}}(t)\mid^{2}\right)=2\sum_{\mu=1}^{n}\sum_{\bf K}\left(\mid X^{\mu}_{\bf
K}(t)\mid^{2}+\omega_{\bf K}^{2}\mid\alpha_{\mu,{\bf K}}(t)\mid^{2}\right)$
(40)

where Eqs (37) has been taken into account. This expression is identical to
the Hamiltonian of a harmonic oscillator if we set the value of the $mass=4$
and we identify the following terms: position$\rightarrow\alpha_{\mu,K}$,
momemtum$\rightarrow 2X^{\mu}_{K}$, and frequency$\rightarrow\omega_{K}$ (see,
for example, griffiths , liboff ). So, we first define two new dimensionless
operators ${\tilde{X}}^{\mu}_{K}$ and ${\tilde{\alpha}}_{\mu,K}$, as follows:

${\tilde{X}}^{\mu}_{\bf K}=\sqrt{\frac{2}{{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf
K}}}X^{\mu}_{\bf K}\qquad;\qquad{\tilde{\alpha}}_{\mu,{\bf
K}}=\sqrt{\frac{2\omega_{\bf K}}{{/\\!\\!\\!k}_{\\!{B}}}}\alpha_{\mu,{\bf K}}$
(41)

In terms of these new variables, the expression for $\sigma_{T}(t)$ reads

$\sigma_{T}(t)=\sum_{\mu=1}^{n}\sum_{\bf K}{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf
K}\left(\mid{\tilde{X}}^{\mu}_{\bf K}\mid^{2}+\mid{\tilde{\alpha}}_{\mu,{\bf
K}}\mid^{2}\right)$ (42)

As for the case of the harmonic oscillator, we assume the validity of the
following Canonical Commutation Rules (ccr):

$[{\tilde{\alpha}}_{\mu,{\bf K}},{\tilde{X}}^{\nu}_{\bf
K^{\prime}}]=i\delta_{\mu\nu}\delta_{{\bf K}{\bf K^{\prime}}}\qquad{\rm
so}\quad[\alpha_{\mu,{\bf K}},X^{\nu}_{\bf
K^{\prime}}]=i\frac{{/\\!\\!\\!k}_{\\!{B}}}{2}\delta_{\mu\nu}\delta_{{\bf
K}{\bf K^{\prime}}}$ (43)

The two operators of creation ”${\rm a}_{\bf K}^{(\mu)+}$” and destruction
”${\rm a}_{\bf K}^{(\mu)}$” can be introduced and defined as usual (see, for
example, weinberg , maiani ):

$\displaystyle{\rm a}_{\bf
K}^{(\mu)}=\frac{1}{\sqrt{2}}\left({\tilde{\alpha}}_{\mu,{\bf
K}}+i{\tilde{X}}^{\mu}_{\bf K}\right)$ (44) $\displaystyle{\rm a}_{\bf
K}^{(\mu)+}=\frac{1}{\sqrt{2}}\left({\tilde{\alpha}}_{\mu,{\bf
K}}-i{\tilde{X}}^{\mu}_{\bf K}\right)$

so $[{\rm a}_{\bf K}^{(\mu)},{\rm a}_{\bf
K^{\prime}}^{(\mu^{\prime})+}]=\delta_{\mu\mu^{\prime}}\delta_{{\bf K}{\bf
K^{\prime}}}$. We finally get the discretization of the total entropy
production strength in Onsager’s region

${\boldsymbol{\sigma}}_{T}(t)=\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf K}\left({\rm a}_{\bf K}^{(\mu)+}{\rm
a}_{\bf K}^{(\mu)}+\frac{1}{2}\right)$ (45)

Previous expression may also be written in terms of the number operator ${\bf
n}_{\bf K}^{(\mu)}\equiv{\rm a}_{\bf K}^{(\mu)+}{\rm a}_{\bf K}^{(\mu)}$:

${\boldsymbol{\sigma}}_{T}(t)=\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf K}\left({\bf n}_{\bf
K}^{(\mu)}+\frac{1}{2}\right)$ (46)

To sum up, in the space of the thermodynamic forces the total entropy
production strength behaves as the sum of ”${\bf K}$” (discretized)
independent one-dimensional harmonic oscillators, each oscillating with
frequency $\omega_{\bf K}$.

Comments

i) As we shall see in section V, the constant term

$\frac{1}{2}\sum_{\mu=1}^{n}\sum_{\bf K}{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf K}$
(47)

corresponds to the entropy production produced by fluctuations very close to
the thermodynamic equilibrium or to the local non-equilibrium state. In
principle, this additional term is divergent. This problem will be solved in
the forthcoming section by applying the correspondence principle which states
that in the limit of large (macroscopic) systems, our predictions must
coincide with those of Einstein-Prigogine’s fluctuations theory.

ii) The discretized total entropy production strength does not depend
explicitly on time, so the entropy production strength of mode K is
stationary, as are the eigenfunctions of the operator, which are also
stationary in time. This unit of information of entropic production strength
may briefly be referred to as TUI (Thermodynamic Unit of Information). Also,
being

$[{\boldsymbol{\sigma}}_{T},{\bf n}_{\bf K}^{(\mu)}]=0$ (48)

the number of TUI in each mode for the total entropy production strength
remains constant over time.

iii) Given the form of total entropy production strength, all the results
found for the oscillator remain valid. In particular, the states of the total
entropic production strength are of the type $\mid n_{1},n_{2},\cdots>$ with
$n_{1},n_{2},\cdots$ positive integers. These states are obtained from the
vacuum state (corresponding to the thermodynamic equilibrium or to the local
non-equilibrium steady state) through the application of the creation
operator, and there is no limit to the population of the modes.

###  IV.2 Probabilistic Treatment of the Entropy Production

Entropy production is a statistical quantity that characterizes the behavior
of a system over time. It involves considering the fluctuations and
probabilities associated with different trajectories and events. In many
nonequilibrium systems, the dynamics and behavior of the system involve
probabilistic transitions between different states or configurations. So, the
probabilistic mathematical framework is well-suited for studying entropy
production because it captures the stochastic nature of nonequilibrium systems
and provides a means to analyze and quantify the statistical properties and
fluctuations associated with entropy production. The operators’ formalism
introduced in the previous section lends in a natural way to a probabilistic
treatment of entropy production. Eq. (46) shows that the total entropy
production is a self-adjoint operator having eigenvectors of the type:

$\mid n_{1},n_{2},\cdots,n_{\kappa},\cdots>$ (49)

for which we have

${\bf n}_{\bf K}^{(\mu)}\mid
n_{1},n_{2},\cdots,n_{\kappa},\cdots>=n_{\kappa}^{(\mu)}\mid
n_{1},n_{2},\cdots,n_{\kappa},\cdots>$ (50)

We interpret the eigenvector Eq. (49) as the state with $n_{1}$ TUI with mode
$K_{1}$, $n_{2}$ TUI with mode $K_{2}$, $\cdots$, $n_{\kappa}$ TUI with mode
$K_{\kappa}$, $\cdots$. The eigenvalue $\sigma_{T}$ of the operator
${\boldsymbol{\sigma}}_{T}$ reads

$\sigma_{T}=\sum_{\bf
K}\sum_{\mu=1}^{n}\left(n^{(\mu)}_{\kappa}+\frac{1}{2}\right){/\\!\\!\\!k}_{\\!{B}}\omega_{\bf
K}$ (51)

In accordance with our expectation, Eq. (51) shows that the eigenvalues of the
entropy production $\sigma_{T}$ can never be negative. Since a self-adjoint
operator has a complete orthonormal system of eigenvectors, we may identify
the space of states of the total entropy production strength as the one
generated by all vectors of the type Eq. (49). In this formalism, Eq. (51) may
be written as

$\sigma_{T}=<n_{1},n_{2},\cdots,n_{\kappa},\cdots\mid{\boldsymbol{\sigma}}_{T}\mid
n_{1},n_{2},\cdots,n_{\kappa},\cdots>=\sum_{\bf
K}\sum_{\mu=1}^{n}\left(n^{(\mu)}_{\kappa}+\frac{1}{2}\right){/\\!\\!\\!k}_{\\!{B}}\omega_{\bf
K}$ (52)

So, the probabilistic interpretation related to entropic production strength
is the following:

The probabilities $p_{\kappa}$ that $n_{\kappa}$ TUI of value
${/\\!\\!\\!k}_{\\!{B}}\omega_{K}$ are detected in a fixed eigenstate, is
described by the coefficients of the development of the state on the complete
orthonormal system of the eigenvectors of ${\boldsymbol{\sigma}}_{T}$.

The above somewhat brings the entropy production formalism presented here
closer to the one currently in use, which is also based on a probabilistic
approach.

###  IV.3 Discretization of the Thermodynamic Variable Conjugate to the
Thermodynamic Force

We have already noted that, in Thermodynamic Field Theory, a thermodynamic
flux $J_{\mu}$ that is conjugate to the thermodynamic force $X^{\mu}$ can be
seen as the covariant aspect of the same quantity as the two variables are
related each with other by the metric tensor $g_{\mu\nu}$ according to the
relation $J_{\nu}=g_{\mu\nu}X^{\mu}$. In other words, $X^{\mu}$ and $J_{\mu}$
are two different aspects of a unique entity. For this reason, these two
quantities commute. A different thing is the relationship between the
thermodynamic force $X^{\mu}$ and the thermodynamic variable $\alpha_{\mu}$
conjugate to it; these two quantities do not commute. We can easily check that
also the quantity $\alpha_{\mu}$ is discretized according to the relation (29)
reported in section III. In the thermodynamic space, the local entropy
production strength reads.

$\sigma({\bf X},t)=\sum_{\bf K}\left(\sigma_{\bf K}(t)e^{i({\bf K}\cdot{\bf
X})}+\sigma^{\star}_{\bf K}(t)e^{-i({\bf K}\cdot{\bf X})}\right)$ (53)

Then,

$J_{\mu}=\frac{\partial\sigma}{\partial X^{\mu}}=i\sum_{\bf K}{\bf
K}_{\mu}\left(\sigma_{\bf K}(t)e^{i({\bf K}\cdot{\bf X})}-\sigma^{\star}_{\bf
K}(t)e^{-i({\bf K}\cdot{\bf X})}\right)$ (54)

On the other hand, we also have

$\displaystyle J_{\mu}=\frac{d\alpha_{\mu}}{dt}\qquad{\rm so}$ (55)
$\displaystyle\alpha_{\mu}({\bf X},t)=i\sum_{\bf K}\omega_{\bf
K}\left(\alpha_{\mu,{\bf K}}(t)e^{i({\bf K}\cdot{\bf
X})}-\alpha^{\star}_{\mu,{\bf K}}(t)e^{-i({\bf K}\cdot{\bf X})}\right)$

By comparing Eq. (54) with Eq. (55) (or, equivalently, by using the
orthogonality relations), we get

$\alpha_{\mu,{\bf K}}\omega_{\bf K}=\sigma_{\bf K}{\bf K}_{\mu}$ (56)

or

$\alpha_{\mu,{\bf K}}={/\\!\\!\\!k}_{\\!{B}}{\bf K}_{\mu}$ (57)

where Eq. (52) has been taken into account. As an example, let us examine a
system where ”$n$” chemical reactions take place simultaneously. We want to
investigate the case where small fluctuations from thermodynamic equilibrium
occur 444For simplicity we shall examine the argument by using the degree of
advancements as parameters. The generalization is trivial.. The total entropy
production due to the fluctuations $\alpha_{\mu}$ reads

$\Delta_{I}S=\int_{\boldsymbol{\xi}_{eq.}}^{\boldsymbol{\xi}}d_{I}S=\sum_{\mu=1}^{n}\int_{{{\boldsymbol{\xi}}_{eq.}}}^{\boldsymbol{\xi}}\frac{A^{\mu}}{T}d{\boldsymbol{\xi}}\qquad{\rm
with}\quad{\boldsymbol{\xi}}\equiv\xi_{1},\cdots,\xi_{n}$ (58)

where $A^{\mu}$ and $\xi_{\mu}$ denoting the affinity and the extent of the
chemical reaction $\mu$, respectively, and $T$ is the temperature of the
system. $A^{\mu}$ may be expanded in a Taylor series and, since
$A({\boldsymbol{\xi}}_{eq.})=0$, we finally get

$\Delta_{I}S\simeq\frac{1}{2T}\left(\frac{\partial
A^{\mu}}{\partial{\xi}_{\nu}}\right)_{eq.}\alpha_{\mu}\alpha_{\nu}$ (59)

where the summation convention on the repeated indexes $\mu$ and $\nu$ has
been adopted. Using the abbreviation

$q^{\mu\nu}\equiv\frac{1}{2T}\left(\frac{\partial
A^{\mu}}{\partial{\xi}_{\nu}}\right)_{eq.}$ (60)

Eq. (59) can be brought into the form

$\Delta_{I}S=q^{\mu\nu}\alpha_{\mu}\alpha_{\nu}$ (61)

Notice that expression (61) is valid in general and coefficient $q^{\mu\nu}$
depend on the type of fluctuations we are considering prigogine1 . We also
have

$\frac{A^{\mu}}{T}=\frac{1}{T}\left(\frac{\partial
A^{\mu}}{\partial\alpha_{\nu}}\right)_{eq.}\\!\\!\\!\\!\alpha_{\nu}=q^{\mu\nu}a_{\nu}=\frac{\partial\Delta_{I}S}{\partial\alpha_{\mu}}=X^{\mu}$
(62)

Hence,

$\alpha_{\mu}=\widehat{q}_{\mu\nu}X^{\mu}$ (63)

with $\widehat{q}_{\mu\nu}$ denoting the reciprocal matrix of $q^{\mu\nu}$
i.e., $\widehat{q}_{\nu\kappa}q^{\kappa\mu}=\delta^{\mu}_{\nu}$, with
$\delta^{\mu}_{\nu}$ denoting the Kronecker delta tensor. By taking into
account Eq. (57) and Eq. (15), we finally get

$\frac{A^{\mu}}{T}={/\\!\\!\\!k}_{\\!{B}}q^{\mu\nu}{\bf K}_{\nu}\qquad;\qquad
v_{\mu}={/\\!\\!\\!k}_{\\!{B}}L_{\mu\rho}q^{\rho\nu}{\bf
K}_{\nu}={/\\!\\!\\!k}_{\\!{B}}q^{\nu}_{\mu}{\bf K}_{\nu}$ (64)

where $q^{\nu}_{\mu}\equiv L_{\mu\rho}q^{\rho\nu}$ and $v_{\mu}$ is the rate
of the chemical reaction $\mu$. Expressions (64) are of a fairly general
nature and, in general, we have

$X^{\mu}={/\\!\\!\\!k}_{\\!{B}}q^{\mu\nu}{\bf K}_{\nu}\qquad;\qquad
J_{\mu}={/\\!\\!\\!k}_{\\!{B}}q^{\nu}_{\mu}{\bf K}_{\nu}$ (65)

Eqs (65) are the expressions for the discretized thermodynamic forces and the
thermodynamic fluxes, respectively, and matrix $q_{\mu\nu}$ depends on the
nature of the fluctuating parameters.

###  IV.4 Discretization of the Glansdorff-Prigogine Dissipative Quantity

By taking into account Eq. (20), in Onsager’s region the rate of the local
entropy production strength reads

$\partial_{t}\sigma=J_{\mu}\partial_{t}X^{\mu}+X^{\mu}\partial
J_{\mu}=L_{\mu\nu}X^{\mu}\partial_{t}X^{\nu}+L^{\mu\nu}J_{\mu}\partial_{t}J_{\nu}=2P$
(66)

where $P$ is the Glansdorff-Prigogine dissipative quantity. By performing
transformation (34) we get

$\partial_{t}\sigma=\sum_{\mu=1}^{n}(X^{{}^{\prime}\mu}\partial_{t}X^{{}^{\prime}\mu}+J^{\prime}_{\mu}\partial_{t}J^{\prime}_{\mu})$
(67)

Without loss of generality, we may assume that during the relaxation,
$X^{{}^{\prime}\mu}$ decreases at a rate proportional to its current value
i.e.

$\partial_{t}X^{{}^{\prime}\mu}=-\lambda X^{{}^{\prime}\mu}$ (68)

where $\lambda$ is a positive rate constant. Since in Onsager’s region
$J^{\prime}_{\mu}=L_{\mu\nu}X^{{}^{\prime}\nu}$, we also have

$\partial_{t}J^{\prime}_{\mu}=-\lambda J^{\prime\mu}$ (69)

where we have assumed that the transport coefficients are independent of time.
So, in terms of the new variables, the rate of the local entropy production
strength reads 555Recall that after transformation (34) $\sqrt{L}\rightarrow
1$.

$\frac{1}{2}\frac{1}{{\Omega}^{\prime}}\int_{\Omega^{\prime}}\partial_{t}\sigma({\bf
X^{\prime}},t)d^{n}X^{\prime}=\frac{1}{{\Omega}^{\prime}}\int_{\Omega^{\prime}}P({\bf
X^{\prime}},t)d^{n}X^{\prime}=-\frac{\lambda}{2\Omega^{\prime}}\int_{\Omega^{\prime}}\sum_{\mu=1}^{n}\left(\mid
X^{{}^{\prime}\mu}\mid^{2}+\mid J^{\prime}_{\mu}\mid^{2}\right)d^{n}X$ (70)

By following the same procedure we used to discretize the total entropy
production strength, we finally get

${\bf P}\equiv\frac{1}{{\Omega}^{\prime}}\int_{\Omega^{\prime}}P({\bf
X^{\prime}},t)d^{n}X^{\prime}=-\lambda\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf K}\left({\rm a}_{\bf K}^{(\mu)+}{\rm
a}_{\bf K}^{(\mu)}+\frac{1}{2}\right)$ (71)

The eigenvalues ${\mathcal{P}}$ of the operator ${\bf P}$ are

${\mathcal{P}}=<n_{1},n_{2},\cdots,n_{\kappa},\cdots\mid{\bf P}\mid
n_{1},n_{2},\cdots,n_{\kappa},\cdots>=-\lambda\sum_{\mu=1}^{n}\sum_{\bf
K}\left(n^{(\mu)}_{\kappa}+\frac{1}{2}\right){/\\!\\!\\!k}_{\\!{B}}\omega_{\bf
K}\leq 0$ (72)

where the inequality is saturated at the steady state.

##  V Correspondence with the Einstein-Prigogine Theory of Fluctuations in the
Limit of Large Systems

The (partial) reconciliation with the classical theory of fluctuations can be
obtained by stating the following correspondence principle:

The entropy production of the system computed with the formalism of the
canonical commutation rules must recover the expression derived by Einstein-
Prigogine’s fluctuations theory in the limit of macroscopic systems.

We have already noted that, in principle, the constant term

$\sigma_{0}=\frac{1}{2}\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}\omega_{\bf K}$ (73)

diverges. In the literature, this kind of divergence is referred to as the
ultraviolet divergence. More in general, an ultraviolet divergence is a
situation in which a sum (or an integral) diverges in physical phenomena at
infinitesimal distances. Since an infinite result is unphysical, ultraviolet
divergences often require special treatment to remove unphysical effects
inherent in the perturbative formalisms 666In this work, we limit ourselves to
studying systems in the linear region of thermodynamics assuming that the
nonlinear contributions are of lower order.. In our specific case, this
drawback can easily be solved by applying the above-mentioned correspondence
principle. As previously mentioned, in nonequilibrium systems entropy
production arises from the irreversible processes and fluctuations occurring
at the microscopic level. Let us then consider a macroscopic system in
thermodynamic equilibrium where very small fluctuations around this state
occur. According to Einstein-Prigogine’s fluctuations theory the probability
$P$ of fluctuation $\Delta\xi_{\mu}$ is proportional to the exponential of the
corresponding entropy production (with negative sign) $-\Delta_{I}S$ divided
by Boltzmann’s constant $k_{B}$ tolman , fowler , greene , prigogine7 . Hence,
the probability of finding a state in which the values of $\alpha_{\mu}$ lie
between $\alpha_{\mu}$ and $d\alpha_{\mu}$ is

$Pd\alpha_{1}\cdots d\alpha_{n}=P_{0}\exp(-\Delta_{I}S/k_{B})d\alpha_{1}\cdots
d\alpha_{n}$ (74)

where $P_{0}$ ensures normalization to unity. Let us now calculate the
following average

${\overline{\overline{X^{\mu}\alpha_{\nu}}}}=\int\cdots\int
X^{\mu}\alpha_{\nu}Pd\alpha_{1}\cdots d\alpha_{n}$ (75)

It should be immediately specified that expression (74) is valid for small
spontaneous fluctuations around the thermodynamic equilibrium only and not for
systematic deviations from equilibrium. Noting that

$X^{\mu}=\frac{\partial\Delta_{I}S}{\partial\alpha_{\mu}}=-k_{B}\frac{\partial}{\partial\alpha_{\mu}}\left(\log\frac{P}{P_{0}}\right)=-\frac{k_{B}}{P}\frac{\partial
P}{\partial\alpha_{\mu}}$ (76)

we have

${\overline{\overline{X^{\mu}\alpha_{\nu}}}}=-k_{B}\int\frac{\partial
P}{\partial\alpha_{\mu}}\alpha_{\nu}d\alpha_{1}\cdots d\alpha_{n}$ (77)

Partial integration over the $\alpha_{i}$ gives

${\overline{\overline{X^{\mu}\alpha_{\nu}}}}=k_{B}\delta^{\mu}_{\nu}$ (78)

where we have taken into account that $P$ vanishes at the boundary, i.e. $P=0$
for $\alpha_{\nu}=\pm\infty$, and that the probability is normalized to unity.
Hence, the fluctuation phenomena are accompanied by a variation in entropy
production. According to Eq. (78) and by taking into account Eq. (59), we get

${\overline{\overline{\Delta_{I}S}}}=\frac{1}{2}{\overline{\overline{X^{\mu}\alpha_{\mu}}}}=\frac{n}{2}k_{B}$
(79)

with $n$ denoting the number of the independent thermodynamic forces. We
recall that Eq. (79) is valid for small spontaneous fluctuations around
thermodynamic equilibrium only. Expression (73) corresponds to the total
entropy production for a macroscopic system generated by very small
fluctuations around the thermodynamic equilibrium. Now we calculate the
eigenvalues of the ${\boldsymbol{\Delta}}_{\bf I}{\bf S}$ operator by using
the canonical commutation rules. By taking into account Eq. (61) and Eq. (63),
the expression for the ${\boldsymbol{\Delta}}_{\bf I}{\bf S}$ operator reads

${\boldsymbol{\Delta}}_{\bf I}{\bf
S}=\frac{1}{4\Omega}\int\left({\hat{q}}_{\mu\nu}X^{\mu}X^{\nu}+q^{\mu\nu}\alpha_{\mu}\alpha_{\nu}\right)\sqrt{L}\
d^{n}X$ (80)

Notice that for the second law of thermodynamics ${\hat{q}}_{\mu\nu}$ is a
definite positive matrix (see Eq. (61)). Therefore, there is a linear
coordinates transformation such that

${\alpha^{\prime}}_{\lambda}=A^{\kappa}_{\lambda}\alpha_{\kappa},\quad{\rm
with}\ \ A^{\mu}_{\nu}\ {\rm such\ that}\ \
A^{\alpha}_{\lambda}q^{\lambda\kappa}A^{\beta}_{\kappa}={\rm I}^{\alpha\beta}$
(81)

With a suitable definition of a new set of dimensionless variables and
following the same procedure adopted in subsection IV.1., we finally obtain

${\boldsymbol{\Delta}}_{\bf I}{\bf S}=\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}\left(\mid{\tilde{X}}^{\mu}_{\bf
K}\mid^{2}+\mid{\tilde{\alpha}}_{\mu,{\bf K}}\mid^{2}\right)$ (82)

or, introducing the operators of creation and destruction ${\rm a}_{\bf
K}^{(\mu)+}$ and ${\rm a}_{\bf K}^{(\mu)}$, and the number operator ${\bf
n}_{\bf K}^{(\mu)}$, we get

${\boldsymbol{\Delta}}_{\bf I}{\bf S}=\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}{\bf n}_{\bf K}^{(\mu)}+\frac{n}{2}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}$ (83)

where the canonical commutation rules have been taken into account. The
components of the hyper-vector $K_{\mu}$ must be compatible with the
conditions of periodicity at the boundaries of a hyper-cuboid with sides
$l_{\mu}$. Hence,

$K_{\mu}=\frac{2\pi m_{\mu}}{l_{\mu}}\qquad{\rm with}\quad m_{\mu}=0,\pm 1,\pm
2,\cdots\qquad\mu=1,\cdots,n$ (84)

where $m_{\mu}$ is a vector with integer components. Therefore, the eigenvalue
$\Delta_{I}S|_{0}$ related to the ground state of ${\boldsymbol{\Delta}}_{\bf
I}{\bf S}$ is

$\Delta_{I}S|_{0}=\frac{n}{2}{/\\!\\!\\!k}_{\\!{B}}M\qquad{\rm where}\quad
M\equiv\Pi_{\mu=1}^{n}m^{(Max)}_{\mu}$ (85)

with $m^{(Max)}_{\mu}$ denoting the maximum number of terms that the vector
number $m_{\mu}$ can assume and $\Pi$ is the product notation, respectively.
Eq. (85) coincides with the ground state calculated by the Einstein-Prigogine
fluctuations theory by setting the following value for $M$ (see Eq. (79)):

$M=\frac{k_{B}}{{/\\!\\!\\!k}_{\\!{B}}}\simeq 8.62\ 10^{7}$ (86)

The expression for the operator ${\boldsymbol{\Delta}}_{\bf I}{\bf S}$ finally
reads

${\boldsymbol{\Delta}}_{\bf I}{\bf S}=\sum_{\mu=1}^{n}\sum_{\bf
K}{/\\!\\!\\!k}_{\\!{B}}{\bf n}_{\bf K}^{(\mu)}+\frac{n}{2}k_{B}$ (87)

having discrete eigenvalues

$\Delta_{I}S=\sum_{\mu=1}^{n}\sum_{\bf K}{/\\!\\!\\!k}_{\\!{B}}n_{\bf
K}^{(\mu)}+\frac{n}{2}k_{B}$ (88)

###  V.1 Exemple

Consider a copper rod with length $L=6cm$ and cross-section $A=9cm^{2}$. The
rod is placed between two ideal reservoirs $T_{1}=10^{\circ}C$ and
$T_{2}=0^{\circ}C$ and laterally is thermally insulated so that it can only
exchange heat at its ends (see Fig. 3). Our goal is to estimate the number of
$TUI$ present in the copper rod.

Figure 3: Entropy production due to heat transfer. A rod of copper is placed
between two reservoirs at temperature $T_{1}$ (the hot one) and $T_{2}$ (the
cold one). The rod is thermally isolated and can exchange heat only at its
ends. For $T_{1}=10^{\circ}C$, $T_{2}=0^{\circ}C$, the length of the rod
$6cm$, and the cross-sectional area $9cm^{2}$, after one second the order of
magnitude of the number of $TUI$ is about $n^{(1)}\simeq 5.40\ 10^{20}$ and
the total number of TUI amounts to $Mn^{(1)}\simeq 4.66\ 10^{28}$.

Denoting with ${\dot{Q}}$ the heat flow rate from $T_{1}$ to $T_{2}$, the
total entropy production strength reads

$\sigma_{T}={\dot{Q}}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)$ (89)

The heat flow rate flowing in a rod is given by Fourier’s law:

${\dot{Q}}=\kappa_{c}\frac{A}{L}(T_{1}-T_{2})$ (90)

with $\kappa_{c}\simeq 385\ W/(mK)$ denoting the thermal conductivity of
copper young . Hence, the total entropy production strength is

$\sigma_{T}=\kappa_{c}\frac{A}{L}\frac{(T_{1}-T_{2})^{2}}{T_{1}T_{2}}=7.46\
10^{-3}W/K$ (91)

To get the order of magnitude of the number of $TUI$, we set $n=1$ in Eq. (88)
(as the system is subject to only one thermodynamic force) and, for
simplicity, we consider the case where each mode contains the same number
$n^{(1)}$ of $TUI$ i.e.,

$\sum_{\bf K}{/\\!\\!\\!k}_{\\!{B}}n_{\bf k}^{(1)}\sim
M{/\\!\\!\\!k}_{\\!{B}}n^{(1)}$ (92)

From Eq. (88) we get

$\Delta_{I}S\simeq
M{/\\!\\!\\!k}_{\\!{B}}n^{(1)}+\frac{1}{2}k_{B}=k_{B}\left(n^{(1)}+\frac{1}{2}\right)$
(93)

Comparing Eq. (93) with Eq. (91), we obtain

$n^{(1)}=\frac{\sigma_{T}}{k_{B}}\Delta t-\frac{1}{2}$ (94)

If we apply typical ”human-scale” values, $\Delta t\sim 1\ sec.$, then
$n^{(1)}\simeq 5.50\ 10^{20}$, and the total number of TUI amounts to
$Mn^{(1)}\simeq 4.66\ 10^{28}$. These are very big numbers, so the system is
indeed within the correspondence limit. We can easily also check why we
perceive a continuum of energy production strength in this limit. With
$\omega=1\ rad/sec$, the difference between two discretized entropy production
strengths is ${/\\!\\!\\!k}_{\\!{B}}\omega\simeq 1.6\ 10^{-31}$ W/K, well
below what we normally resolve for macroscopic systems. One then describes
this system through an emergent classical limit. Notice that Eq. (94) tells us
that we have no $TUI$ for

$\sigma_{T}\Delta t=\frac{1}{2}k_{B}$ (95)

This result is in line with our expectations as, for the Einstein-Prigogine
fluctuations theory, the ground state of the entropy production is $1/2k_{B}$
and, in accordance with our formalism, this corresponds to the vacuum state of
the entropy production strength $|0,\cdots,0>$, i.e., no $TUI$.

##  VI Microscopic Reversibility and the Onsager Reciprocity Relations

The transport flux-force relation given by Eq. (15) involves the transport
coefficients $\tau_{0\mu\nu}$ where the symmetric coefficients of this matrix,
denoted by $L_{\mu\nu}$, are called phenomenological coefficients. An
important theorem due to Onsager states that onsager1 , onsager2

$L_{\mu\nu}=L_{\nu\mu}$ (96)

These relations, referred to as Onsager’s reciprocity relations, express that
when the flux, corresponding to the irreversible processes $\mu$, is
influenced by the thermodynamic forces $X^{\nu}$ of the irreversible processes
$J_{\nu}$ through the interference $L_{\nu\mu}$, then the flux $J_{\nu}$ is
also influenced by the thermodynamic forces $X^{\mu}$, through the same
interference $L_{\mu\nu}$ i.e., $L_{\mu\nu}=L_{\nu\mu}$. Prigogine showed the
validity of the Onsager reciprocity relations, starting from the property of
microscopic reversibility and by using the Einstein-Prigogine fluctuations
theory prigogine1 . However, his proof is valid only in case of (very) small
fluctuations around the ground state i.e., around the thermodynamic
equilibrium or the local non-equilibrium state. In this section, we shall
demonstrate that, under the assumption of the microscopic reversibility
property, the formalism of the canonical commutation rules preserves the
validity of the Onsager reciprocity relations in the whole linear region of
thermodynamics (and not only in the region very close to the ground state).
For Microscopic reversibility we mean the symmetry of all mechanical equations
of motion of single particles with respect to time. Let us consider a
fluctuation $\alpha_{\mu}({\bf X},t)$ in ${\bf X}$ at time $t$ and a
fluctuation $\alpha_{\nu}({\bf X},t+\tau)$ in ${\bf X}$ after a time interval
$\tau$, and form the average (in the thermodynamic space) of the product of
both quantities:

$\frac{1}{2\Omega}\int\left(\alpha_{\mu}({\bf X},t)\alpha_{\nu}({\bf
X},t+\tau)+\alpha_{\nu}({\bf X},t+\tau)\alpha_{\mu}({\bf
X},t)\right)\sqrt{L}d^{n}X$ (97)

We also consider the value of the average product

$\frac{1}{2\Omega}\int\left(\alpha_{\nu}({\bf X},t)\alpha_{\mu}({\bf
X},t+\tau)+\alpha_{\mu}({\bf X},t+\tau)\alpha_{\nu}({\bf
X},t)\right)\sqrt{L}d^{n}X$ (98)

where fluctuation $\alpha_{\mu}({\bf X},t+\tau)$ occurs after the time
interval $\tau$. The two expressions (97) and (98) differ only in the temporal
order of the two fluctuations or, say in other words, by substitution
$t\rightarrow-t$. So, under the assumption of microscopic reversibility, we
get

$\displaystyle\frac{1}{2\Omega}\int\left(\alpha_{\mu}({\bf
X},t)\alpha_{\nu}({\bf X},t+\tau)+\alpha_{\nu}({\bf
X},t+\tau)\alpha_{\mu}({\bf X},t)\right)\sqrt{L}d^{n}X=$ (99)
$\displaystyle\frac{1}{2\Omega}\int\left(\alpha_{\nu}({\bf
X},t)\alpha_{\mu}({\bf X},t+\tau)+\alpha_{\mu}({\bf
X},t+\tau)\alpha_{\nu}({\bf X},t)\right)\sqrt{L}d^{n}X$

Subtracting the same quantity

$\frac{1}{2\Omega}\int\left(\alpha_{\nu}({\bf X},t)\alpha_{\mu}({\bf
X},t)+\alpha_{\mu}({\bf X},t)\alpha_{\nu}({\bf X},t)\right)\sqrt{L}d^{n}X$
(100)

from both members of Eq. (99) and dividing the expressions by $\tau$, as
$\tau\rightarrow 0$ we get

$\frac{1}{2\Omega}\int\left(\alpha_{\mu}{\dot{\alpha}}_{\nu}+{\dot{\alpha}}_{\nu}\alpha_{\mu}\right)\sqrt{L}d^{n}X=\frac{1}{2\Omega}\int\left(\alpha_{\nu}{\dot{\alpha}}_{\mu}+{\dot{\alpha}}_{\mu}\alpha_{\nu}\right)\sqrt{L}d^{n}X$
(101)

or

$\frac{1}{2\Omega}\int\left(\alpha_{\mu}J_{\nu}+J_{\nu}\alpha_{\mu}\right)\sqrt{L}d^{n}X=\frac{1}{2\Omega}\int\left(\alpha_{\nu}J_{\mu}+J_{\mu}\alpha_{\nu}\right)\sqrt{L}d^{n}X$
(102)

By applying the flux-force linear transport relations
$J_{\mu}=L_{\mu\nu}X^{\nu}$ we find

$\L_{\nu\kappa}\frac{1}{2\Omega}\int\left(\alpha_{\mu}X^{\kappa}+X^{\kappa}\alpha_{\mu}\right)d^{n}X=L_{\mu\kappa}\frac{1}{2\Omega}\int\left(\alpha_{\nu}X^{\kappa}+X^{\kappa}\alpha_{\nu}\right)d^{n}X$
(103)

As for the integrals, if $\kappa=\mu$ or $\kappa=\nu$ we get operator
${\boldsymbol{\Delta}}_{\bf I}{\bf S}$ given by Eq. (87). However, their
eigenvalues vanish for $\kappa\neq\mu$ or $\kappa\neq\nu$ (all commutators
$[a_{\mu},a_{\nu}]=[a_{\mu},a^{+}_{\nu}]=[a^{+}_{\mu},a^{+}_{\nu}]=0$ for
$\mu\neq\nu$). Hence, in terms of eigenvalues, Eq. (103) gives

$2L_{\nu\kappa}\delta_{\mu}^{\kappa}\Delta_{I}S=2L_{\mu\kappa}\delta_{\nu}^{\kappa}\Delta_{I}S$
(104)

or

$L_{\nu\mu}=L_{\mu\nu}$ (105)

i.e., the Onsager relation which we sought. To sum up, Prigogine in prigogine1
showed the validity of Onsager’s reciprocity relation for small spontaneous
fluctuations around the thermodynamic equilibrium only. Here, we demonstrated
that Onsager’s relations are satisfied even for systematic deviations from the
ground state, as long as the flux-force transport relations remain linear.

Notice that having checked the validity of Onsager’s reciprocity relations
over the whole range of linear thermodynamics through the canonical
commutation relations (ccr) is not a mere mathematical exercise. It is a very
important result as it demonstrates that the ccr can only be formulated as
given by Eq. (31).

##  VII Effect of the Volume of the System on the Simultaneous Measurement of
Entropy Production Strength and Time

In terms of the local thermodynamic variables ${\bar{\sigma}}$,
${\bar{X}}_{\bf K}^{\mu}$, and ${\bar{\alpha}}_{\mu,{\bf K}}$, the uncertainty
relations read

$\displaystyle[t,{\bar{\sigma}}]=i\frac{{/\\!\\!\\!k}_{\\!{B}}}{2V}$ (106)
$\displaystyle[{\bar{\alpha}}_{\mu,{\bf K}},{\bar{X}}_{\bf
K^{\prime}}^{\nu}]=i\frac{{/\\!\\!\\!k}_{\\!{B}}}{2V}\delta_{\mu\nu}\delta_{{\bf
K}{\bf K^{\prime}}}$

i.e., the uncertainty relations increase as the volume of the system
decreases. This result is in line with our expectations: as the volume
decreases and we move to the mesoscopic level, the simultaneous measurement of
entropy production strength and time becomes more indeterminate. As already
remarked, this is due to our increased ability to see more and more details of
collisions and to the influence of fluctuations. These are the effects that
cause the inherent uncertainty associated with the smallest systems. The
transport coefficients (and then the entropy production) arise from particle
collisions, so discrete behaviors may be observed on a smaller scale.
Fluctuations arise due to the probabilistic nature of individual particle
interactions and movements. Fluctuations become more significant as we move to
the mesoscopic scale, where systems are smaller and composed of a limited
number of particles. In smaller systems, these fluctuations have a greater
impact on the system’s overall behavior as they can lead to transient changes
in entropy production strength. Summing up, the limited number of particles in
the system leads to a greater sensitivity to individual events, making it more
challenging to precisely determine the entropy production strength at any
specific moment.

##  VIII Conclusions and Future Works

The physical world can exhibit different behaviors and be described by
different laws depending on the scale of observation. The laws governing
macroscopic systems may not necessarily apply directly to mesoscopic or
microscopic systems, and vice versa. Overall, the physical world is governed
by a hierarchy of laws that apply at different scales, ranging from classical
mechanics and thermodynamics at the macroscopic scale to quantum mechanics and
emergent behavior at the mesoscopic and microscopic scales. Understanding
these scale-dependent laws is crucial for comprehending the diverse range of
phenomena observed in the universe. Emergent phenomena, as seen at the
mesoscopic scale, can arise due to the collective behavior of a large number
of particles. These phenomena may not have direct counterparts in either
classical or quantum mechanics alone, and they often require new theoretical
frameworks or models to describe them. One of the main objectives of the
Brussels School of Thermodynamics, directed by Prof. Ilya Prigogine, was to
investigate systems on a mesoscopic scale with the aim to discover the
fundamental laws governing them. In this work, inspired by this goal and
encouraged by recent experimental results, we established the canonical
commutation rules (ccr) valid at the mesoscopic scale. The ccr (106) shows
that the closer we get to the mesoscopic level, the more indeterminate becomes
the simultaneous measurement of the canonically conjugate variables. For
example, the uncertainties $\Delta\sigma$ and $\Delta t$ in simultaneously
existing values of the entropy production strength and time are related by the
expression $\Delta t\Delta\sigma\geq{/\\!\\!\\!k}_{\\!{B}}/2$; the greater the
accuracy with which one of these quantities is measured, the less the accuracy
with which the other can be measured at the same time. We have seen that
fundamental quantities such as the total entropy production, the thermodynamic
variables conjugate to the thermodynamic forces, and the Glansdorff-
Prigogine’s dissipative variable are discretized at the mesoscopic scale. The
ultraviolet divergence problem has been solved by applying the correspondence
principle to Einstein-Prigogine’s fluctuations theory in the limit of
macroscopic systems. Incidentally, we have also shown that the formalism based
on the canonical commutation rules confirms the validity of the Onsager
reciprocity relations over the entire linear region of thermodynamics.

Future works are devoted to the discretization of the total entropy
production, the thermodynamic variables conjugate to the thermodynamic forces,
and the Glansdorff-Prigogine’s dissipative variable for systems out of
Onsager’s region. This task will be accomplished by using the field theory
developed in sonnino3 , sonnino7 , and sonnino8 . This will involve, in the
expression for the entropy production, terms not only quadratic but cubic or
even of a higher degree i.e., terms, for example, of the type ($a+a^{+})^{4}$,
etc. The new expression for the total entropy production will then be the sum
of two terms: the expression found for systems in Onsager’s region and a sort
of interaction term (which does not commute with the previous one) multiplied
by a coupling constant $\epsilon$. The discretized entropy production is
obtained by a perturbative expansion by assuming that the constant $\epsilon$
is small enough.

##  IX Acknowledgments

Preliminary discussions on this matter were held with the late Prof. Enrique
Tirapegui. I would like to pay a special tribute to his memory. Enrique always
supported and encouraged me to pursue my research in the field of
Thermodynamics of Irreversible Processes. I would also like to pay tribute to
Prof. Ilya Prigogine. My strong interest in this domain of research is due to
him, who promoted the Brussels School of Thermodynamics at the Université
Libre de Bruxelles (U.L.B.), where I took my doctorate in Physics. I am
grateful to Prof. Pasquale Nardone and Dr Philippe Peeters, from the U.L.B.,
and Prof. Carlo Maria Becchi of the University of Genoa (Italy) for the useful
discussions and suggestions.

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# Debunking Fake News One Feature at a Time

Melanie Tosik1
<EMAIL_ADDRESS>Antonio Mallia2
<EMAIL_ADDRESS>Kedar Gangopadhyay1
<EMAIL_ADDRESS>1Department of Computer Science
New York University
60 Fifth Avenue
New York, NY 100112Computer Science and Engineering
New York University
6 MetroTech Center
Brooklyn, NY 11201

###### Abstract

Identifying the stance of a news article body with respect to a certain
headline is the first step to automated fake news detection. In this paper, we
introduce a 2-stage ensemble model to solve the stance detection task. By
using only hand-crafted features as input to a gradient boosting classifier,
we are able to achieve a score of 9161.5 out of 11651.25 (78.63%) on the
official Fake News Challenge (Stage 1) dataset. We identify the most useful
features for detecting fake news and discuss how sampling techniques can be
used to improve recall accuracy on a highly imbalanced dataset.

##  1 Introduction

Fake news is fueled in part by advances in technology, from automated bots
fabricating headlines and entire stories, to sophisticated software that can
create seemingly authentic videos. The spread of fake news is generally
considered a threat to media outlets, and to the democratic order. We aim to
use current techniques in natural language processing (NLP) and machine
learning (ML) to help combat the fake news problem.

Several professional fact-checking outlets111Popular fact-checking outlets
include snopes.com, truthorfiction.com, and fullfact.org. exist that employ
humans to resolve questionable claims. However, given the increasing rate of
fake news articles being generated each day, it is unlikely that even
professional fact checkers can evaluate every new headline without the
assistance of intelligent machines.

The _Fake News Challenge Stage 1_ (FNC-1) (Pomerleau and Rao, 2017) was
launched with the explicit goal of fostering the development of automated
tools to help address the spread of fake news. The main task is stance
detection, i.e. identifying the viewpoint of a given news article body with
respect to a given news headline. As part of the competition, a dataset of
49,972 of headline-body pairs was released.

50 competing teams developed systems for stance and fake news detection.
Systems were designed to classify whether a given article body was related to
a given headline, and if so, whether it agreed with, disagreed with, or
discussed the headline. Many of the top scoring systems, including the winning
system, at least partially relied on neural network (NN) architectures to
classify the headline-body pairs (Riedel et al., 2017).

While deep learning models have proven highly effective at a variety of text
classification tasks (Goldberg and Hirst, 2017), including stance detection,
they also severely lack interpretability. However, the ability to inspect and
evaluate the contribution of individual, interpretable features is
particularly desirable for an emergent task like fake news detection, because
it can provide valuable insight into the nature of the task and the inherent
idiosyncrasies that might exist in the data.

For this reason, we introduce a new system developed using only traditional
feature engineering. By incorporating hand-crafted features, we hope to gain a
better understanding of what exactly constitutes fake news, and which kind of
text-based features contribute the most to successful stance detection.
Overall, our system outperforms the official baseline by 3.43% and would rank
#13/50 on the official leaderboard.

##  2 Related work

The FNC-1 dataset is inspired by _Emergent_ , a dataset derived from a digital
journalism project at Columbia University (Ferreira and Vlachos, 2016). The
original dataset contains 300 rumored claims and 2,595 news articles collected
and labeled by journalists, with an estimate of their veracity (true, false,
or unverified). Each associated article was summarized into a headline and
labeled to indicate whether its stance is for, against, or observing the
claim. The FNC-1 dataset extends the _Emergent_ dataset, assigning one of the
following labels to each headline-body pair: agree, disagree, discuss, or
unrelated. The goal is to automatically predict the labels in a supervised
classification task.

Riedel et al. (2017) introduce a NN architecture based on term frequency-
inverse document frequency (TF-IDF) transformed bag-of-words (BOW)
representations as input to a multi-layer perceptron (MLP). With a final score
of 81.72%, their system won third place. The second place winners also employ
a suite of MLPs (Hanselowski et al., 2017), scoring 81.98% accuracy on the
FNC-1. The winners of the competition were a team from Talos Intelligence, a
threat intelligence subsidiary of Cisco Systems. Their system uses an equally-
weighted ensemble model of a deep convolutional NN and gradient boosted
decision trees (Pan et al., 2017).

In this paper, we introduce two supervised classifiers to perform the task of
stance detection on the FNC-1 dataset. All of our code and data are publicly
available in our GitHub repository222github.com/NYU-FNC/FakeNewsChallenge.

##  3 Dataset

We employ the publicly available FNC-1 dataset333github.com/FakeNewsChallenge.
The dataset consists of two data splits for training and testing, each
containing a number of annotated headline-body pairs. The original sample
distribution of the dataset is shown in Table 1.

Total | agree | disagree | discuss | unrelated
---|---|---|---|---
49,972 | 3,678 | 840 | 8,909 | 36,545
Table 1: Data distribution of the FNC-1 dataset.

Due the high imbalance of class labels in the original data set, we also
experiment with simple oversampling of the disagree category to match the
number of agree samples in the original dataset. Table 2 shows the data
distribution of the dataset after resampling.

Total | agree | disagree | discuss | unrelated
---|---|---|---|---
52,810 | 3,678 | 3,678 | 8,909 | 36,545
Table 2: Data distribution after oversampling. Name | Description | Output type
---|---|---
Overlap
dep_object_overlap | Overlap between headline/body grammatical objects | Integer
dep_subject_overlap | Overlap between headline/body grammatical subjects | Integer
ngram_overlap | Intersection over union of headline/body n-gram set | Real
ngram_overlap_intro | Intersection over union of headline/intro n-gram set | Real
word_overlap | Intersection over union of headline/body word set | Real
word_overlap_intro | Intersection over union of headline/intro word set | Real
Distance and similarity measures
cosine_count | Cosine similarity between binary headline/body count vectors | Real
cosine_tdidf | Cosine similarity between headline/body TF-IDF vectors | Real
doc_similarity | Cosine similarity between averaged headline/body Common Crawl vectors | Real
doc_similarity_intro | Cosine similarity between averaged headline/intro Common Crawl vectors | Real
hamming_distance | Hamming distance between binary headline/body count vectors | Real
wmdistance | Word Mover’s Distance (WMD) between headline/body (Kusner et al., 2015) | Real
Miscellaneous
len_stance | Length of headline | Integer
len_body | Length of body | Integer
KL_pk_qk | Kullback–Leibler divergence between headline/body topic probability distribution | Real
KL_qk_pk | Kullback–Leibler divergence between body/headline topic probability distribution | Real
refute | Occurrence of {fake, fraud, hoax, not, deny, fabricate, authenticity} in body | Binary
refute_intro | Occurrence of {fake, fraud, hoax, not, deny, fabricate, authenticity} in intro | Binary
sentiment_body | Average CoreNLP sentiment score of body | Real
sentiment_stance | Average CoreNLP sentiment score of headline | Real
Table 3: List of features in the XGBoost classifiers, along with descriptions
and output types.

##  4 Classifier architecture

Our system is built with XGBoost444github.com/dmlc/xgboost, an optimized
distributed gradient boosting library. Gradient boosting is a popular
technique that can solve complex regression or classification tasks by
producing and combining a number of weaker and smaller prediction models in
the form of decision trees. The model is built in stages and generalized by
optimizing a differentiable loss function. As a result, gradient boosting
combines a number of weak learners into a single, strong learner on an
iterative basis. In contrast to linear classifiers (such as logistic
regression) decision tree models are capable of capturing non-linear
relationships in data as well.

Initially, we develop a 1-stage classifier to predict all four class labels
(agree, disagree, discuss, or unrelated) in a single pass. To address the high
class imbalance in the dataset, we also implement a 2-stage ensemble model to
first classify samples into related and unrelated before replacing all related
predictions with a more granular category of agree, disagree, or discuss in
the second stage.

We estimate the best hyperparameter settings for each model using a grid
search with cross-validation on the training set. Carefully tuning the tree-
related hyperparameters (such as the maximum depth of a tree) results in the
largest increase of cross-validation accuracy. Tuning the learning rate is
effective to prevent overfitting on the training data. Using a large number of
estimators (approx. 1,000) results in the best performance overall, with
training time increasing proportionally.

##  5 Evaluation

The FNC-1 evaluates predictions using a weighted metric that assigns +0.25
points for each correctly predicted related / unrelated sample, and +0.75
points for each correctly predicted agree / disagree / discuss sample. The
rationale behind the 2-level scoring system is that the second stage of
classification is both more difficult _and_ more relevant to fake news
detection which should be reflected in the scoring schema.

##  6 Feature engineering

We experiment with the following features: overlap between token attributes,
distance and similarity measures between vector representations, relative
entropy between topic model probability distributions, headline and body
lengths, sentiment scores, as well as a short list of explicit words used to
refute or debunk a false claim. The complete set of features is described in
detail in Table 3.

“Intro” features are computed on the first 250 character of the original body
text. Most of the features assume basic preprocessing of the input text,
including removing stop words and punctuation and lemmatizing all remaining
tokens in the resulting sequence. All preprocessing is done using _spaCy_
555spacy.io, an emerging open source NLP toolkit written in Python. The only
exception is the sentiment analysis component, which relies on the _Stanford
CoreNLP_ 666stanfordnlp.github.io/CoreNLP/ Java library. We use latent
Dirichlet allocation (LDA) (Blei et al., 2003) to train a topic model on the
first 100,000 documents of The New York Times Newswire Service portion of the
English Gigaword777catalog.ldc.upenn.edu/ldc2003t05 corpus.

##  7 Results

Our best models achieves a FNC-1 score of 9161.5 out of 11651.25 (78.63%),
compared to the official baseline score of 8761.75 out of 11651.25 (75.20%).
Our score would correspond to rank 13 out of 50 on the official FNC-1
leaderboard888https://competitions.codalab.org/competitions/16843#results. Our
best-performing model is the 2-stage ensemble model trained on the original,
imbalanced dataset.

In total, we share results for 3 different model architectures. Table 4 and 5
contain results for our 1-stage and 2-stage classifiers trained on the
original dataset. Table 6 also shows results for our 2-stage classifier when
trained on the resampled training data.

Overall, we find that the 2-stage ensemble model outperforms the 1-stage
classifier. We assume this is due to the enhanced flexibility when it comes to
finding the optimal set of features and parameters at each stage.

Confusion matrix
---
| agree | disagree | discuss | unrelated
agree | 144 | 4 | 1,607 | 148
disagree | 12 | 1 | 522 | 162
discuss | 190 | 2 | 3,874 | 398
unrelated | 2 | 0 | 246 | 18,101
Accuracy: | 0.870
FCN-1 score: |  9128.5 (78.35%)
Table 4: FNC-1 results for the 1-stage classifier. Confusion matrix
---
| agree | disagree | discuss | unrelated
agree | 27 | 0 | 1,733 | 143
disagree | 9 | 0 | 533 | 155
discuss | 45 | 0 | 4,060 | 359
unrelated | 5 | 0 | 366 | 1,7978
Accuracy: | 0.868
FCN-1 score: |  9161.5 (78.63%)
Table 5: FNC-1 results for the 2-stage classifier. Confusion matrix
---
| agree | disagree | discuss | unrelated
agree | 25 | 21 | 1,718 | 139
disagree | 4 | 7 | 529 | 157
discuss | 33 | 84 | 3,993 | 354
unrelated | 6 | 3 | 366 | 1,7974
Accuracy: | 0.866
FCN-1 score: |  9115.75 (78.24%)
Table 6: FNC-1 results for the 2-stage classifier, on the resampled dataset.

##  8 Analysis

In order to understand both the task and best features in our system, we
conducted posthoc and abalation analyses of our results.

All of the simple overlap features between words and n-grams of the
headline/body text prove surprisingly effective. In contrast, only some of the
defined distance and similarity measures improve classification accuracy.
Specifically, useful distance measure features include the cosine similarity
TF-IDF feature, document similarity features on intro and body text, and most
importantly, the Word Mover’s Distance (WMD) feature (Kusner et al., 2015).
Hamming distance and cosine similarity between binary count vectors on the
other hand are not helpful for stance detection.

Somewhat surprisingly, sentiment features also did not improve classification
accuracy. Our hypothesis is that the overall polarity of of the body text in
particular is weakened by simply averaging the sentiment scores for each
sentence in the article. As a result, the sentiment features exhibit low
discriminative power at classification time.

Similarly, we find that computing the relative entropy between topic model
probability distributions does not contribute to the predictive accuracy of
the classifier. It is possible that this feature could be improved by
increasing the amount of training data, fine-tuning the number of latent
topics, or employing a different metric to measure the divergence of the
output probability distributions. On the other hand, introducing a simple list
of words used to debunk a claim or statement improves model accuracy.
Incorporating the word count of each heading-body pair is helpful as well.

By resampling the disagree category, we are able to successfully introduce a
bias towards the minority class into the ensemble model. As shown in Table 5,
the 2-stage model is unable to predict any disagree examples when trained on
the original FNC-1 dataset. After resampling, recall accuracy on the disagree
category increases significantly. Unfortunately, this improvement comes at the
expense of precision accuracy on agree and discuss labels, indicating a
classic precision-recall trade-off. However, we feel that this finding also
illustrates a major shortcoming of the evaluation metric: our best-performing
model fails to predict disagree examples entirely, yet is able to achieve the
highest weighted score overall.

Among the 50 challenge participants, even the top scoring systems were only
able to achieve accuracy scores of around 80%. We believe that there are two
main reasons for the perceived difficulty of the challenge.

First, the dataset is highly imbalanced. In addition, many of the labeled
examples are ambiguous upon manual inspection. While the FNC-1 did not release
any statistics on the inter-annotator agreement or the correlation with human
adequacy judgments on this dataset, we believe that the stance detection task
could benefit from additional labeled training data.

Second, the stance detection task requires the design of features that capture
the degree and nature of the relatedness between two text paragraphs of highly
varying length. This is significantly more challenging than most traditional
NLP tasks, which commonly require the analysis of only a single word,
sentence, or document during feature extraction.

##  9 Conclusion

In this paper, we present our approach to stance detection, which relies on
hand-crafted features as input to a gradient boosted ensemble classifier. Our
strategy outperforms the FNC-1 baseline while adopting a well-defined and easy
to interpret methodology.

Despite the promising results, there is room for improvement. The most
challenging aspect of the FNC-1 dataset is the high imbalance of class labels
in addition to having to relate two text bodies of varying lengths.

Given the recent advances in automated text summarization (Bharti and Babu,
2017), one ambitious continuation of this project could involve the
summarization of article bodies before relating them to their headline. This
approach could help decrease random noise in the text, thereby improving the
robustness of our features. Similarly, only including sentiment scores that
are highly polarizing might help obtain more discriminative sentiment
features. Implementing a more advanced sampling technique could improve
handling of the imbalance in the dataset.

Finally, we could experiment with employing different feature sets at
different stages of the multi-stage ensemble classifiers.

## Acknowledgements

We are grateful to our professor, Ralph Grishman, for the opportunity to
pursue this project as part of his NLP graduate course at NYU.

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