# Fluctuation theorem for irreversible entropy production in electrical
conduction

Marcus V. S. Bonança<EMAIL_ADDRESS>Instituto de Física ‘Gleb
Wataghin’, Universidade Estadual de Campinas, 13083-859, Campinas, São Paulo,
Brazil  Sebastian Deffner<EMAIL_ADDRESS>Department of Physics, University
of Maryland, Baltimore County, Baltimore, MD 21250, USA  Instituto de Física
‘Gleb Wataghin’, Universidade Estadual de Campinas, 13083-859, Campinas, São
Paulo, Brazil

###### Abstract

Linear, irreversible thermodynamics predicts that the entropy production rate
can become negative. We demonstrate this prediction for metals under AC-
driving whose conductivity is well-described by the Drude-Sommerfeld model. We
then show that these negative rates are fully compatible with stochastic
thermodynamics, namely, that the entropy production does fulfill a fluctuation
theorem. The analysis is concluded with the observation that the stochastic
entropy production as defined by the surprisal or ignorance of the Shannon
information does not agree with the phenomenological approach.

##  I Introduction

The only processes that are fully describable by means of traditional
thermodynamics are infinitely slow successions of equilibrium states Callen
(1985). While considering such idealized situations is well-suited to
formulate universal statements, its practical insight is somewhat limited. All
real processes occur at finite rates, and therefore entropy is irretrievably
lost into the environment. Historically the first attempt to quantify this
entropy “production” was developed in _linear, irreversible thermodynamics_
Prigogine (1961); De Groot and Mazur (2013). The central assumption of this
generalized theory is that fluxes depend only linearly on the forces driving
the system away from equilibrium. Its results are obtained from a combination
of the local equilibrium hypothesis with conservation laws. The textbook
example is heat conduction generated by a temperature gradient, which is
described by Fourier’s law Zwanzig (2001).

An even more widely-known example of such linear problems is electrical
conduction, that is described by Ohm’s law. Nevertheless, there are still
subtleties to be unveiled. Only recently, we showed in Ref. Bonança  _et al._
(2021) that the entropy production in noble metals under AC driving exhibits
non-monotonic growth as a function of time. In other words, in electrical
conduction the entropy production rate can become negative. Such negative
rates occur in situations, in which the external driving is too fast for the
system to react, and hence the response lacks behind the overall dynamics.
Similar observations have been reported in, e.g., viscoelastic fluids Williams
_et al._ (2007) and single-level quantum dots Thingna  _et al._ (2017) under
oscillatory driving and in freely expanding ideal gases Chakraborti  _et al._
(2021). Interestingly, negative entropy production rates in open systems
undergoing non-Markovian dynamics are somewhat commonplace Bhattacharya  _et
al._ (2017); Marcantoni  _et al._ (2017); Xu  _et al._ (2018); Popovic  _et
al._ (2018); Strasberg and Esposito (2019). However, our analysis is entirely
based on the lag of response Zwanzig (1961); McLennan (1964), which also gives
rise to negative rates in Markovian settings Nazé and Bonança (2020); Bonança
_et al._ (2021).

Even more remarkably, negative entropy production rates are not indicative of
scenarios operating far from thermal equilibrium, but they can be observed
fully within the regime of linear, irreversible thermodynamics Nazé and
Bonança (2020). Since this approach fully rests on phenomenological and
macroscopic arguments, the considered entropy production is often considered
as genuinely thermodynamic.

A more modern approach to nonequilibrium problems is _stochastic
thermodynamics_ Peliti and Pigolotti (2021). Arguably, the most central notion
is the stochastic entropy production, which is defined as a statistical, or
rather information theoretic quantity. Whereas in linear, irreversible
thermodynamics the entropy production is expressed as a bilinear form of
fluxes and forces, in stochastic thermodynamics we have the “surprisal”, i.e.,
the logarithm of the system’s distribution in state space Seifert (2005). The
main results of stochastic thermodynamics are the fluctuation theorems, which
quantify that negative fluctuations of the entropy production are
exponentially suppressed Crooks (1999); Seifert (2005); Esposito and Van den
Broeck (2010a).

The natural question arises, whether the two paradigms are consistent with
each other, or rather whether the two “versions” of entropy production are
equivalent. In particular, the occurrence of negative rates in irreversible
thermodynamics Williams  _et al._ (2007); Nazé and Bonança (2020); Bonança
_et al._ (2021) appears incompatible with the strict positivity of the
stochastic entropy production rate often discussed in the literature Spohn
(1978); Prigogine and Géhéniau (1986); Ruelle (1996); Maes  _et al._ (2000);
Belandria (2005); Esposito and Van den Broeck (2010b); Van den Broeck and
Esposito (2010); Tomé and Oliveira (2010); Bauer  _et al._ (2016); Brandner
and Seifert (2016); Deffner (2017); Jabraoui  _et al._ (2020); Landi and
Paternostro (2021); Deffner and Bonança (2020).

In the present letter, we show that the entropy production as defined in
linear, irreversible thermodynamics indeed fulfills a fluctuation theorem. For
pedagogical reasons and for specificity we focus on electrical conduction in
Drude-Sommerfeld metals Drude (1900a, b); Sommerfeld (1928); Bardeen (1940)
under AC-driving. Despite its somewhat crude approximations, experiments have
shown that the Drude model, combined with Fermi-Dirac statistics, does
describe properties of real metals such as gold, copper and silver Olmon  _et
al._ (2012); Yang  _et al._ (2015) at room temperature and low photon
energies. However, the validity of our results is not restricted to electric
conduction. Rather, it is easy to see that our analysis remains valid for
other situations that can be described with the linear framework. Finally, we
will also see that the irreversible entropy production and the stochastic
entropy production are typically different, and that they only become
identical in the limit of infinitely slow driving.

##  II Irreversible thermodynamics

To keep the discussion self-contained, we begin by briefly reviewing the
approach, and by establishing notions and notations. In our analysis, we use
irreversible thermodynamics to derive an expression for the entropy production
(EP) rate for a situation in which monochromatic and polarized light is shined
on a piece of metal. In this case, the balance equation for the
electromagnetic energy density $u_{\mathrm{EM}}(\mathbf{r},t)$ reads,

$\frac{\partial u_{\mathrm{EM}}}{\partial
t}+\nabla\cdot\mathbf{S}=-\mathbf{J}_{e}\cdot\mathbf{E}\,,$ (1)

where $\mathbf{S}$ is the Poynting vector. Equation (1) contains a source term
which describes the power density lost to the charge carriers, which is the
product of the electrical current density $\mathbf{J}_{e}$ and the electric
field $\mathbf{E}$.

Accordingly, the balance equation for the internal energy density
$u(\mathbf{r},t)$ of the charge carriers becomes,

$\frac{\partial u}{\partial
t}+\nabla\cdot\mathbf{J}_{u}=\mathbf{J}_{e}\cdot\mathbf{E}\,,$ (2)

where $\mathbf{J}_{u}$ denotes the flux of internal energy and, due to energy
conservation, the source term has the opposite sign. Further assuming a
constant number of charge carriers, we also have the balance equation for the
entropy density $s(\mathbf{r},t)$,

$\frac{\partial s}{\partial
t}+\nabla\cdot\mathbf{J}_{s}=\dot{\overline{\Sigma}}\,,$ (3)

whose source term is the EP rate we want to obtain.

Combining Eqs. (2) and (3) with the local equilibrium hypothesis (see Eq. (41)
and App. A), we obtain,

$\dot{\overline{\Sigma}}=\mathbf{J}_{u}\cdot\nabla\left(\frac{1}{T}\right)-\mathbf{J}_{n}\cdot\nabla\left(\frac{\mu}{T}\right)+\frac{\mathbf{J}_{e}\cdot\mathbf{E}}{T}\,,$
(4)

whose last term is the contribution from electrical conduction. The quantities
$T$ and $\mu$ denote, respectively, the temperature and the chemical potential
imposed on the set of charge carriers as scalar fields, and $\mathbf{J}_{n}$
is the particle flux.

Temperature, chemical potential and pressure gradients are constraint by the
Gibbs-Duhem equation Reichl (2016) (see App. A), which implies that under
uniform temperature and pressure, the EP rate simply becomes

$\dot{\overline{\Sigma}}=\frac{\mathbf{J}_{e}\cdot\mathbf{E}}{T}\,.$ (5)

Additionally, phenomenological linear relations between currents and forces
are assumed to hold. In electrical conduction, such a relation exists in the
frequency domain Kubo  _et al._ (2012),

$\hat{J}_{e,i}(\omega)=\sum_{j}\sigma_{ij}(\omega)\hat{E}_{j}(\omega)\,,$ (6)

i.e., between the Fourier transforms of the electrical current,
$\hat{J}_{e,i}(\omega)$, and the electric field, $\hat{E}_{j}(\omega)$, along
the $i$ and $j$ direction, respectively. Note that Eq. (6) is nothing but
Ohm’s law with the conductivity tensor $\sigma_{ij}(\omega)$.

To evaluate the EP rate $\dot{\overline{\Sigma}}$ (5), an expression for the
current in time domain is required. The inverse Fourier transform of Eq. (6)
reads

$J_{e,i}(t)=\int_{-\infty}^{t}dt^{\prime}\,\sum_{j}\Phi_{ij}(t-t^{\prime})\,E_{j}(t^{\prime})\,,$
(7)

where $\Phi_{ij}(t)$ denotes the response function Kubo  _et al._ (2012).
Equation (7) describes possible memory effects since current and electric
field are not evaluated at the same instant of time Kubo  _et al._ (2012);
Zwanzig (1961). Combining Eq. (7) and the bilinear form (5) for
$\dot{\overline{\Sigma}}$, we obtain

$\dot{\overline{\Sigma}}=\frac{1}{T}\,\sum_{i,j}E_{i}(t)\,\int_{-\infty}^{t}dt^{\prime}\,\Phi_{ij}(t-t^{\prime})\,E_{j}(t^{\prime})\,.$
(8)

This expression contrasts the more common case in which thermodynamic forces
are evaluated at the same instant of time and the EP rate remains strictly
positive, see for instance Ref. Maes  _et al._ (2000); Bauer  _et al._ (2016);
Brandner and Seifert (2016).

##  III Drude-Sommerfeld entropy production

In the Drude model the charge carriers in an ideal metal are described as a
classical free electron gas which obeys Maxwell-Boltzmann statistics Drude
(1900a, b). In its extension to the quantum domain, the equilibrium statistics
is upgraded to the Fermi-Dirac one and the quantum effects are encoded in the
density $N$ of charge carriers at the Fermi level, in the behavior of the
relaxation time with temperature, and in the effective (band) mass $m$ of the
electrons Sommerfeld (1928); Bardeen (1940); Ashcroft and Mermin (1976).

Whereas an ideal gas is strictly free of collisions, assuming a finite
relaxation time in the Drude model implies a finite mean free path. This mean
free path is justified by considering the scattering of the charge carriers
with phonons and impurities Bardeen (1940). It can be phenomenologically
introduced expressing the collision term of the Boltzmann equation as Ashcroft
and Mermin (1976), $\left(\partial\rho/\partial
t\right)_{coll}=-\delta\rho/\tau_{R}\,$, where $\tau_{R}$, $\rho$ and
$\delta\rho=\rho-\rho_{0}$ denote the relaxation time, the non-equilibrium
distribution and the deviation from the initial equilibrium distribution
$\rho_{0}$, respectively.

Hence, we obtain for the conductivity

$\sigma(\omega)=\frac{\sigma_{0}}{1-i\omega\,\tau_{R}}\,,$ (9)

where $\sigma_{0}=Nq^{2}\tau_{R}/m$ is the DC conductivity in the zero-
frequency limit (Ashcroft and Mermin, 1976) and $q$ denotes the electric
charge.

Computing the inverse Fourier transform of Eq. (9), we obtain the response
function

$\Phi(t)=\Theta(t)\,\frac{\sigma_{0}}{\tau_{R}}\exp{\left(-|t|/\tau_{R}\right)}\,,$
(10)

and $\Theta(t)$ is the Heaviside step function. Hence, the EP rate (8) becomes

$\dot{\overline{\Sigma}}=\frac{\sigma_{0}}{T\tau_{R}}\,E(t)\int_{0}^{t}dt^{\prime}\,\exp{\left(-|t-t^{\prime}|/\tau_{R}\right)}\,E(t^{\prime})\,.$
(11)

Assuming a monochromatic electric field,
$E(t)=E_{0}\sin\left(\omega_{0}t\right)$, and employing Eq. (11), the entropy
production rate (8) reads,

$\begin{split}&\dot{\overline{\Sigma}}=\frac{1}{T}\frac{\sigma_{0}\,E^{2}_{0}\,\sin{(\omega_{0}t)}}{1+(\omega_{0}\tau_{R})^{2}}\\\
&\times\left[\omega_{0}\tau_{R}\,\exp{\left(-t/\tau_{R}\right)}-\omega_{0}\tau_{R}\,\cos{(\omega_{0}t)}+\sin{(\omega_{0}t)}\right].\end{split}$
(12)

Figure 1 illustrates Eq. (12) for different values of $\omega_{0}\tau_{R}$. As
discussed in Ref. Bonança  _et al._ (2021), negative values of
$\dot{\overline{\Sigma}}$ persist within a small vicinity of
$\omega_{0}t=n\pi$, $n=1,2,3,\ldots$, as $\omega_{0}\tau_{R}$ decreases and
vanish only in the limit $\omega_{0}\tau_{R}\to 0$.

The emergence of negative values of $\dot{\overline{\Sigma}}$ can be
understood qualitatively from Eq. (11). $\dot{\overline{\Sigma}}$ is given by
the product of a time-dependent electric field and a convolution between the
same field (evaluated at a previous time) and the response function. This
describes a delay between the response of the system and the electric field.
Thus, if the function $E(t)$ is non-monotonic and acquires negative values,
negative values of $\dot{\overline{\Sigma}}$ occur 111Note, however, that it
has been shown that the entropy production,
$\overline{\Sigma}=\int_{0}^{t}dt^{\prime}\,\dot{\overline{\Sigma}}$, itself
remains positive at all times Nazé and Bonança (2020); Bonança  _et al._
(2021). .

Figure 1: (Color online) Entropy production rate (12) as a function of
$t/\tau_{R}$ for different values of $\omega_{0}\tau_{R}$. Solid-blue, dashed-
red and dotted-brown lines correspond to $\omega_{0}\tau_{R}=0.8$, $0.5$ and
$0.2$, respectively. Units are $\dot{\Sigma}_{0}\equiv\sigma_{0}E_{0}^{2}/T$.

##  IV Fluctuation theorem

As mentioned before, Drude originally considered the charge carriers as a
classical gas of particles following Maxwell-Boltzmann statistics. Although
this leads to wrong predictions of the DC conductivity $\sigma_{0}$, it does
not change the form of $\sigma(\omega)$ given by Eq. (9). This means that the
qualitative features of the EP rate $\dot{\overline{\Sigma}}$ and its
expression (12) remain unaltered in the original Drude model, although with a
different value of $\sigma_{0}$.

For the next part of the analysis, we restrict ourselves to the classical
Drude model and develop the corresponding stochastic thermodynamics. Here, $N$
denotes the _number_ (instead of density) of non-interacting charge carriers
and by $J_{e}$ we denote the electrical current (instead of current density).
We start by combining Ohm’s law (6) with the Drude conductivity (9),

$\hat{J}_{e}(\omega)=\frac{\sigma_{0}}{1-i\omega\tau_{R}}\hat{E}(\omega)\,,$
(13)

which in time domain takes the form of the following equation of motion for
$J_{e}(t)$ ,

$\frac{dJ_{e}}{dt}+\frac{J_{e}}{\tau_{R}}=\frac{\sigma_{0}}{\tau_{R}}E(t)\,.$
(14)

Recalling that $J_{e}(t)$ is defined as $J_{e}(t)=Nq\langle p(t)\rangle/m$,
where $\langle p(t)\rangle$ is the average momentum of a charge carrier along
the direction of the electric field, Eq. (14) yields

$\frac{d\langle p(t)\rangle}{dt}+\frac{\langle
p(t)\rangle}{\tau_{R}}=qE(t)\,,$ (15)

and $\sigma_{0}=Nq^{2}\tau_{R}/m$.

Fluctuations can be introduced by removing the average in Eq. (15) and
introducing a Gaussian-distributed white noise $f_{i}(t)$ with zero mean
acting on the $i$th charged particle. The equation of motion for the
fluctuating linear momentum of the $i$th charge along the direction of
$\mathbf{E}$(t) then reads

$\frac{dp_{i}(t)}{dt}+\frac{p_{i}(t)}{\tau_{R}}=qE(t)+f_{i}(t)\,,$ (16)

which is nothing but Brownian motion under the external force $qE(t)$. The
fluctuating current $j_{e}(t)$ then is,

$j_{e}(t)=\frac{q}{m}\sum_{i=1}^{N}p_{i}(t)\,,$ (17)

and we define the fluctuating entropy production,

$\Sigma(t)=\frac{1}{T}\int_{0}^{t}dt^{\prime}\,j_{e}(t^{\prime})E(t^{\prime})\,,$
(18)

inspired by irreversible thermodynamics (see Eq. (5)).

Since the $p_{i}(t)$ are Gaussian distributed and $\Sigma(t)$ is a linear
transformation of their sum, the probability distribution of $\Sigma(t)$ will
be Gaussian, as well. See Refs. Zon and Cohen (2003); Jiménez-Aquino (2010)
for similar arguments. From the solution of Eq. (16) and expression (18), it
can be shown that (see App. B)

$\langle\Sigma(t)\rangle=\frac{Nq^{2}}{mT}\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\,E(t^{\prime})\,e^{-(t^{\prime}-t^{\prime\prime})/\tau_{R}}\,E(t^{\prime\prime})\,.$
(19)

when the linear momenta of the charge carriers are distributed according to
Maxwell’s distribution at $t=0$.

Under the same assumptions, we have (see App. B)

$\displaystyle\langle\Sigma^{2}(t)\rangle=\langle\Sigma(t)\rangle^{2}$ (20)
$\displaystyle+$ $\displaystyle
k_{B}\frac{Nq^{2}}{mT}\int_{0}^{t}dt^{\prime}\int_{0}^{t}dt^{\prime\prime}\,E(t^{\prime})\,e^{-|t^{\prime}-t^{\prime\prime}|/\tau_{R}}\,E(t^{\prime\prime})\,,$

and we obtain

$\mathrm{Var}\left[\Sigma(t)\right]=\langle\Sigma^{2}(t)\rangle-\langle\Sigma(t)\rangle^{2}=2k_{B}\langle\Sigma(t)\rangle\,.$
(21)

The probability distribution of $\Sigma$ then reads

$P(\Sigma)=\frac{1}{\sqrt{2\pi\mathrm{Var}\left[\Sigma\right]}}\exp{\left(-\frac{\left[\Sigma-\langle\Sigma\rangle\right]^{2}}{2\mathrm{Var}\left[\Sigma\right]}\right)}\,,$
(22)

which implies the detailed Fluctuation Theorem,

$\frac{P(\Sigma)}{P(-\Sigma)}=\exp{\left[\frac{2\langle\Sigma\rangle\Sigma}{\mathrm{Var}\left(\Sigma\right)}\right]}=e^{\Sigma/k_{B}}\,.$
(23)

and its integral version,
$\left\langle\exp{\left(-\Sigma/k_{B}\right)}\right\rangle=1$

Note that the average EP rate is,

$\langle\dot{\Sigma}(t)\rangle=\frac{Nq}{mT}\langle p(t)\rangle E(t)\\\ $ (24)

whose time integral is identical to Eq. (19), and with different values and
units of $\sigma_{0}$ in Eq. (11). Hence, we conclude that negative values of
the EP rate are not incompatible with the Fluctuation Theorem (23) and, hence,
the second law.

##  V Entropy production from microreversibility

As a next step, we show that the expression (18) for the EP appears naturally
from microreversibility Crooks (1999), i.e., from the comparison of
probabilities of observing a given trajectory and its time-reversed conjugated
twin Chernyak  _et al._ (2006); Imparato and Peliti (2006); Deffner  _et al._
(2011); Pal and Deffner (2020).

Considering Eq. (16) for a single charge carrier, its corresponding Fokker-
Planck equation reads Risken (1996) (we have dropped the index $i$)

$\displaystyle\frac{\partial}{\partial t}\mathcal{W}(p,t)$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial
p}\left[\left(-\frac{p}{\tau_{R}}+qE(t)\right)\mathcal{W}(p,t)\right]$ (25)
$\displaystyle+$ $\displaystyle\frac{\partial^{2}}{\partial
p^{2}}\left(\frac{mk_{B}T}{\tau_{R}}\mathcal{W}(p,t)\right)\,,$

where $\mathcal{W}(p,t)$ denotes the probability density of linear momentum at
a given $t$. Following the standard procedures Risken (1996), it is
straightforward to obtain the path-integral solution of Eq. (25) Chernyak  _et
al._ (2006); Imparato and Peliti (2006); Risken (1996).

In this formulation, the time evolution of $\mathcal{W}(p,t)$ is expressed in
terms of probabilities of trajectories. Denoting by
$\mathcal{P}[\Gamma|p_{0}]$ the conditional probability of observing a
trajectory $\Gamma$ that starts at $p_{0}$, its expression reads

$\displaystyle\mathcal{P}[\Gamma|p_{0}]=\mathcal{N}\exp{\bigg{\\{}-\int_{0}^{t}dt^{\prime}\,\mathcal{L}(p(t^{\prime}),\dot{p}(t^{\prime});E(t^{\prime}))\bigg{\\}}}\,,$
(26)

where

$\mathcal{L}(p(t^{\prime}),\dot{p}(t^{\prime});E(t^{\prime}))=\frac{\left[\dot{p}(t^{\prime})-(-p(t^{\prime})/\tau_{R}+qE(t^{\prime}))\right]^{2}}{(4mk_{B}T/\tau_{R})}\,,$
(27)

and $\mathcal{N}$ is a normalization factor Risken (1996).

Following Ref. Chernyak  _et al._ (2006), we define $\Gamma^{\dagger}$ as the
time-reversed conjugated twin of $\Gamma$ for the time-reversed electric field
$E^{R}(t)$. Thus, the probability of observing $\Gamma^{\dagger}$ is given by

$\displaystyle\mathcal{P}[\Gamma^{\dagger}|-p_{t}]$ $\displaystyle=$
$\displaystyle\mathcal{N}$
$\displaystyle\exp{\bigg{\\{}-\int_{0}^{t}dt^{\prime}\,\mathcal{L}(p^{\dagger}(t^{\prime}),\dot{p}^{\dagger}(t^{\prime});E^{R}(t^{\prime}))\bigg{\\}}}\,,$

where the initial condition $-p_{t}$ of $\Gamma^{\dagger}$ is the time-
reversal of the final point of $\Gamma$.

Combining Eqs. (26) and (LABEL:eq.probtrajr) we have the microreversibility
condition (see App. C),

$\displaystyle\frac{\mathcal{P}[\Gamma|p_{0}]}{\mathcal{P}[\Gamma^{\dagger}|-p_{t}]}=$
(29)
$\displaystyle\exp{\bigg{\\{}-\frac{(p_{t}^{2}-p_{0}^{2})}{2mk_{B}T}+\frac{q}{mk_{B}T}\int_{0}^{t}dt^{\prime}\,p(t^{\prime})E(t^{\prime})\bigg{\\}}}\,,$

where the second term in the exponent is proportional to the single-particle
version of Eq. (18). This term can be understood as proportional to the
fluctuating power input due to the electric field. On the other hand, the
first term in the exponent is proportional to the variation of the stochastic
internal energy (which in the present case is purely kinetic) between the
final and initial states. Hence, energy conservation implies that the entire
exponent is equal to the heat absorbed by the heat bath in units of $k_{B}T$
Chernyak  _et al._ (2006); Imparato and Peliti (2006); Crooks (1999); Pal and
Deffner (2020).

##  VI Entropy production from Shannon information

In the preceding sections, we showed how the fluctuating entropy production
can be found in full consistency with linear irreversible thermodynamics. In
stochastic thermodynamics, however, the EP is often obtained from the Shannon
information Peliti and Pigolotti (2021),

$S(t)=-k_{B}\int dp\,\mathcal{W}(p,t)\ln{\mathcal{W}(p,t)}\,,$ (30)

for the solution $\mathcal{W}(p,t)$ of Eq. (25) Seifert (2005). Following
standard arguments Tomé and Oliveira (2010); Peliti and Pigolotti (2021), we
have

$\displaystyle\frac{1}{k_{B}}\frac{dS}{dt}$ $\displaystyle=$
$\displaystyle\frac{\tau_{R}}{mk_{B}T}\int
dp\,\frac{\mathcal{J}(p,t)^{2}}{\mathcal{W}(p,t)}+\frac{1}{mk_{B}T}\int
dp\,p\mathcal{J}(p,t)$ (31) $\displaystyle=$
$\displaystyle\Pi(t)-\Omega(t)\,,$

where

$\displaystyle\mathcal{J}(p,t)=-\frac{p}{\tau_{R}}\mathcal{W}(p,t)-\frac{\partial}{\partial
p}\left(\frac{mk_{B}T}{\tau_{R}}\mathcal{W}(p,t)\right)\,.$ (32)

The term denoted by $\Omega(t)$,

$\Omega(t)=-\frac{1}{mk_{B}T}\int dp\,p\mathcal{J}(p,t)\,,$ (33)

is typically called the entropy flux Peliti and Pigolotti (2021) and the non-
negative term,

$\Pi(t)=\frac{\tau_{R}}{mk_{B}T}\int
dp\,\mathcal{W}(p,t)\left(\frac{\mathcal{J}(p,t)}{\mathcal{W}(p,t)}\right)^{2}\,,$
(34)

is taken as the EP rate Tomé and Oliveira (2010); Van den Broeck and Esposito
(2010); Peliti and Pigolotti (2021).

The terminology and interpretation of the terms in Eq. (31) make contact with
the balance equation (3) introduced in the macroscopic approach of
irreversible thermodynamics. However, the fact that $\Pi(t)$ is always non-
negative already shows a strong disagreement with the EP rates discussed in
the previous sections. Additionally, it is easy to see that $\Pi(t)$ does not
have the bilinear form discussed in Sec. II.

Consider the solution of Eq. (25) Jiménez-Aquino (2010); Ferrari (2003),

$\displaystyle\mathcal{W}(p,t)=\left(2\pi
mk_{B}T\right)^{-1/2}\exp{\left(-\frac{\left[p-\langle
p(t)\rangle\right]^{2}}{2mk_{B}T}\right)}\,,$ (35)

for an initial Maxwell-Boltzmann distribution. Then, the average momentum
$\langle p(t)\rangle$ is,

$\langle
p(t)\rangle=\int_{0}^{t}dt^{\prime}e^{-(t-t^{\prime})/\tau_{R}}qE(t^{\prime})\,,$
(36)

which is obtained from Eq. (15). Plugging expression (35) in Eq. (34), it is
straightforward to show that,

$k_{B}\Pi(t)=\frac{\langle p(t)\rangle^{2}}{mT\tau_{R}}\,,$ (37)

which is clearly different from current, $q\langle p(t)\rangle/m$, times
thermodynamic force, $E(t)/T$, which we had before in Eq. (24). However, note
that Eqs. (24) and (37) do become identical in the limit of infinitely slow
driving, i.e, $\omega_{0}\tau_{R}\to 0$. Equation (13) tells us that the delay
between $J_{e}(t)$ and $E(t)$ vanishes in this limit, i.e., the current
becomes simply proportional to the electric field.

Additionally, the solution (35) leads to the following expression for the
entropy flux $\Omega(t)$,

$k_{B}\Omega(t)=\frac{\langle p(t)\rangle^{2}}{mT\tau_{R}}\,.$ (38)

Together with Eq. (37), this implies that the time derivative of the Shannon
information (30) is zero whether the system is in the transient or long-time
state.

##  VII Concluding remarks

Motivated by the macroscopic approach of irreversible thermodynamics, we have
defined an expression for the fluctuating EP in the classical Drude model. We
have shown that this quantity fulfills a fluctuation theorem despite the
existence of negative values of its corresponding rate. Hence, this shows, for
a paradigmatic example, that negative entropy production rates are not
incompatible with the fluctuation theorem and the second law. Moreover, we
have shown that the entropy production rate obtained from the Shannon entropy
and the Fokker-Planck equation contrasts with our expression and does not have
the bilinear form predicted by irreversible thermodynamics. In addition, we
have shown that the Shannon information remains constant along the transition
from the initial equilibrium state to long-time nonequilibrium one and that
the entropy production obtained from it is simply proportional to the power
related to the friction force. Our expression, however, is a measure of the
total amount of energy absorbed by the insulated system composed of the set of
charge carriers and its heat bath.

##  Appendix A Irreversible thermodynamics

In this appendix, we give a short derivation of Eq. (5) for the EP rate in
irreversible thermodynamics. The hypothesis of local equilibrium allows to
extend standard relations of equilibrium thermodynamics to the non-equilibrium
regime Reichl (2016). For instance, the combination of the first with the
second law,

$Tds=du-\mu dn\,,$ (39)

and the Gibbs-Duhem equation,

$sdT=-dP-nd\mu\,,$ (40)

are assumed to hold when the temperature $T$, the pressure $P$ and the
chemical potential $\mu$ along with the densities of entropy $s$, internal
energy $u$ and particles $n$, are treated as scalar fields.

Using Eqs. (39) and (40), it is possible to write Reichl (2016)

$\frac{du}{dt}-T\frac{ds}{dt}-\mu\frac{d\mu}{dt}=\frac{\partial u}{\partial
t}-T\frac{\partial s}{\partial t}-\mu\frac{\partial\mu}{\partial t}=0\,,$ (41)

through which the balance equations can be connected. Considering Eqs. (2),
(3) and conservation of particles,

$\frac{\partial n}{\partial t}+\nabla\cdot\mathbf{J}_{n}=0\,,$ (42)

it is possible to find for the entropy flux,

$\mathbf{J}_{s}=\frac{\mathbf{J}_{u}}{T}-\mu\frac{\mathbf{J}_{n}}{T}\,,$ (43)

and the EP rate,

$\dot{\overline{\Sigma}}=\mathbf{J}_{u}\cdot\nabla\left(\frac{1}{T}\right)-\mathbf{J}_{n}\cdot\nabla\left(\frac{\mu}{T}\right)+\frac{\mathbf{J}_{e}\cdot\mathbf{E}}{T}\,.$
(44)

Under the conditions of uniform temperature and pressure, the Gibbs-Duhem
equation in the form,

$\nabla P+s\nabla T+n\nabla\mu=0\,,$ (45)

implies that $\nabla\mu=0$ and Eq. (44) reduces to

$\dot{\overline{\Sigma}}=\frac{\mathbf{J}_{e}\cdot\mathbf{E}}{T}\,.$ (46)

##  Appendix B The first two moments of $\Sigma(t)$

To calculate the first moment $\langle\Sigma(t)\rangle$, we start with Eq.
(18) and the solution $p(t)$ of Eq. (16),

$\displaystyle p_{i}(t)=p_{i}(0)e^{-t/\tau_{R}}$ $\displaystyle+$
$\displaystyle\int_{0}^{t}dt^{\prime}e^{-(t-t^{\prime})/\tau_{R}}qE(t^{\prime})$
(47) $\displaystyle+$
$\displaystyle\int_{0}^{t}dt^{\prime}e^{-(t-t^{\prime})/\tau_{R}}f_{i}(t^{\prime})\,.$

Assuming that the system is in equilibrium at $t=0$, we insert the previous
expression in Eq. (18) and, after taking the average, Eq. (19) is obtained
using that $\langle p_{i}(0)\rangle=0$ and $\langle f_{i}(t)\rangle=0$.

To obtain the second moment $\langle\Sigma^{2}(t)\rangle$, we take the average
of Eq. (18) squared, which reads

$\displaystyle\langle\Sigma^{2}(t)\rangle=\left(\frac{q}{mT}\right)^{2}\int_{0}^{t}ds\int_{0}^{t}duE(s)E(u)\sum_{i,j}\langle
p_{i}(s)p_{j}(u)\rangle\,.$ (48)

Using again the solution (47), the initial condition $\langle
p_{i}(0)p_{j}(0)\rangle=\delta_{ij}mk_{B}T$ and the white-noise property

$\langle
f_{i}(t)f_{j}(t^{\prime})\rangle=\frac{2mk_{B}T}{\tau_{R}}\delta_{ij}\delta(t-t^{\prime})\,,$
(49)

where $\delta_{ij}$ is Kronecker’s delta and $\delta(t)$ is Dirac’s delta
function, it is possible to show that

$\displaystyle\langle
p_{i}(s)p_{j}(u)\rangle=\delta_{ij}mk_{B}Te^{-|s-u|/\tau_{R}}$ (50)
$\displaystyle+q^{2}\int_{0}^{s}dt^{\prime}\int_{0}^{u}dt^{\prime\prime}\,e^{-(s+u-t^{\prime}-t^{\prime\prime})/\tau_{R}}E(t^{\prime})E(t^{\prime\prime}).$

Plugging the previous expression into Eq. (48), we finally obtain

$\displaystyle\langle\Sigma^{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle\Sigma(t)\rangle^{2}$ (51) $\displaystyle+$
$\displaystyle
k_{B}\frac{Nq^{2}}{mT}\int_{0}^{t}ds\int_{0}^{t}du\,e^{-|s-u|/\tau_{R}}E(s)E(u)$
$\displaystyle=$
$\displaystyle\langle\Sigma(t)\rangle^{2}+2k_{B}\langle\Sigma(t)\rangle\,.$

##  Appendix C Conditional probability of time-reversed trajectories

In this appendix, we explain what is meant by the “action” appearing in Eq.
(LABEL:eq.probtrajr). Firstly, we clarify that
$p^{\dagger}(t^{\prime})=-p(t-t^{\prime})$,
$\dot{p}^{\dagger}(t^{\prime})=\dot{p}(t-t^{\prime})$ and
$E^{R}(t^{\prime})=E(t-t^{\prime})$, where $p(t)$ refers to the trajectory
$\Gamma$ whose evolution is subjected to $E(t)$. Using then Eq. (27), we
obtain

$\displaystyle\mathcal{L}(p^{\dagger}(t^{\prime}),\dot{p}^{\dagger}(t^{\prime});E^{R}(t^{\prime}))=$
(52)
$\displaystyle\qquad\;\frac{\left[\dot{p}(t-t^{\prime})-(p(t-t^{\prime})/\tau_{R}+qE(t-t^{\prime}))\right]^{2}}{(4mk_{B}T/\tau_{R})}\,.$

which implies that,

$\displaystyle\int_{0}^{t}dt^{\prime}\mathcal{L}(p^{\dagger}(t^{\prime}),\dot{p}^{\dagger}(t^{\prime});E^{R}(t^{\prime}))=$
(53)
$\displaystyle\qquad\;\int_{0}^{t}ds^{\prime}\,\frac{\left[\dot{p}(s^{\prime})-(p(s^{\prime})/\tau_{R}+qE(s^{\prime}))\right]^{2}}{(4mk_{B}T/\tau_{R})}\,,$

after the change of variables $s^{\prime}=t-t^{\prime}$. Finally, the
conditional probability of observing the trajectory $\Gamma^{\dagger}$ under
the time-reversed field $E^{R}(t)$ reads

$\displaystyle\mathcal{P}[\Gamma^{\dagger}|-p_{t}]$ $\displaystyle=$
$\displaystyle\mathcal{N}$
$\displaystyle\exp{\bigg{\\{}-\int_{0}^{t}ds^{\prime}\,\frac{\left[\dot{p}(s^{\prime})-(p(s^{\prime})/\tau_{R}+qE(s^{\prime}))\right]^{2}}{(4mk_{B}T/\tau_{R})}\bigg{\\}}}\,,$

###### Acknowledgements.

The authors acknowledge comments by J. Thingna and K. Ptaszyński. M. V. S.
Bonança acknowledges financial support from FAPESP (Fundação de Amparo à
Pesquisa do Estado de São Paulo) (Brazil) (Grant No. 2020/02170-4).

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