# The radial variation of the solar wind turbulence spectra near the kinetic
break scale from Parker Solar Probe measurements

S. Lotz11affiliation: Equal contribution authors 22affiliation: South African
National Space Agency, Hermanus, 7200, South Africa 33affiliation: MUST,
Faculty of Engineering, North-West University, South Africa , A. E.
Nel11affiliation: Equal contribution authors 22affiliation: South African
National Space Agency, Hermanus, 7200, South Africa , R. T.
Wicks11affiliation: Equal contribution authors 44affiliation: Department of
Mathematics, Physics and Electrical Engineering, Northumbria University,
Newcastle upon Tyne, NE1 8ST, UK , O. W. Roberts55affiliation: Space Research
Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, Graz, 8042,
Austria , N. E. Engelbrecht66affiliation: Center for Space Research, North-
West University, Potchefstroom, 2522, South Africa 77affiliation: National
Institute for Theoretical and Computational Sciences (NITheCS), South Africa ,
R. D. Strauss66affiliation: Center for Space Research, North-West University,
Potchefstroom, 2522, South Africa 77affiliation: National Institute for
Theoretical and Computational Sciences (NITheCS), South Africa , G. J. J.
Botha44affiliation: Department of Mathematics, Physics and Electrical
Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK , E. P.
Kontar88affiliation: School of Physics and Astronomy, University of Glasgow,
Glasgow, G12 8QQ, UK , A. Pitn̆a99affiliation: Department of Surface and
Plasma Science, Faculty of Mathematics and Physics, Charles University, V
Holes̆ovic̆kách 2, 180 00 Prague, Czech Republic , S. D. Bale1010affiliation:
University of California, Berkeley, USA

###### Abstract

In this study we examine the radial dependence of the inertial and dissipation
range indices, as well as the spectral break separating the inertial and
dissipation range in power density spectra of interplanetary magnetic field
fluctuations using Parker Solar Probe data from the fifth solar encounter
between $\sim$0.1 and $\sim$0.7 au. The derived break wavenumber compares
reasonably well with previous estimates at larger radial distances and is
consistent with gyro-resonant damping of Alfvénic fluctuations by thermal
protons. We find that the inertial scale power law index varies between
approximately -1.65 and -1.45. This is consistent with either the Kolmogorov
(-5/3) or Iroshnikov-Kraichnan (-3/2) values, has a very weak radial
dependence with a possible hint that the spectrum becomes steeper closer to
the Sun. The dissipation range power law index, however, has a clear
dependence on radial distance (and turbulence age), decreasing from -3 near
0.7 au (4 days) to -4 [$\pm$0.3] at 0.1 au (0.75 days) closer to the Sun.

Solar wind — interplanetary turbulence — Space plasmas

=3 The Friends of AASTeX Collaboration

##  1 Introduction

High-cadence magnetic field observations made by the Parker Solar Probe (PSP)
during its various perihelia provide us with an unprecedented opportunity to
study the radial evolution of various quantities associated with the
dissipation of solar wind turbulence (see, e.g., Woodham et al., 2019; Perrone
et al., 2020; Chhiber et al., 2021a; Alexandrova et al., 2021), motivated by
the fact that a greater understanding of the processes of kinetic dissipation
in magnetized plasma is essential for explaining the physical origin and
evolution of the solar wind (e.g. Bruno & Carbone, 2016; Matteini et al.,
2020; Duan et al., 2020; Telloni et al., 2021; Zhao et al., 2022a). This, in
turn, also informs modelling efforts of turbulence transport (e.g. Engelbrecht
& Strauss, 2018; Chhiber et al., 2019; Adhikari et al., 2020, 2021a) and
energetic particle transport (e.g. Strauss et al., 2017; Engelbrecht, 2019;
Laitinen & Dalla, 2019; Chhiber et al., 2021b) and relates to density
turbulence that is important for solar radio burst interpretation (e.g. Krupar
et al., 2020; Kontar et al., 2019). Magnetic field fluctuations in the solar
wind are commonly observed to follow a power-law spectrum. The inertial range
is created by an energy-conserving spectral cascade (e.g. Smith et al., 2006)
where interactions between fluctuations can still be described by fluid
dynamics (however, for an exception, see Bian et al., 2010). The spectral
index in the inertial range has been observed to be close to both the
Kolmogorov (1941) value of $-5/3$, as well as the Iroshnikov-Kraichnan value
of $-3/2$ (see, e.g., Smith et al., 2006; Podesta, 2011; Zhao et al., 2020;
Chen et al., 2020). This range is followed by a break and then a steepening in
the magnetic field power spectrum, where the MHD description breaks down and
kinetic effects of individual particles, and thermal heating, start playing a
role (Alexandrova et al., 2008). The dissipation range spectral index is
dependent on the type of turbulent fluctuation present, being either Alfvén
waves or coherent structures, and can vary significantly, ranging between
$\sim-1$ and $-4$ (Leamon et al., 1998; Smith et al., 2006; Markovskii et al.,
2006; Alexandrova, 2008; Bruno et al., 2014; Bruno & Trenchi, 2014; Lion et
al., 2016; Vech et al., 2018; Franci et al., 2020).

The physical mechanisms responsible for the break between the inertial and
dissipation range at these ion-kinetic scales are still not fully understood
(Matteini et al., 2020). The processes at play in energy transport in the
transition range and how it affects the properties of plasma is still an open
question (e.g. Goldstein et al., 2015; Bruno & Carbone, 2016; Terres & Li,
2022). The break frequency $f_{b}$, where the transition from inertial to
dissipation range is observed, has been shown to display a radial dependence
in previous studies. Analyzing observations taken in the 0.42 to 5.3 au range
during radial alignments respectively between MESSENGER and WIND, and WIND and
ULYSSES, Bruno & Trenchi (2014) found that the break frequency increased as
the heliocentric distance decreased such that $f_{b}\propto r^{-1.09\pm
0.11}$. In another study Duan et al. (2020) used data from the cruise phase of
the second orbit of PSP, ranging from 0.17 to 0.63 au, to measure the spectral
break that those authors interpret as the transition to kinetic turbulence.
They found that the break frequency increased with a decrease in heliocentric
distance, following a power law of $f_{b}\propto r^{-1.11\pm 0.01}$.

A wide array of studies examine time periods for the break scale and find
evidence for a link to the kinetic plasma physics of protons (e.g. Goldstein
et al., 1994; Hamilton et al., 2008; Markovskii et al., 2008; Smith et al.,
2012; Chen et al., 2014; Woodham et al., 2018). At 1 au the ion scale spectral
break tends to occur near $f\sim 0.1-1$ Hz in the spacecraft frame (e.g.
Leamon et al., 1998; Smith et al., 2001; Bale et al., 2005; Markovskii et al.,
2008; Bourouaine et al., 2012). The break is located near the spacecraft frame
frequencies that correspond to either the proton gyroradius $l_{g}$ where
damping of kinetic Alfvén waves becomes significant or the proton inertial
length $l_{i}$ where protons decouple from the turbulent magnetic field. It
may also be associated with magnetic reconnection via the so-called disruption
scale (for more detail see Vech et al., 2018; Terres & Li, 2022, and
references therein). The break frequency has been observationally associated
with these various characteristic plasma lengthscales, for example Leamon et
al. (1998) and Leamon et al. (2000) report from analyses of WIND data that
this break occurs at spatial scales in the plasma frame near the proton
gyroradius. Alternatively the break may be related to the combined scale
$l_{i}+l_{g}=2\pi k_{c}^{-1}$, which is associated with cyclotron resonance of
Alfvén waves propagating along the mean field direction (e.g Leamon et al.,
1998; Bruno & Trenchi, 2014; Woodham et al., 2018; Engelbrecht & Strauss,
2018). At 1 au it is often the case that plasma $\beta\sim 1$ implying
$l_{g}\sim l_{i}$, which makes it difficult to determine which of spatial
scales is related to the break (see, e.g., Terres & Li, 2022). To overcome
this limitation, Chen et al. (2014) investigated intervals with extreme values
of plasma $\beta$ where the two scales are well separated. It was found that
the break tends to be associated with the larger of the two scales, which is
consistent with the break being near the combined scale. Bruno & Trenchi
(2014) performed a study of how the spectral break changes with heliocentric
distance and found the best agreement to be with the combined scale, with a
radial dependence of the wavenumber at which this break occurs of
$k_{b}\propto r^{-1.08\pm 0.08}$. A study of Wind data also confirmed that the
best agreement is with the combined scale Woodham et al. (2018).

In this work we determine the spectrum near the break wavenumber and spectral
indices using high-cadence PSP magnetic field measurements taken during its
fifth orbit, in an extension of the study of Duan et al. (2020). The fifth
orbit includes observations from 07.05.2020 to 19.06.2020, with PSP radial
distance from the Sun varying between 0.1 and 0.7 au. These values are then
compared to the theoretical estimates for this quantity by employing in situ
observations for the various plasma quantities they are a function of in an
attempt to determine which is the dominant turbulence dissipation process in
the inner heliosphere. The break wavenumber is then quantitatively compared to
previous studies and the radial dependence across the widest range of
heliocentric distances yet measured is calculated. Furthermore, the radial
evolution of the inertial and dissipation range spectral indices are also
investigated. The next section details the analysis method employed in this
study. Sections 3 and 4 present the results of this study with regards to the
dissipation range onset frequency/wavenumber, and spectral indices,
respectively. Finally, the results are discussed in Section 5.

##  2 Data Analysis

Figure 1: An overview of the solar wind plasma measurements during PSP orbit
5. The black stepped curves indicate the local mean for 50 equispaced radial
distance bins.

PSP is a 3-axis-stabilized Sun-pointing spacecraft in a elliptical
heliocentric orbit, with aphelia between Earth and Venus (Whittlesey et al.,
2020). The Solar Wind Electrons, Alphas, and Protons (SWEAP) instrument suite
onboard the PSP primarily measures solar wind thermal plasma. The suite
consists of three Electrostatic Analyzer instruments, called the Solar Probe
ANalyzers (SPANs): A sun-pointing Faraday Cup (SPC) that primarily measures
protons, alpha particles, and periodically electrons and the Solar Probe
Analysers (SPAN) that are situated at either side of the spacecraft bus and
measures protons, alpha particles, heavy ions (SPAN-Ion), and electrons (SPAN-
Electron). We use density and thermal speed data from SPC and temperature from
the SPAN-Ion instrument. These instruments were designed to overlap their
fields of view and capabilities, and make complementary measurements (Kasper,
2021). We restricted the analysis to periods when data quality (quantified by
the quality flag parameters) was at it’s highest level for the SPAN, SPC and
MAG instruments. The three-component heliocentric ($R$, $T$ and $N$ where $R$
is in the radial direction, $T$ is perpendicular to $R$ and lying in the
equatorial place, and $N$ is normal to this plane and completes the right-
handed coordinate system) solar wind magnetic field is measured by the MAG
fluxgate magnetometer from the FIELDS instrument (Bale et al., 2016). During
this period the spacecraft moved, in terms of radial distance, between $\sim$
0.1 and 0.7 au. Fig. 1 shows, from top to bottom, the magnetic field
magnitude, the solar wind proton density, the effective temperature, and the
solar wind speed against radial distance for the inward and the outward part
of the orbit; note the logarithmic y-axis necessary to capture the wide change
in parameters over the orbit in the top three panels.

Figure 2: Magnetic field spectrum from a solar wind interval measured by PSP.
Blue dotted line: initial spectrum. Grey lines: individual slopes measured
over 20 points, and used to determine the 1) green slope which is the average
fit for the inertial range, and 2) red slope which is the average fit for the
dissipation range. The vertical black line represents the break frequency
approximation. The green highlighted section defines the cutoff for the
inertial range estimation and the red-highlighted section the cutoff for the
dissipation range estimation.

The magnetic field is observed at data rate $d$ quantified by samples per
instrument cycle. For this study we only utilised intervals with data rate
$d\geq 64$ samples / cycle. The instrument cycle duration is $L\approx 0.874$s
(Bale et al., 2016), i.e. $d=64$ would translate to 55 samples per second. The
time between samplings is not always consistent and to rectify this we
interpolate the $B$ time series data to a fixed rate of $L/d$. We estimate the
break frequency in the magnetic field spectra ($f_{b}$) using a procedure
similar to the method employed by Woodham et al. (2019). The time series
spanning 44 days is divided into non-overlapping windows of 128s each, and
$f_{b}$ is estimated for each of these windows. The approximately 1025 hours
of Orbit 5 equates to roughly 28,000 128-second intervals; however only a
fraction of these intervals results in reliable $f_{b}$ estimates. The $f_{b}$
estimation procedure is illustrated in Figure 2 for one such interval.

For every 128 second interval of $\mathbf{B}=[B_{R},B_{T},B_{N}]$ ($N\times 3$
matrix with $N$ the number of magnetic field observations) the power spectral
density of $\mathbf{B}$ is computed by taking the trace
$P(f)=tr\\{\mathbf{P}(f)\\}$ of the matrix
$\mathbf{P}(f)=\tilde{\mathbf{B}}(f)\tilde{\mathbf{B}}(f)^{*}$, where
$\tilde{\mathbf{B}}(f)$ denotes the Fourier components of $\mathbf{B}$ at
frequency $f$ and $\tilde{\mathbf{B}}(f)^{*}$ its complex conjugate. Since the
shape of the spectral break is more pronounced in log space than linear space
we linearly interpolate the power terms along log-spaced frequencies, denoted
$P_{l}(f_{l})$. This is the blue dotted curve in Figure 2. Overlapping linear
fits are applied to $P_{l}(f_{l})$ and the average slope and intercept are
determined within the inertial and dissipation ranges (green and red shaded
regions in Fig. 2). These average fits are indicated by the red and green
lines and their intersection determines the break frequency estimate $f_{b}$
(black vertical dashed line in Fig. 2). The mean standard deviation of the
linear fits are used to estimate the uncertainty, denoted by the shaded
regions around the average linear fits. We define the uncertainty in the
$f_{b}$ estimate as the frequencies where the upper and lower bounds of the
linear fit uncertainty regions overlap (see Fig. 2).

This yields a set of 26,581 estimates of $f_{b}$ along with an upper and lower
uncertainty ($f_{b}\epsilon^{-}$, $f_{b}\epsilon^{+}$) for the orbit 5 data
set. The uncertainty estimates allow us to discard problematic $f_{b}$
estimates. All estimates with (i) $f_{b}\epsilon^{-}>f_{b}\epsilon^{+}$, or
(ii) $f_{b}$ outside the range of frequencies in the signal, or (iii)
$f_{b}\epsilon^{-}$ or $f_{b}\epsilon^{+}$ beyond the dissipation and inertial
ranges are discarded. This results in a final set of 8551 reliable estimates
of $f_{b}$.

From the estimated break frequency, and the measured solar wind speed in each
interval, a corresponding wavenumber is calculated as

$k_{b}=\frac{2\pi f_{b}}{V_{sw}},$ (1)

with associated break scale $l_{d}=2\pi/k_{b}$. As such, the Taylor hypothesis
is explicitly used in this study. For turbulence analyses performed on
observations taken specifically near the Alfvén critical point, this could be
problematic (see, e.g., Bourouaine & Perez, 2018, 2019, 2020). In terms of the
dispersive regime specifically, Howes et al. (2014) report that such a regime,
were it primarily supported by Whistler waves, would violate Taylor’s
hypothesis, while Klein et al. (2014) expect a flattening of the dissipation
range due to this issue. Note also that we do not take the angle between the
flow and magnetic fields into account (Chen et al., 2014; Bourouaine et al.,
2012; Duan et al., 2018).

In order to compare our results with the various proposed break wavenumbers,
these quantities are calculated directly from PSP observations.

As a first approach, temperature isotropy ($T_{\text{eff}}=T_{||}=T_{\perp}$)
is assumed, allowing the effective plasma proton temperature to be calculated
from the equipartition theorem as $mu_{\text{eff}}^{2}=3kT_{\text{eff}}$,
where $u_{\text{eff}}$ is the effective thermal speed. From this, the proton
gyro-scale can be estimated as

$l_{g}=2\pi\frac{u_{\text{eff}}}{\Omega_{ci}}=\frac{2\pi}{k_{g}},$ (2)

with $\Omega_{ci}=qB/m_{i}$ the proton gyro-frequency. The proton inertial
length now follows as

$l_{i}=2\pi\frac{c}{\omega_{ci}}=2\pi\frac{V_{A}}{\Omega_{ci}}=\frac{2\pi}{k_{i}},$
(3)

with $V_{A}=B/\sqrt{\mu_{0}n_{i}m_{i}}$ the Alfvén speed. Thermal particles
can also resonate with circularly polarized waves when, in the guiding centre
frame, the following resonance condition is met

$\omega^{*}=n\Omega^{*},$ (4)

where $\omega^{*}$ is the wave frequency, $\Omega^{*}$ the particle cyclotron
frequency, and $n=\pm 1$ labels left-hand and right-handed waves,
respectively. Transforming back to the bulk flow frame, the resonance
condition becomes

$\omega-\vec{k}\cdot\vec{v}=n\Omega.$ (5)

Assuming left-handed, parallel propagating Alfvén waves with
$\omega^{2}=k_{\parallel}^{2}V_{A}^{2}$, resonating with thermal protons,
Doppler shifted by their (parallel) thermal speeds $u_{\text{eff}}$, one
obtains the so-called proton cyclotron resonance as (see, e.g., Engelbrecht &
Strauss, 2018, and references therein)

$k_{c}=\frac{\Omega_{ci}}{V_{A}+u_{\text{eff}}}=\frac{2\pi}{l_{g}+l_{i}}.$ (6)

The following sections outline the results of the analyses discussed above.

##  3 Dissipation Range Break Frequency and Onset Wavenumber

Figure 3: Power spectral distributions at different heliocentric distances.
The blue spectrum is from an interval measured at 0.13 au, the orange spectrum
from 0.4 au, and the green spectrum as measured at 0.65 au. Vertical dashed
lines indicate the estimated break frequency for each spectrum and the shaded
regions the error estimate.  Figure 4: Break frequency, as a function of
radial distance, compared with the Duan et al. (2020) power law estimate
($f_{b}\propto r^{-1.11\pm 0.01}$) determined from PSP data in the range 0.17
to 0.63 au. The dashed line shows a power law fit to our results
($f_{b}\propto r^{-1.08\pm 0.04}$).

Figure 3 shows the magnetic field spectral densities at three different
heliocentric distances, 0.13, 0.40 and 0.67 au. The dashed vertical lines
represent the estimated break frequency and it’s estimated error is indicated
by the shaded band for each of these examples. Sharp peaks in power spectral
densities at frequencies above 1 Hz are visible in all of the traces. These
are caused by the spacecraft attitude control system and change in amplitude
and frequency slowly over time. The increase in total power of the
fluctuations and break frequency towards the Sun is expected. This behaviour
can also be seen over wider radial distance in Figure 4, showing binned
estimates of the break frequency as a function of radial distance. A power law
fit to the data is indicated by a black dashed line ($f_{b}\propto r^{-1.08\pm
0.04}$). This agrees reasonably well with the fit performed by Duan et al.
(2020), shown in blue ($f_{b}\propto r^{-1.11\pm 0.01}$).

Figure 5: The breakscale wavenumber estimations are binned, according to
radial heliocentric distance, into 5 intervals and shown in the top panel,
with the wavenumber given in rad/km and radial distance in au. The
corresponding distributions obtained for each respective binned dataset are
shown in the lower panels as a function of wavenumber. The mean (dashed line)
and median (dotted line) of each histogram is indicated by vertical lines.
Figure 6:  The estimated breakscale wavenumber $k_{b}$, in rad.km-1, versus
the radial distance for 10 intervals ranging from 0.1 to 0.7 au. The red
markers represent the mean $k_{b}$ values, and the blue markers the median
$k_{b}$ values. The vertical error bars indicate the error derived from the
break frequency estimates. Horizontal error bars indicate the radial range
covered. Comparison is made with $k_{b}$ reported by Bruno & Trenchi (2014)
for radial distances 0.42 to 5.3 au (green squares), as well as the average
value for this quantity reported at $1$ au by Smith et al. (2012) (black
triangle), where the error bar indicates the standard deviation of the Smith
et al. (2012) measurements.  Figure 7: The estimated breakscale wavenumber
$k_{b}$ and a function of the corresponding scale size, calculated from Eqs. 2
– 6, using in situ observed plasma data. The dashed line indicates perfect
linear correlation with a slope of unity and going through the origin; the
coefficient of determination $R^{2}$ between $k_{b}$ and the modelled
wavenumbers are indicated in the legend.

The wavenumbers change with radial distance and we show this in Fig. 5. The
set of $k_{b}$ estimates are divided into 5 bins, depicted by the top panel.
Lower panels show the distribution of $k_{b}$ for each radial bin. As function
of radial distance, these distributions become steeper further away from the
sun, so that median and mean averages (denoted by short and long dashed lines,
respectively) shift further away from distribution peaks at smaller radial
distances. Overall, distribution peaks shift towards larger values of $k_{b}$
as radial distances increase, with a corresponding increase in the median and
mean average values for this quantity.

The PSP results presented in this work enables us to extend earlier estimates
of wavenumber to smaller radial distances. Figure 6 shows $k_{b}$
corresponding to the median (blue markers) and mean (red markers) values of
the distributions illustrated in Fig. 5. The green markers correspond to
earlier results for this quantity reported by Bruno & Trenchi (2014), who
employ observations from MESSENGER, WIND, and ULYSSES to calculate values of
$k_{b}$ corresponding to an overall radial range spanning 0.42 and 5.3 au. The
units of these data points are here adjusted for comparison with the values
acquired in the present analysis. This range overlaps with the range of radial
distances considered here and, as can be seen from Fig. 6, the break
wavenumbers calculated by Bruno & Trenchi (2014) tie in reasonably well with
estimates from the present study. An average of the values for $k_{b}$
reported by Smith et al. (2012) from an analysis of spacecraft observations at
$1$ au is also shown, and estimates the study of Bruno & Trenchi (2014) fall
well within the uncertainty of that value. Not unexpectedly, the radial
behaviour of the observations shown in Fig. 6 suggests a power law radial
dependence for $k_{b}$, similar to that seen for the break frequency.
Accordingly, the figure also shows a power law fit to the observations, with
exponent $-1.18\pm 0.02$, a value steeper than that reported by Bruno &
Trenchi (2014).

In order to directly compare the dissipation range onset wavenumbers
calculated here with those wavenumbers corresponding to the various
lengthscales that have been proposed previously, Fig. 7 shows $k_{b}$ as
function of wavenumbers corresponding to the cyclotron resonance scale, the
proton gyroscale, and the ion inertial length, as calculated from the
corresponding in situ measurements, with corresponding correlation
coefficients indicated in the legend. Of all three lengthscales, the cyclotron
resonance wavenumber best fits the dissipation range onset wavenumber
calculated here, although the considerable scatter in the data points leads to
a relatively coefficient of determination ($R^{2}$).

The dashed line in the figure indicates where $k_{b}$ is equal to the model
wavenumber. When using the cyclotron wavelength model wavenumber for
comparison (blue circles), the results follow this trend extremely well.

##  4 Spectral indices

Figure 8: Similar to Fig. 5, but now the value of the inertial range power law
index (left column) and dissipation range power law index (right column) are
binned into several radial intervals. Figure 9: Computed inertial (top) and
dissipation range (bottom) power law indices as a function of radial distance.
Figure 10: The inertial (top) and dissipation range (bottom) power law indices
as a function of solar wind age.

In a similar method to that use in the previous section, Fig. 8 shows the
estimated power law indices of the inertial and dissipation range turbulence,
along with their distributions, binned into different radial intervals. The
median and standard deviation of each binned histogram is calculated and shown
in Fig. 9 as a function of radial distance. The inertial range index
distribution does not show a change in width, and remains relatively constant
with an increase in radial distance. The dissipation range index shows a
constant distribution width, and generally decreases as radial distances get
larger. This is shown more explicitly in Figure 9, where both spectral
indices, with their accompanying uncertainties, are shown as function of the
average radial distances corresponding to the bins for which they were
calculated. The inertial range spectral indices appear to remain relatively
constant as function of radial distance (within uncertainty). Furthermore, it
is not clear whether either the Kolmogorov or Iroshnikov-Kraichnan values for
this quantity, indicated respectively by the black and red lines on the
figure, is favoured, once more due to the range of uncertainty. Although this
result is similar to the inertial range spectral indices calculated for the
5th and 7th PSP orbits calculated by Zhao et al. (2022b), there are some
differences, as the averaged indices reported by those authors steepen from a
value roughly between the Iroshnikov-Kraichnan and Kolmogorov values, to a
value approximately equal to the Kolmogorov index beyond $\sim 0.35$ au for
the 5th orbit data (see also Sioulas et al., 2022).

Dissipation range spectral indices, however, display a clear radial
dependence, increasing in absolute value as radial distance increases, similar
to what is reported by Franci et al. (2020). It is interesting to note that,
were the violation of the Taylor hypothesis to play a significant role, a
flatter dissipation range spectrum would be expected (Klein et al., 2014). The
radial behaviour of this quantity may be related to the increase in power of
the turbulence with decreasing radial distance (see Figure 3), in qualitative
agreement with what was reported by Smith et al. (2006) in their analysis of
dissipation range spectral indices at $1$ au, by Bruno et al. (2014), who
found a correlation between steeper dissipation ranges, and enhanced
turbulence levels, and by Huang et al. (2021) in their analysis of first orbit
PSP data. The radial decrease in the dissipation range spectra index is,
however, not uniform, as indicated by the slight drop in this quantity between
$\sim 0.45$ au and $\sim 0.6$ au in Fig. 9. This behaviour corresponds to a
marked increase in the solar wind speed shown in Fig. 1, with the implication
that the behaviour of the dissipation range spectral index at these radial
distances may be a reflection of the behaviour of this quantity at smaller
radial distances due to the fact that the solar wind is ‘younger’ here. To
investigate this, the age of the solar wind $\tau=r/V_{sw}$ corresponding to
each point in Fig. 9 was calculated, and the spectral indices were plotted as
function of this age in Fig. 10. The inertial range spectral indices behave in
a relatively uniform manner as function of solar wind age, while the
dissipation range spectral indices clearly become steeper as solar wind age
decreases.

##  5 Discussion and conclusions

This work extends the study of Duan et al. (2020), by considering the radial
evolution of the spectral break between the inertial and dissipation ranges of
turbulence power spectra calculated for PSP data taken during the 5th
perihelion of that spacecraft, as well as the spectral indices associated with
these ranges. The radial dependence of the spectral break reported on here
closely resembles that found by Duan et al. (2020) for the the cruise phase of
the 2nd PSP orbit, as well as tying up with that reported at larger radial
distances by Bruno & Trenchi (2014), as well as an average value for this
quantity calculated from the results of Smith et al. (2012). We find a radial
dependence of $k_{b}\propto r^{-1.18\pm 0.02}$. Furthermore, a comparison of
the break frequencies calculated here with frequencies corresponding to the
proton gyroradius, ion inertial length, and the cyclotron resonance scale, all
computed using in situ PSP observations of the various plasma quantities these
scales depend on, found that these break frequencies correspond most closely
with those corresponding to the cyclotron resonance scale. This finding is in
agreement with that of Woodham et al. (2018) in their analysis of Wind data.
These results provide a valuable benchmark against which the results of
various turbulence transport models (e.g. Engelbrecht & Strauss, 2018;
Adhikari et al., 2021a) can be tested, as well as a valuable input for solar
energetic particle and cosmic ray transport models.

The present study finds, in contrast to previous studies (e.g. Chen et al.,
2020; Shi et al., 2021; Zhao et al., 2022b; Sioulas et al., 2022) that the
inertial range spectral index, within uncertainties, remains relatively
constant as function of heliocentric radial distance. However, this index does
steepen for intervals of greater age, corresponding to relatively slower solar
wind speeds. This discrepancy with the results of prior studies may be due to
the fact that the present study does not distinguish between intervals of
greater or lesser Alfvénicity. The ambiguity in the results presented here may
then be due to the fact that that inertial range spectral indices have been
observed to be steeper for intervals of low Alfvénic content (characterized by
Sioulas et al. (2022) as intervals of low normalized cross-helicity), and vice
versa (see also Shi et al., 2021). However, the steepening of inertial range
spectral indices reported here for older turbulence intervals does agree with
the findings of Sioulas et al. (2022), who report such a steepening for slow
solar wind intervals.

The dissipation range spectral indices reported on here show a clear radial
dependence, becoming less steep at larger radial distances, in agreement with
previous studies (e.g. Franci et al., 2020). This could be related to the
increase in turbulence levels closer to the Sun (e.g. Adhikari et al., 2021b;
Zank et al., 2021, and references therein), and would be in accordance with a
correlation between enhanced turbulence levels at $1$ au and steeper
dissipation range spectral indices reported by Smith et al. (2006). It is
interesting to note that, when intervals are binned according to the age of
the turbulence, dissipation range spectral indices almost uniformly decrease
as radial distance increases.

Another possibility for the mechanism that steepens the dissipation range
spectral index is the presence of Ion Cyclotron Waves (ICWs). Bowen et al.
(2020) showed that the number, amplitude and duration of ICW packets increases
closer to the Sun. The presence of ICWs causes a bump in the spectrum at the
proton cyclotron resonance scale (Lion et al., 2016; Wicks et al., 2016;
Woodham et al., 2018; Telloni et al., 2019) and so can appear as a steepening
of the dissipation range spectrum. Our method rejects spectra with large
standard deviation of the fitted lines and so spectra with large peaks due to
ICWs will be rejected, but we cannot rule out that low-amplitude ICW effects
on the spectra have not been measured as steepening of the dissipation range.

Future work aims to extend the current analysis in two ways. Firstly, by
considering longer data intervals, so as to include a portion of the energy-
containing range of the turbulence power spectrum, thereby allowing for the
calculation of the inertial range outerscale. This, in turn will enable us to
include the disruption scale (e.g. Terres & Li, 2022, and references therein)
in our comparative analyses. Secondly, the analysis will be extended to other
PSP perihelia, taking into account various additional factors, such as solar
wind speed and plasma-$\beta$, that are known from previous studies (e.g. Chen
et al., 2014; Wang et al., 2018; Sioulas et al., 2022) to influence the
dissipation range spectral break frequency. It should also be noted that the
influence of Taylor’s hypothesis on results calculated for the dispersive
regime should be investigated, in the manner proposed by e.g. Bourouaine &
Perez (2019), in more detail. Future measurements with the MeerKAT radio
telescope and the Square Kilometer Array (SKA) are also planned to get
information about solar wind density fluctuations at very small scales inside
the Alfvén radius, and thus close the gap between the Sun and $\sim$10
$R_{\odot}$.

Software repository at https://bitbucket.org/stefansansa/pspanalysis/.

This work is based on the research supported in part by the National Research
Foundation of South Africa (NRF grant numbers 119424, 120345, 120847, and
137793). Opinions expressed and conclusions arrived at are those of the
authors and are not necessarily to be attributed to the NRF. The
responsibility of the contents of this work is with the authors. Figures
prepared with Matplotlib (Hunter, 2007) and certain calculations done with
NumPy (Harris et al., 2020). RTW is funded by STFC Grant ST/V$006320$/$1$. EPK
was supported by STFC grant ST/T000422/1. Parker Solar Probe was designed,
built, and is now operated by the Johns Hopkins Applied Physics Laboratory as
part of NASA’s Living with a Star (LWS) program (contract NNN$06$AA$01$C).
Support from the LWS management and technical team has played a critical role
in the success of the Parker Solar Probe mission. Thanks to the FIELDS team
for providing data (PI: Stuart D. Bale, UC Berkeley). Thanks to the Solar Wind
Electrons, Alphas, and Protons (SWEAP) team for providing data (PI: Justin
Kasper, BWX Technologies).

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# Entropy Production in the Inflationary Epoch Using the Gouy-Stodola Theorem

R. H<EMAIL_ADDRESS>Federal University of São Carlos,
Sorocaba CEP 18052780, Brazil  S. D<EMAIL_ADDRESS>Applied
Mathematics Laboratory-CCTS/DFQM,
Federal University of São Carlos, Sorocaba CEP 18052780, Brazil

###### Abstract

In this work, we use the Gouy-Stodola theorem to calculate the entropy
production rate in the inflationary epoch of the universe. This theorem allows
us the simple calculation of entropy and entropy production rate occasioned by
the decaying of the inflaton scalar field. Both the entropy and entropy
production rate achieve large values, agreeing with the expected values
present in the literature.

##  I Introduction

The laws of thermodynamics are one of the cornerstones of physics, and the
second law possesses the same importance as the energy conservation law.
Nonetheless, correctly defining entropy is a difficult task because it depends
strongly on the physical properties of the system under study. For example,
the entropy can be related to one-quarter of the black hole event horizon area
[1, 2], or Shannon’s information disorder approach [3, 4], or energy
dissipation in the classic thermodynamics approach. Moreover, the correct
definition of entropy is a problem that permeates several issues, for example,
the study of entropy production in rotating black holes [5], in the so-called
Ekpyrotic universe [6], in Szekeres spacetimes [7], quark-gluon plasma [8],
light clusters [9], and decoherence processes [10], among others.

In classical mechanics, the existence of non-conservative forces interferes
with entropy production, resulting in they should be taken into account for
the correct system description. According to literature, there are different
ways to include entropy calculation in classical mechanics, such as the
canonical Hamiltonian or Lagrangian formalism. These formalism allows
calculating the rate of entropy production [11] as well as considering
dissipative processes [12] or the method of molecular dynamics by modifying
the Newtonian dynamics [13, 14].

In mechanical systems, entropy generation can be computed through the
Silverberg and Widom approach [11] or by the Gouy-Stodola theorem [15, 16,
17], for instance. This theorem is a less known result even in engineering,
where its formal applicability conditions are not clearly stated. However, its
original formulation is mathematically quite simple: The work lost in an open
system is directly proportional to the amount of entropy generated in the
process [18]. The work lost by the system, in its turn, is given by the
difference between the work done by the reversible (free of damping forces)
and the irreversible processes, where different kinds of drag forces can be
introduced. It should be noted this theorem hold even for adiabatic systems
[18].

In the present paper, we first introduce the rate of entropy production given
by the Gouy-Stodola theorem by considering two different systems. The first
system is a simple pendulum subject to a damping force. After that, we modify
the system introducing the possibility of the parametric resonance mechanism
but still considering the same damping force. In both mechanical conditions,
one obtains the entropy produced as stated by the Gouy-Stodola theorem.

As the second and most important step, we apply the Gouy-Stodola theorem for a
system taking into account parametric resonance to the particle production in
the early universe. In the simple case of the scalar inflaton field $\phi$
decaying into a scalar $\chi$ particle, the entropy production for $\phi$ can
be calculated using the damping term of the equation of motion for the
inflaton. For the $\chi$ particles, is used the same method. As well-known,
exponential particle production due to the parametric resonance mechanism can
fastly populate the universe [19, 20]. Thus, entropy production due to this
effect may be an important ingredient in explaining the huge amount of entropy
observed in the universe [22, 21].

It is important to stress that we are not interested in the details of the
particle production mechanism, which has been exhaustively studied over
decades as shown in the literature [19, 20, 23, 24, 25, 26, 27, 28, 29]. Our
main goal is to show that the Gouy-Stodola theorem can be used as a simple
tool, allowing the calculation of entropy production even for complex physical
situations. We are interested in showing that entropy generation in the
inflationary epoch depends on the considered scalar field. Furthermore,
entropy and entropy production due to $\phi$ depends strongly on both its
mass, $m_{\phi}$, and its self-coupling constant, $\lambda$. However, for some
values, entropy associated with $\phi$ can achieve $>10^{98}$. Thus, it is
important to understand the role played by each scalar field in entropy
generation during the inflationary era.

This paper is organized as follows. Section II presents the simple pendulum
subject to a damping force, with and without the parametric resonance
mechanism, and calculates the entropy produced in both cases. In Section III,
one calculates the entropy produced according to the Gouy-Stodola theorem.
Section IV presents our final remarks and criticism.

##  II Entropy Production and the Simple Pendulum

For the sake of clarity, the Gouy-Stodola theorem is applied to the simple
pendulum in two situations. In the first step, one considers only the simple
pendulum subject of a drag force. In the second step, we introduce the
parametric resonance mechanism with the same drag force as before.

###  II.1 Entropy Production

Entropy production depends on specific details of the physical system (its
dynamics), and there are several frameworks developed to define entropy
production in some open systems [30, 31, 32, 33, 34, 35]. For our purposes, it
is sufficient to show how the entropy production rate is related to the work
done in reversible and irreversible systems. Firstly, notice that the Clausius
inequality can be written as

$\displaystyle\oint_{i}^{f}\frac{\delta Q}{T}\leq 0,$ (1)

where one assumes the system undergoes some process from an initial state $i$
up to the final state $f$. It is important to stress that the integral above
defined is calculated on the system boundary, and for convenience, one writes

$\displaystyle\oint_{i}^{f}\frac{\delta Q}{T}=-\Sigma$ (2)

where one defines that $\Sigma=\Sigma(t)$ is zero for reversible processes as
well as positive for irreversible processes, and $\delta Q$ is the heat
transfer due to a specific process (isothermic, isobaric or isocoric), and $T$
is system temperature. For a reversible process, which ensure the equality in
(2), $\Sigma$ is just the entropy of the system

$\displaystyle S_{i}-S_{f}=\oint_{f}^{i}\frac{\delta Q}{T},$ (3)

which is, in fact, zero since the process is reversible. Observe that the
temperature which occurs the heat transfer from the reservoir to the system is
constant.

Now, consider a system consisting of two cycles. The first cycle is
reversible, while the second is irreversible. Then, one writes

$\displaystyle\Sigma=\Delta S-\oint_{f}^{i}\frac{\delta Q}{T}\geq 0,$ (4)

where the integral takes into account the irreversible process. From the above
result, one observes that $\Sigma$ is due to the irreversible process given by
the integral. Thus, one defines $\Sigma$ as the entropy production, which is
always a non-negative quantity. From definition (4), one introduces the
entropy production rate

$\displaystyle\dot{\Sigma}=\dot{S}-\frac{\dot{Q}}{T},$ (5)

where the dot means the usual time derivative $d/dt$.

Of course, energy conservation holds for reversible and irreversible systems.
For general open systems composed of $n$ independent processes, one can write
energy conservation rate as (following close Ref. [36])

$\displaystyle\dot{E}=\sum_{i=0}^{n}\dot{Q}_{i}-\dot{W}+c_{1}(m)=\dot{Q}_{0}+\sum_{i=1}^{n}\dot{Q}_{i}-\dot{W}+c_{1}(m),$
(6)

where $c_{1}(m)$ is the sum of the contribution of the finite mass rate to the
energy rate, and $W$ is the work due to the irreversible forces acting on the
system. Moreover, one writes the entropy production rate for open systems

$\displaystyle\dot{\Sigma}=\dot{S}-\sum_{i=0}^{n}\frac{\dot{Q}_{i}}{T_{i}}+c_{2}(m)=\dot{S}-\frac{\dot{Q}_{0}}{T_{0}}-\sum_{i=1}^{n}\frac{\dot{Q}_{i}}{T_{i}}+c_{2}(m)\geq
0,$ (7)

where, now, $c_{2}(m)$ represents the sum of the contribution of the finite
mass rate to the entropy production rate. The rate of the heat transfer
$\dot{Q}_{0}$ can be eliminated by combining the above results, implying the
work rate for open systems is given by

$\displaystyle\dot{W}=-(\dot{E}-T_{0}\dot{S})+\sum_{i=1}^{n}\left[1-\frac{T_{0}}{T_{i}}\right]\dot{Q}_{i}+c_{1}(m)+T_{0}c_{2}(m)-T_{0}\dot{\Sigma}.$
(8)

For reversible systems, the last term on the r.h.s of the above result is
null, resulting

$\displaystyle\dot{W}_{r}=-(\dot{E}-T_{0}\dot{S})+\sum_{i=1}^{n}\left[1-\frac{T_{0}}{T_{i}}\right]\dot{Q}_{i}+c_{1}(m)+T_{0}c_{2}(m).$
(9)

Subtracting (8) from (9), one obtains the entropy production rate in terms of
the difference between the work rate for the reversible process and the
irreversible one

$\displaystyle T_{0}\dot{\Sigma}=\dot{W}_{r}-\dot{W}.$ (10)

The above result is known as the Gouy-Stodola theorem [15, 16, 17]. In the
next subsections, one discusses this result for both a simple pendulum and the
one subject to the parametric resonance phenomenon.

###  II.2 The Simple Pendulum

One considers a damped pendulum, i.e. a weight of mass $m$ (bob) suspended
from a pivot so that it can oscillate freely subject to the restoring force of
gravity $\vec{g}$. In this example, one uses a very simple damping force given
by

$\displaystyle D(v)=-bv=-b\dot{s}(t),$ (11)

where the motion of the bob is along the arc $s(t)=l\theta(t)$, being $l$ the
inextensible string length of negligible mass. The drag parameter $b$ measures
the strength of the damping force. Of course, the presence of a drag force
implies the system is non-conservative (irreversible). The work $W_{D}$ done
by the drag force $D(\theta)$ is given by

$\displaystyle
W_{D}={\color[rgb]{0,0,1}-}\int_{\theta_{0}}^{\theta}D(\theta^{\prime})ld\theta^{\prime}.$
(12)

The resulting total energy $E$ describing the system depends on $\theta$,
being given by

$\displaystyle
E=K+U+W_{D}=\frac{ml^{2}}{2}\dot{\theta}^{2}+mgl(1-\cos\theta)-bl^{2}\dot{\theta}(\theta-\theta_{0}),$
(13)

where $K$ is the kinetic energy, $U$ is the potential energy, and $\theta_{0}$
is the initial angle.

For mechanical systems, the Gouy-Stodola theorem [15, 16] relates the entropy
production to the difference between the time derivative of the reversible and
irreversible work realized by the forces acting in the system. In the present
case and according to Eq. (10), this difference is given by

$\displaystyle\dot{\Sigma}=\frac{\dot{W}_{r}-\dot{W}}{T}=\frac{\dot{W}_{r}-\dot{W}_{D}}{T}=\frac{bl^{2}}{T}[\dot{\theta}^{2}-(\theta-\theta_{0})\ddot{\theta}],$
(14)

where $T$ is the system temperature taken as constant. Considering clockwise
oscillation as initial condition, (14) can be rewritten,

$\displaystyle\dot{\Sigma}=\frac{bl^{2}}{T}[\dot{\theta}^{2}+(\theta-\theta_{0})\ddot{\theta}].$
(15)

It should be noted that $\dot{\Sigma}$ shown above only depends on the work
done by the drag force given by equation (11). Thus, entropy production is
strictly related to energy dissipation. Therefore, it is important to keep in
mind the difference between definitions of entropy and entropy production:
Entropy can be calculated for any physical system, and it is an extensive
property of the system (state function) and, therefore, it is process-
independent. As the dissipative force characterizes entropy production,
entropy production is process-dependent. Both entropy and entropy production
exists for reversible and irreversible processes. However, in the first
process both remain constant while not for the latter. Notice that if the drag
force is neglected, then the entropy related to the conservative system
(reversible) achieves its maximum at the minimum of the kinetic energy (the
maximum entropy state) and the entropy production, accordingly to relation
(15), is zero.

The equation of motion for $\theta(t)$ can help us to understand the time-
dependent behavior of $\dot{\Sigma}$. Considering the drag force, one has the
well-known equation

$\displaystyle\ddot{\theta}(t)+\kappa\dot{\theta}+\omega^{2}\sin\theta=0,$
(16)

where $\kappa=b/m$ and $\omega^{2}=g/l$. Considering the small-angle solution,
$\sin\theta\approx\theta$, the simplest solution is given by

$\displaystyle\theta(t)=\alpha e^{-\gamma_{1}t}+\beta e^{-\gamma_{2}t},$ (17)

where $\alpha$ and $\beta$ depends on the initial conditions for the system.
Moreover, one defines $\gamma_{1}$ and $\gamma_{2}$ as

$\displaystyle-\gamma_{1}=-(\kappa+\sqrt{\kappa^{2}-\omega^{2}}),$ (18)
$\displaystyle-\gamma_{2}=-(\kappa-\sqrt{\kappa^{2}-\omega^{2}}).$ (19)

Regarding the above calculations and taking into account the solution for
$\theta(t)$ given by (17), one obtains the total entropy produced by the drag
force in the result (14). One adopts here the SI system and, for the sake of
simplicity, one sets $l=1$ m, $|\vec{g}|=10$ m/s2, $m=1$ kg, and $T=298$ K.
The parameter $b$ is defined according to the damped cases from classical
mechanics. For $\kappa^{2}<\omega^{2}$, one has the underdamped case, while
for $\kappa^{2}>\omega^{2}$ and $\kappa^{2}=\omega^{2}$ one obtains the over-
and critically damped cases, respectively. It is important to point out that
due to the oscillatory character of the solutions for $\theta(t)$, entropy and
entropy production also oscillate, assuming positive and negative values as
$t$ rises. Therefore, in all figures, we display the absolute value of entropy
and entropy production.

Figure 1(a) shows $\theta(t)$ according to $\kappa^{2}=1$ s-2 (solid line),
and $\kappa^{2}=50$ s-2, for dashed and dotted lines, respectively. Figure
1(b) shows the absolute value for the entropy production until the pendulum
motion ceases, reaching its equilibrium state. The inner panel in Figure 1(b)
represents the entropy production rate in the underdamped case. It should be
noted that in the over- and critically damped cases shown by dashed and dotted
curves in both panels, the entropy production is positive since there is no
inversion in the pendulum motion, i.e. the values adopted for $\kappa$ does
not allow oscillatory motion.

Figure 1: Panel (a) shows $\theta(t)$ for the simple pendulum. Solid line
represents the underdamped while dashed and dotted lines describes the over-
and critically damped cases, respectively. Panel (b) shows the absolute value
for the entropy production rate, $\dot{\Sigma}$, considering the underdamped
(solid line), overdamped (dashed line), and critically damped (dotted line)
cases. The parameters used are displayed along the text.

The sum of entropy production rates occurring at each period should be a
growing function since it corresponds to the accumulated entropy production
rate. Figure 2 shows the numerical calculation for the accumulated entropy
production rate, $\dot{\Sigma}_{T}$, for the oscillatory case, panel (a), and
for the overdamped case, panel (b). For both situations, the accumulated
entropy production is a growing function of $t$. Also notice that since the
entropy production rate tends to zero due to the damped term, the accumulated
result approaches a constant value.

Figure 2: Accumulated entropy production rate for the oscillatory, panel (a),
and overdamped case, panel (b).

From Eq. (14), we can observe that the zeros of $\dot{\Sigma}(t)$ are given by

$\displaystyle\dot{\theta}^{2}(t)=-(\theta(t)-\theta_{0})\ddot{\theta}(t),$
(20)

which implies the product in the r.h.s of result (20) is always positive for
all $t$ since $\theta(t)\leq\theta_{0}$. The equality is valid for the simple
harmonic oscillator where non-conservative forces are absent and the system
can return to the same energy state ($\Delta S=0$), and the resulting entropy
production variation, according to result (14), should also be zero.
Therefore, the result (20) establishes conditions for the oscillatory motion
when considering non-conservative forces described in (11). We can realize
that the unavailable energy for the weight force to do work, which means the
dissipated energy by the drag force, is responsible for the entropy
production. Due to the inversion in the pendulum motion, the entropy
production rates oscillate, and the zeros of the entropy production rate
correspond to the intermediates equilibrium states of the pendulum motion.

###  II.3 The Simple Pendulum: Parametric Resonance

Parametric resonance is a well-known phenomenon associated with time-dependent
frequency, which is the parameter of the problem. The general case without a
drag force can be written as

$\displaystyle\ddot{f}(t)+\omega_{0}(t)f(t)=0,$ (21)

where, for convenience, one defines the time-dependent frequency as

$\displaystyle\omega_{0}(t)=\omega_{0}^{2}[1+\zeta\cos(\omega_{1}t)],$ (22)

being $\omega_{0}$, $\omega_{1}$, and $\zeta$ real parameters. The result
given by (21) is the so-called Mathieu equation [37] commonly used to explain
particle production processes in cosmology [19, 20]. In general, the amplitude
of the solution depends on the relation between $\omega_{1}$ and $\omega_{0}$.
When $\omega_{1}\approx 2\omega_{0}$, the amplitude of the oscillations grows
exponentially with time, leading to an explosive particle production. It is
convenient to write

$\displaystyle\omega_{1}=2\omega_{0}+\epsilon,$ (23)

where $\epsilon<\\!\\!<1$ is a small perturbation. Parametric resonance
condition is achieved if [37]

$\displaystyle\epsilon<\left|\frac{\zeta\omega_{0}}{2}\right|.$ (24)

Considering the presence of a drag force, one has the modified Mathieu
equation for the simple pendulum written as

$\displaystyle\ddot{\theta}(t)+\kappa\dot{\theta}+\omega_{0}(t)\sin\theta=0,$
(25)

where one uses

$\displaystyle\omega_{0}(t)=\omega_{0}^{2}\left[\frac{g}{l}+\zeta\cos(2\omega_{0}+\epsilon)t\right].$
(26)

Of course, for $\zeta=0$, one recovers the non-parametric resonance case
taking $\omega_{0}=1$. Considering $\sin\theta\approx\theta$, one writes

$\displaystyle\ddot{\theta}(t)+\kappa\dot{\theta}(t)+\omega_{0}(t)\theta(t)=0.$
(27)

For the sake of simplicity, one writes a possible solution for the equation
(27) as

$\displaystyle\theta(t)\approx(\cos(\omega_{0}+\epsilon/2)t+\sin(\omega_{0}+\epsilon/2)t)e^{-\kappa
t}.$ (28)

It should be stressed the proper solution of equation (27) encompasses Mathieu
functions, defined through a combination of sine and cosine functions.
However, for our purposes in this section, the ansatz (28) is sufficient to
show the entropy production in the parametric resonance case.

Figure 3 displays the results for $\theta(t)$ and the absolute value of
$\dot{\Sigma}(t)$ according to solution (28). As expected, the parametric
resonance case is similar to the result given by the damped pendulum.

Figure 3: Results taking into account the parametric resonance for $\theta(t)$
(panel a), and the absolute value of $\dot{\Sigma}$ (panel b).

##  III Particle Production in the Inflationary Era

One assumes the inflationary period can provide a causal mechanism for
generating several structures observed in the present-day Universe, such as
the primordial perturbations responsible for the formation of galaxies [38]
and the homogeneity problem [39]. The inflaton scalar field $\phi$ is the
leading contribution to the energy-momentum tensor possessing here the
following Lagrangian

$\displaystyle\mathcal{L}(\phi)=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi),$
(29)

where $V(\phi)$ is the effective potential of the scalar field $\phi$. The
choice of such potential depends on the interactions we want to assume. From
now on, one adopts the natural unit system where $c=h=k_{B}=1$.

The Einstein field equations control the evolution of the flat Friedman-
Robertson-Walker universe,

$\displaystyle\left(\frac{\dot{a}}{a}\right)H^{2}=\frac{8\pi}{3M_{p}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right),$
(30)

where $a=a(t)$ is the scale factor, $H=\dot{a}(t)/a(t)$ is the Hubble
parameter, and $M_{p}$ is Planck mass. The Klein-Gordon equation for $\phi$
depends on the choice of $V(\phi)$ and can be written as

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0,$ (31)

where $V^{\prime}(\phi)=dV(\phi)/d\phi$ and $3H\dot{\phi}$ act as a ”friction
term”. In general, the interest resides on the decay of $\phi$ into scalar
$\chi$ particles, avoiding the Pauli Exclusion Principle. Then, it is
necessary to know the total decay rate, and for the case where $\phi$ decays
into two scalar particles, one writes [40]

$\displaystyle\Gamma=\Gamma(\phi\rightarrow\chi\chi)=\frac{h^{4}\sigma^{2}}{8m_{\phi}},$
(32)

where $h$ is coupling constant and $\sigma$ is a constant with dimension of
mass. Of course, we are supposing the mass of $\phi$ scalar field
$m_{\phi}>m_{\chi}$, where $m_{\chi}$ is mass of $\chi$ particles. Moreover,
during the oscillating period, one also suppose $m_{\phi}>\\!\\!>H$. Thus, one
adds the term $\Gamma\dot{\phi}$ to equation (31), representing the back
reaction of produced particles. The decay rate act as a friction term,
resulting in a phenomenological description for $\phi$, written as

$\displaystyle\ddot{\phi}+(3H+\Gamma)\dot{\phi}+V^{\prime}(\phi)=0.$ (33)

The effective potential $V(\phi)$, chosen here by its simplicity, is given
here by the usual $\lambda\phi^{4}$

$\displaystyle
V(\phi)=\frac{m_{\phi}^{2}}{2}\phi^{2}-\frac{\lambda}{4}\phi^{4},$ (34)

where $\lambda$ is the $\phi$ self-coupling constant and should be chosen in
order to keep the flatness of $V(\phi)$. Of course, there are several
potential to $\phi$, each one depending on the approach performed [41]. The
value of $\phi$ at the minimum of $V(\phi)$ is given by
$\phi_{m}=m_{\phi}/\lambda^{1/2}$ (retaining only the positive value), and the
resulting equation of motion for $\phi$ can be written as

$\displaystyle\ddot{\phi}+(3H+\Gamma)\dot{\phi}+m_{\phi}^{2}\phi-\lambda\phi^{3}=0.$
(35)

The presence of the damping term $(3H+\Gamma)\dot{\phi}$ add some
complications to the solution of 35. In order to propose a manageable solution
for equation (35), one first assumes the Hubble parameter during the inflation
is given by [22]

$\displaystyle
H=\frac{t_{0}}{t^{2}}\left[1-\ln\left(\frac{t}{t_{0}}\right)\right]+\frac{1}{t},$
(36)

where $t\geq t_{0}$, being $t_{0}$ the time when inflation starts. Figure 4a)
shows the evolution of $Ht_{0}$, $|\dot{H}t_{0}^{2}|$ and $\ddot{H}t_{0}^{3}$
for $t\geq t_{0}$. It is possible to see that for $t_{0}\leq t\lesssim
5t_{0}$, the Hubble parameter given by (36) fastly decreases, becoming almost
flat for $5t_{0}<t$. Therefore, in some situations, the time-dependent
behavior of $H$ can be neglected.

Figure 4:  Panel a) shows the evolution of $Ht_{0}$ (solid line),
$|\dot{H}t_{0}^{2}|$ (dashed line) and $\ddot{H}t_{0}^{3}$ (dotted line), for
$H$ given by (36). Panel b) shows the ”residue” $r(t)/\phi_{m}$ (solid line)
and the approximate solution $\phi/\phi_{m}$ (dashed line) given by (28).

Analytic solutions for the case where one can neglect the damping term
$(3H+\Gamma)\dot{\phi}$ are well-known, involving Jacobi elliptic functions.
For our purpose, one supposes the approximate solution for equation (35) can
be written as

$\displaystyle\phi(t)\approx\phi_{m}e^{-(3H+\Gamma)t}\sin(m_{\phi}t),$ (37)

which depends strongly on the behavior of $H$. Replacing the approximate
solution (37) into differential equation (35), then it furnishes the ”residue”

$\displaystyle\begin{split}r(t)&=e^{-3t(3H+\Gamma)}[\lambda\phi_{m}^{3}\sin^{3}(m_{\phi}t)+e^{2t(3H+\Gamma)}\phi_{m}(-m_{\phi}\cos(m_{\phi}t)(3H+\Gamma+6t\dot{H})+\\\
&+\sin(m_{\phi}t)(-2m_{\phi}^{2}+3\dot{H}(-2+\Gamma
t+3t(H+t\dot{H}))-3t\ddot{H}))].\end{split}$ (38)

The behavior of $r(t)/\phi_{m}$ is shown by the solid line in Figure 4(b)
while the dashed line corresponds to the behavior of the approximate solution
$\phi/\phi_{m}$. For both curves, one uses $m=0.1$ GeV, $\lambda=0.1$,
$h=0.01$ GeV1/2 and $\phi_{e}=0.1\phi_{m}$. As can be viewed, for $t_{0}\leq
t<5t_{0}$ the Hubble parameter dominates whereas for $t\gtrsim 5t_{0}$ this
parameter tends to vanish. Thus, as $5t_{0}\leq t$ grows, the “residue”
becomes less important, resulting the ansatz (37) tend to be an asymptotic
solution for (35) when the Hubble parameter turns to be almost constant.

To apply the Gouy-Stodola theorem, one should analyse the possible mean free
path for $\phi$, which is not an easy task. One defines here $l$ as the
distance traveled by $\phi$ without any collision or decay, allowing us to
calculate the work due to the scalar field $\phi$.

The required knowledge of initial physical conditions of the inflationary
epoch remains unknown, but it seems reasonable to suppose these conditions are
close to the Quark-Gluon Plasma (QGP) than the classical plasma, for example.
The QGP regime is characterized by a high-temperature regime,
$T>\\!\\!>\Lambda_{QCD}$, where $\Lambda_{QCD}$ is the energy scale of Quantum
Chromodynamics (QCD). Moreover, for this energy regime, the quarks and gluons
are free (nonconfinement of QCD), and the running coupling constant of QCD is
much smaller than 1, $\alpha_{s}=\alpha_{s}(T)<\\!\\!<1$, where $Q_{s}\approx
T$ at the saturation scale ($Q_{s}$ is the transferred momentum in the gluon
frame). It is important to emphasize that the temperature in the inflationary
epoch is, at least, of the order of the electroweak? epoch $T\approx 100\sim
200$ GeV [42], higher than the saturation scale.

The mean free path in such a system may depend on the energy available, the
particle mass, the system temperature, the running coupling constant of QCD,
among others parameters (density medium and volume, for example). The
viscosity coefficients [43, 44] can also be considered to the free mean path
calculation. However, in general, only the bulk viscosity is considered
because the usual assumption of isotropy of the universe [45, 46] and,
consequently, the shear viscosity is neglected. The bulk viscosity, on the
other hand, acts to enhance the entropy in the Friedmann-Robertson-Walker
evolution [47, 48, 49].

The mean free path is temperature-dependent and its functional form varies
according to the constituents of the system. For example, the neutrino mean
free path depends on $T^{-5}$ in the early universe, while in the limit
$T\lesssim m_{e}$ where Klein-Nishima corrections are small and, the photon
mean free path depends on $(n_{e}\sigma_{T})^{-1}\sim T^{-1}$, where $n_{e}$
is the number density of electron and positrons in the medium as well
$\sigma_{T}$ is the Thomson cross section.

At the very beginning of inflation, the mean free path for $\phi$ may be
comparable to the scale factor $a(t)$, implying the decay of $\phi$ starts
when $l<a(t)$. Each oscillation of $\phi$ around the minimum of $V(\phi)$ is
damped, reducing the mean free path due to the decaying rate of $\phi$ into
$\chi$ particles. Thus, the number of particles in the medium is enhanced as
$\phi$ oscillates and, consequently, $l$ diminishes. Observe the decaying rate
$\Gamma$ is taken into account in the equation of motion for $\phi$.

For a system composed of heavy quarks in the QGP regime, the mean free path
depends on the energy loss and for $l$ smaller than the size of the system,
one has [50]

$\displaystyle l\simeq\frac{\mu^{2}}{\alpha_{s}^{2}T^{3}}$ (39)

where $\mu$ is the Debye screening mass of the particle. The Debye screening
mass depends on the proper definition of $\alpha_{s}$ in the nonconfinement
phase of QCD, which is a subject far from agreement [51]. Recently, a mass
generation mechanism was carried out for particles in QED using the Debye
screening [52].

The energy of the scalar field $\phi$ is proportional to the square of wave
amplitude, which allows us to assume the following ansatz to the mean free
path in the inflationary epoch

$\displaystyle l=\frac{\phi^{2}}{\alpha_{s}^{2}T_{0}^{3}},$ (40)

where $l$ explicitly depends on the solutions for $\phi$, and $T_{0}$ is the
initial temperature taken as constant during the process. In a more realistic
situation, $T_{0}$ should be released; the inflaton decreases due to the
damping term in (35). Hence, at the end of the inflationary epoch, $l$
vanishes independently of the value attributed to $T_{0}$. In other words, the
dumped solution (37) is a consequence of both the decaying rate $\Gamma$ and
the growth of $H$. Therefore, the number of $\phi$ particles vanishes as $T$
grows, also implying $l\rightarrow 0$ since there is no physical meaning in
the existence of a mean free path for $\phi$ in a system where it does not
exist anymore.

By the nature of the scalar field $\phi$, which has an energy unit, the
damping term in the equation of motion (35) does not represent a true force
since $\dot{\phi}$ posses a square energy unit, while $(3H+\Gamma)$ has an
energy unit. To recover the square energy unit, one simply write the damping
force as $-(3H+\Gamma)\dot{\phi}/m$, where $m=1$ GeV is a mass scale.

Then, taking into account all the above discussion, the irreversible work done
by the damping force during the mean free path $l$ is given by

$\displaystyle
W_{ir}=-\frac{l(3H+\Gamma)\dot{\phi}}{m_{\phi}}=-\frac{\phi^{2}}{m_{\phi}\alpha_{s}^{2}T_{0}^{3}}(3H+\Gamma)\dot{\phi},$
(41)

which can be used to find out the entropy production using the Gouy-Stodola
theorem

$\displaystyle\dot{\Sigma}=\frac{2(3H+\Gamma)\dot{\phi^{2}}\phi+\phi^{2}[3\dot{H}\dot{\phi}+(3H+\Gamma)\ddot{\phi}]}{m_{\phi}\alpha_{s}^{2}T_{0}^{4}}.$
(42)

Figure 5: Absolute value of entropy $\Sigma$ due to the damped scalar field
$\phi$ during the inflationary epoch. Panel a) and b) $m_{\phi}=10^{14}$ GeV
and $\lambda=10^{-9}$ and $10^{-7}$, respectively. Panel c) and d)
$m_{\phi}=10^{12}$ GeV and $\lambda=10^{-9}$ and $10^{-7}$, respectively. In
all cases, $T_{0}=200$ GeV, $t_{0}=1$ GeV-1, and $\sigma=0.01$.

Figure 6: Absolute value of entropy production $\dot{\Sigma}$ due to the
damped scalar field $\phi$ during the inflationary epoch. Panel a) and b)
$m_{\phi}=10^{14}$ GeV and $\lambda=10^{-9}$ and $10^{-7}$, respectively.
Panel c) and d) $m_{\phi}=10^{12}$ GeV and $\lambda=10^{-9}$ and $10^{-7}$,
respectively. In all cases, $T_{0}=200$ GeV, $t_{0}=1$ GeV-1, and
$\sigma=0.01$.

The inflaton mass ($m_{\phi}$) has been the subject of a long debate in the
literature with a particular revival in recent years [53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64]. Depending on the approach performed, it can vary from
$10^{-4}$ GeV [65] up to $10^{14}$ GeV [66]. The temperature in our
calculations is set as $T_{0}=200$ GeV. Notice, however, the strong dependence
of $\Sigma$ and $\dot{\Sigma}$ on $T_{0}$ may reduce drastically both values
since upping in one order the temperature implies a decrease of four orders in
$\Sigma$ and $\dot{\Sigma}$ (fine-tuning problem).

According to the Planck Collaboration results [67], the inflaton must follow a
slow-roll trajectory in a plateau-like potential for large values of $\phi$.
Then, the higher the mass of $\phi$, fewer the value of $\lambda$ to keep the
flatness of $V(\phi)$ around its minimum. The value of $\lambda$ can vary
according to the approach, of course. For the string theory approach called
M-flation [68], one can observe very small values, $\lambda\sim 10^{-14}$.
Also, the gauge-invariant inflaton approach [69] has a very small self-
coupling constant, $\lambda\sim 10^{-12}$. However, for the Higgs inflation
model, where the mass for the decaying field is of the order of the top quark
mass [57], the self-coupling has a large value, $\lambda\sim 10^{-2}\sim
10^{-1}$ [57, 70, 63]. Due to the necessary flatness of the potential and the
large value expected for the entropy in the inflationary epoch, $\Sigma\sim
10^{88}\sim 10^{100}$ [22, 71], one assumes small values for $\lambda$ and
large for $m_{\phi}$.

Taking into account the above discussion, one assumes $h=0.01$ and $T_{0}=200$
GeV, resulting in the two free adjusting parameters, $m_{\phi}$ as well
$\lambda$. Figure 5 shows the entropy production given by result (42). As
aforementioned, the functional form of the Hubble parameter depends on the
specific features of the scale factor $a(t)$, leading to a strongly dependence
in the entropy and entropy production in the choice of $a(t)$. Figure 5 shows
the entropy due to $\phi$: in panel a) and b), one uses $m_{\phi}=10^{14}$ GeV
and $\lambda=10^{-9}$ and $\lambda=10^{-7}$, respectively. On the other hand,
panel c) and d), one keeps $m_{\phi}=10^{12}$ GeV and uses $\lambda=10^{-9}$
and $10^{-7}$, respectively. One obtains a large entropy due to the inflaton
decay during the inflationary epoch, where larger values are given by
$\lambda=10^{-9}$ for both masses adopted for $\phi$. Of course, the
accumulated entropy grows as $t/t_{0}$ rises.

Figure 6 shows the entropy production using the same parameters and the same
sequence of results as for Figure 5. The large entropy production, in any
case, occurs for $t<10t_{0}$, confirming the results shown in Figure 5. The
accumulated entropy production rate also grows as $t$ rises.

##  IV Final Remarks

The laws of thermodynamics are cornerstones of physics. In principle, every
physics system should be explained by using these laws. Of course, the
complexity of some systems may prevent a thermodynamic treatment.

The Gouy-Stodola theorem may be a useful tool to evaluate the entropy
production rate in irreversible systems by analyzing its thermodynamics. As a
toy model, one uses this theorem to study the thermodynamics of a simple
pendulum subject to a damping force.

In a more complex situation, one applies this theorem to study the entropy
production rate in the inflationary epoch. In general, one expects huge values
for the entropy in this regime, $\Sigma\approx 10^{88}$. In our calculations,
high values for the entropy as well as the entropy production rate can be
reached, showing that the Gouy-Stodola theorem may lead to similar results.

## Acknowledge

RHL and SDC thanks to UFSCar for the financial support.

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