1]Space Research Institute, Austrian Academy of Sciences, Graz, Austria
2]International Space Science Institute, Bern, Switzerland111Visiting
3]Department of Geophysics, Munich University, Germany

_Correspondence to_<EMAIL_ADDRESS>
# Entropic Force in MHD, MHD turbulence, and the Entropy-Kinetic Equation

W. Baumjohann  R. A. Treumann  [  [  [

###### Abstract

Following earlier work, reference is made to the classical entropic force
which results from spatially variable disorder, an exclusively repulsive
force. In terms of macroscopic variables it is applied to
magnetohydrodynamics, causing minor changes on the dispersion of
magnetohydrodynamic waves. More important is its effect in magnetohydrodynamic
turbulence. Here the entropic force affects the ion inertial-range scales on
the electron-dominated perpendicular spectrum causing a steeper than
Kolmogorov $\kappa_{\perp}=-8/3$ spectral slope in agreement with gyro-kinetic
simulations. In kinetic theory, inclusion of the ensemble-averaged entropic
force weakly modifies the Langmuir wave dispersion. It leads to re-formulation
of kinetic theory in terms of Gibbs-Boltzmann-entropy. Liouville’s equation in
this case becomes an entropy-kinetic equation under Hamiltonian action, with
the entropic force-density appearing as an additional force term. This
entropy-kinetic equation governs the self-consistent kinetic evolution of the
classical entropy including self-modulation.

###### keywords:

Entropy Force, MHD, MHD turbulence, Entropy Kinetic Equation

##  1 Introduction

In recent work (Treumann & Baumjohann, 2019) we put forward the notion of a
_classical_ entropic force as the result of spatial differences in disorder
or, if one wants, internal information, the negative of Shannon’s external
information. We applied it in passing to the entropy of a Schwarzschild black
hole, whose entropy is proportional to its surface and can only increase. This
was intuitively natural to be understood as entropy (Bekenstein, 1972, 1973),
an insight that had profound consequences in black hole theory.

In the present note we remain in the realm of less spectacular objects asking
for the effects caused by the entropic force in a fluid, more specifically in
magnetohydrodynamics (MHD). The entropic force results from the definition of
entropy in thermodynamics. Multiplied by temperature $T$ (in energy units) the
dimensionless entropy $S$ is encoded in the thermodynamic identity (Kittel &
Kroemer, 1980)

$dU=TdS-PdV+\mu dN+\vec{m}\cdot d\vec{B}$ (1)

with $V$ volume, $P$ pressure, $N$ particle number, $\mu$ chemical potential,
$\vec{m}$ (elementary) magnetization, $\vec{B}$ magnetic field, as the
irrecoverable part of energy which results in internal disorder (heat) or, if
one wants, gain of internal information. This suggests that inhomogeneity in
disorder is equivalent to the action of a force, which (in contrast to the
proposal of a rather different entropic quantum force that has been identified
with gravity (Verlinde, 2011), with consequences in fundamental physics and
cosmology) is much more mundane and which we modestly called _entropy force_
(Treumann & Baumjohann, 2019). In that case the function $TdS(\vec{x})$ is
understood as a mechanical potential whose gradient, the spatial differences
in disorder (information content) may exert a mechanical force on matter.
Here, we adopt the conventional name of an entropic force, however applying it
just to classical systems.

As for an example, we first investigate some consequences of this notion on a
magnetohydrodynamic fluid, in particular its expected effect on
magnetohydrodynamic turbulence. We then turn to the kinetic definition of
entropy which, as we demonstrate, also allows the definition of a more
fundamental kinetic form of the entropic force. Since this form implies the
phase-space distribution we straightly turn to the derivation of a kinetic
equation which determines the evolution of entropy in an $N$-particle system.
Including the entropic force, we provide its simplified one-particle version
which is written in the form of a fundamental classical kinetic equation
governing the evolution and self-interaction of entropy in any system subject
to an Hamiltonian dynamics. Based on this equation the evolution of
(Boltzmann) entropy or its equivalent (Shannon) information it will become
possible to develop a general entropic (information) dynamics within the
bounds of classical kinetic theory. Here we do not attempt its transformation
into the domain of field and quantum theory.

##  2 Entropic force density in MHD

Asking for the effect of entropy it makes little sense to refer to total
entropy, which is just an unknown number however large it may become. It
accumulates with time with unknown total action in the universe, for instance.
Its accumulation may be a driving force, the force that disorder (accumulating
heat) exerts. What counts is the local space-time difference $\Delta S$
generated by the considered process.

For its explicit determination one needs the entropy as function of the state
variables.

A way to proceed has been opened long ago (by Einstein (1905) in his famous
work on Brownian motion) where, from thermodynamic considerations, the
Hamiltonian density $h_{E}(\delta\rho)=(\delta\rho/\rho)^{2}/2\kappa_{T}$ was
obtained. Here $\rho(\vec{x}),T(\vec{x}),\kappa_{T}(\vec{x})$ are the
respective density, temperature, and compressibility and their fluctuations,
indicated by the prefix $\delta$, averaged over a small though sufficiently
large volume $\Delta V$. In Einstein’s case this Hamiltonian density is given
just by the fluctuations of density $\rho$. Remaining in the Gibbs-
Boltzmann222We will be using the name Gibbs-Boltzmann throughout even though
we remain in the kinetic frame where the entropy is defined just by the
logarithm of the distribution function. Gibbs entropy is the phase-space
probability-weighted Boltzmann entropy used in the expectation value of
entropy. picture of entropy, the probability $P\propto\Omega$ of any
macroscopic configuration is proportional to the phase space volume
$\Omega=\mathrm{e}^{S}$. In order to obtain it, one calculates the
dimensionless entropy $S_{B}=\log\Omega$ and its fluctuation induced variation
in the non-relativistic approximation, dealing only with classical systems.
Then configuration and momentum spaces decouple, with the momentum dependence
becoming intrinsic to the entropy, allowing for the use of thermodynamic and
statistical mechanical arguments which later can be extended to kinetic
theory.

###  2.1 Entropic force potential

With these remarks in mind the change $\Delta S$ of the entropy in the chosen
spatial volume $\Delta V$ is written

$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle\\!\\!\int_{\Delta
V}\\!\\!d\vec{x}\,\rho(\vec{x})\int_{0}^{1}\\!d\alpha\,\delta s(\vec{x})$ (2)
$\displaystyle=$ $\displaystyle\\!\\!\int_{\Delta
V}\\!\\!d\vec{x}\,\rho(\vec{x})\int_{0}^{1}\\!d\alpha\,T^{-1}\big{[}\delta
u+p\,\delta V-\vec{m}\cdot\delta\vec{B}\big{]}$ (3)

Here $\delta s$ is the spatial entropy-density variation which equals the
thermodynamic expression in brackets. A magnetization density $\vec{m}$ has
been added which contributes to the energy in an external magnetic field
$\vec{B}$. In a classical high temperature fluid/plasma the microscopic
magnetizations caused by spins and microscopic atomic orbital motions are of
no importance. The magnetic energy density is $B^{2}/2\mu_{0}$. However, the
macroscopic diamagnetic moment density $\rho\vec{m}=-\rho T/B$ of the
particles enters instead, whenever the scale exceeds the gyroscale and the
fluctuation induced magnetic moment is adiabatically conserved. In this case
the magnetization $\vec{m}$ depends on the magnetic field and the variation
also applies to it.

The clever introduction of the parameter $\alpha$ is due to Einstein who with
it took into account all the normalized realizations of the entropy
fluctuations which have to be summed over. Evaluating the expression on the
right when varying the intensive variables $T(\vec{x})\simeq
T(\vec{x},0)+\alpha\delta T,\ p(\vec{x})\simeq p(\vec{x},0)+\alpha\delta p,\
\vec{m}(\vec{x})\simeq\vec{m}(\vec{x},0)+\alpha\delta\vec{m}$, integrating
over all realizations $\alpha$, making use of some thermodynamic relations as
well as Maxwell-identities (cf., e.g. (Landau & Lifshitz, 1994; Kittel &
Kroemer, 1980)) yields for the entropy at constant total energy $U$ and volume
$V$, up to second order in the fluctuations,

$\Delta S=-\int_{\Delta V}d\vec{x}\frac{1}{2}\Big{[}\rho
C_{v}\Big{(}\frac{\delta
T}{T}\Big{)}^{2}+\frac{1}{\kappa_{T}T}\Big{(}\frac{\delta\rho}{\rho}\Big{)}^{2}+\rho\frac{\delta
B_{\|}}{B}-\rho(1-2\cos^{2}\theta)\Big{(}\frac{\delta B}{B}\Big{)}^{2}\Big{]}$
(4)

where in the scalar product of magnetization density $\vec{m}$ and field
$\vec{B}$ we included the projection angle $\theta$, and
$\vec{B}\simeq\vec{B}_{0}$ is the stationary external magnetic field. It
should be stressed again that the magnetic terms appear only because in the
external magnetic field $\vec{B}_{0}$ the particles perform a macroscopic
gyration with field-dependent diamagnetic magnetic moment $\vec{m}$. We also
assume that the number of particles $N$ in the volume is unchanged as usual in
the microcanonical case. The integrand of this expression is the variation
$\delta s(\vec{x})$ of the spatial entropy density in terms of the variations
of density, temperature, and field. $C_{v}(\vec{x})$ is the specific heat as
function of space, and similarly $\kappa_{T}(\vec{x})=-V^{-1}(\partial
V/\partial p)_{T,B}=\rho^{-1}(\partial\rho/\partial p)_{T,B}$ is the thermal
compressibility. Each of the fluctuations contributes a separate additive term
to the entropy. Usually the fluctuations in temperature are less important for
the slowly variable temperature. Integration adds just an unimportant constant
to the entropy which in most cases can be neglected. Then only the density and
magnetic terms remain. We nevertheless provisionally and for completeness
retain the temperature fluctuation in the following.

The integrand in the above expression is the entropy density in space. One may
note that this density and its spatial variation is immune against local
temporally isentropic variations. Multiplication with temperature $T$ then
yields the potential $-\phi_{s}(\vec{x})$

$T\delta s(\vec{x})=-\frac{1}{2}\Big{[}\rho TC_{v}\Big{(}\frac{\delta
T}{T}\Big{)}^{2}+\frac{1}{\kappa_{T}}\Big{(}\frac{\delta\rho}{\rho}\Big{)}^{2}-\rho
T(1-2\cos^{2}\theta)\Big{(}\frac{\delta B}{B}\Big{)}^{2}+\rho T\frac{\delta
B_{\|}}{B}\Big{]}=\phi_{s}(\vec{x})$ (5)

whose negative gradient is the mechanical entropic force density

$\vec{f}_{s}(\vec{x})=-\nabla\phi_{s}(\vec{x})=\frac{1}{2}\nabla\Big{[}\rho
TC_{v}\Big{(}\frac{\delta
T}{T}\Big{)}^{2}+\frac{1}{\kappa_{T}}\Big{(}\frac{\delta\rho}{\rho}\Big{)}^{2}-\rho
T(1-2\cos^{2}\theta)\Big{(}\frac{\delta B}{B}\Big{)}^{2}+\rho T\frac{\delta
B_{\|}}{B}\Big{]}$ (6)

which is quadratic and thus nonlinear in the temperature and density
fluctuations but has a linear term in the fluctuation of the field. This is to
be used in any macroscopic fluid description where densities $\rho$ (per mass)
are well defined in a given volume $\Delta V$. As argued above, the latter
results from the macroscopic magnetization which develops in the presence of a
magnetic field $\vec{B}$. It is obvious that, under the assumptions made, the
second order density and temperature fluctuations can be neglected in linear
theory. The entropic force enters only in a more elaborate nonlinear
treatment. However, in magnetohydrodynamic fluids the last linear term plays a
minor role. It depends on the projection of the magnetic field variation onto
the external field. In linear theory it comes in only as a compressive low
frequency magnetic effect.

That the entropic force potential is a nonlinear function of the fluctuations
was expected, because the kinetic definition of entropy $S=\log
f(\vec{x},\vec{v},t)$ through the (dimensionless) distribution function $f$,
i.e. the phase-space probability $\Omega\propto f$, is nonlinear. Shannon
entropy/information is just the negative of this. It implies that it affects
any nonlinear evolution of the fluid where it should be taken into account in
the contribution of spatially dependent disorder to the dynamics unless other
arguments arise which allow its neglect. Its contribution to nonlinear
correlations may become particularly interesting in turbulence theory.

###  2.2 Entropic force-modified mhd-waves

Let us briefly discuss the possible role of the entropic force on low
frequency MHD waves. The only linear term in the entropic force is the
magnetic component induced via the orbital magnetic moments of the charges.
This should weakly affect low frequency waves the electrically conducting
fluid plasma, where at those low frequencies adiabatic conservation of the
magnetic moment is natural. These are linear solutions of the
magnetohydrodynamic equations. The fluctuating magnetic field component
$\delta B_{\|}$ along the stationary ambient field $\vec{B}$ is susceptible in
Alfvén and magnetosonic waves. Adding the entropic force to the linearized
momentum equation yields, with $T/m=c_{s}^{2}$ the square of sound speed,

$\frac{\partial\delta\vec{v}}{\partial
t}=v_{A}^{2}\partial_{\|}\Big{(}\frac{\delta\vec{B}_{\perp}}{B}\Big{)}-(v_{A}^{2}-{\textstyle\frac{1}{2}}c_{s}^{2})\nabla_{\perp}\Big{(}\frac{\delta
B_{\|}}{B_{0}}\Big{)}-\nabla\Big{(}\frac{\delta
p}{m\rho}\Big{)}+{\textstyle\frac{1}{2}}c_{s}^{2}\partial_{\|}\Big{(}\frac{\delta
B_{\|}}{B}\Big{)}\vec{e}_{\|}$ (7)

(Remember that $\rho$ is just the particle number density.) To this one adds
the unchanged induction equation

$\frac{\partial}{\partial
t}\Big{(}\frac{\delta\vec{B}}{B}\Big{)}=\partial_{\|}\delta\vec{v}_{\\!\\!\perp}-\big{(}\nabla_{\perp}\\!\\!\cdot\delta\vec{v}_{\perp}\big{)}\vec{e}_{\|}$
(8)

Combination with continuity and equation of state leads to the slightly
modified dispersion relation of magnetosonic waves

$\Big{[}\omega^{2}-k_{\|}^{2}v_{A}^{2}-k_{\perp}^{2}\big{(}v_{A}^{2}+{\textstyle\frac{1}{2}}c_{s}^{2}\big{)}\Big{]}\big{(}\omega^{2}-c_{s}^{2}k_{\|}^{2}\big{)}-{\textstyle\frac{3}{2}}c_{s}^{4}k_{\|}^{2}k_{\perp}^{2}=0$
(9)

The isentropic alfvénic dispersion relation $\omega^{2}-k_{\|}^{2}v_{A}^{2}=0$
remains of course unchanged. The entropic force just causes a minor change in
the speed $c_{ms}$ of infinitesimal-amplitude magnetosonic waves in
magnetohydrodynamics and introduces a factor $\frac{3}{2}$ in the last term.
These lesser modifications are caused by the linear term in the entropic force
when accounting for the induced magnetic moments $m=-T/B$ on the particles in
the presence of a sufficiently strong magnetic field $\vec{B}$. It is clear
that in sufficiently strong magnetic fields with $v_{A}^{2}\gg c_{s}^{2}$, the
usual case in the solar wind or near Earth space physics, the entropic
contribution is widely negligible as neither sound nor sound velocity play any
susceptible role. However, in weakly magnetized fluid dynamics their
contribution is not negligible.

Otherwise the entropic force is a nonlinear force. Thus, in the nonlinear
evolution and theory of magnetohydrodynamic waves it is expected to play some
more important role this time then not through the magnetic fluctuations but
the density modulation. Thus, when going nonlinear, its neglect of the
nonlinear terms in the entropy potential can hardly be justified how difficult
its inclusion may ever become.

In general, however, when considering high frequency waves in plasmas the
entropic force will have to be included either as an additional fluid force
through the above potential or, on the kinetic level, expressing the density
and temperature fluctuations through their moments with respect to the kinetic
distribution function. In that case the magnetic contribution at high
frequencies vanishes as the magnetic moments of the charges will not be
conserved anymore. Any magnetic wave component then contributes as usual
solely through electrodynamic coupling.

###  2.3 Entropic force in MHD Turbulence

The most interesting contribution of the entropic force to
magnetohydrodynamics is expected to occur in turbulence theory. Turbulence is
a prominent nonlinear phenomenon (Biskamp, 2003). One expects that the
entropic force will intervene ever when turbulence evolves and turbulent
disorder arises, heat is exchanged, and the entropy exhibits local
fluctuations.

Since in turbulence theory the correlation moments dominate, the intrinsically
nonlinear entropy potential necessarily contributes. In a fluid system, the
fluctuations are given by (4). Turbulence evolves on different scales.
Measurements provide statistical averages of the turbulent energy transport
through the scales. Theory thus deals with the energy transport equation which
describes the flow of energy density and its distribution onto the sequence of
decreasing scales. Accounting in addition for the entropic force implies
completing the energy-density flow with entropy-density flow
$\vec{v}Ts(\vec{x})=-\vec{v}\phi_{s}(\vec{x})$ in the energy transport
equation, causing additional complications which so far have been neglected.

Turbulence theory in magnetohydrodynamics is based on the full system of
dissipative low-frequency hydrodynamic equations including the Lorentz-force
and the electrodynamic fields. Including the entropic force the equation of
motion of turbulent magnetohydrodynamic fluctuations becomes

$\displaystyle\frac{d\delta\vec{v}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{\rho}\Big{\\{}-\nabla\Big{[}P_{\mathit{th}}+\frac{\delta
B^{2}}{2\mu_{0}}\Big{(}1-\frac{v_{\mathit{th}}^{2}}{2v_{A}^{2}}(1-2\cos^{2}\theta)\Big{)}+\frac{\vec{B}\cdot\delta\vec{B}}{\mu_{0}}\Big{(}1-\frac{v_{\mathit{th}}^{2}}{4v_{A}^{2}}\Big{)}+\rho
TC_{v}\Big{(}\frac{\delta T}{T}\Big{)}^{2}$ (11)
$\displaystyle-\frac{1}{2\kappa_{T}}\Big{(}\frac{\delta\rho}{\rho}\Big{)}^{2}\Big{]}+\frac{1}{\mu_{0}}(\vec{B}+\delta\vec{B})\cdot\nabla\delta\vec{B}\Big{\\}}+\nu\nabla^{2}\delta\vec{v}+{\textstyle\frac{2}{3}}\nu\nabla\nabla\cdot\delta\vec{v}$

which is to be supplemented by the low frequency electrodynamic equations. We
retain the temperature variations for completeness, because they contribute to
higher order entropy-modified transport. Here they contribute just a constant
which drops out when applying the gradient. Similarly, the heat or energy
transport equation would have to be rewritten to include the above defined
entropy-density flow. This becomes necessary when expressing the correlations
and developing the full renormalization group theory of turbulence in
configuration space (Forster et al., 1977) as, for instance, has been
initiated in (Verma, 2004). Since we will not refer to it below, we do not
write it down at this place.

Assume for simplicity an ideal gas equation of state $P_{\mathit{th}}=\rho T$,
with $T=\frac{1}{2}v_{\mathit{th}}^{2}$ (remember that all quantities are
understood here as per mass). The coefficient $\nu$ is the kinematic
viscosity. From the last expression it is seen that the magnetic terms simply
add to magnetic pressure.

An interesting problem arises with the density fluctuations $\delta\rho$.
Before discussing it we rewrite the total pressure term in the last equation
as

$P_{\mathit{tot}}\equiv
P_{\mathit{th}}+\frac{\vec{B}\cdot\delta\vec{B}}{\mu_{0}}\Big{(}1-\frac{v_{\mathit{th}}^{2}}{4v_{A}^{2}}\Big{)}+\frac{(\delta
B)^{2}}{2\mu_{0}}\Big{[}1-\frac{v_{\mathit{th}}^{2}}{2v_{A}^{2}}(1-2\cos^{2}\theta)\Big{]}+\rho
TC_{v}\Big{(}\frac{\delta
T}{T}\Big{)}^{2}-\frac{1}{2\kappa_{T}}\Big{(}\frac{\delta\rho}{\rho}\Big{)}^{2}$
(12)

The entropic force causes a modification of the magnetic pressure. It
apparently also adds a contribution from the turbulent density. However, this
density contribution contains the compressibility $\kappa_{T}=-V^{-1}(\partial
V/\partial P_{\mathit{th}})_{T}$, which is a transport coefficient, indicating
that the corresponding term is dissipative and should be related to the
fluctuation $\delta\vec{v}$ of the velocity instead the pressure.

To see this more clearly, consider the Lorentz force electric field
$\vec{E}=-\vec{v}\times\vec{B}-\eta\vec{j}$ where
$\vec{j}=\nabla\times\vec{B}/\mu_{0}$ is the low-frequency electric current,
and $\eta$ resistivity, assumed constant here.333Constancy of $\eta$ in
turbulence is not given a priori. Nonlinear reaction of turbulence on
transport is a normal process which, however, is of higher order in the
fluctuations and needs not to be considered here. It should be accounted for
in a renormalization group theory of turbulence. We first note that stationary
turbulence implies that $\nabla\times\delta\vec{E}=0$ which, from Faraday’s
law and incompressible turbulence $\nabla\cdot\delta\vec{v}=0$ and vanishing
average flow speed $\langle\vec{v}\rangle=0$, for instance, yields that to
lowest order

$\vec{B}\cdot\nabla\delta\vec{v}=\frac{\eta}{\mu_{0}}\nabla^{2}\Big{(}\frac{\delta\vec{B}}{B}\Big{)}$
(13)

which relates the magnetic fluctuations in incompressible turbulence to the
velocity fluctuation field with magnetic diffusion coefficient $\eta/\mu_{0}$,
an expression used to obtain the turbulent magnetic density when the
stationary turbulent velocity field density is known. As usually, however, in
collisionless ideal flows with $\eta=0$ this vanishes identically.

Density fluctuations can, in magnetohydrodynamic turbulence theory, mostly be
neglected at low frequencies for the assumption of quasi-neutrality.
Continuity of flow $\delta\rho/\rho\sim\delta v/\langle v\rangle$ suggests
that the spectrum of density fluctuations along the mean flow
$\langle\vec{v}\rangle\neq 0$ should resemble that of the velocity
fluctuations. In the absence of a mean flow this relation becomes obsolete,
though distinctions have been made between large and small scale turbulence,
when the large scales represent mean flows for the small scales (Tennekes,
1975). Nevertheless, in spite of the quasineutrality which usually is
considered valid to hold, with the exception of the high frequency range on
the respective electron plasma $\omega_{e}^{-1}$ or cyclotron
$\omega_{ce}^{-1}$ time scales, various observations of turbulence in the
solar wind (cf., e.g. (Podesta et al., 2007; Podesta, 2009)) have identified
turbulent density spectra in the solar wind. An explanation of the modulated
spectral slope has been given (Treumann et al., 2019) based on divergence of
the Lorentz force. There is good reason to assume that this is the case in the
ion inertial range where the ions drop out from being magnetized just
following their inertia, and the electron behaviour is quite different.
Currents are then carried solely by the remaining magnetized electrons. These
scales are subject to kinetic Alfvén wave generation, propagation and
dissipation, and it is, presumably even much more important, subject to
reconnection in electron-scale current filaments, the most probable mechanism
of kind of violent (spontaneous, in contrast to moderate weakly-turbulent
wave-induced) dissipation of mechanical energy in turbulence (Treumann &
Baumjohann, 2013, 2015) far above the molecular scale. In a completely ionized
collisionless fluid on scales $L\gg\lambda_{D}$ exceeding the Debye length,
all electrons participate in the divergence of the induced electric Lorentz
force field $\vec{E}$, which is equivalent to an electronic charge density on
those scales.

Taking the divergence of the Lorentz field
$\vec{E}=-\vec{v}\times\vec{B}-(\eta/\mu_{0})\vec{j}$ eliminates the current
$\nabla\cdot\vec{j}=0$. Poisson’s equation gives the relation between charge
density, field, and velocity fluctuations

$\displaystyle\frac{eq\delta\rho}{\epsilon_{0}}$ $\displaystyle=$
$\displaystyle-\big{(}\vec{B}+\delta\vec{B}\big{)}\cdot\big{(}\nabla\times\delta\vec{v}\big{)}+\big{(}\langle\vec{v}\rangle+\delta\vec{v}\big{)}\cdot\nabla\times\delta\vec{B}$
(14) $\displaystyle\longrightarrow$ $\displaystyle\
-B[\vec{z}\cdot(\nabla\times\delta\vec{v})]=-B(\nabla\times\delta\vec{v})_{\|}$
(15)

with electron charge $-e$, average velocity $\langle\vec{v}\rangle$, and
$\vec{z}$ the unit vector along the ambient magnetic field $\vec{B}$. Still,
we cautiously included a fraction $q\lesssim 1$ of electrons here in order to
maintain the freedom that a fraction $1-q$ of high energy electrons with large
gyroradii exceeding the ion inertial length escape the Lorentz force on these
scales. Only the linear term needs to be retained, here, thus neglecting the
average flow velocity $\langle\vec{v}\rangle$ and the mixed terms in the
density expression (14). Applying the gradient to the entropic force potential
$\phi_{s}(\vec{x})$ one obtains for the contribution of the density
fluctuation

$\frac{1}{2\kappa_{T}\rho}\nabla\Big{(}\frac{\delta\rho}{\rho}\Big{)}^{\\!2}=\frac{1}{q^{2}\omega_{i}^{2}}\Big{(}\frac{v_{A}}{c}\Big{)}^{\\!2}\Big{(}\frac{\partial
P_{\mathit{th}}}{\partial\rho}\Big{)}_{\mathit{\\!\\!T\\!,\\!B}}\big{(}\nabla\times\delta\vec{v}\big{)}_{\|}\big{[}\nabla\big{(}\nabla\times\delta\vec{v}\big{)}_{\|}\big{]}$
(16)

where we inserted for the compressibility
$\kappa_{T}=\rho^{-1}(\partial\rho/\partial P_{\mathit{th}})_{T,B}$. This
expression is of the form of the last viscous term in Eq. (11) with velocity
dependent factor which has the correct dimension of a viscosity. Only the
field-parallel component of the vector product appears here both in the first
and second order differentials. Thus the whole expression maintains the scalar
properties while applying to the perpendicular turbulent velocity
fluctuations. Note that this term does not vanish even in incompressible
turbulence $\nabla\cdot\delta\vec{v}=0$ as it affects only the perpendicular
velocity component $\delta\vec{v}_{\perp}$. It will always be present whenever
velocity fluctuations evolve coupling to density fluctuations $\delta\rho$,
here through Faraday’s law respectively the divergence of the Lorentz force.
Comparing with the above equation of motion in viscous magnetohydrodynamics,
its factor formally defines a second turbulent entropic viscosity through

$\nu_{\mathit{S}}(\delta\vec{v})=\frac{3}{2q^{2}\omega_{i}^{2}}\Big{(}\frac{v_{A}}{c}\Big{)}^{\\!2}\Big{(}\frac{\partial
P_{\mathit{th}}}{\partial\rho}\Big{)}_{\mathit{\\!\\!T\\!,\\!B}}\big{|}\nabla\times\delta\vec{v}\big{|}_{\|}\
\propto\ \nu_{S0}\,\big{|}\nabla\times\delta\vec{v}\big{|}_{\|}$ (17)

This viscosity is not constant but depends on the (transverse) velocity
fluctuations $\delta\vec{v}_{\perp}$. Up to a dimensional constant it however
defines a velocity independent viscosity $\nu_{S0}$. It thus through the
velocity depends on the scale. In collisionless magnetohydrodynamics
$\nu=\eta=0$ this will actually become the only and thus non-negligible
viscous term, produced solely by the entropic force, i.e. by space dependent
turbulent disorder respectively internal heat manifesting itself in density
fluctuations. Still it needs to be represented through the averaged
fluctuations in density and/or velocity. Fourier expanding according to

$\delta\vec{v}(\vec{k})=(2\pi L)^{-d}\int
d^{d}{x}\,\delta\vec{v}(\vec{x})\,\mathrm{e}^{-i\vec{k}\cdot\vec{x}}$ (18)

with $d$ dimension and $L$ length, we have, with $\delta(k_{\|})$ accounting
for the absence of the parallel wavenumber,

$\langle\big{|}\delta\rho^{2}(\vec{x})\big{|}\rangle\sim\langle\big{|}(\nabla\times\delta\vec{v})_{\|}\big{|}^{2}\rangle=\int
d\vec{k}\,k_{\perp}^{2}\mathcal{S}_{\perp}(k)\delta(k_{\|})$ (19)

Here $\mathcal{S}_{\perp}(k)$ is the spectral energy density of the
perpendicular velocity fluctuations from $\int
d\vec{k}\,S(k)=\frac{1}{2}\langle|\delta\vec{v}|^{2}\rangle$. With $\nu_{S0}$
the above dimensional factor, we obtain for the expectation value of the
viscosity

$\langle|\nu_{S}|^{2}\rangle=\nu_{S0}^{2}\int
d\vec{k}\,k_{\perp}^{2}\mathcal{S}_{\perp}(k)\delta(k_{\|})$ (20)

This is proportional to the spectral density in the transverse velocity
fluctuations and clearly scale-dependent. It folds into the diffusive term in
the equation of motion. We may reasonably assume that the spectral interaction
is local, depending only on the differences $\vec{k}^{\prime}-\vec{k}$ in wave
number. Then the right hand side of this equation becomes proportional to the
expectation value of the dissipated energy $\varepsilon_{S}$ per unit of time,
which identifies it as

$\varepsilon_{S}\ =\nu_{S0}\int
d\vec{k}\,k^{2}_{\perp}\mathcal{S}(k_{\perp})\delta(k_{\|})\
=\frac{\langle|\nu_{S}|^{2}\rangle}{\nu_{S0}}$ (21)

We are less interested neither in the turbulent viscosity itself, which by the
above expression for $\nu_{S0}$ is not a large number, nor in the amount of
energy dissipated due to the presence of the entropic force and its
dissipative potential. Our interest is in any possible effect on the shape of
the wave number spectrum of turbulence $\mathcal{S}(\vec{k})$. To infer about
this, a simple dimensional analysis suffices when following the adopted
procedure, assuming that the energy $\varepsilon_{S}$ is dissipated at a
certain dissipation scale $\ell_{d}$ and that possibly an inertial range
exists, where the viscosity $\nu_{S}$ does not depend on scale and dissipation
consists merely in the nonlinear transport of energy from large to small
scales or vice versa, according to the action of the entropy, the second law,
and the entropy-generated viscosity. If such an inertial region exists at all,
then the viscosity $\nu_{S}$ in this region should not have any influence on
the shape of the spectrum, which means that the spectrum should be self-
similar exhibiting some constant spectral slope.

Adopting this approach, Eq. (21) gives dimensionally when multiplying with
powers of the dissipation length $\ell_{d}$ and identifying the dissipated
energy $\varepsilon_{S}=\varepsilon_{Sd}$ per unit time with the energy
dissipated at the dissipation scale

$\mathcal{S}(k_{\perp})\sim\frac{\varepsilon_{Sd}}{\nu_{S0}}\ell_{d}^{2+d}(k_{\perp}\ell_{d})^{\kappa}F(k_{\perp}\ell_{d})$
(22)

Here $F(k_{\perp}\ell_{d})$ is the scale-free function that cares for self-
similarity. Dimensionally we have
$[\varepsilon_{S}]\sim\mathrm{L}^{2}/\mathrm{T}^{3}$ and
$[\nu_{S0}]\sim\mathrm{L}^{2}/\mathrm{T}$, and after eliminating the time
dimensions in the energy flow and viscosity, also for
$\mathrm{L}\sim\ell_{d}=(\nu_{S0}^{3}/\varepsilon_{Sd})^{1/4}$. Inserted into
the last equation and requiring that the power of the viscosity vanishes
(because in the inertial-scale region viscous dissipation has no effect on the
spectral shape), we obtain for the inertial slope of the spectrum

$\kappa=-{\textstyle\frac{1}{3}}(2+3d)$ (23)

In one dimension with $d=1$ this would recover Kolmogorov’s spectrum. However,
the entropic viscosity in our approach works exclusively in the two
perpendicular dimensions only as prescribed by the Lorentz force. Thus we have
$d=2$ obtaining a spectral shape of the perpendicular turbulent velocity
spectrum

$\mathcal{S}(k_{\perp})\sim k_{\perp}^{-8/3}F(k_{\perp}\ell_{d})$ (24)

This perpendicular spectrum is steeper than Kolmogorov being close to
$\mathcal{S}(k_{\perp})\sim k_{\perp}^{-3}$. It should, however, be noted that
it applies to the ion inertial range scale region
$\lambda_{e}<k_{\perp}^{-1}<\lambda_{i}$ (with $\lambda_{e,i}=c/\omega_{e,i}$
the skin depths of electrons and ions), where electrons and ions magnetically
decouple. Here, the electrons still provide a dissipationless non-viscous
scale-invariant spectral energy flux from large to small scales while the ions
do not anymore participate. They already act dissipating their turbulent
energy contributed through the turbulent viscosity $\nu_{S0}$. The scale-free
spectrum thus necessarily should become anisotropic here and decay at a
steeper rate in wave numbers, an effect due to the entropic force in otherwise
completely collisionless magnetohydrodynamics.

Perpendicular spectra of slope $\kappa_{\perp}\approx 2.7$ very close to the
above prediction have indeed been obtained in numerical gyro-kinetic
simulations of solar wind turbulence in the ion inertial scale range (Howes et
al., 2011) where they have tentatively been attributed to the sole dissipative
action of kinetic Alfvén waves. Here they arise from fundamental theory
without reference to any waves or instabilities as the direct signature of the
entropic force respectively the action of entropy in the ion inertial scale
range. It should be noted in addition that the entire theory presented here is
done under the assumption of absence of any viscosity other than that
introduced by the entropic force.

One may again repeat that independence of slope from viscosity does not imply
$\nu_{S0}=0$. It just means that the spectral slope is not deformed by the
constant viscosity. It is instead maintained by its uniform action
transporting energy from larger to shorter scales which is nothing else but
just sort of some dissipation which acts in resolution of scales into smaller
elements though not transforming energy into heat. However, outside the
entropic inertial range the entropic viscosity acts like a normal viscosity,
transforms energy into heat and thus will necessarily deform the spectrum. Its
mere presence, if at all remarkable, outside the entropic inertial range, for
instance in the Kolmogorov or Kraichnan ranges, should cause deviations from
the inertial spectral slope and possibly lead to a reduction of the extension
of the inertial range, an effect observed both in nature and simulations.

Does one expect a similar though separate effect leading to a magnetic
spectrally different slope when considering the contribution of magnetic
fluctuations to the entropic force? The answer is no, whenever the turbulence
is stationary with vanishing average flow $\langle\vec{v}\rangle=0$. Then
under stationary conditions and incompressibility Faraday’s law does not
couple the additional magnetic entropy contribution to the turbulent velocity
fluctuations. Thus the magnetic spectrum should simply mirror the velocity
spectrum. This might supposedly change in flowing turbulence (like the solar
wind) where Faraday’s law imposes a relation between the divergence of the
perpendicular velocity and the parallel field fluctuations

$(\langle\vec{v}\rangle\cdot\nabla)\delta
B_{\|}=B\nabla_{\perp}\cdot\delta\vec{v}_{\perp}$ (25)

which spectrally translates into

$k_{\perp}^{2}\mathcal{S}(k)=\langle v^{2}\rangle
k_{v}^{2}\mathcal{S}_{B_{\|}}(k)/B^{2}$ (26)

where $k_{v}$ is the wave vector of the magnetic fluctuations along the
average flow. As usually, this suggests that the spectrum of parallel magnetic
fluctuations simply translates into that of (perpendicular) velocity
fluctuations, just depending on the angle of flow and constant ratio $\langle
v^{2}\rangle/v_{A}^{2}$. The average over the angle with respect to the
direction of $k_{v}$ merely gives a number while maintaining the spectral
slope.

The result of this section which says that in collisionless MHD with $\nu=0$
and absence of any collisional dissipation on scales far above the molecular,
is that the entropic force itself provides a finite viscosity $\nu_{S0}$.
However small, it causes a feedback of the turbulent fluctuations on
turbulence whose nature through it becomes viscous and dissipative. Generation
of this viscosity should be a general phenomenon in MHD though its effect
might mostly be not remarkable or within the range of uncertainty. In the
presence of any otherwise generated anomalous sufficiently large viscosity,
its effect on the shape of the inertial range spectrum will probably become
buried below the general spectral trend. The inertial range will remain
determined by kind of Kolmogorov spectral decay as is mostly observed in the
solar wind where it is determined up to the ion inertial scale by fluid
turbulence. When entering the ion inertial scale, however, the turbulence
should become anisotropic because the inertial range spectrum is continued
only into the perpendicular spectrum where, in the absence of any otherwise
generated dissipation, the spectrum decays more steeply than Kolmogorov until
the turbulence reaches the electron inertial scale where it should abruptly
end in dissipation.

##  3 General entropic force in kinetic theory

After this brief excursion into magnetohydrodynamics and turbulence, we return
to the main problem of the definition of the entropic force. On the kinetic
level, the entropy has been defined by Gibbs-Boltzmann in its famous
logarithmic form $S_{B}$ as given above. Assume there is a classical
probability distribution in phase space $f(t,\vec{v},\vec{x})$ which obeys the
Liouville equation

$\partial_{t}f(t,\vec{x},\vec{v})+[\mathcal{H},f(t,\vec{x},\vec{v})]_{P}=\mathcal{C}$
(27)

where $\mathcal{H}$ is the Hamiltonian of the system from that the equations
of motion in momentum/velocity and configuration space can be derived, and
$\mathcal{C}$ some source (or collision) term on the right which is positive
if acting as source, negative if dissipating and causing losses, and the
brackets are Poisson brackets (as indicated by the subscript $P$) defining the
particle dynamics. We are dealing here for simplicity with particles not with
fields. The Liouville equation has to be completed with the equations which
describe the interaction between the particles in the volume, and these
interactions are mediated by fields if not simply by collisions. Then, in
terms of the phase-space probability distribution, the kinetic Gibbs-Boltzmann
entropy is defined as $S_{B}=\log f$. The total entropy of the entire system
which would become involved into the thermodynamic potentials is of course the
integral all over phase space

$\langle\,S\,\rangle=\int d\vec{x}d\vec{v}\,f(\vec{x},\vec{v})\log
f(\vec{x},\vec{v})$ (28)

as the average moment of the entropy (or in other terms its statistical
expectation value). Here $\log f=S_{B}$ is again Gibbs-Boltzmann’s
microcanonical entropy, which is to be multiplied by $f(\vec{x},\vec{v})$ and
integrated in order to provide the total entropy. The momentum space integral
defines the entropy density in real space

$s(\vec{x})=\int d\vec{v}\,f(\vec{x},\vec{v})\log f(\vec{x},\vec{v})$ (29)

Assume that the temperature $T$ of the system is known. Multiplying by $T$ and
applying the spatial gradient we obtain the microcanonical Gibbs-Boltzmann
entropic force density

$\vec{f}(\vec{x})=-\nabla[Ts(\vec{x})]$ (30)

In this expression the temperature can still be a function of space and time.
From this definition follows for the entropic force-potential

$\phi_{s}(\vec{x})=Ts(\vec{x})$ (31)

which is a mechanical potential that is to be added to the potential either in
the Hamiltonian $\mathcal{H}$ or the equivalent Lagrangean $\mathcal{L}$. Its
sign depends upon the sign of the logarithm. Since $f<1$ usually is a
normalized probability density, this sign is negative, and the entropic force
is repulsive. It acts against gravity causing spatial expansion of any
inhomogeneities in disorder respectively expansion of internal information.

The important difference between this and any other force potential is that
the entropy potential is defined solely through the phase space density
itself, not by other external fields. More precisely, it is defined through
its integral over the chosen volume $\Delta V_{\vec{v}}$ in momentum space. In
this sense it describes the self-interaction and feedback of the phase-space
density on itself through the changes of the phase-space density induced by
the fields which determine the particle dynamics.

Clearly this self-interaction is highly non-linear, and it becomes even more
complicated by the fact that the temperature itself is determined as
$T=(\partial_{S}U)_{V}$, the derivative of the total energy with respect to
the entropy. However, even in the case of global equilibrium with constant
overall temperature $T=$ const, which does still allow for local non-
equilibria, the intrinsic nonlinearities remain. Through them, the entropy
density of the system affects itself, something well known from field theory
for instance. It moreover says that the _entropy_ , at least under certain so
far unspecified conditions, _behaves as its proper source_.

The above definition of the entropic force density through the distribution
function and integral over the momentum space volume identifies the entropic
force density and its potential as independent of the momentum and therefore
as a conservative force. However, its definition through an integral over
momentum space turns the Liouville equation (27) into an integro-differential
equation for the phase space distribution

$\partial_{t}f(t,\vec{x},\vec{v})+[\mathcal{H}_{0},f(t,\vec{x},\vec{v})]_{P}-\nabla\Big{(}T\int
d\vec{v}\,f(\vec{x},\vec{v})\log
f(\vec{x},\vec{v})\Big{)}\cdot\partial_{\vec{v}}f(\vec{x},\vec{v},t)=\mathcal{C}$
(32)

where $\mathcal{H}_{0}=\mathcal{H}-\phi_{s}$ is the (entropic
potential)-reduced Hamiltonian. One may note that the term in round brackets
is independent on momentum/velocity as this dependence is integrated over,
such that the momentum gradient can be put in front. For constant temperature
this simplifies to become

$\partial_{t}f(t,\vec{x},\vec{v})+[\mathcal{H}_{0},f(t,\vec{x},\vec{v})]_{P}-\partial_{\vec{v}}f(\vec{x},\vec{v},t)\cdot\int
d\vec{v}\ [T\,\nabla f(\vec{x},\vec{v},t)]\big{(}1+\log
f(\vec{x},\vec{v},t)\big{)}=\mathcal{C}$ (33)

These equations show the self-interaction of the phase space distribution
through its self-generated entropy. In all cases when the distribution
function develops an inhomogeneity there arises an unavoidable collisionless
reaction of the phase space distribution on its own evolution, with nonlinear
self-interaction term. It results from the nonlinearity of the entropy and
thus enters quite naturally. However, due to the nature of the entropy, the
local self-reaction of the distribution function through the entropy is a non-
local effect in momentum space as it involves a finite momentum space volume
which has to be integrated over, whereas in configuration space it is local in
the infinitesimal environment of the local volume as selected by the spatial
gradient.

We note that any of these equations can be taken as an evolution equation of
entropy where the entropy turns out as its own source. This evolution equation
of the entropy can be obtained when multiplying the Liouville equation with
the logarithm of the distribution function.

In spite of the fact that the entropy regulates itself, the last expression
(33) is interesting when applied in linear theory. Only the last term needs to
be considered when introducing $f=f_{0}+\delta f$. Neglecting all nonlinear
terms and observing that under the integral such products like $f_{0}\log
f_{0}$ when integrated just produce the constant global entropy which vanishes
when $\nabla$ is applied to it. The only two terms remaining, one the velocity
integral over the fluctuation of the distribution function which thus just
produces the fluctuating number density $\int
d\vec{v}\,\delta\\!f(\vec{x},\vec{p},t)=\delta\rho(\vec{x})$. The other
remaining term is mixed, with integrand $(f_{0}\log f_{0})\delta\\!f/f_{0}$
where the term in brackets is the overall density of the entropy. Partial
integration then lets the stationary term vanish. In the remaining integral
then the density of entropy $s_{0}$ is independent of velocity and can be
taken out of the integration leaving a total derivative of $\delta f$ unter
the integral. Hence this term vanishes as well and thus does not contribute
linearly. In linear approximation on the kinetic level, the entropic force
then adds only the additional force term

$-\nabla(T\delta\rho/\rho_{0})\cdot\partial_{\vec{v}}f_{0}(\vec{v})$ (34)

to the linear kinetic equation. The linearized kinetic equation becomes
finally

$\displaystyle\partial_{t}\delta\\!f+[\mathcal{H}_{0},\delta\\!f]_{P}$
$\displaystyle=$
$\displaystyle\nabla\big{(}T\delta\rho[\delta\\!f]/\rho_{0}\big{)}\cdot\partial_{\vec{v}}f_{0}(\vec{v})$
(35) $\displaystyle\delta\rho(\vec{x},t)$ $\displaystyle=$
$\displaystyle\int_{\Delta V_{\vec{v}}}d\vec{v}\,\delta f(\vec{x},\vec{v},t)$
(36)

The spatial gradient of the density fluctuation is multiplied by the momentum
space gradient of the equilibrium distribution function $f_{0}(\vec{v})$. Here
the density fluctuation $\delta\rho$ arises from the momentum space integral
of the fluctuation of the distribution function $\delta\\!f$. In any
particular problem the density fluctuation is to be expressed through the
field equations which, in the electrodynamic case are the Maxwell equations
where the density appears in Poisson’s law and becomes re-expressed through
its above integral form which thus can be used directly.

As for an example, let us briefly consider electrostatic electron (Langmuir)
waves in one dimension. Inclusion of the entropic force modifies the
linearized Fourier-transformed Vlasov equation to become

$-i(\omega-kv)\delta f_{k\omega}=\big{[}(e/m)\delta
E_{k\omega}+ikT(\delta\rho_{k\omega}/\rho_{0})\big{]}\partial_{v}f_{0}(v)$
(37)

where we reintroduced the electron mass $m$. This is to be completed by
Poisson’s law for the Fourier transformed electric field $E_{k\omega}$ as

$ik\delta E_{k\omega}=-(e/\epsilon_{0})\delta\rho_{k\omega}$ (38)

which is used to eliminate the density fluctuation $\delta\rho_{k\omega}$ to
yield an expression for $\delta f_{k\omega}$. Inserting this into (36) to
express $\delta\rho_{k\omega}$ and then again into Poisson’s law eliminates
the electric field from both sides and yields the slightly modified one-
dimensional complex dispersion relation of electrostatic unmagnetized-electron
plasma (Langmuir) waves with plasma frequency $\omega_{e}$:

$\mathcal{D}(k,\omega)=1-\frac{\omega_{e}^{2}(1+k^{2}\lambda_{D}^{2})}{\rho_{0}k^{2}}\int_{-\infty}^{\infty}dv\frac{\partial
f_{0}(v)/\partial v}{v-\omega/k+i0}$ (39)

which here is extended to include an entropy-generated factor
$(1+k^{2}\lambda_{D}^{2})$. The entropic effect in this case occurs at short
wavelengths the order of the Debye length $\lambda_{D}$. Solution of
$\mathcal{D}(k,\omega)=0$ is prescribed by the Landau contour in the complex
$v$-plane to yield the collisionless Landau damping $\gamma_{L}(k)$. In
thermal plasma with Maxwellian distribution $f_{0}(v)$ it causes a minor
modification of the Langmuir wave frequency

$\omega_{L}\approx\pm\omega_{e}\big{(}1+2k^{2}\lambda_{D}^{2}\big{)}+i\gamma_{L}(k)$
(40)

replacing the familiar ratio $\frac{3}{2}$ in the brackets by the integer 2,
while leaving Landau damping about unaffected. Accounting for the entropy
effect thus causes a slightly stronger increase of the frequency with
wavenumber which also will slightly affect the wavenumber dependence of the
thermal fluctuation spectrum. Similar minor changes can be expected in the
dispersion relations of other plasma waves.

The collision term on the right in (33), which in this Liouville-Vlasov
approach is neglected, is in fact a rather complicated integral which in its
simplest binary collisional form gave rise to the definition of the Gibbs-
Boltzmann entropy we used in the above. In a stricter theory, $\mathcal{C}$ is
a huge extended sum over a long series of integrals (Bogoliubov, 1962)
accounting for all higher interactions between increasing numbers $N$ of
particles involved. It is so far not known what effects the inclusion of the
$N$-particle entropy $S_{N}=\log
f_{N}(\vec{x}_{N},\vec{v}_{N},\vec{x}_{N-1},\vec{v}_{N-1},\dots\vec{x}_{1},\vec{v}_{1})$
would cause.

##  4 Entropy-kinetic equation

Remaining on the level of the above used one-particle phase-space distribution
$f_{1}(\vec{x},\vec{v},t)\equiv f$, and referring to Gibbs-Boltzmann’s kinetic
definition of the local entropy $S=S_{B}=\log f$ at a given phase-space point
$(\vec{x},\vec{v})$, an evolution equation for the kinetic entropy
$S(\vec{x},\vec{v},t)$ can be derived directly when dividing the one-particle
Liouville equation by the one-particle distribution function
$f(\vec{x},\vec{v},t)$. Since the Hamiltonian $\mathcal{H}$ is a linear
operator, this generates a kinetic equation for $S$, the entropy itself:

$\partial_{t}\,S+[\mathcal{H}_{0},S]_{P}+(\nabla\,\phi_{s}[S])\cdot\partial_{\vec{p}}S=\mathcal{C}/f$
(41)

(written in terms of the momentum $\vec{p}$ instead velocity $\vec{v}$) which
can be understood as the wanted one-particle evolution equation of the kinetic
Gibbs-Boltzmann entropy $S$ in the underlying one-particle phase space
$(\vec{p},\vec{x})$.444It should be noted here that we prefer using
Boltzmann’s kinetic definition of entropy. Gibbs definition as the sum of the
probability-weighted Boltzmann entropies $\int d\vec{p}\ f\log f$ refers to
the canonical level. Though it is also possible to derive a kinetic equation
for the Gibbs entropy-density $f\log f$ in phase space when multiplying the
Liouville equation by $1+\log f$, little is gained then instead some more
severe complications with the collision term. Here $\phi_{s}[S]$ is the above
given entropic force-functional as integral over the chosen relevant momentum-
space $V_{\vec{p}}=\int_{\Delta V_{\vec{p}}}d\vec{p}$. It thus contains the
total entropy the momentum-space volume contributes and, in this way, reacts
on itself through the last term. The right-hand side of this equation is well
defined for any normalized non-zero distribution function $f\neq 0$, the
normal case of course. As usual, the Hamilton equations of motion are implicit
to this expression and of course contain/require the prescription of the
dynamics. Spelled out in momentum $\vec{p}$ instead velocity, the last
equation becomes explicitly

$\partial_{t}\,S(t,\vec{x},\vec{p})+\frac{\vec{p}}{m}\cdot\nabla\,S(t,\vec{x},\vec{p})+\dot{\vec{p}}\cdot\partial_{\vec{p}}S(t,\vec{x},\vec{p})=\mathcal{C}\mathrm{e}^{-S(t,\vec{x},\vec{p})}\equiv\mathcal{C}^{\prime}$
(42)

where the force term $\dot{\vec{p}}=\vec{F}([S],\vec{x},\vec{p},t)$ in the
equation contains the external forces and includes the internal entropic
force, and on the right we have taken advantage of the definition of the
kinetic Gibbs-Boltzmann entropy to replace the one-particle kinetic
distribution function $f=\exp S$ through it. (One may remember that this means
that the local phase-space probability-density is proportional to the local
phase-space volume $f\propto\Omega=\exp S$.) This replacement also applies to
the collision term in order to close the equation as a function of $S$ alone.
The same replacement must also be done in any of the dynamical field equations
which govern the dynamics of the particles and hence also the dynamics of the
entropy.

In this sense the above kinetic equation completely describes the generation
and phase-space evolution of entropy (or information) under the considered
Hamiltonian dynamical process $\mathcal{H}(\vec{x},\vec{p},[S])$ with the
accessible volume-entropy (or information) to be obtained as its Gibbs-
Boltzmann moment $\Delta S(\vec{x},t)=\int_{V_{\vec{p}}}d\vec{p}\,f\log f$. It
is clear from here that taking the moment of this kinetic equation one arrives
at an average or even fluid equation for the evolution of entropy in real
space-time under general non-equilibrium conditions.

In a more general sense, one can consider Eq. (42) as a fundamental equation
in physics and probably also, in greater generality, as a fundamental equation
governing the dynamics of information and information transport affected by
the extraordinarily complicated collision term on the right and by
entropy/information itself.

There are two difficulties to overcome when attempting to solve the kinetic
entropy equation in application to real problems. The first is that, though
being fundamental to the evolution of entropy, Eq. (42) is an _integro-
differential_ equation through the entropic force acting in phase space. The
second difficulty lies in the complicated collision term, which is defined
through higher-order interactions and thus through higher-order entropies. To
lowest order in the local deviation of entropy the inconvenient exponential
can be cautiously expanded and replaced by555One would write
$S=S_{0}+\delta\,S$. However, $S_{0}=$ const, the total entropy in the global
volume plays no role. So we can either retain it which in the exponential
expansion just produces the exponential e${}^{S_{0}}$ and normalizes $\delta
S/S_{0}$. Or, since $S_{0}$ is an unmeasurable constant, we can also put
$S_{0}=1$ with the entropy change $\delta S$ in all expressions understood as
being re-normalized.

$\exp\,S\approx\mathrm{e}^{S_{0}}\big{(}1+\delta\,S/S_{0}\big{)}$ (43)

to be used conveniently in the underlying Hamiltonian and the field equations.

The philosophy behind this approach is the usual probabilistic picture of
defining the probability density in a certain small though finite volume
$\Delta V_{\vec{p},\vec{x}}=(d\vec{p}\,d\vec{x})$ of phase space. This volume
may be small but must contain sufficiently many particles/constituents at any
time to be integrated over, in order to be able to define a probability
density $f(\vec{p},\vec{x},t)$ at a given time $t$ (a sufficiently short time
interval $\Delta t$ prescribed by the time resolution of measurement).
Considering the equivalent picture drawn in chaos theory which deals with the
nonlinear interaction between just a few particles and their dependence on the
initial conditions and measurement errors, so in the statistical Liouvillian
phase-space approach, these interactions are averaged over; they are sub-
kinetic (microscopic) to it and contribute only in as far as they leave a
trace or signature on the probability in the kinetic phase space density
$\Omega\propto f$. Since the Hamiltonian in principle contains all these
interactions, it takes them into account in the size of the phase space
volume, at least in an average sense. Their sub-microscopic (chaotic)
resolution would require to solve the $N$-particle phase space Liouville
equation for all the $N-1$ particles interacting with the $N$th particle,
which includes all the $N-1$ single-particle phase-space volumina including
all their mutual interactions. This problem can be solved only by coarse
graining and thus only approximately. It shows however that chaos theory
without application of such a coarse graining procedure describes reality only
if it considers a countably and manageably small number of particles such that
a reasonable particle _density_ for the phase-space unit can be defined, not
the rare case of a collection of a small number of single mutually interacting
particles. Poincaré-chaos theory describes reality on the sub-kinetic level
where it hardly is accessible to measurement in any system that is built up of
very many constituents, that is of most real systems. The chaotic approach of
few particles only is from this point of view to be considered as most
interesting but below accessible reality in most physical problems if not
leading to observable kinetic effects whose detection requires reference to
renormalization-group techniques (Wilson, 1971).

The overall integrated entropy $\langle S\rangle=\int d\vec{x}d\vec{p}\,f\log
f$ under stationary conditions is of little interest as it is just a number, a
huge constant of integration and in any dynamical process can be neglected.
The entropy-kinetic equation can be reduced to an equation for the entropy
fluctuations in momentum space $\delta S$, allowing for functional
linearization

$\partial_{t}\delta S+\big{(}\vec{p}/m\big{)}\cdot\nabla\delta
S+\big{\\{}\vec{F}_{0}+\delta\vec{F}[S]\big{\\}}\cdot\partial_{\vec{p}}[S+\delta
S]=\delta\mathcal{C}^{\prime}$ (44)

which in this form accounts for the local variation of the entropy while still
containing all the higher order products of the variation $\delta S$. Here
$\vec{F}_{0}$ is the external force involving the fields, and $\delta\vec{F}$
is the variation of the force with respect to the entropy. The last term
maintains the stationary spatially homogeneous entropy $S(\vec{p})$ because,
as a function of momentum $\vec{p}$ like, for instance under stationary and
homogeneous conditions it gives $\partial_{t}S_{B}=\nabla S_{B}=0$, but
$\partial_{\vec{p}}S_{B}(\vec{p})\neq 0$.

Formally the kinetic equation is an identity (the total time derivative) of
similar kind as the Liouville equation. It describes the phase-space evolution
of the one-particle entropy $S(\vec{x},\vec{p},t)\equiv
S_{1}(\vec{x}_{1},\vec{p}_{1},t)$ under the dynamical action imposed by the
Hamiltonian $\mathcal{H}_{0}$ which contains the involved fields (for instance
the electromagnetic field). In this general form, it is so far restricted just
to the one-particle entropy $S$ and the one-particle distribution function
$f_{1}=f$ but could formally become generalized to the $N$-particle case
(developed in (Bogoliubov, 1962) and somewhere else, the famous BBGKY
hierarchy). The space-dependent expectation value of the entropy (which enters
the entropic force) is obtained, in the usual way, as the statistical
momentum-space average

$\displaystyle\langle S(\vec{x},t)\rangle$ $\displaystyle=$
$\displaystyle\int_{\Delta
V_{\vec{p}}}d\vec{p}\,f(\vec{x},\vec{p},t)\,S(\vec{x},\vec{p},t)$ (45)
$\displaystyle=$ $\displaystyle\int_{\Delta
V_{\vec{p}}}d\vec{p}\,S(\vec{x},\vec{p},t)\exp[S(\vec{x},\vec{p},t)]\,$ (46)

The integration is over the sufficiently large momentum-space volume $\Delta
V_{\vec{p}}$. Given an underlying dynamics, it yields the momentum-space
average of the entropy $S(\vec{x},\vec{p},t)$ with retained dependence on
configuration space. Once again this expression demonstrates the self-
determination of the entropy.

The entropy in this representation just exchanges its place in the integral,
becoming its own probability distribution which weighs its inverse dependence,
the exponential of itself. In this form the entropy appears as the momentum-
space probability which weighs the kinetic phase space volume element
$\Omega=\mathrm{e}^{S}$. Interpreted that way, it says that the _entropy
controls the evolution of the phase space volume_ occupied by Hamiltonian
dynamics. The entropy affects its hosting phase space volume $\Omega$. Locally
however this affection does not simply proceed in a smooth expansion of phase
space as this is the product of momentum and configuration spaces.

Given the importance of information in the modern world, an equation like the
one derived here from fundamental considerations including the entropic force
should have some application and thus could be of practical use.

Having derived the kinetic evolution equation for the kinetic entropy, a most
interesting problem arises when asking for the second law and its maintenance
on the kinetic level. The second law offers a constraint on the entropy
evolution which has to be incorporated into a full theory. It requires
discussing what, on the kinetic level, the notion of an _exclusively growing_
entropy means and how it has to be understood or re-interpreted. Clearly, the
_global over-all_ entropy in configuration space $S_{0}(t)=\int
d\vec{v}d\vec{x}\,f\log f$ will only grow at $\Delta S_{0}(t)>0$. The same
should also hold for any _isolated_ spatial volume $\Delta V$. However, in
momentum space permanent increase of $S$ is clearly not an obligation as
energy and heat can be distributed with respect to the moments $\vec{p}$ in
phase space. Here we do not enter into a deeper investigation of the related
questions but relegate it instead to an own consideration.

This derivation of the kinetic entropy equation remains in the realm of
classical physics.666A quantum approach is beyond this work but should make
use of von Neumann entropy. There are two basic classical theories,
electrodynamics and general relativity where it could be applied. In
electrodynamics this does not look impossible as the Hamiltonian of the
underlying charged particles is known. We have demonstrated its effect on the
example of Langmuir waves above. General relativity, on the other hand, is not
a particle but a field theory to which our particle approach does primarily
not apply. Its kinetic consideration should refer to some non-quantum version
of _Geometrodynamics_ but requires the field-theoretical version of kinetic
theory which is beyond the present investigation.

The electrodynamic field is a gauge field $\vec{A}$ which enters the
generalized momentum $\vec{p}^{\prime}=\vec{p}\pm ie\vec{A}$, where $e$ is the
elementary charge. It is to be accomplished with the field equations including
sources, charge and current. Replacing the distribution function $f_{\pm}=\exp
S_{\pm}$ it becomes formally possible to rewrite the kinetic theory in terms
of the entropies $S_{\pm}$ under the action of the electrodynamic gauge field
$\vec{A}$. Since the field equations couple the entropy distributions of the
positive and negative charges, it is clear that, similar to ordinary kinetic
theory, separate though coupled kinetic equations apply to $S_{\pm}$.
Ultimately, in electrodynamics (plasma physics) the relevant entropy at
location $\vec{x}$ in real space is obtained as the sum

$S_{\mathit{tot}}(\vec{x},t)=\sum_{+,-}\int_{\Delta V_{{p}}}d\vec{p}\
S_{\pm}(\vec{x},\vec{p},t)\exp{S_{\pm}}(\vec{x},\vec{p},t)$ (47)

It is obvious that, in principle, it is the entropy which on the kinetic level
determines the dynamics of the system, with the probability distribution
$f(\vec{x},\vec{p},t)$ having occurred in an intermediate and incomplete step.
It will be interesting to infer which consequences this insight might have in
quantum physics, where one probably would need to refer to von Neumann instead
Gibbs-Boltzmann entropy, and the density matrix instead the classical
distribution function.

##  5 Summary

The present note started from the definition of a classical non-quantum
entropic force as was proposed in (Treumann & Baumjohann, 2019). This force
acts like any mechanical force on particles or fluids. We gave it an explicit
expression for a fluid system and applied it in passing to
magnetohydrodynamics where it, in linear theory, causes just small
modifications in the magnetosonic wave dispersion relation. Applied to
magnetohydrodynamic turbulence it leads to a modified inertial range spectral
slope on the transverse velocity fluctuation spectrum which might be
remarkable just in the magnetohydrodynamic ion-inertial range. A more
elaborate theory might be useful to be put forward in the context of electron
magnetohydrodynamics (Gordeev et al., 1994; Lyutikov, 2013) when applying the
same or similar reasoning there. Numerical electron-MHD simulations favourably
reproduce the proposed spectral shape.

We then proceeded to the more fundamental question of the entropic force in
kinetic theory. This becomes a rather involved problem. However, skipping
those difficulties, we chose to attempt the derivation of a kinetic equation
for Gibbs-Boltzmann’s entropy, not the phase space distribution function,
following Gibbs-Boltzmann’s philosophy that the entropy is the logarithm of
the distribution function in classical physics. The equation obtained is the
phase-space evolution equation of the kinetic entropy under the action of a
given Hamiltonian which includes the entropic force-potential.

This equation should provide the basis of the evolution, self-interaction and,
possibly, also the self-generation of entropy (or information) in any
classical Hamiltonian system. Such a self-generation would arise if taking
into account the self-consistent version of the collision term $\mathcal{C}$
on the right of the equation as this includes all the nonlinear interactions
in an $N$-dimensional phase space as suggested in BBGKY theory (Bogoliubov,
1962) of classical many-particle kinetic theory. A similar quantum approach in
many-body theory (Fetter & Walecka, 1971) will probably face severe hurdles.
Nevertheless, it awaits its construction.

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