# Nonequilibrium Phase Transitions and Pattern Formation as Consequences of
Second Order Thermodynamic Induction

S. N. Patitsas  University of Lethbridge,
4401 University Drive, Lethbridge AB, Canada, T1K3M4

###### Abstract

Development of thermodynamic induction up to second order gives a dynamical
bifurcation for thermodynamic variables and allows for the prediction and
detailed explanation of nonequilibrium phase transitions with associated
spontaneous symmetry breaking. By taking into account nonequilibrium
fluctuations, long range order is analyzed for possible pattern formation.
Consolidation of results up to second order produces thermodynamic potentials
that are maximized by stationary states of the system of interest. These new
potentials differ from the traditional thermodynamic potentials. In particular
a generalized entropy is formulated for the system of interest which becomes
the traditional entropy when thermodynamic equilibrium is restored. This
generalized entropy is maximized by stationary states under nonequilibrium
conditions where the standard entropy for the system of interest is not
maximized. These new nonequilibrium concepts are incorporated into traditional
thermodynamics, such as a revised thermodynamic identity, and a revised
canonical distribution. Detailed analysis shows that the second law of
thermodynamics is never violated even during any pattern formation, thus
solving the entropic coupling problem. Examples discussed include pattern
formation during phase front propagation under nonequilibrium conditions and
the formation of Turing patterns. The predictions of second order
thermodynamic induction are consistent with both observational data in the
literature as well as the modeling of this data.

###### pacs:

05.70.-a, 05.40.-a, 05.65.+b, 68.43.-h

##  I Introduction

Thermodynamic induction (TI) has recently been put forward as a general
approach towards the study of nonequilibrium systems and has been used to
explain some important particular details regarding the manipulations of atoms
and molecules by STM Patitsas (2014, 2015). A new type of thermoelectric
cooling by TI has also been studied and proposed as a good test for the
existence of TI Patitsas (2016). In short, TI can result in a thermodynamic
variable being influenced in a surprising way when it plays the role of a
gate, or control, variable. The variable may be pushed away from equilibrium
when there is no apparent force to do so, thus giving the appearance of
violating the second law of thermodynamics (SLT). The direction of this
influence is always in such a way that facilitates the approach to equilibrium
of the entire system. When the entire systems reaches equilibrium, the
influence disappears.

So far, the TI theory applies to the case of a conductance coefficient (or
kinetic coefficient) that depends on a thermodynamic variable in a linear
fashion, i.e., the case of first order TI (TI1), ex. the electrical
conductivity of a channel depending on the temperature of the channel Patitsas
(2016). What is missing in the theory is a treatment of second order TI (TI2).
This step is important in this development of nonequilibrium thermodynamics as
it will allow treatment of spontaneous symmetry breaking, pattern formation,
as well as a description on nonequilibrium phase transitions (PT). This
approach also allows for a thermodynamic way view of bifurcations, a
phenomenon usually approached in the realm of pure, zero temperature,
mechanics.

By establishing TI up to second order, I will be able to answer a very old and
important scientific question, which I term the entropic coupling problem.
Since the establishment of the laws of thermodynamics, it has been noted that
many systems in nature are highly ordered and seem to break the second law of
thermodynamics. The often given explanation is simply that even though entropy
might decrease in a given region, somewhere else the entropy must increase by
at least that much. This is surely the case but a rigorous theory for this has
proved elusive, until now. In fact, to my knowledge, no theoretical work on
this problem exists, beyond the qualitative explanation just given.

There must exist some other system that increases its entropy by at least as
much, and this must be true at all times. This means that the entropy
production of this system must always exceed any negative rate that may occur
in the given region. Here I explain both what this other system is, as well as
the details of the coupling. The approach I outline here for solving the
entropic coupling problem is novel. The coupling is not energetic in the same
way mechanical systems are often coupled by adding a term to a Hamiltonian
that depends on the variables of both systems. Instead, the coupling between
variables occurs through the conductance.

Much of the considerations here are under circumstances I refer to as well
away from equilibrium. What is meant by well away is far enough from
equilibrium where kinetic and transport coefficients will have significant
deviations from constancy, but not so far that destructive or catastrophic
events occur, i.e., the system can be repeatedly cycled well away from
equilibrium and back again. This variation of kinetic coefficients is not
merely for convenience of definition, but plays a critical role in my
analysis. As it turns out, this is a modest step beyond the basic approach of
near-equilibrium thermodynamics, such as used for calculation of transport
properties. Here, one is not dealing with far-from-equilibrium physics where
the concepts of equilibrium statistical mechanics break down. In particular,
the local temperature, pressure, and chemical potential are still well-defined
thermodynamic parameters.

By combining TI results at both first and second order, I construct a
thermodynamic potential that is maximized when the gate is well away from
equilibrium and finds itself in a stationary state. Finding such a potential
has remained an open question since the laws of thermodynamics were
established. Maximizing the entropy is not helpful because this is known to
happen at equilibrium. This new potential differs from the entropy in general
but does become the entropy when the system is returned to equilibrium.
Maximizing this potential will, under certain circumstances, produce a PT,
which may or may not spontaneously create interesting patterns.

The time is right now to use this new potential towards the establishment of
governing principles for nonequilibrium thermodynamics. An abundant amount of
data has been taken from observations on a widely varying set of
nonequilibrium systems over a period of many decades now. These systems have
been described in a lengthy review article Cross and Hohenberg (1993) as well
as textbooks including Refs. Cross and Greenside (2009); Desai and Kapral
(2009) as good examples. Moreover, a great deal of modeling has been reported
on these results and much understanding has been gained from this. In this
work, I make a concerted effort to link general TI results to this modeling.

I begin, in Sec. II, by developing a theory for thermodynamic induction up to
second order for one gate, or control, variable. This includes showing that
TI2 produces a nonequilibrium PT with a well-defined order parameter. In Sec.
III this theory is extended to the case of more than one gate variable. This
includes the description of a nonequilibrium front and pattern formation
during chemical reactions. Review and comparison is made to various important
models in the literature.

##  II General Theory for Thermodynamic Induction up to Second Order

Considered here is the coupled dynamics of two thermodynamic variables,
referred to as the dynamical reservoir (DR) and the gate. After some initial
considerations with both variables on an equal footing, emphasis will then be
placed on the gate variable. The gate is the system capable of displaying
interesting behaviour such as pattern formation and self-organization. The DR
is simply a thermodynamic variable with a large capacity so that when not in
equilibrium, the relaxation is slow. The relaxation of the DR is always
considered as slowly varying compared to all other time scales. In fact the DR
may be held static in many systems, for example by continuously
replacing/feeding in reactants into a reactor.

The DR thermodynamic variable $x_{DR}$, normally considered as slowly
approaching an equilibrium value of $x_{DR0}\equiv x_{DR}-a$, has a conjugate
force $X_{DR}=-g_{DR}a$, where $g_{DR}^{-1}$ can be thought of as a
generalized capacitance and would be large for this type of reservoir
variable. The dynamics for approaching equilibrium is described by

$\frac{da}{dt}=\dot{a}=M_{DR}X_{DR}\,.$ (1)

So far the analysis closely follows standard textbook material for
nonequilibrium dynamics Reif (1965); de Groot and Mazur (1984). Ordinarily,
the Onsager coefficient $M_{DR}$ is considered as constant and describes
strict proportionality between the flux and force, ex. Fourier’s heat transfer
law. The key idea behind TI is that $M_{DR}$ is not constant, and may depend
on thermodynamic variables other than $x_{DR}$. (A dependence on $x_{DR}$
creates nonlinear dynamics but fails to create the interesting coupling.) The
coefficient $M_{DR}$ is assumed to depend on these other thermodynamic (gate)
variables so that $M_{DR}$ can be broken into a sum of a constant term
$L_{DR}$ and a variable component $W_{DR}$ which has a functional dependence
on the gate variables. For simplicity, I first consider the case where the
dependence of $M_{DR}$ on these variables is very weak and negligible, except
for one gate variable, $x_{GT}$. The term $W_{DR}$ couples $x_{DR}$ and
$x_{GT}$ and plays a role similar to the perturbation potential in the
Hamiltonian description of mechanical systems. The coupling does not occur
through a Hamiltonian but instead through entropy production rates. Both the
DR and gate entropy production rates play an important role here. The entropy
production rate for the DR is

$\sigma_{DR}=X_{DR}\dot{x}_{DR}=L_{DR}X_{DR}^{2}+W_{DR}X_{DR}^{2}\,.$ (2)

For the gate I define the difference variable $b\equiv x_{GT}-x_{GT0}$, so
that the conjugate force is $X_{GT}=-g_{GT}b$ and the change in entropy from
equilibrium for the gate variable is

$S_{GT}-S_{GT,eq}=\frac{1}{2}X_{GT}b=-\frac{1}{2}g_{GT}b^{2}\,.$ (3)

The $g_{GT}$ parameter sets the extent of fluctuations with $\langle
b^{2}\rangle_{0}=k_{B}/g_{GT}$ (the brackets $\langle\rangle_{0}$ denoting
equilibrium ensemble averaging). Since the gate is often thought of a small
system, fluctuations play an important role. The entropy production rate for
the gate which also plays an important role in the formulation of variational
principles is given by

$\sigma_{GT}=L_{GT}X_{GT}^{2}=L_{GT}g_{GT}^{2}b^{2}\,.$ (4)

I will show below that important potentials may be formed as linear
combinations of $\sigma_{DR}$ and $\sigma_{GT}$.

###  II.1 TI1

In previous work I considered the case where $W_{DR}$ depends in a linear
fashion on one or more gate, or control, variables Patitsas (2014, 2015,
2016). This resulted in a type of TI that is classified as first order, i.e.,
TI1. If the TI1 gate variable is $b$ then $M_{DR,1}=L_{DR}+\gamma b$, with
$\gamma$ as the TI1 coefficient, and the induction effect gives dynamics for
the gate variable that is not merely dissipative but is instead given by:

$\dot{b}=\gamma
X_{DR}^{2}L_{GT}\tau^{*}-L_{GT}g_{GT}b~{}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(1st
order, uncorrected)}$ (5)

where $\tau^{*}$ is the characteristic time for the fastest fluctuations in
the gate variable. For TI1 the induction term in Eq. (5) is constant and this
constant term pushes $b$ away from its equilibrium value of zero.

Before proceeding to TI2, I point out that essentially the same result, Eq.
(5), can be found by considering the two variables, $a$ and $b$ as random
walkers. As pointed by Wigner in his analysis of various proofs for the
(linear) Onsager relations, a random walker where the result of each step is
weighted by $\exp(\Delta S/k_{B})$ will give a mean value that relaxes
properly towards equilibrium Wigner (1954). For the two variables considered
here, $\Delta S=-g_{DR}a^{2}-g_{GT}b^{2}$. I introduce an interesting coupling
by making the step length for variable $a$ depend on $b$. In Appendix A, the
two random walker problem is analyzed and for linear dependence the analytic
results for the mean values for $a$ and $b$ are given by

$\dot{a}=-\frac{q_{1}}{\tau_{0}}\left(l_{0}^{2}+\delta
b\right)a\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(random walk, 1st order)}$ (6)

and

$\dot{b}=\frac{q_{1}\delta}{2\tau_{0}}l_{2}^{2}\left(q_{1}a^{2}-1\right)-\frac{q_{2}}{\tau_{0}}l_{2}^{2}b\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(random
walk, 1st order)}$ (7)

To compare this result with Eq. (1) and Eq. (5) one notes that
$q_{1}=g_{DR}/k_{B}$, $q_{2}=g_{GT}/k_{B}$, $X_{DR}=-g_{DR}a$,
$X_{GT}=-g_{GT}b$, and relaxation times,
$\tau_{DR}=1/L_{DR}g_{DR}=\tau_{0}/q_{1}l_{0}^{2}$,
$\tau_{GT}=1/L_{GT}g_{GT}=\tau_{0}/q_{2}l_{2}^{2}$. This means
$L_{DR}=l_{0}^{2}/k_{B}\tau_{0}$ and $L_{GT}=l_{2}^{2}/k_{B}\tau_{0}$. The
Onsager coefficients $L_{DR}$ and $L_{GT}$ are simply the random walk
diffusion coefficients divided by $k_{B}$. Equation (6) is the same as Eq. (1)
as long as $\delta=\gamma k_{B}\tau_{0}$. With this substitution Eq. (7)
becomes

$\dot{b}=\frac{\gamma}{2}L_{GT}\tau_{0}\left(X_{DR}^{2}-k_{B}g_{DR}\right)-L_{GT}g_{GT}b\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(random
walk, 1st order)}$ (8)

Thus, it is natural to identify the random walk time step as twice $\tau^{*}$.
This gives

$\dot{b}=\gamma L_{GT}\tau^{*}\left(X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}\right)-L_{GT}g_{GT}b~{}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(1st
order, corrected)}$ (9)

which closely resembles Eq. (5) except for the extra term $\langle
X_{DR}^{2}\rangle_{0}=k_{B}g_{DR}$ which represents the variance of the DR
force under equilibrium conditions. The extra term guarantees that the mean
value of $\dot{b}$ is zero in equilibrium, as it should be. Evidently the
random walk analysis is more accurate than the derivation of Eq. (5) presented
in Ref. Patitsas (2014). When the DR is large then fluctuations play less of a
role and the correction term is small. The assumption made in Ref. Patitsas
(2014) is that the DR is very large and slow, so it makes sense that terms
like $\langle X_{DR}^{2}\rangle_{0}$ are missed in this analysis. In contrast,
Eqs. (6) and (7) hold regardless of how large and slow the DR is compared to
the gate.

Figure 1: Random walk simulations for 1st order TI. A single random walk is
shown with 1st order TI parameter $\delta=0.04$, showing the DR (GT) variable
$a$, solid red circles ($b$, green crosses) initially at 1.0 (0.0). An average
for $b$ over 10000 walks is displayed as the thin solid black curve. Also
shown is the relaxation of $a$, thick solid blue curve, when $\delta=0$, $a$
and $b$ uncoupled. Both parameters $q_{1}$ and $q_{2}$ were set to 1000.

Figure 1 shows the result of a single, representative, random walk (solid red
circles for $a$, green crosses for $b$) which clearly shows variable $b$ being
pushed away from zero. In the numerical simulation, which uses very simple
code, it is the square of the DR step length that has the form
$l_{0}^{2}+\delta b$. In the simulation $l_{0}=0.01$, and $\delta=0.04$. The
thin, black, solid curve shows $b$ after being averaged over 10000 walks.
After about 20 timesteps $a$ relaxes to zero so that the induction on $b$ is
greatly reduced, Afterwards, $b$ relaxes towards equilibrium. The thick,
solid, blue curve shows the averaged response of $a$ when $\delta=0$, (and
hence $b=0$) i.e., with the coupling between $a$ and $b$ removed (no
induction). The approach to equilibrium for $a$ is substantially faster with
$\delta=0.04$ than it is for the case where $\delta$ is set to zero.

Though the essence of the results displayed in Fig. 1 was already established
in previous work it is reassuring to see a different approach based on random
walk simulations confirm the expected TI1 predictions. This confirmation will
also be displayed for TI2. Before moving to TI2, I briefly discuss a potential
that is maximized under TI1.

####  II.1.1 TI1 Principle of Maximum Entropy Production

Since a correction for DR fluctuations has been added to the dynamics, Eq.
(9), for TI1, as compared to what was derived in Ref. Patitsas (2014), a
slightly updated form of the Principle of Maximum Entropy Production (PMEP) is
required here. Towards this end I define the following thermodynamic potential
function, a type of entropy production rate:

$\Phi_{1}\equiv\sigma_{DR}-\lambda_{1}\sigma_{GT}=L_{DR}X_{DR}^{2}+X_{DR}^{2}\gamma
b-\lambda_{1}\left[-g_{GT}\gamma L_{GT}\tau^{*}\left(X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}\right)b+L_{GT}g_{GT}^{2}b^{2}\right]~{}\,,$ (10)

where the constant $\lambda_{1}$ is a Lagrange multiplier. One maximizes the
DR rate of entropy production, $\sigma_{DR}$, with respect to $b$, subject to
the stationary state constraint. Equivalently, one also maximizes
$\sigma_{T}$, subject to the same stationary state constraint. Setting the
first derivative of $\Phi_{1}$ to zero gives a way to specify $\lambda_{1}$ as

$\lambda_{1}=\frac{\tau_{GT}}{\tau^{*}}\frac{X_{DR}^{2}}{X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}}~{}\,,\,\,\,\,\,$ (11)

which is positive definite. Explicitly, $\partial\Phi_{1}/\partial
b=2\lambda_{1}g_{GT}\dot{b}$. At the stationary state
$\partial^{2}\Phi_{1}/\partial
b^{2}=-\frac{2g_{GT}}{\tau^{*}}\frac{X_{DR}^{2}}{X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}}<0$, verifying that $\Phi_{1}$ is maximized.

###  II.2 TI2

In many systems, symmetry considerations preclude the linear form of $M_{DR}$
discussed above, and the quadratic term becomes the leading term with
$W_{DR}=\alpha b^{2}$:

$M_{DR}=L_{DR}+\alpha b^{2}\,,$ (12)

where again $L_{DR}$ is constant and $\alpha$ is the coefficient for TI2. In
these systems, one might expect spontaneous symmetry breaking to occur. In
TI1, the gate variable is pushed away from equilibrium in a direction
determined by the sign of $\gamma$. In TI2, the gate can potentially veer away
from equilibrium in either direction, so a bifurcation may occur. This would
not occur when the DR is at or very near equilibrium so one might expect a
threshold needs to be exceeded before any bifurcation occurs.

In Fig. 2, I present random walk results for the case where the DR step
length, $l$, for variable, $a$, depends quadratically on $b$ as
$l^{2}=l_{0}^{2}+\beta b^{2}$. Three individual random walks for $b$ are shown
depicting no bifurcation (black crosses, trace 1) when the DR is at
equilibrium, and two representative walks (traces 2, 3) showing bifurcation
when the DR is pushed well away from equilibrium. As in the case of TI1, the
induction has significant impact on the gate variable. Without induction the
simulations always resemble trace 1. If induction is suddenly introduced at
$t=0$ then after executing a random walk for some time before $t=0$ which
resembles trace 1, at $t=0$ the random walk resembles one of traces 2 and 3,
and after compiling many runs, one obtains the well-known pitchfork
bifurcation. In the second order case the variable $b$ averages to zero over
many random walks but as one would expect, $b^{2}$ does show the bifurcation.

The analytic solution for the random walks is given in Appendix A as

$\dot{a}=-\frac{q_{1}}{\tau_{0}}\left(l_{0}^{2}+\beta
b^{2}\right)a\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(random walk, 2nd order)}$
(13)

and

$\dot{b}=\frac{q_{1}\beta}{\tau_{0}}l_{2}^{2}\left(q_{1}a^{2}-1\right)b-\frac{q_{2}}{\tau_{0}}l_{2}^{2}b\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(random
walk, 2nd order)}$ (14)

Equation (14), with the first term on the right-hand side having a positive
coefficient of $b$, shows a simple (pitchfork) bifurcation, confirming the
numerical random walk results. Making a similar mapping to the thermodynamic
problem as done above with the 1st order, one obtains:

$\dot{a}=(L_{DR}+\alpha
b^{2})X_{DR}\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(2nd order TI)}$ (15)

and

$\dot{b}=2\alpha L_{GT}\tau^{*}\left(X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}\right)b-L_{GT}g_{GT}b~{}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(2nd
order TI, unconstrained)}$ (16)

Equation (16) constitutes an important result of this paper as it shows how
unstable dynamics may occur in a purely dissipative system. The same result,
though missing the $\langle X_{DR}^{2}\rangle_{0}$ term, is derived in a
different way in Appendix B, using the same approach used in Ref. Patitsas
(2014) to establish TI1.

When the force $X_{DR}$ is small, the gate variables continue to relax to
equilibrium with the only effect of TI being that the relaxation time is
lengthened. However when the DR is pushed harder, a critical value for
$X_{DR}$ can be reached which results in a bifurcation and unstable growth of
the gate variable:

$X_{c}=\sqrt{\frac{g_{GT}}{2\alpha\tau^{*}}+\langle
X_{DR}^{2}\rangle_{0}}~{}\,.$ (17)

For $X_{DR}$ fixed and above critical, Eq. (16) predicts unabated growth in
time of the form $b(t)=b(0)\exp(t/\tau)$ with

$\tau^{-1}=2\alpha L_{GT}\tau^{*}(X_{DR}^{2}-X_{c}^{2})~{}\,.$ (18)

The second order result expressed in Eq. (16) differs significantly from TI1
which adds a constant positive term to Eq. (16) Patitsas (2014). There is no
sudden transition (bifurcation) in first order TI and also no exponential
growth; the gate variable always relaxes to a nonequilibrium stationary state.

Figure 2: Random walk simulations for TI2. Three individual random walks for
the gate variable $b$ are displayed, one with $X_{DR,0}=0$ (black crosses, 1),
and two with $X_{DR,0}>X_{c}$ (red circles, 2, and blue squares, 3). For the
random walk with $X_{DR,0}=0$, the DR $a$ variable (not shown) initial value
was 0.0. The two walks with $X_{DR,0}>X_{c}$ are representative of
bifurcations going up and down. For these two random walks, the $a$ variable
initial value was 1.0. Both parameters $q_{1}$ and $q_{2}$ were set to 1000
and $\beta=0.1$. Note the pitchfork shape would be evident over the range from
-20 to +20 timesteps.

The form of Eq. (16) shows that TI can produce what is effectively a negative
relaxation time. In some systems this could happen in the form of a negative
(effective) diffusion coefficient as in the physical examples discussed below
in Sec. III.4 and Sec. III.5.

Introducing terms leading to negative relaxation times is the basic approach
in formulating the Swift-Hohenberg (SH) model Cross and Hohenberg (1993);
Cross and Greenside (2009). This mathematical model is quite sophisticated
with extra terms added to constrain unstable solutions. Following this
approach, I add a cubic term to Eq. (16) to produce:

$\dot{b}=2\alpha L_{GT}\tau^{*}\left(X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}\right)b-L_{GT}g_{GT}b-c^{2}b^{3}~{}\,.\,\,\,\,\,\,\,\,\,\,\,\text{(2nd
order, constrained)}$ (19)

The cubic term constrains what would otherwise be unabated exponential growth.
The physics behind the added term would have to be justified on a case-by-case
basis and there may be more than one effect contributing to this cut-off.

For example, in the realm of purely dissipative systems, treating the small
$b$ expansion for $\Delta S$ to higher order will produce terms that will
limit this growth. Replacing $\Delta S=-g_{GT}b^{2}/2$ with

$\Delta S=-\frac{1}{2}g_{2}b^{2}-\frac{1}{2}g_{4}b^{4}~{}\,,$ (20)

is equivalent to replacing $g_{GT}$ with $g_{2}+g_{4}b^{2}$ with the result of
producing Eq. (19) as long as $c^{2}=g_{4}L_{GT}$. This analysis is only
preliminary and a more thorough treatment should be developed. Nevertheless
this provides a simple mechanism for limiting the unstable growth. This
connection between the entropy and the cut-off term will be invoked below in
Sec. II.7.

The new term in Eq. (19), cubic in $b$, modifies the dynamics in a way that
checks the exponential growth and allows for stationary states ($\dot{b}=0$)
above critical:

$b_{ac}=\sqrt{\frac{2\alpha
L_{GT}\tau^{*}(X_{DR}^{2}-X_{c}^{2})}{c^{2}}}~{}\,.\,\,\,\,\,\,\,\,\text{(above
critical)}$ (21)

Strictly speaking, this state is quasistationary and is only meaningful when
the relaxation time of the gate is much shorter than that of the DR. In this
case the $X_{DR}$ is slowly changing and $b_{ac}$ adjusts according to Eq.
(21). The stationary state value of $b$ can be interpreted as an order
parameter, which spontaneously springs up from zero at the critical point. The
critical exponent of this nonequilibrium PT is defined by the dependence on
$X_{DR}$ just above the critical point, and here takes the value 0.5.

Equation (19) does not account for fluctuations in $b$ and these will be dealt
with below. The results, Eqs. (19) and (21) are still useful in a certain
limit which I refer to as the infinite $Q$ limit. In this limit, induction
does nothing when $X_{DR}<X_{c}$, and has a perfectly sharp transition at
$X_{c}$. This will change in the finite $Q$ case.

###  II.3 TI2 PMEP

In this section a principle of maximum entropy production (PMEP) will be
established and will complement the same principle which holds for TI1. For
TI2 the PMEP will closely resemble the Ginzberg-Landau-Wilson free energy
functional Desai and Kapral (2009) which has been successful in modeling
equilibrium PTs. Establishing a similar thermodynamic potential should aid in
understanding nonequilibrium PTs.

####  II.3.1 TI2 PMEP - Infinite $Q$

In the case of TI2, the starting point for a PMEP potential is

$\Phi_{2,\infty
Q}(b)\equiv\sigma_{DR}-\lambda\sigma_{GT}=L_{DR}X_{DR}^{2}+X_{DR}^{2}\alpha
b^{2}-\lambda\left(-2\alpha\left(X_{DR}^{2}-X_{c}^{2}\right)L_{GT}g_{GT}\tau^{*}b^{2}+c^{2}g_{GT}b^{4}\right)~{}\,,$
(22)

where $\lambda$ above critical is found to be

$\lambda_{2,\infty
Q}=\frac{\tau_{GT}}{2\tau^{*}}\frac{X_{DR}^{2}}{X_{DR}^{2}-X_{c}^{2}}~{}\,.\,\,\,\,\,\text{(above
critical)}$ (23)

Above critical, at the stationary state, $\frac{d^{2}\Phi_{2,\infty
Q}}{db^{2}}=-8\alpha X_{DR}^{2}$ which is always negative and shows that
stationary states maximize $\Phi_{2,\infty Q}$. As will be shown below, Eq.
(22) is only valid above critical in the limit of infinite quality factor.

Above critical, $\Phi_{2,\infty Q}(b)$ resembles an upside down sombrero with
two maxima at nonzero $b=\pm b_{ac}$ which in turn closely resembles the free
energy functional found in Landau theory Plischke and Bergerson (1986). Far
enough above critical, transitions between $+b_{ac}$ and $-b_{ac}$ become rare
and the system essentially freezes into one branch, i.e., a proper bifurcation
in the sense of nonlinear dynamics theory. As one varies $X_{DR}$ and
approaches the critical point from above, the two maxima soften and coalesce
into one maximum at $b=0$.

####  II.3.2 TI2 PMEP - Finite $Q$

The form of $\lambda_{2,\infty Q}$ is similar to that of $\lambda_{1}$, with
$\langle X_{DR}^{2}\rangle_{0}$ replaced by the larger $X_{c}^{2}$. This means
that the possibility of $\langle X_{DR}^{2}\rangle_{0}$ less than $X_{c}^{2}$
should be considered. Below critical, $\lambda$ is indeterminate and could be
set to zero. From the mathematics alone it seems reasonable to consider the
function that is the magnitude of $X_{1}^{2}/(X_{1}^{2}-X_{c}^{2}+i\epsilon)$
as a function that could cover cases both above and below critical. This can
be justified by analysis that takes fluctuations in $b$ into better account.
From this analysis comes a lineshape function to replace $\lambda$ with
$\lambda_{2}(X_{DR})=(\tau_{GT}/2\tau^{*})\mu$, with $\mu$ given by:

$\mu=\frac{X_{DR}^{2}}{\sqrt{(X_{DR}^{2}-X_{c}^{2})^{2}+X_{DR}^{2}X_{c}^{2}/Q^{2}}}~{}\,,\,\,\,\,\,\text{(lineshape)}$
(24)

where the quality factor is specified by $Q^{-2}=4c^{2}\tau_{GT}\langle
b^{2}\rangle_{0}$. The lineshape function $\mu$ is identical to that which
comes about from the frequency response of a damped harmonic oscillator. The
familiarity of this lineshape function helps to make the thermodynamics of
bifurcation quite intuitive. Loosely speaking, the thermodynamic force
$X_{DR}$ plays a role similar to frequency or energy of excitation, and
$X_{c}$ is like the resonant frequency. Fluctuations broaden the excitation
peak and produce the interesting consequence of the system bifurcating (in a
statistical manner) to some extent, below the critical point. Unlike the case
of pure classical mechanics at zero temperature where bifurcation occurs with
perfect sharpness, here the bifurcation is broadened over a range $\approx
X_{c}/Q$. This constitutes an important result in the thermodynamics of
bifurcations.

The lineshape function reaches below critical, suggesting that Eq. (22) might
be applicable below critical if $\lambda$ is replaced by $\lambda_{2}$. If so,
then dynamics below critical would be specified by making $\dot{b}$
proportional to $\partial\Phi_{2}/\partial b$. However, further analysis
requires care in treating fluctuations and in distinguishing $b^{2}$ from
$\langle b^{2}\rangle$. In Eq. (22) it is the dispersion of $b$,
$u\equiv\Delta b^{2}=\langle b^{2}\rangle-\langle b^{2}\rangle_{0}$, that is
the argument and not the square of the mean, $\langle b^{2}\rangle$. In
equilibrium it is $u$ that is zero, not $\langle b^{2}\rangle$.

Since focusing on fluctuations in the gate is the more important issue for
TI2, I’ll assume from now on that fluctuations in the DR are negligible. This
is justified by the DR being physically much larger than the gate. Multiplying
Eq. (19) through by $b$ gives

$\frac{db^{2}}{dt}=2b\dot{b}=4\alpha
L_{GT}\tau^{*}\left(X_{DR}^{2}-X_{c}^{2}\right)b^{2}-2c^{2}b^{4}~{}\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{(uncorrected)}$
(25)

Taking into account fluctuations in $b$ means that a term $\langle
b^{2}\rangle_{0}$ must be added into the dynamics for $b^{2}$. This ensures
that $b^{2}$ takes the limit $\langle b^{2}\rangle_{0})$ well below critical.
Equation (25) gets modified to become

$\frac{db^{2}}{dt}=4\alpha
L_{GT}\tau^{*}\left(X_{DR}^{2}-X_{c}^{2}\right)b^{2}+4\alpha
L_{GT}\tau^{*}X_{c}^{2}\langle b^{2}\rangle_{0}-2c^{2}b^{4}~{}\,.$ (26)

The dynamics for the difference $u=b^{2}-\langle b^{2}\rangle_{0}$ requires a
small adaptation for the quartic cutoff term:

$\frac{du}{dt}=4\alpha
L_{GT}\tau^{*}\left(X_{DR}^{2}-X_{c}^{2}\right)u+4\alpha
L_{GT}\tau^{*}X_{DR}^{2}\langle b^{2}\rangle_{0}-2c^{2}u^{2}~{}\,.$ (27)

In terms of $z\equiv X_{DR}/X_{c}$ the stationary state for the dispersion of
$b$ is

$\left(\Delta b^{2}\right)_{ss}=2Q^{2}\langle
b^{2}\rangle_{0}\left[z^{2}-1+\sqrt{(z^{2}-1)^{2}+z^{2}/Q^{2}}\right]\,.$ (28)

In terms of the function $h_{+}$ defined by $h_{+}(x)\equiv
x-1+\sqrt{(x-1)^{2}+x/Q^{2}}$, $\left(\Delta b^{2}\right)_{ss}=2Q^{2}\langle
b^{2}\rangle_{0}h_{+}(z^{2})$. For positive argument $h_{+}$ is always
positive and resembles a hockey-stick with a sharp upwards bend at $x=1$. The
function $h_{+}(z^{2})$ differs very little from the infinite $Q$ limit except
within the interval centered at $z^{2}=1$ and with a width of several
$Q^{-1}$. For small $x$, $h_{+}(x)\approx x/2Q^{2}$, while for $x\gg 1$,
$h_{+}(x)\approx 2x$. Far above critical, $\left(\Delta
b^{2}\right)_{ss}=b_{ac}^{2}$, showing that this approach connects seamlessly
with Eqs. (22) and (23) when well above critical. This also is consistent with
classical chaos theory (where $Q$ is infinite) and one expects the classical
particle to strictly go to either $+b_{ac}$ or $-b_{ac}$ after bifurcation.
Near critical the level of fluctuation is substantially higher than it is in
equilibrium. This enhancement motivates defining the term, nonequilibrium
fluctuations, as an interpretation of Eq. (28).

To describe conditions both above and below critical for TI2 one must use a
potential $\Phi_{2}$ that is a function of $u$ since the mean value of $b$ is
technically zero. In this case the Lagrange multiplier is
$\lambda_{2}=\tau_{GT}\mu/2\tau^{*}$, and the correct form for this potential
is

$\Phi_{2}(u)=L_{DR}X_{DR}^{2}+\frac{g_{GT}\mu}{2\tau^{*}}\left[z^{2}\langle
b^{2}\rangle_{0}+uh_{+}(z^{2})-\frac{u^{2}}{4Q^{2}\langle
b^{2}\rangle_{0}}\right]~{}\,.$ (29)

For all $u\geq 0$, $\Phi_{2}$ is maximized, with respect to $u$, at the
stationary state Eq. (28). As in the first order case, one maximizes
$\sigma_{DR}$ (or equivalently $\sigma_{T}$) subject to the stationary state
constraint.

The second order PMEP nicely complements the first order PMEP first discussed
in Ref. Patitsas (2014). The second order result is quite a bit more involved;
Attempts to use the same variable for the arguments of both $\Phi_{1}$ and
$\Phi_{2}$ are not successful. In the end, it is the statistical moments of
the statistical variable $b$ that are used as the arguments, i.e., $\langle
b\rangle$ for $\Phi$, and $\langle u\rangle=\langle b^{2}\rangle-\langle
b^{2}\rangle_{0}$ for $\Phi_{2}$. The two moments are generated from a
probability distribution. It is helpful to think then of a Gaussian
distribution for $b$. The distribution is centered at $\langle b\rangle$, and
is shifted by TI1. For the case of TI2, with $\langle b\rangle=0$, the
variance of the distribution is given by $\langle b^{2}\rangle$. Again for
TI2, as one raises $X_{DR}$ upwards through the critical point, the Gaussian
distribution spreads suddenly to describe the initiation of bifurcation.

For convenience, the symbols $\langle\rangle$ for ensemble averaging will be
dropped in the discussion below, and it is understood that $b$ and $u$
actually represent $\langle b\rangle$ and $\langle u\rangle$.

In principle, third and higher orders of induction would be dealt with in a
similar way as TI1 and TI2. There may be systems with both $\alpha=0$ and
$\gamma=0$ which would require treatments at higher order. These systems are
expected to be rare, however, and given the success demonstrated in Sec. III
below in using TI2 on real systems, leaving higher order TI for future work is
justified for now.

###  II.4 Generalized Entropy with Focus on the Gate

At this point, I move from a principle that maximizes the entropy production
of the entire system to a principle that focuses on the gate. As I will show,
the DR still plays an important role, but only in supplying parameters for the
theory focused on describing the gate. I will treat first and second order
cases separately before combining them. The procedure merely amounts to
multiplying $\Phi_{1}$ and $\Phi_{2}$ by constants, thus preserving their
maximization properties. These multiplicative constants will be chosen in such
a way as to produce the entropy when the DR is restored to equilibrium.

####  II.4.1 First Order

Multiplying $\Phi_{1}$ by a term independent of $b$ also gives a potential
that is maximized when the gate is stationary. I define a new thermodynamic
potential as

$\Psi_{1}\equiv\left(\frac{\tau^{*}}{2}\right)\left(\frac{X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}}{X_{DR}^{2}}\right)\Phi_{1}=\left(\frac{\tau^{*}}{2}\right)\left(\frac{X_{DR}^{2}-\langle
X_{DR}^{2}\rangle_{0}}{X_{DR}^{2}}\right)\sigma_{DR}-\frac{\tau_{GT}}{2}\sigma_{GT}~{}\,.$
(30)

Though $\Psi_{1}$ has the same units as entropy, it is not the same as the
total system entropy. It is a nonequilibrium potential since it depends on
$X_{DR}$. In the limit where $X_{DR}^{2}$ is small, and approaches $\langle
X_{DR}^{2}\rangle_{0}$, $\Psi_{1}$ takes the limit of
$-\frac{1}{2}\tau_{GT}\sigma_{GT}=-\frac{1}{2}g_{GT}b^{2}$ which is equal to
the difference $\Delta S_{GT}=S_{GT}-S_{GT,eq}$ when this difference is taken
up to second order in $b$. At, or near, $X_{DR}=0$, the guiding principle of
maximizing $\Psi_{1}$ for determining $b$, is to maximize $S_{GT}$, which of
course is the governing thermodynamic principle for the gate at, or near,
equilibrium. Thus, maximizing $\Psi_{1}$ has the attractive feature of
determining the (stationary) state of the gate when the DR is driven well away
from equilibrium and also seamlessly determining the equilibrium state of the
gate when the DR is at equilibrium.

The potential function $\Psi_{1}$ is an excellent tool for determining
nonequilibrium thermodynamics specifically for the gate. The DR is effectively
separated away from the gate, much like how in equilibrium thermodynamics, the
thermal reservoir (heat bath) is mathematically separated away from the system
of interest to produce the Helmholtz free energy of that system. With the
traditional heat bath the only process that matters for the system of interest
is the transfer of thermal energy with the bath. Transferring thermal energy
will increase the entropy of one of either that system or the bath, at the
expense of the other; There is then one energy value that maximizes the total
entropy. The rate at which the bath increases entropy with energy is described
in a simple manner by one parameter, $\beta=1/k_{B}T$.

Here, at a fixed $X_{DR}$, adjusting $b$ will increase one of $\sigma_{DR}$ or
$\sigma_{GT}$ at the expense of the other (while the gate is stationary). One
particular value $b_{ss}$ will maximize $\Psi_{1}$. From Eq. (30), $\Psi_{1}$
may be rewritten as:

$\Psi_{1}(b)=\Psi_{1,0}+k_{B}\beta_{1}b-\frac{1}{2}g_{GT}b^{2}~{}\,,$ (31)

where $\Psi_{1,0}=\frac{1}{2}L_{DR}X_{DR}^{2}\tau^{*}$, $\beta_{1}=\gamma
X_{DR}^{2}\tau^{*}/k_{B}$, and with fluctuations in $X_{DR}$ ignored. The rate
at which the DR increases entropy production with $b$ is described by one
parameter, $\beta_{1}$. Theoretically, $\beta_{1}$ for the DR is analogous to
$\beta$ for the heat bath, or the chemical potential $\mu$ for a particle
bath. Since $\beta_{1}$ vanishes when $X_{DR}=0$ it is not surprising that
this parameter was not discovered during investigations of equilibrium
thermodynamics. Because of the similarities with how the Helmholtz free energy
is formulated to account for interactions of a system with a heat bath, the
term free entropy is apt for $\Psi_{1}$. One thinks then of some amount of
entropy as being available in some sense; In particular the gate (dynamically)
lends entropy to the dynamical reservoir.

####  II.4.2 Second Order

In terms of $\Delta b^{2}$, the thermodynamic potential
$\Psi_{2}\equiv\frac{\tau^{*}}{\mu}\Phi_{2}$ becomes:

$\Psi_{2}(u)=\Psi_{2,0}+k_{B}\beta_{2}u-\frac{g_{GT}}{8Q^{2}\langle
b^{2}\rangle_{0}}u^{2}~{}\,,$ (32)

where $\Psi_{2,0}=L_{DR}X_{DR}^{2}\tau^{*}/\mu+\frac{1}{2}g_{GT}z^{2}\langle
b^{2}\rangle_{0}$, and $\beta_{2}=g_{GT}h_{+}/2k_{B}$. This free entropy
function is maximized both above and below critical when $u=u_{ss}$, and the
second derivative at $u=u_{ss}$ is always negative. Near the maxima, well
above critical, the dependence for $\Psi_{2}$ near $b=\pm b_{ss}$ goes as
$-g_{GT}z^{2}(b\pm b_{ss})^{2}$ which is on the same order as the dependence
of $S_{GT}-S_{GT,ss}=-\frac{1}{2}g_{GT}b^{2}$ at equilibrium. Well below
critical, terms in $\Psi_{2}$ that go as $u$ disappear, so the potential
flattens out, and the induction essentially disappears. Also,

$\lim_{X_{DR}\to 0}\Psi_{2}(u)=\Psi_{0}-\frac{1}{2}g_{4}u^{2}~{}\,,$ (33)

where $\Psi_{0}=L_{DR}X_{c}^{2}\tau^{*}$.

Given the definition of the second order variable $u$, with inspection of Eq.
(20), the best expression for describing small changes from equilibrium of the
gate entropy, up to second order, is:

$\Delta
S_{GT}=S_{GT}-S_{GT,eq}=-\frac{1}{2}g_{2}b^{2}-\frac{1}{2}g_{4}u^{2}~{}\,.$
(34)

Equation (34) allows for the connection of the first and second order
potentials, simply by adding $\Psi_{1}$ and $\Psi_{2}$. This is consistent
with the $z=0$ limit:

$\lim_{X_{DR}\to
0}\left(\Psi_{1}(b)+\Psi_{2}(u)\right)=\Psi_{0}+S_{GT}-S_{GT,eq}~{}\,.$ (35)

####  II.4.3 Combining up to Second Order

Establishment of both $\Psi_{1}$ and $\Psi_{2}$ places the PMEP on a solid
footing for a great many physical systems; For those thermodynamic systems
where the gate symmetry is already broken in equilibrium, one would use and
maximize $\Psi_{1}$, and for systems where symmetry in thermodynamic
equilibrium is not broken, $\Psi_{2}$ is the potential chosen to be maximized.
If one forms a potential function $\Psi\equiv\Psi_{1}(\langle
b\rangle)+\Psi_{2}(\Delta b^{2})$ as the sum of first and second order
potentials then one must consider the free entropy $\Psi$ as a function of two
variables, the mean, and the dispersion of $b$ and maximize with respect to
each variable separately. In practice, one may very well use only one of the
$\Psi_{1}$ and $\Psi_{2}$ potentials in a given problem. This usage would be
determined by symmetry considerations.

Since $\Psi$ is a linear combination of $\Phi_{1}$ and $\Phi_{2}$, it
satisfies the stationary state PMEP with regards to both $b$ and $u$. This
alone is a key result and allows for the formulation of variational principles
when order parameter fields are discussed below. The specific linear
combination that I have chosen will connect nicely to the traditional entropy
when the DR is in equilibrium. Equation (35) shows that, apart from the
constant term $\Psi_{0}$, $\Psi_{1}+\Psi_{2}$ approaches $S_{GT}-S_{GT,eq}$ as
$X_{DR}$ approaches zero. Thus the quantity $S_{GT,eq}+\Psi-\Psi_{0}$ is the
gate entropy at or near equilibrium and would be maximized according to
equilibrium thermodynamics. Well away from equilibrium, I have found that
$S_{GT,eq}+\Psi$ is also maximized, not by the equilibrium state, but by the
stationary state. Of course, the stationary state becomes the equilibrium
state as $X_{DR}$ is turned down to zero. Thus, the potential,

$\mathpzc{S}_{NE}\equiv S_{GT,eq}+\Psi-\Psi_{0}\,,$ (36)

differs from $\Psi$ by only constants and makes a seamless transition to the
entropy as $X_{DR}$ goes to zero, i.e.,

$\lim_{X_{DR}\to 0}\mathpzc{S}_{NE}=S_{GT}~{}\,.$ (37)

This new potential, $\mathpzc{S}_{NE}$, is not the entropy; The standard
entropy function provides an incomplete description well away from
equilibrium. The potential $\mathpzc{S}_{NE}$ is what the relevant
thermodynamic potential becomes when the DR is well away from equilibrium, and
$\mathpzc{S}_{NE}$ can be thought of as the generalized entropy.

Equivalently, one may define the excess entropy,
$\Xi\equiv\Psi-\Psi_{0}+\frac{1}{2}g_{2}b^{2}+\frac{1}{2}g_{4}u^{2}$, which
leaves the generalized entropy expressed as

$\mathpzc{S}_{NE}\equiv S_{GT}+\Xi\,.$ (38)

This bookkeeping allows one to identify $\Xi$ as something completely outside
of the realm of equilibrium thermodynamics, i.e., $\Xi=0$ when $X_{DR}=0$ even
if some other agent pushes $b$ away from zero. Explicitly then,

$\Xi(b,u)=\Xi_{0}+k_{B}\beta_{1}b+k_{B}\beta_{2}u\,,$ (39)

where
$\Xi_{0}=L_{DR}X_{DR}^{2}\tau^{*}\left(\frac{1}{2}+\frac{1}{\mu}-\frac{1}{z^{2}}\right)+\frac{1}{2}g_{GT}z^{2}\langle
b^{2}\rangle_{0}$. Since each term in Eq. (39) contains a factor of
$\tau^{*}$, $\Xi$ is essentially an entropy created during the time span of a
scattering event so it may be referred to as a dynamical entropy. The term
rheological entropy may also be apt. It’s also important to note the additive
nature of Eq. (38); This is not a trivial result, as there are many ways to
mathematically extend $S$ to the nonequilibrium realm ($S$ being shortform for
$S_{GT}$).

At this point it is helpful to summarize physical laws as a three-tier
structure: (1) standard mechanics, both classical and quantum, taking place at
zero temperature where $S=0$ (assuming no degeneracy in the ground state) and
$\Xi=0$, (2) equilibrium thermodynamics where the temperature is raised above
zero, $S\neq 0$, and $\Xi=0$, and (3) nonequilibrium thermodynamics where
$X_{DR}$ is adjusted from zero, $S\neq 0$, and $\Xi\neq 0$. In tier 1, the
guiding principle is finding extrema in the Lagrangian. In tier 2, the guiding
principle is maximizing $S$, and in tier 3, the guiding principle is
maximizing $\mathpzc{S}_{NE}$.

This three-tier structure bears a strong resemblance to the laws of
thermodynamics, especially how maximizing a generalized entropy
$\mathpzc{S}_{NE}$ is so similar to the SLT, i.e., maximizing entropy in
equilibrium thermodynamics. Also, having $\Xi=0$ when $X_{DR}=0$ bears strong
resemblance to having $S=0$ when $T=0$. These results are compelling and lead
to the following principles for governing nonequilibrium systems:

Nonequilibrium Principle 1: The generalized entropy for the gate is additive:
$\mathpzc{S}_{NE}=S+\Xi$,

Nonequilibrium Principle 2: $\mathpzc{S}_{NE}$ is maximized when the gate is
in a stationary state,

Nonequilibrium Principle 3: $\Xi=0$ when the DR is in equilibrium.

It’s important to point out that the gate is not the entire system; For the
entire system the guiding principle is still the SLT, i.e., the system always
evolves so that $S_{T}$ is eventually maximized. What these three new
principles do is describe key aspects of the dynamics as the entire system
evolves towards equilibrium. If these principles are in time to become
established as laws there must always exist a function $\Xi$ that satisfies
these three laws, no matter how complicated the system. This function exists
and the three laws work for a pitchfork bifurcation. The question remains
about whether they also apply to more complicated instabilities, ex. Hopf and
Takens-Bogdanov bifurcations Pearson and Horsthemke (1989); Castets  _et al._
(1990).

It also remains to be seen if these results apply to systems that are not
purely dissipative, i.e., for systems that have inertial degrees of freedom as
well as damping. Some discussion of the physical significance of the variable
$b$ in Eqs. (9) and (19) is warranted. If $b$ is a mechanical variable then
under certain circumstances the dynamics can be purely dissipative, for
example the LR circuit and mechanical equivalent, an otherwise free particle
with damping. If a small spring constant is added, the system is highly
overdamped and is still well-approximated by purely dissipative dynamics.
However, when the damped harmonic oscillator becomes underdamped, it becomes
difficult to define a stationary state in the pure sense ($\dot{b}=0$) because
of the natural oscillations. In this case the dynamics described in Eq. (19)
would be for the envelope of $b(t)$. This approach is quite effective for the
highly underdamped case where the oscillation period is much less than the
relaxation time. This, essentially, is coarse-graining in time, and it is
understood that some detailed information about the system is filtered out of
the dynamical equations.

A fundamental assumption for thermodynamic variables is that they provide an
incomplete description; Describing a macroscopic system with only several
variables means that there are many distinct microstates corresponding to a
given thermodynamic state, in this case for a given value of $b$. It is
reasonable then that after coarse graining, the dynamics of thermodynamic
variables would be purely dissipative even when the dynamics for internal
microscopic dynamics is not. Stationary states would then always be achievable
and the principle of maximizing $\mathpzc{S}_{NE}$ will always be valid.

Given the paucity of general thermodynamic results in the nonequilibrium realm
Nicolis and Prigogine (1977), I anticipate these laws as building on the
Onsager symmetry relations to constitute the current extent of such general
knowledge. Indeed the Onsager symmetry relations might be referred to as the
initial (or perhaps zeroth) law of nonequilibrium thermodynamics.

###  II.5 Nonequilibrium Le Chatelier’s Principle

In this section I establish a nonequilibrium version of Le Chatelier’s
principle, for the case of second order TI. For TI1 this principle has already
been established Patitsas (2014) and this treatment follows my previous one
closely.

If the DR is pushed away from equilibrium, while the gate is held fixed in its
equilibrium state, then the total rate of entropy production $\sigma_{T}$
coincides with $\sigma_{DR}|_{\\{X_{GT}=0\\}}=L_{DR}X_{DR}^{2}$. The question
then is in determining how this rate compares with that when the gate is
allowed to relax. If the gate is released at $t=0$ then right after $t=0$, as
$u$ grows from zero, the gate would appear to violate the SLT if an observer
was unaware of the DR, since $\sigma_{GT}=-g_{4}u\dot{u}$. By Eq. (27)

$\sigma_{GT}=-g_{4}u\left[4\alpha
L_{GT}\tau^{*}\left(X_{DR}^{2}-X_{c}^{2}\right)u+4\alpha
L_{GT}\tau^{*}X_{DR}^{2}\langle b^{2}\rangle_{0}-4c^{2}u^{2}\right]~{}\,.$
(40)

As $u=0$ at $t=0$, for small $t$, $u=4\alpha L_{GT}\tau^{*}X_{DR}^{2}\langle
b^{2}\rangle_{0}t$ to leading order, and
$\sigma_{GT}=-\frac{\tau^{*}}{\tau_{GT}Q^{2}}\alpha X_{DR}^{2}u$. In the
meantime $\sigma_{DR}=L_{DR}X_{DR}^{2}+\alpha X_{DR}^{2}u$ and the net rate is
$\sigma_{T}=L_{DR}X_{DR}^{2}+\alpha
X_{DR}^{2}u(1-\frac{\tau^{*}}{\tau_{GT}Q^{2}})$. The relaxation time
$\tau_{GT}$ is always larger than $\tau^{*}$ which is typically the fastest
time scale in the analysis. Also, $Q>>1$ typically, so there is no risk of
$\sigma_{T}$ being negative for short times. Indeed, the relaxation of the
gate increases $\sigma_{T}$.

For longer times, $u^{2}$ becomes significant, but only if $z\geq 1$. The rate
$\sigma_{GT}$ increases for some time before it eventually decreases and falls
to zero when the gate becomes stationary. The maximum for $\sigma_{GT}$ occurs
at $u_{max}=2\langle b^{2}\rangle_{0}(z^{2}-1)$ at which
$\sigma_{GT}=-(\tau^{*}/\tau_{GT})\alpha
X_{DR}^{2}u_{max}[(z-1/z)^{2}+1/Q^{2}]$. With $Q$ normally much greater than
unity and $\sigma_{GT}$ does not overcome the $\alpha X_{DR}^{2}u$ term in
$\sigma_{DR}$, unless $z^{2}$ is very large, i.e., on the order of
$\tau_{GT}/\tau^{*}$. Achieving $z$ values of 1 or 2 is not easy in real
systems, and making $z^{2}$ on the order of $\tau_{GT}/\tau^{*}$ could lead to
catastrophic conditions for real systems. Also, even if $z^{2}$ could get this
large there is still the $L_{DR}X_{DR}^{2}$ term (from $\sigma_{DR}$) to
overcome, making violation of the SLT unlikely. Nevertheless it is interesting
to note that under conditions with very large $X_{DR}$ there is a possibility
of briefly violating the SLT when the gate is a very small system with a
relaxation time approaching $\tau^{*}$.

For even later times, the gate settles into its stationary state where
$\sigma_{GT}=0$ and the positive definite nature of $\sigma_{DR}$ guarantees
the total entropy always increases. More careful inspection reveals an even
stronger result: If $\sigma_{DR}|_{\\{X_{GT}=0\\}}$ is the total entropy
production with the gate at equilibrium, and $\sigma_{DR}|_{\\{J_{GT}=0\\}}$
is the total entropy production with the gate stationary (so the flux
$J_{GT}=\dot{b}$ is zero) then one always has the condition

$\sigma_{DR}|_{\\{J_{GT}=0\\}}\geq\sigma_{DR}|_{\\{X_{GT}=0\\}}~{}\,.\,\,\,\,\,\,\,\,\,\,\,\,\text{(2nd
order noneq. Le Chatelier's principle)}$ (41)

The same result was previously established for TI1 Patitsas (2014).

Similarly to TI1, the physical meaning of this result is that the DR produces
entropy faster when the gate variable is allowed to relax by bifurcating away
from equilibrium and becoming stationary. This result constitutes a
nonequilibrium version of Le Chatelier’s principle. In the traditional Le
Chatelier’s principle, when a given thermodynamic variable is pushed away from
equilibrium, other thermodynamic variables relax to new equilibrium values, so
that the total entropy is again maximized, and the new relaxed entropy is
always greater than the unrelaxed entropy Landau and Lifshitz (1980). Here,
when the DR variable is pushed away from equilibrium, the gate variable will
temporarily move away from equilibrium to stationary states, and the new rate
of entropy production is always greater. The actual difference is

$\sigma_{DR}|_{\\{J_{GT}=0\\}}-\sigma_{DR}|_{\\{X_{GT}=0\\}}=\frac{\tau^{*}L_{GT}}{c^{2}}\alpha^{2}X_{DR}^{4}\left(\frac{h_{+}}{z^{2}}\right)~{}\,.$
(42)

The right hand side of Eq. (42) is always positive definite. Since the entropy
production rate $\sigma_{GT}$ of the gate is zero when the gate is stationary,
the nonequilibrium Le Chatelier’s principle also applies to the total rate of
entropy production, $\sigma_{T}=\sigma_{DR}+\sigma_{GT}$, then the principle
works also for $\sigma_{T}$, i.e.,
$\sigma_{T}|_{\\{J_{GT}=0\\}}\geq\sigma_{T}|_{\\{X_{GT}=0\\}}$. In fact, the
principle will work for any quantity formed by the sum of $\sigma_{DR}$ and
any multiple of $\sigma_{GT}$. The induction effect always causes the DR to
approach equilibrium faster, thus leading to the conclusion that the gate will
always facilitate the DR’s approach to equilibrium.

###  II.6 Entropic Coupling Problem

In the second order facilitation of the approach to equilibrium, the gate will
be shifted away from its own thermodynamic equilibrium, and if $X_{DR}$ is
large enough compared to $X_{c}$, the disruption of the gate equilibrium may
be detected. The stationary state result Eq. (28) suggests that for TI2 the
gate entropy change from equilibrium is given by

$\Delta S_{GT,ss}=-\frac{1}{2}k_{B}Q^{2}h_{+}^{2}<0\,.$ (43)

For consistency this does coincide at large $z^{2}$ with the infinite $Q$
result using the scheme $g_{GT}=g_{2}+g_{4}b^{2}$. This change is important
regarding the entropic coupling problem, which will be further discussed in
Sec. II.6. One verifies that the entropy is continuous through the transition
at $z=1$ and therefore the physics described here is similar to that of a
second order (equilibrium) PT. Equation (43) provides a sort of budget that is
available for forming patterned structures in the gate.

Equation (43), along with the corresponding first order result discussed in
Ref. Patitsas (2014) and the nonequilibrium Le Chatelier’s principle, up to
second order, solves the entropic coupling problem. The gate can achieve a
state that may be considered to be patterned, or self-organized/assembled, by
having its entropy lowered relative to its equilibrium, and during the entire
time of this self-assembly, the SLT is never violated. The result holds for
both TI1 and for TI2, both below and above the critical bifurcation point. The
DR plays a key role as that system which always increases it entropy by at
least as much as the decrease in the entropy of the gate. I stress the
dynamical nature; If the DR returns to equilibrium, the entropy budget $\Delta
S_{GT,ss}$ goes to zero.

This type of coupling between the DR and the gate is unconventional.
Ordinarily one couples two systems mechanically via a potential energy term.
When the potential energy depends on variables from the two systems then the
dynamical differential equations become coupled. Here, the physics is purely
dissipative and potential and kinetic energies do not play a role. Instead, it
is the entropy production that depends on both DR and gate variables, and
produces the interesting couplings in the dynamics. More specifically, the
ultimate source of the coupling is that the DR conductance coefficient
$M_{DR}$ depends on the gate variable. One can say that the conductance
becomes paramount in nonequilibrium systems, playing a role similar to the
potential energy in equilibrium systems.

I will resume discussion on this problem below in Sec. III.8.

###  II.7 Nonequilibrium Canonical Distribution

For (isolated) systems in equilibrium, all accessible microstates receive
equal statistical weight, according to the postulate of equal a priori
probabilities Reif (1965); Tolman (1979). In this section I show how this
changes in nonequilibrium and I derive revised weighting functions.

The thermodynamic potentials $\Psi_{1}$ and $\Psi_{2}$ may be used to
calculate nonequilibrium partition functions for the gate specifically. These
would be multiplied by the regular equilibrium partition functions to provide
the full picture. I define the first order partition function as

$Z_{1,noneq}\equiv
c^{\prime}\int_{-\infty}^{\infty}e^{\Psi_{1}(b)/k_{B}}db~{}\,,$ (44)

where $c^{\prime}$ is a constant to be determined below. Using Eq. (31) the
integral is easily evaluated as:

$Z_{1,noneq}=c^{\prime}~{}e^{\Psi_{1,0}/k_{B}}\sqrt{\frac{2\pi
k_{B}}{g_{GT}}}e^{k_{B}\beta_{1}^{2}/2g_{GT}}~{}\,.$ (45)

The same result for $Z_{1,noneq}$ can be found by focusing on microstates of
the gate and it is instructive to do this analysis. The variable $b$
represents a macrostate to which there are $\Omega_{GT}(b)$ gate microstates
all corresponding to the same $b$. The probability of the gate having value
$b$ is proportional to the total multiplicity:
$\Omega_{T}=\Omega_{DR}\Omega_{GT}(b)$. The assumption is that all microstates
for the entire system are accessed equally. The DR multiplicity $\Omega_{DR}$
does not depend directly on $b$, but its rate of change does because the
Onsager coefficient $M_{DR}$ depends on $b$. The way to physically interpret
$\Psi$ is to not evaluate $\Omega_{DR}$ at time $t$ but rather at
$t+\tau^{*}/2$, i.e., to reach forward in time the time required for a
scattering event. This means using not $\Omega_{DR}$ but
$\Omega_{DR}\exp{(\sigma_{DR}\tau^{*}/2k_{B})}$ when attempting to derive a
probability distribution. The probability of the gate having value $b$ is
proportional to $\exp{(\Psi_{1}(b)/k_{B})}$. Noting that

$\Omega_{GT}(b)=\Omega_{GT,eq}\exp{(-g_{GT}b^{2}/2k_{B})}~{},$ (46)

which is independent of any TI effects, and inspecting Eq. (31), the
probability of the GT having value $b$ is found to be proportional to
$\exp{(\beta_{1}b)}\Omega_{GT}(b)$.

In equilibrium, the probability of any particular gate microstate being
occupied is equal to any other. Away from equilibrium, this is not the case;
For each gate microstate $i$, with corresponding $b_{i}$, the probability of
occupancy is the nonequilibrium canonical distribution,

$P_{1,i}=c_{1}e^{\beta_{1}b_{i}}~{}\,.$ (47)

The normalization constant $c_{1}$ is evaluated in the standard manner,
invoking a partition function:

$c_{1}^{-1}=Z_{1,noneq}=\sum_{i}e^{\beta_{1}b_{i}}=\int_{-\infty}^{\infty}{db\,e^{\beta_{1}b_{i}}\Omega_{GT}(b)}~{}\,.$
(48)

Inserting $\Omega_{GT}(b)$ from Eq. (46) into Eq. (48), allows the integral to
be evaluated, leading to:

$Z_{1,noneq}=\Omega_{GT,eq}\sqrt{2\pi
k_{B}/g_{GT}}\exp{(k_{B}\beta_{1}^{2}/2g_{GT})}\,.$ (49)

After comparing to Eq. (45), one identifies the constant
$c^{\prime}=\Omega_{GT,eq}e^{-\Psi_{1,0}/k_{B}}$.

From Eq. (49) one readily verifies that $\langle
b\rangle=\partial\ln{Z_{1,noneq}}/\partial\beta_{1}$, which equals the
expected value $b_{ss}$. Also $\langle(\Delta
b)^{2}\rangle=\partial^{2}\ln{Z_{1,noneq}}/\partial\beta_{1}^{2}=k_{B}/g_{GT}$.
From Eq. (48) one notes that to a good approximation:

$k_{B}\ln Z_{1,noneq}=S_{GT,eq}+k_{B}\beta_{1}\langle b\rangle-g_{GT}\langle
b\rangle^{2}/2\,,$ (50)

which can be expressed as $k_{B}\ln Z_{1,noneq}=S_{GT}+k_{B}\beta_{1}\langle
b\rangle$.

Using Eq. (32), the second order partition function is then

$Z_{2,noneq}\equiv\int_{-\infty}^{\infty}e^{\Psi_{2}(u)/k_{B}}du=e^{\Psi_{2,0}/k_{B}}\sqrt{\frac{8\pi
Q^{2}\langle b^{2}\rangle_{0}}{g_{GT}}}\exp{\left(\frac{2k_{B}Q^{2}\langle
b^{2}\rangle_{0}\beta_{2}^{2}}{g_{GT}}\right)}~{}\,.$ (51)

To a good approximation $Z_{2,noneq}\approx e^{\Psi_{2}(\langle
u\rangle)/k_{B}}$.

In order to obtain the probability distribution for gate states, $i$, with
values $u_{i}$ one notes that the multiplicity function for the gate is a
function of both $b$ and $u$:
$\Omega_{GT}(b,u)=\Omega_{GT,eq}\exp{(-g_{2}b^{2}/2k_{B}-g_{4}u^{2}/2k_{B})}$.
Proceeding similarly to the first order case the gives:

$P_{2,i}=\frac{1}{Z_{2,noneq}}e^{\beta_{2}u_{i}}~{}\,.$ (52)

For $z<<1$ this probability distribution is flat, coinciding with equilibrium
expectations. Near and above the critical point at $z=1$ the situation changes
and larger values of $\Delta b^{2}$ are favoured, leading to the bifurcation.
Combining first and second order for a total nonequilibrium partition function
gives $Z_{noneq}=Z_{1,noneq}Z_{2,noneq}=\Omega_{GT}\exp(\beta_{1}\langle
b\rangle+\beta_{2}\langle u\rangle)$.

In equilibrium thermodynamics the partition function is simply related to the
Helmholtz free energy. The analysis presented so far in this section leads
naturally to the definition of a generalized free energy as

$\mathpzc{F}\equiv F-k_{B}T\beta_{1}\langle b\rangle-k_{B}T\beta_{2}\langle
u\rangle\,,$ (53)

where $F=U-TS$ is the Helmholtz free energy, with $U$ and $S$ being understood
as the internal energy and entropy, respectively, of that part of the gate
that is removed from the heat bath. Equation (39) can be used to express this
generalized free energy as $\mathpzc{F}=F-T(\Xi-\Xi_{0})$. There is a great
deal of physics built into the function $\mathpzc{F}$ which has a general,
all-encompassing, quality with application to all scientific problems. The
internal energy $U$ accounts for the mechanics of inter-particle interactions,
the entropy $S$ uses the state-counting multiplicity function, often using
combinatorial methods, and finally $\Xi$ keeps track of the nonequilibrium
fluxes from thermodynamic forces, including TI.

###  II.8 Nonequilibrium Phase Transitions

Well away from equilibrium, $U$ may not have interesting behaviour as $X_{DR}$
changes, even through a nonequilibrium PT. The internal energy may provide
minor corrections, but the key elements of the PT are determined by $\Xi$, not
$U$. The nonequilibrium PT is clearly distinct from equilibrium PTs at which a
free energy $F$ is minimized. While it is true that at the critical
temperature for an equilibrium PT there may be large fluctuations with a
nonequilibrium character, this is qualitatively very different from a
nonequilibrium PT. In a nonequilibrium PT, a thermodynamic variable is pushed
well away from equilibrium, where $F$ is nowhere near its minimum value and
fluctuations from equilibrium would reach only extremely rarely. When this
variable reaches its critical value, the thermodynamic potential $\Psi$ is
maximized (and $\mathpzc{F}$ is minimized). In the above analysis, energetic
considerations play no significant role and the free energy $F$ would display
no interesting behaviour at the transition.

I also make a note on terminology; Since the nonequilibrium PT occurs at the
bifurcation point and the bifurcation can now be described thermodynamically,
then in the context of the thermodynamics of dissipative systems, bifurcation
and nonequilibrium PT are interchangeable terms.

It is instructive to compare nonequilibrium PTs to the second order PTs of
equilibrium thermodynamics. Second order PTs are well described by minimizing
the (Helmholtz) free energy function of Landau,
$F_{L}=F_{0}+\xi_{1}b^{2}+\xi_{2}b^{4}$, with the transition occurring at
$\xi_{1}=0$ as $\xi_{1}$ varies with an external parameter such as
temperature. Below (above) critical, $\xi_{1}>0$, ($\xi_{1}<0$), and $F_{L}$
resembles a parabola (sombrero). For the nonequilibrium PT well above
critical, $\mathpzc{F}$ will resemble the sombrero function because of the
form of $\Phi_{2}$ in Eq. (22). With $\gamma=0$ (TI2 only), $\Phi_{1}$ gives
the desired parabolic appearance for $\mathpzc{F}$. The similarity ends there
though as for the function $\Phi_{2}$ to be useful above and below critical it
is actually a function of $u$, and $\mathpzc{F}$ is a function of two
variables, $\langle b\rangle$ and $u$. Unlike the equilibrium case, one must
minimize $\mathpzc{F}$ with respect to two variables. The simple graphical
pedagogy used for Landau theory with a varying $\xi_{1}$ does not work so
easily in the nonequilibrium case.

These nonequilibrium PTs occur when a balance between (gate) entropy and TI is
achieved. In contrast, equilibrium PTs occur when a balance is created between
entropy and energy. Without the energetics, the analysis is of course very
different. Very close to an equilibrium PT it is known that many higher-order
terms in the perturbative analysis are required which creates a challenging
problem Patashinskii and Pokrovskii (1979). While it may be true that a more
accurate description of nonequilibrium PTs would require expanding
$\exp(\Delta S/k_{B})$ to higher than first order, this analysis would differ
from the energetic approach that would be described in terms of particle-
particle interactions.

Finally, I point out that a second class of nonequilibrium PTs is possible,
where a balance is achieved between energy and induction only. As a possible
example, consider a percolation network Kirkpatrick (1973) that is non-
conductive at or near equilibrium, but which becomes conductive well away from
equilibrium when the applied bias $X_{DR}$ reaches a certain threshold. The
induced changes in the network that make it conductive may be subtle and
purely mechanical. This indeed may be behind the type of nonequilibrium PTs
reported as directed percolation Hinrichsen (2006).

###  II.9 Auxiliary Potential for TI2

The thermodynamic potentials $\Phi_{2}(u)$ and $\Psi_{2}(u)$ were devised
after the dynamics for $u$ was established in Eq. (27). It turns out that
proceeding in the other direction has a complication that needs explanation;
Simply differentiating one of the potentials wrt $u$ does not give Eq. (27).
Instead an auxiliary potential is also required. Before doing so it is
convenient to define $h_{-}(x)=x-1-\sqrt{(x-1)^{2}+x/Q^{2}}$,
$\beta_{-}=g_{2}h_{-}/2k_{B}$, $u_{+}=2Q^{2}\langle b^{2}\rangle_{0}h_{+}$,
and $u_{-}=2Q^{2}\langle b^{2}\rangle_{0}h_{-}$. Both $u_{+}$ and $u_{-}$ are
solutions to the quadratic equation formed when setting $du/dt=0$. However,
only $u_{+}$ makes physical sense as the stationary state. The auxiliary
potential $\Psi_{-}$ is defined as
$\Psi_{-}=\Psi_{2,0}+k_{B}\beta_{-}u-\frac{g_{GT}}{8Q^{2}\langle
b^{2}\rangle_{0}}u^{2}$. Creating $\Psi_{-}$ poses no extra difficulties after
$\Psi_{2}$ is created. Towards obtaining $du/dt$ from potentials, noting that
$\partial\Psi_{-}/\partial u=-\frac{g_{2}}{4Q^{2}\langle
b^{2}\rangle_{0}}(u-u_{-})$ which is positive definite for the range of
physically allowable $u$. Using both potentials gives the dynamics as

$\frac{du}{dt}=-\frac{2L_{GT}^{2}}{c^{2}(\tau^{*})^{2}}\left(\frac{\partial\Psi_{-}}{\partial
u}\right)\left(\frac{\partial\Psi_{2}}{\partial u}\right)~{}\,.$ (54)

In contrast, an auxiliary potential is not required in the infinite $Q$
approach where the dynamics for $b$ comes directly from $\Psi_{2,\infty
Q}\equiv\tau^{*}(1-z^{-2})\Phi_{2,\infty Q}$ and

$\frac{db}{dt}=\frac{L_{GT}}{2}\left(\frac{\partial\Psi_{2,\infty Q}}{\partial
b}\right)~{}\,.$ (55)

###  II.10 Nonequilibrium Free Energy Functional

The above results are easily transferable to the treatment of order parameter
fields, where $b(t)$ is replaced by $\phi(\mathbb{r},t)$, which varies
spatially. For TI1, one uses the variable $\langle\phi(\mathbb{r},t)\rangle$.
For example, for the first order free entropy potential, one uses
$\Psi_{1}(\langle\phi(\mathbb{r},t)\rangle)$, which gets abbreviated for
convenience to $\Psi_{1}(\phi(\mathbb{r},t))$. For second order, the variable
to use is
$\psi(\mathbb{r},t)\equiv\langle\phi^{2}(\mathbb{r},t)\rangle-\langle\phi^{2}(\mathbb{r})\rangle_{0}$,
and the TI2 free entropy potential is $\Psi_{2}(\psi(\mathbb{r},t))$.

The nonequilibrium free entropy is written as a functional of two functions
$\phi$ and $\psi$ as
$\Psi[\phi(\mathbb{r},t),\psi(\mathbb{r},t)]=\int{d^{3}r\mathpzc{f}[\phi(\mathbb{r},t),\psi(\mathbb{r},t)]}$
with free entropy density
$\mathpzc{f}(\phi(\mathbb{r},t),\psi(\mathbb{r},t))$. For the special case
where $\gamma=0$ and $\langle b\rangle=0$, a prototype form of

$\mathpzc{f}_{p}=\mathpzc{f}_{0}-k_{B}T\beta_{2}\psi(\mathbb{r},t)+\frac{Tg_{GT}}{8Q^{2}\langle
b^{2}\rangle_{0}}\psi^{2}(\mathbb{r},t)\,$ (56)

borrowed from Eq. (32) will work.

For TI2 in the infinite $Q$ scheme, the $\phi$ field is used with
$\Psi_{2,\infty Q}[\phi(\mathbb{r},t)]=\int{d^{3}r\mathpzc{f}_{2,\infty
Q}[\phi(\mathbb{r},t)]}$ where

$\mathpzc{f}_{2,\infty
Q}[\phi(\mathbb{r},t)]=L_{DR}(X_{DR}^{2}-X_{c}^{2})\tau^{*}+g_{2}(z^{2}-1)\phi^{2}-\frac{1}{2}g_{4}\phi^{4}\,.$
(57)

Modifying Eq. (55) for a functional derivative gives the dynamics for the
order parameter in this scheme is

$\frac{\partial\phi}{\partial
t}=\frac{(z^{2}-1)}{\tau_{GT}}\phi-\frac{1}{4Q^{2}\langle\phi^{2}\rangle_{0}\tau_{GT}}\phi^{3}~{}\,.$
(58)

Equation (58) resembles the SH equation. The SH equation is an important tool
for researchers of nonequilibrium systems and has, for example, been
successfully utilized to model patterns formed in laminar flame fronts,
certain types of Poiseuille flow, trapped ion modes in plasmas, and systems
with Eckhaus instabilities such as Rayleigh-Bénard convection Kuramoto and
Tsuzuki (1976); Sivashinsky (1977); Michelson (1986); Manneville (1988); Cross
and Hohenberg (1993); Desai and Kapral (2009); Cross and Greenside (2009). The
most important feature of the SH equation is a tunable parameter $r$ which can
produce a negative relaxation rate, and instability, at $r=1$ in

$\frac{\partial\phi}{\partial
t}=(r-1)\phi-2\nabla^{2}\phi-\nabla^{4}\phi-\phi^{3}~{}\,.\,\,\,\,\,\,\,\,\,\text{(SH
equation)}$ (59)

In Eq. (58) the first term on the right-hand side is the vital component in
the SH equation, with $X_{DR}$ being the tunable parameter. A nonequilibrium
PT occurs when $X_{DR}^{2}$ exceeds $X_{c}^{2}$, or equivalently, $r>1$.

The key differences between Eq. (58) and the SH equation are the derivative
terms, in particular a type of negative diffusion term, $-2\nabla^{2}\phi$.
Dealing with such terms requires an analysis for more than one gate variable
which is the topic of Sec. III. Before doing so, one can add some simple
physics for the order parameter as either diffusion or as a surface tension
(or interface energy) term to modify Eq. (58) to

$\frac{\partial\phi}{\partial
t}=\frac{(z^{2}-1)}{\tau_{GT}}\phi+D\nabla^{2}\phi-\frac{1}{4Q^{2}\langle\phi^{2}\rangle_{0}\tau_{GT}}\phi^{3}~{}\,.$
(60)

The form of Eq. (60) is identical to the Langevin Model A equation (without
noise), under the Hohenberg-Halperin classification scheme used to describe
the (nonconserved) order parameter critical dynamics, including quench
dynamics, near an equilibrium PT Desai and Kapral (2009). This comparison
works right down to the term $z^{2}-1$ playing the role of $T-T_{c}$, and
changing sign at the critical point. I point out that the physics of this
comparison is unclear, as far whether or not TI2 would play any role near an
equilibrium PT. At the very least, the TI2 approach presented here could play
a useful role in modeling quench dynamics near equilibrium PTs. Using $\psi$
instead of $\phi$ will account for fluctuations in the order parameter field.
The following adaptation of Eq. (27) (plus adding in diffusion):

$\frac{\partial\psi}{\partial
t}=2\frac{\left(z^{2}-1\right)}{\tau_{GT}}\psi+2\frac{z^{2}}{\tau_{GT}}\langle
b^{2}\rangle_{0}+D_{u}\nabla^{2}\psi-2c^{2}\psi^{2}~{}\,,$ (61)

will provide just as good a model for describing quenching as the Langevin
Model A equation. Near the critical point $\psi$ becomes large and thus
describes well the fluctuations that would be created by adding in a noise
term to the Langevin Model A.

##  III Case of more than one gate variable

Many of the results presented in Sec. II are easily transferable to the case
of more than one gate variable. For the case where there are $N$ identical
copies of the gate system considered in Sec. II, each gate would have the same
values of $\alpha$, $X_{c}$, and $Q$. Having more gate systems has the obvious
advantage of boosting the entropy budget available for creating patterns. The
expression in Eq. (43) is multiplied by $N$; In particular at critical,
$(z=1)$, $\Delta S_{GT,ss}=-2NQk_{B}$. If one is interested in patterns being
used to store information, then this entropy budget is sufficient to store up
to $2NQ/\ln 2$ bits of information. How the storing of this information occurs
is not a trivial matter and will depend on the pattern quality which involves
characteristics such as the long range order. If $q$ physically represents a
wavevector then all length scales will be excited with equal weight. If all
modes are excited equally then long range order is not expected and clear
patterns are unlikely to form. This analysis is carried out below in Sec.
III.6. Thus, pattern formation requires $\alpha$ to depend on $q$.

When variables $\alpha$, $g_{GT}$, and $c$ depend on $q$, the bifurcation
point $X_{c}$ depends on $q$ with a relationship I call the dispersion
relation. Equation (17) becomes

$X_{c,q}=\sqrt{\frac{g_{q}}{2\alpha_{q}\tau^{*}}}~{}\,.$ (62)

Since the parameters $g_{q}$ and $\alpha_{q}$ must always remain positive it
is natural to define the excitation gap, $X_{eg}$, as the minimum value of
$X_{c,q}$. As one ramps up $X_{DR}$ from zero there will be an interval, this
excitation gap, over which little happens before $X_{DR}$ reaches the first of
a set of discrete levels at $X_{c,q}$, each one creating a transition at a
given mode $q$ that makes $b_{q}$ nonzero. This is analogous to quantum
transitions in molecular systems when the excitation energy matches or exceeds
the energy gap between highest occupied and lowest unoccupied molecular
orbitals, or in solid state insulators where energy levels coalesce to form
bands, and the excitation energy meets or exceeds the band gap energy. This
point of view is in the infinite $Q$ limit. For finite $Q$, there will be
enhanced fluctuations as $X_{DR}$ approaches the gap. In either case, when the
excitation gap is surmounted one might expect some type of pattern to form in
the system with a Fourier transform peaked around $q_{eg}$. This patterned
system could also be referred to as a self-assembled or self-organized
structure.

The potentials $\Phi_{1}$, $\Phi_{2,Q\infty}$, and $\Phi_{2}$ are
straightforward to generate from Eqs. (10) , (22), and (29)

$\Phi_{1}=L_{DR}X_{DR}^{2}+\sum_{q}\left[X_{DR}^{2}\gamma_{q}b_{q}-\lambda_{1,q}\left(-g_{q}\gamma_{q}L_{q}\tau^{*}X_{DR}^{2}b_{q}+L_{q}g_{q}^{2}b_{q}^{2}\right)\right]~{}\,,$
(63) $\Phi_{2,\infty
Q}=L_{DR}X_{DR}^{2}+\sum_{q}\left[X_{DR}^{2}\alpha_{q}b_{q}^{2}-\lambda_{Q\infty,q}\left(-2\alpha_{q}\left(X_{DR}^{2}-X_{c,q}^{2}\right)L_{q}g_{q}\tau^{*}b_{q}^{2}+c_{q}^{2}g_{q}b_{q}^{4}\right)\right]~{}\,,$
(64)
$\Phi_{2}(u)=L_{DR}X_{DR}^{2}+\sum_{q}\left[\frac{g_{q}\mu}{2\tau^{*}}\left(z_{q}^{2}\langle
b_{q}^{2}\rangle_{0}+u_{q}h_{+}(z_{q}^{2})-\frac{u_{q}^{2}}{4Q_{q}^{2}\langle
b_{q}^{2}\rangle_{0}}\right)\right]~{}\,,$ (65)

where $\lambda_{1,q}=\tau_{q}/\tau^{*}$,
$\lambda_{Q\infty,q}=\tau_{q}X_{DR}^{2}/2\tau^{*}(X_{DR}^{2}-X_{c,q}^{2})$,
and $z_{q}^{2}=X_{DR}^{2}/X_{c,q}^{2}$. When constructing the free entropy
potentials it is best to first drop the $L_{DR}X_{DR}^{2}$ terms in the $\Phi$
potentials and then divide each term by the required factors including the
Lagrange multipliers. Dropping the $L_{DR}X_{DR}^{2}$ terms makes no
difference in equilibrium and also when differentiating by any variables
$b_{q}$ and $u_{q}$. This construction yields:

$\Psi_{1}=\sum_{q}\left[k_{B}\beta_{1,q}b_{q}-\frac{1}{2}g_{q}b_{q}^{2}\right]~{}\,,$
(66) $\Psi_{2,\infty
Q}=\sum_{q}\left[(z_{q}^{2}-1)g_{q}b_{q}^{2}-\frac{1}{2}g_{4,q}b_{q}^{4}\right]~{}\,,$
(67)

and

$\Psi_{2}=\frac{1}{2}\sum_{q}\left[g_{q}z_{q}^{2}\langle
b_{q}^{2}\rangle_{0}+g_{q}u_{q}h_{+}(z_{q}^{2})-g_{4,q}u_{q}^{2}\right]~{}\,.$
(68)

As discussed below, these $q$ sums may or may not include $q=0$, depending on
the physical details of the DR. Replacing $h_{+}$ with $h_{-}$ in Eq. (68)
gives the auxiliary potential $\Psi_{-}$ from which one may derive the
dynamics for each $u_{q}$, following Eq. (54):

$\frac{du_{q}}{dt}=-\frac{2}{(cg_{q}\tau_{q}\tau^{*})^{2}}\left(\frac{\partial\Psi_{-}}{\partial
u_{q}}\right)\left(\frac{\partial\Psi_{2}}{\partial u_{q}}\right)~{}\,.$ (69)

Equation (69) accounts for fluctuations of many modes $q$ near a PT, and
represents the most complete description to date of nonequilibrium dynamics in
a system that breaks symmetry at a nonequilibrium PT.

From $\Psi_{1}$ and $\Psi_{2}$, one constructs $\Psi=\Psi_{1}+\Psi_{2}$,
$\mathpzc{S}_{NE}$ and $\mathpzc{F}$, which obey general extremum principles
by stationary states (one for each $q$).

###  III.1 Case where $\alpha_{q}=\alpha_{0}+\alpha_{2}q^{2}+\alpha_{4}q^{4}$

I now show that the SH equation can be derived from first principles via the
infinite $Q$ approach inside of TI2, if the induction parameter $\alpha_{q}$
has the following small $q$ expansion up to fourth order:

$\alpha_{q}=\alpha_{0}+\alpha_{2}q^{2}+\alpha_{4}q^{4}~{}\,.$ (70)

The starting point is the linearized (unconstrained) Eq. (16) modified for
each mode $q$:

$\dot{b}_{q}=\frac{z_{0}^{2}-1}{\tau_{GT}}b_{q}+D_{TI}q^{2}b_{q}+d_{4}q^{4}b_{q}~{}\,,$
(71)

where $z_{0}^{2}=2\tau^{*}X_{DR}^{2}\alpha_{0}/g_{GT}$,
$D_{TI}=2L_{GT}\tau^{*}X_{DR}^{2}\alpha_{2}$, and
$d_{4}=2L_{GT}\tau^{*}X_{DR}^{2}\alpha_{4}$. As the $b_{q}$ are Fourier
components of $\phi(\mathbb{r})$ then the Fourier transform of Eq. (71) gives
the desired result after diffusion with coefficient $D$ is taken into account,
and the cubic cut-off term is returned:

$\frac{\partial\phi}{\partial
t}=\frac{z_{0}^{2}-1}{\tau_{GT}}\phi+(D-D_{TI})\nabla^{2}\phi+d_{4}\nabla^{4}\phi-\frac{1}{4Q^{2}\langle\phi^{2}\rangle_{0}\tau_{GT}}\phi^{3}~{}\,.$
(72)

With negative $d_{4}$, Eq. (72) is the SH equation, Eq. (59). The induction
parameter $D_{TI}$ acts like a negative diffusion coefficient. This means that
instabilities can be created by making $X_{DR}$ large enough and this can lead
to either a negative effective relaxation rate, a negative effective diffusion
coefficient, or both. This has been shown to produce patterns in simulations
Cross and Greenside (2009). When returning to Fourier components, the
condition for instability is

$\frac{z_{0}^{2}-1}{\tau_{GT}}-(D-D_{TI})q^{2}+d_{4}q^{4}>0~{}\,.$ (73)

Solving for $X_{DR}$ at the instability threshold gives the nonequilibrium
dispersion relation:

$X_{DR}(q)=X_{c0}\sqrt{\frac{1+\tau_{GT}Dq^{2}}{\alpha_{q}/\alpha_{0}}}~{}\,.$
(74)

The dynamics for the SH equation can be obtained from a free entropy
functional as

$\frac{\partial\phi}{\partial
t}=\frac{1}{2}\frac{\delta}{\delta\phi}\int{d^{3}r\left[\frac{z_{0}^{2}-1}{\tau_{GT}}\phi^{2}-(D-D_{TI})(\nabla\phi)^{2}+d_{4}(\nabla^{2}\phi)^{2}-\frac{1}{4Q^{2}\langle\phi^{2}\rangle_{0}\tau_{GT}}\phi^{4}\right]}~{}\,.$
(75)

For the case of dissipative systems, any phenomenon successfully modeled with
the SH equation is also successfully modeled by TI2. When $D-D_{TI}<0$, the SH
model on its own violates the SLT, but under TI2 the solution to the entropic
coupling problem applies and the SLT is never violated.

When dealing with fluctuations, Eq. (27) may be modified to:

$\frac{1}{2}\frac{du_{q}}{dt}=\left(\frac{z_{0}^{2}}{\tau_{GT}}+D_{TI}q^{2}+d_{4}q^{4}\right)\left(u_{q}+\langle
b^{2}\rangle_{0}\right)-\frac{u_{q}}{\tau_{GT}}-Dq^{2}u-c^{2}u_{q}^{2}~{}\,.$
(76)

A real space version can be produced as:

$\frac{1}{2}\frac{\partial\psi}{\partial
t}=\left(\frac{z_{0}^{2}}{\tau_{GT}}\psi+(D-D_{TI})\nabla^{2}\psi+d_{4}\nabla^{4}\psi\right)\left(\psi+\langle
b^{2}\rangle_{0}\delta(\mathbb{r})\right)-c^{2}\psi^{2}~{}\,.$ (77)

Equations (76) and (77) would describe the finite-Q broadened resonant
transition as $X_{DR}$ is varied above the excitation gap. Near the gap
fluctuations are large and these equations should be useful in modeling any
nonequilibrium phenomena with large fluctuations. Such systems have been
modeled using the Edwards-Wilkinson (EW), Burgers, complex Ginzberg-Landau,
and Kardar-Parisi-Zhang (KPZ) equations, all of which have noise terms added
in artificially Desai and Kapral (2009); Cross and Greenside (2009); Kardar
_et al._ (1986). I believe that the essential physics contained in all of
these models can be arrived at from the basic TI analysis up to second order,
leading up to Eqs. (76) and (77). In Eqs. (76) and (77) noise is incorporated
more naturally, with noise levels varying sensitively with $Q$ and $X_{DR}$.

The EW and KPZ equations do not have a negative diffusion coefficient or a
negative relaxation time so are not models corresponding to behaviour above a
nonequilibrium critical point. They also do not have higher order quenching
terms. They do, however, possess an external stochastic noise term. The EW
equation is simply the diffusion equation with an added noise term. Since
fluctuations are large near the critical point, $X_{eg}$, the EW equation and
Eqs. (76), (77) are well-suited for describing nonequilibrium systems just
below, and right up to, the critical point. A nonequilibrium PT may be an
explanation for the kinetic roughening transition sometimes observed in film
growth and modeled by the EW and KPZ equations Alon  _et al._ (1998);
Pinnington  _et al._ (1997); Desai and Kapral (2009). A high influx of
adsorbates, and thus a large $X_{DR}$, would be required to achieve this type
of nonequilibrium PT. Above the bifurcation point, the EW equation is not
suitable, whereas Eqs. (76) and (77) are well-suited for this task, as the
adjustable reactant flux in film growth flux is related to $X_{DR}$.

###  III.2 Case of Conserved Order Parameter

In systems with conserved order parameters, direct relaxation has little
influence and Eq. (72) becomes:

$\frac{\partial\phi}{\partial
t}=\nabla^{2}\left[(D-D_{TI})\phi+d_{4}\nabla^{2}\phi-\frac{1}{4Q^{2}\langle\phi^{2}\rangle_{0}\tau_{GT}}\phi^{3}\right]~{}\,.$
(78)

The extra $\nabla^{2}$ operator arises from the relation
$\frac{\partial\phi}{\partial t}=-\overrightarrow{\nabla}\cdot\mathbb{j}$
while the current density is given by
$\mathbb{j}=-M\overrightarrow{\nabla}\mu$. The chemical potential $\mu$ can be
expressed as $\mu(\mathbb{r},t)=\delta\mathpzc{F}/\delta\phi$ where

$\mathpzc{F}=\frac{1}{2M}\int{d^{3}r\left[(D-D_{TI})\phi^{2}-d_{4}(\nabla\phi)^{2}-\frac{1}{4Q^{2}\langle\phi^{2}\rangle_{0}\tau_{GT}}\phi^{4}\right]}~{}\,.$
(79)

Mathematically, Eqs. (78) and (79) coincide with the Langevin B Model used to
model the interfacial structure created after quenches through equilibrium
PTs. This model is built from the Landau-Ginzberg-Wilson free energy
functional, $\mathpzc{F}_{LGW}$. In this case the first coefficient
$(D-D_{TI})$ changes sign at the transition temperature and the physics behind
this is well understood without invoking TI. Even though it is tempting to
associate $X_{DR}$ with $(T_{c}-T)/T_{c}^{2}$ in the specific quenching case
discussed here, the physical reasoning for any induction is unclear. As things
stand, Eqs. (78) and (79) should be only thought of as an effective model for
PTs with conserved order parameters. From the modeling point of view, Eqs.
(76) and (77) may provide better results than Eq. (79) since they better
account for fluctuations.

Returning to fundamental thermodynamics, it is not surprising that the Landau-
Ginzberg-Wilson free energy functional, $\mathpzc{F}_{LGW}$ has been used to
model nonequilibrium phenomena, given the success of using this functional
towards understanding second order equilibrium PTs Desai and Kapral (2009).
However the physical significance of minimizing this functional has never been
established since $\mathpzc{F}_{LGW}$ is the Helmholtz free energy and can
only be minimized in thermodynamic equilibrium. The reality is that
$\mathpzc{F}_{LGW}$ has been used because there were no other options
available. Thermodynamic functionals that have extrema at nonequilibrium PTs
must be different, and I have shown here how these functionals are
constructed, with $\mathpzc{F}$ being the best substitute for
$\mathpzc{F}_{LGW}$.

###  III.3 Case where the DR is the $q=0$ mode

At this point I will emphasize important systems that approach equilibrium
through diffusion, i.e., where the relaxation rate $1/\tau_{GT}$ is
effectively $Dq^{2}$. In particular I focus on systems where the complete set
of $q$ modes acts as both DR and gate. The $q=0$ mode is the DR and all other
wavevectors are gate states. To make a clear demarcation between DR and gate,
the induction must disappear as $q$ approaches zero. For this reason I
consider only systems with $\alpha_{0}=0$ and I consider specifically the
restricted case where $\alpha=\alpha_{2}q^{2}$. These results will be
applicable to two important physical examples discussed below in Sec. III.4
and Sec. III.5.

The DR dynamics is given by

$\dot{b}_{0}=L_{DR}X_{DR}+\frac{1}{2}\alpha_{2}X_{DR}\sum_{q}q^{2}b_{q}^{2}~{}\,.$
(80)

For the variables describing the gate, Eq. (16) becomes,

$\dot{b}_{q}=2\frac{\alpha_{2}}{g_{2}}Dq^{4}\tau^{*}X_{DR}^{2}b_{q}-Dq^{2}b_{q}-\frac{g_{4}}{g_{2}}Dq^{2}b_{q}^{3}~{}\,.$
(81)

Equations (80) and (81) are extensively coupled in general since
$X_{DR}=-g_{DR}b_{0}$. However when the DR is very large and slow, $X_{DR}$ is
considered to be essentially constant in Eq. (81). There is an issue with Eqs.
(80) and (81) having a discontinuity at $q=0$. From Eq. (81) $\dot{b}_{q}$ is
very small for small $q$, while $\dot{b}_{0}$ may in fact be quite large. The
best way to deal with this is to use the well-known procedure of redefining
the real-space variable $\phi$ as following the ”front” as done in the
formulation of the KPZ equation Desai and Kapral (2009). To linear order this
is done by adding the spatially uniform term $L_{DR}X_{DR}t$ to $\phi$. This
works well if $L_{DR}X_{DR}$ varies slowly in time and this is indeed the case
when studying short-term front dynamics. One can refine this by adding the
(also spatially uniform) term
$\frac{1}{2}\alpha_{2}X_{DR}\sum_{q}q^{2}b_{q}^{2}$ as well, to follow the
growth more accurately. Explicitly then the new front-following field is
$\phi^{\prime}=\phi-(L_{DR}X_{DR}+\frac{1}{2}\alpha_{2}X_{DR}\sum_{q}q^{2}b_{q}^{2})t$.
For this front-following procedure, a spatial average is implied, so the
dynamics for the $b_{q}$ modes in Eq. (81) is unaffected, while for the primed
variable Eq. (80) for the $q=0$ mode becomes a trivial identity. This
procedure will work just as well on systems without a physical
front/interface, for example, when $b_{0}$ represents the extent of reaction
during a chemical reaction. When taking the Fourier transform of $b_{q}$ (see
below), Eq. (81) does indeed have $\dot{b}_{q}$ continuously going to zero as
$q$ approaches zero. For further analysis below the prime on $\phi^{\prime}$
is dropped for convenience and $\phi$ is to be understood as tracking the
front.

The appropriate potentials at this level of analysis are the infinite $Q$
potentials $\Phi_{2,\infty Q}$ and $\Psi_{2,\infty Q}$, each functions of the
set $b_{q}$. For $\Phi_{2,\infty Q}(b)$, Eq. (64) becomes:

$\Phi_{2,\infty Q}=L_{DR}X_{DR}^{2}+\sum_{q\neq
0}{\left[\alpha_{2}X_{DR}^{2}q^{2}b_{q}^{2}-\lambda_{q}\left(-2\alpha_{2}\left(X_{DR}^{2}-X_{c,q}^{2}\right)Dq^{4}\tau^{*}b_{q}^{2}+g_{4}Dq^{2}b_{q}^{4}\right)\right]}~{}\,,$
(82)

where $X_{c,q}=\sqrt{g_{2}/2\alpha_{2}\tau^{*}q^{2}}$, $\lambda_{q}=0$ when
$X_{DR}<X_{c,q}$, and
$\lambda_{q}=(1/2Dq^{2}\tau^{*})(X_{DR}^{2}/(X_{DR}^{2}-X_{c,q}^{2})$ when
$X_{DR}\geq X_{c,q}$. Below the threshold for each $q$ there is no induction
in the infinite $Q$ limit. Defining
$q_{c}\equiv\sqrt{g_{2}/2\alpha_{2}\tau^{*}X_{DR}^{2}}$, the sum $\sum_{q\neq
0}$ in Eq. (82) may be replaced by $\sum_{q>q_{c}}$. The corresponding free
entropy, from Eq. (67) is

$\Psi_{2,\infty
Q}=\sum_{q>q_{c}}{\left[2\alpha_{2}X_{DR}^{2}\tau^{*}q^{2}b_{q}^{2}-g_{2}b_{q}^{2}-\frac{1}{2}g_{4}b_{q}^{4}\right]}~{}\,.$
(83)

If the linearized version of Eq. (81) is Fourier transformed and an
appropriate large amplitude cutoff is re-introduced then the following PDE is:

$\frac{\partial\phi}{\partial
t}=2\frac{\alpha_{2}}{g_{2}}D\tau^{*}X_{DR}^{2}\nabla^{4}\phi+D\nabla^{2}\phi+\frac{g_{4}}{g_{2}}D\nabla^{2}\phi^{3}~{}\,.$
(84)

A functional defined as:

$\Psi_{2,\infty
Q}[\phi(\mathbb{r},t)]\equiv-\int{d^{d}\mathbb{r}\left[2\alpha_{2}X_{DR}^{2}\tau^{*}(\overrightarrow{\nabla}\phi)^{2}+g_{2}\phi^{2}+\frac{1}{2}g_{4}\phi^{4}\right]}~{}\,,$
(85)

is appropriate for determining nonequilibrium dynamics by taking one
functional derivative with $X_{DR}$ held constant:

$\frac{\partial\phi}{\partial
t}=\frac{2g_{2}}{D}\nabla^{2}\left(\frac{\delta\Psi_{2,\infty
Q}}{\delta\phi}\right)~{}\,.$ (86)

A time-independent function $\phi(\mathbb{r})$ that maximizes $\Psi_{2,\infty
Q}$ will also be a stationary state. When dealing with fluctuations, one
adapts Eq. (27) to become

$\frac{du_{q}}{dt}=4\frac{\alpha_{2}}{g_{2}}\tau^{*}\left(X_{DR}^{2}-X_{c}^{2}\right)Dq^{4}u_{q}+4\frac{\alpha_{2}}{g_{2}}\tau^{*}X_{DR}^{2}Dq^{2}\langle
b^{2}\rangle_{0}-2\frac{g_{4}}{g_{2}}Dq^{2}u_{q}^{2}~{}\,.$ (87)

In terms of the real space fluctuation function $\psi(\mathbb{r},t)$:

$\frac{\partial\psi}{\partial
t}=2D\nabla^{2}\left[2\frac{\alpha_{2}}{g_{2}}\tau^{*}X_{DR}^{2}\nabla^{2}\psi+\psi+2\frac{\alpha_{2}}{g_{2}}\tau^{*}X_{DR}^{2}\langle
b^{2}\rangle_{0}\nabla^{2}\delta(\mathbb{r})+\frac{g_{4}}{g_{2}}\psi^{2}\right]~{}\,.$
(88)

In this section I have established the first principles theory required to
treat two important physical examples, discussed next. In these systems
relaxation occurs by diffusion with rate $Dq^{2}$, the complete set of $q$
modes acts as both DR and gate, and there is a sound physical basis for having
nonzero $\alpha_{2}$.

###  III.4 Front Propagation

Phase separation is common in nonequilibrium systems and the interface/front
between the two phases will be expected to move as one phase overtakes the
other. These fronts can form after a quench occurs through an equilibrium PT.
The velocity $v$ of the front, when spatially averaged, will follow the
typical dissipative relation:

$v=M_{DR}X_{DR}~{}\,,$ (89)

where $X_{DR}=-\Delta\mu/T$ and $\Delta\mu$ is the chemical potential
difference between the two phases in question. If the mean front velocity is
in the $z$ direction, and $\chi(y,z)$ is the local position of the front then
purely geometric effects arising from the eikonal factor
$\sqrt{1+(\overrightarrow{\nabla}\chi)^{2}}$ play an important role in the
dynamics Desai and Kapral (2009); Kuramoto and Tsuzuki (1976). Incorporating
this effect into the rate of front motion results in the following expression
for the Onsager coefficient for each Fourier mode $q$:

$M_{DR,q}=L_{DR}\left(1+\frac{1}{2}q^{2}b_{q}^{2}\right)~{}\,,$ (90)

where the small amplitude approximation was made, and the $b_{q}$ are Fourier
coefficients of $\chi$. The eikonal factor means that curvature in the
interface will enhance the spatially averaged dynamics. This enhancement
attenuates at long wavelengths. From the definition of $\alpha_{q}$ the
eikonal factor directly results in TI2 with

$\alpha_{q}=\frac{1}{2}L_{DR}q^{2}~{}\,.$ (91)

Thus, the following structure emerges: 1) the $q=0$ mode is the DR variable,
2) all modes with $q\neq 0$ are gate modes, and 3) the DR and gates are
distinct in this scheme because the TI2 effect goes as $q^{2}$ and gate states
near $q=0$ are affected very little. In this system both the DR and gate are
built into the same continuous variable describing the front. The DR variable
is the overall (spatially averaged) position of the front, which is being
driven by the chemical potential difference between the two competing phases.
Equation (80) represents not only the overall motion of the front but also the
considerable rate at which entropy is being created as one phase turns into
another. It is precisely this rate of entropy increase that prevents the SLT
from being violated if any patterns were to form in the shape of the front.
The gate variables in this case describe the shape of the propagating front.
As the front propagates, the front will be exchanging particles with both
phases (the particle reservoir) and possibly heat, with again the phases on
both sides of the front. Thus, it is the shape of the front and all of this
exchanging that is actually the gate, which winds up being almost all of the
system, in this example. Equation (91) represents the entropic coupling
between the DR and the gate modes, $q\neq 0$.

All of Eqs. (80) through to (88) apply to this case with
$\alpha_{2}=\frac{1}{2}L_{DR}$. In particular the DR dynamics is given by Eq.
(80) while in real space the dynamics of the spatially averaged front position
is:

$\frac{d\widetilde{\chi}}{dt}=L_{DR}X_{DR}\left(1+\frac{1}{2}\widetilde{(\overrightarrow{\nabla}\chi)^{2}}\right)~{}\,.$
(92)

As pointed out in the literature, the front velocity is always greater when
undulations in the front, or interface, are taken into account Desai and
Kapral (2009). This is an example of the nonequilibrium Le Chatelier’s
principle.

When treating the dynamics for the gate states, the relaxation of undulations
in the front shape occurs by diffusion only and the relaxation rate is
$Dq^{2}$. When Eq. (81) is linearized and expressed as
$\dot{b}_{q}=(D_{TI}-D)b_{q}$ where
$D_{TI}=L_{DR}Dq^{2}\tau^{*}X_{DR}^{2}/g_{2}$. The result is a variable
effective diffusion coefficient $D-D_{TI}$ which can become negative with
large enough $X_{DR}$, i.e., at
$X_{DR}=X_{c}(q)=\sqrt{g_{2}/L_{DR}\tau^{*}q^{2}}$. This is the main result as
far as how TI affects front propagation. The Fourier transformed linear
equation is:

$\frac{\partial\chi(\mathbb{r})}{\partial
t}=D\nabla^{2}\chi(\mathbb{r})+\frac{L_{DR}D\tau^{*}X_{DR}^{2}}{g_{2}}\nabla^{4}\chi(\mathbb{r})~{}\,.$
(93)

Equation (93) is very similar to the linearized Kuramoto-Sivashinsky (KS)
equation, which also has both $q^{2}$ and $q^{4}$ terms Desai and Kapral
(2009). It is well known that modeling with the KS equation can produce
instabilities and interesting pattern formation when the diffusion coefficient
is made negative.

The KS equation is another important model in the field of nonequilibrium
systems and has been successfully utilized to model patterns formed in laminar
flame fronts, certain types of Poiseuille flow, trapped ion modes in plasmas,
and systems with Eckhaus instabilities such as Rayleigh-Bénard convection
Kuramoto and Tsuzuki (1976); Sivashinsky (1977); Michelson (1986); Manneville
(1988); Cross and Hohenberg (1993); Desai and Kapral (2009); Cross and
Greenside (2009). Though the SH equation has been used widely as a model, no
clear explanation has been provided in the literature for why a diffusion
coefficient can be negative. The TI2 analysis presented here shows precisely
how this negative (effective) diffusion coefficient in the KS equation can
exist.

The essential physics is now in place for understanding how so many
nonequilibrium phenomena may be modeled, i.e., the possibility of tuning user-
controlled parameters such as $X_{DR}$ to create an effective diffusion
coefficient that is negative, all without violating the SLT. Parameters may be
tuned to produce instabilities and patterns at a certain wavevector, $q_{c}$.
These features, along with the enhanced front velocity from Eq. (92) are the
key elements of the KS model. The analysis presented here improves upon the KS
equation, as it properly accounts for an excitation gap, an essential feature,
I believe, for nonequilibrium PTs. Equation (93) can be derived from a free
entropy functional $\Psi_{2,Q\infty}$ in Eq. (85) which is constructed in the
infinite $Q$ limit. This provides a good example of the usefulness of the
infinite $Q$ analysis, and also motivates the finite $Q$ approach as an
improvement which better accounts for fluctuations.

The analysis presented here is a good example of entropic coupling; This
specific solution to the entropic coupling problem shows how real diffusion is
overcome by TI to produce a nonequilibrium PT in such a way as to not violate
the SLT. Any patterns that may form from undulations of the front have an
entropy cost, but while the pattern is forming, the front as a whole is always
moving fast enough to ensure that the total entropy is always increasing. A
similar conclusion is drawn with the reaction-diffusion system discussed in
the next subsection.

The approach given here can be used to successfully model undulations of a
propagating front separating any two phases with a large miscibility gap
$\Delta\mu$ ($X_{DR}=-\Delta\mu/T$), ex. fronts in laminar flames, liquid-gas
PTs, and autocatalytic chemical reactions. These types of dissipative systems,
as well as any others that can be modeled by TI2, present themselves as strong
evidence for the existence of TI.

###  III.5 Turing Patterns in Chemical Systems

In Sec. III.4 I discussed an example of second order TI with a physical
interface (front). In more general terms, a physical interface is not
necessary for spontaneous pattern formation. Turing suggested that a uniform
chemical mixture could, under the right reaction conditions, spontaneously
self-organize and form patterns Turing (1952); Cross and Greenside (2009).
Nonlinearity is required in the reaction-diffusion equations; In theory, under
the right conditions a bifurcation in the dynamics could be reached and the
instability would break symmetry to produce patterns.

Recently, Turing patterns have been observed while observing the chlorite-
iodide-malonic acid (CIMA) reaction and the Belousov-Zhabotinsky (BZ) reaction
which involves bromide and bromous acid Castets  _et al._ (1990); Ouyang and
Swinney (1991a, b); Zhabotinsky (1991); Ouyang  _et al._ (1992); Vanag and
Epstein (2001). For example, the well-studied CIMA reaction is known for
producing two-dimensional patterns possessing clear crystal-like symmetry,
some with hexagonal patterns, others resembling modulated stripes, as well as
mixed states Castets  _et al._ (1990); Ouyang and Swinney (1991a, b); Cross
and Hohenberg (1993); Cross and Greenside (2009). That almost four decades
passed between Turing’s theoretical paper and the first experimental
observations of Turing patterns, shows that in most chemical reactions it is
very difficult to produce $X_{DR}$ values large enough to achieve pattern
formation. In the systems which do show patterns, diffusion coefficients must
be optimized, presumably bringing the bifurcation point closer to
thermodynamic equilibrium and thus easier to access. Clearly these are not
transitions that occur near thermodynamic equilibrium and qualify as
nonequilibrium PTs.

For these interesting reactions a thermodynamic analysis is warranted. If
reaching the threshold for this type of nonequilibrium PT is challenging then
it makes sense to study how the system will behave both near critical and just
below. I consider a chemical reaction much simpler than the CIMA and BZ
reactions. A second order chemical reaction will suffice to produce the TI
effects which will demonstrate the induced pattern formation, so for the sake
of simplicity, the following reaction is considered

$A+2B\longleftrightarrow C\,.$ (94)

The volume concentration $n_{B}$ is assumed to be much smaller than $n_{A}$,
making $n_{B}$ the bottleneck, or gate variable, for the system, i.e., there
is plenty of $A$ but a small amount of $B$ that limits the reaction. The
extensive thermodynamic variable $N_{A}=n_{A}V$ is identified with the
dynamical reservoir (DR), i.e., $x_{DR}=N_{A}$. When driven away from the
equilibrium value $N_{A_{0}}$ by an amount $\Delta N_{A}\equiv a_{DR}$, the
relaxation time for the DR is very long so that over the gate timescales,
relative changes in $N_{A}$ are always small. The thermodynamic force
conjugate to $a_{DR}$ is

$X_{DR}=-\frac{A}{T}\,,$ (95)

expressed in terms of the affinity de Groot and Mazur (1984); Kondepudi and
Prigogine (1998):

$A=\mu_{A}+2\mu_{B}-\mu_{C}\,.$ (96)

The DR capacitance coefficient, given by $g_{DR}^{-1}=N_{A}/k_{B}$, is
considered very large in this analysis.

The DR dynamics is given in terms of standard chemical rate theory with the
reaction rate $v$ given by

$v=R_{F}\left(1-e^{-A/k_{B}T}\right)\,,$ (97) $R_{F}=k_{f}n_{A}n_{B}^{2}\,,$
(98)

where $R_{F}$ is the forward reaction rate and $k_{f}$ is the forward rate
constant. For very small affinity Eq. (97) becomes $v=R_{F}A/k_{B}T$ and the
dynamics can ordinarily be expressed in terms of Eq. (1), with the Onsager
coefficient given by

$M_{DR}=\alpha_{0}N_{B}^{2}\,,$ (99)

with

$\alpha_{0}=\frac{k_{f}n_{A}}{Vk_{B}}\,.$ (100)

In the limit of very small $n_{B}$ the bottleneck effect is severe as there is
so little of component B available. In this case the reaction can proceed
forwards more rapidly by having some regions gaining in component B at the
expense of other nearby regions, i.e., by spontaneously forming undulations in
$n_{B}$. This type of borrowing and lending of B will be difficult at long
wavelengths but rather easy to accommodate at microscopic lengthscales. This
motivates a reaction rate that is proportional to
$(\overrightarrow{\nabla}n_{B})^{2}$ as opposed to simply $n_{B}^{2}$. The
coefficient $\alpha_{0}$ in Eq. (99) is replaced by
$\alpha_{0}q^{2}/q_{0}^{2}$ where $q_{0}^{-1}$ defines a length scale below
which the borrowing/lending becomes significant.

When the concentrations vary spatially, Eq. (99) is modified by replacing
$N_{B}^{2}$ by the spatial average which is denoted with a tilde as
$\widetilde{N_{B}^{2}}$. Any patterns that we seek will be expressed by
Fourier expansion of the density function for component $B$ as

$n_{B}=n_{0}+\sum_{q\neq 0}{n_{q}}\cos(qr-\phi_{q})\,.$ (101)

Spatial averaging after squaring gives

$\widetilde{n_{B}^{2}}=n_{0}^{2}+\frac{1}{2}\sum_{q\neq 0}{n_{q}^{2}}\,.$
(102)

The gate variables are defined as $x_{0}=n_{0}V$, and $b_{q}\equiv Vn_{q}$
with $q\neq 0$. Comparison of Eq. (12) with Eq. (102) shows that
$\alpha_{q}=\alpha_{0}q^{2}/2q_{0}^{2}$ and
$\alpha_{2}=\alpha_{0}/2q_{0}^{2}$.

Systems displaying Turing patterns are often referred to as reaction-diffusion
systems because of the important role played by diffusion V. and J. (1991);
Desai and Kapral (2009). When the system relaxes to equilibrium via diffusion,
$g_{q}=\Delta\mu/\Delta N_{B}=\zeta\mu_{B}/n_{B}V$, where $\zeta$ is a
dimensionless parameter typically close to unity. After incorporating
diffusion as well as second order TI and the cut-off term, all the dynamics
are well-described by Eqs. (80) through to (88). In particular:

$\dot{b}_{q}=\left(\frac{A^{2}}{A_{c,q}^{2}}-1\right)Dq^{2}b_{q}-c_{q}^{2}b_{q}^{3}~{}\,.$
(103)

Similarly to the KS equation the possibility of an effectively negative
diffusion coefficient arises when the affinity is large enough in magnitude.
The critical condition for the affinity is

$|A_{c,q}|=\sqrt{\frac{q_{0}^{2}g_{q}T^{2}}{{q^{2}\alpha_{0}\tau_{q}^{*}}}}~{}\,.$
(104)

Thus, in an experiment where concentration levels for A or C are pushed from
equilibrium far enough, the system will become unstable and Turing pattern
formation could occur. Equation (104) defines the dispersion relation and
shows clearly the existence of a threshold; Control parameters such as
reactant concentrations are adjusted until $A$ is away from equilibrium by a
sufficient amount. This is indeed observed in the reported literature. For
example in one report on the CIMA reaction, a clear threshold at a specific
malonic acid concentration was observed as the system transitioned from the
normal state to a striped pattern Ouyang  _et al._ (1992). Even though the
CIMA reaction has a good deal more complexity than Eq. (94) and there exists
more than one bifurcation point, the essential feature of a threshold for a
nonequilibrium PT has been established. Note that the analysis presented here
does not predict the precise type of patterns that are observed in CIMA and BZ
reactions; These reactions are more complex than Eq. (94). Reported analysis
of the stability equations for the CIMA reaction have been shown to predict
specific patterns Pismen (1980); Cross and Hohenberg (1993); Desai and Kapral
(2009). Fluctuations should play an important role in the specifics of pattern
formation. Indeed, the long range order analysis presented below in Sec. III.6
shows that pattern formation could be more robust in lower-dimensional
systems. Thus, with more work, TI2 could explain multiple bifurcation points
and complex phase diagrams.

The dynamics of Eq. (103) can be derived from a free entropy functional
$\Psi_{2,Q\infty}$ in Eq. (85) which is constructed in the infinite $Q$ limit,
thus providing another application of this useful functional. It is
interesting to go beyond this approach and include effects of nonequilibrium
fluctuations. For example, my analysis shows that just below critical there
should be large fluctuations that could be interpreted as chemical turbulence.
It is interesting that the CIMA reaction does indeed show a transition between
striped patterns and chemical turbulence under certain conditions Ouyang and
Swinney (1991a). This might be the result of frustration between hexagons,
stripes and rhombic patterns, but it might also be the result of thermodynamic
fluctuation just below the instability point. If so then the dynamics would be
described by the following equation for the fluctuations in $b_{q}$:

$\dot{u}_{q}=2\left(\frac{X_{DR}^{2}}{X_{c}^{2}(q)}-1\right)\frac{u_{q}}{\tau_{q}}+2\left(\frac{X_{DR}^{2}}{X_{c}^{2}(q)}\right)\frac{\langle
b_{q}^{2}\rangle_{0}}{\tau_{q}}-2Dq^{2}u_{q}-2c_{q}^{2}u_{q}^{2}~{}\,.$ (105)

Large thermodynamic fluctuations below critical might also explain how for the
case of the BZ reaction, reports make mention of a long-lived and complicated
transient state before a sustained pattern is produced Winfree and Strogatz
(1983); Cross and Greenside (2009).

The long range order analysis presented below in Sec. III.6 shows that
nonequilibrium fluctuations may make pattern formation difficult in 3
dimensions, and likely easier in 2 dimensional systems. Given the typical
design of reaction cells in the literature Winfree and Strogatz (1983);
Castets  _et al._ (1990); Ouyang and Swinney (1991a, b); Ouyang  _et al._
(1992); Vanag and Epstein (2001) as a gel contained between two closely spaced
plates, it makes sense that pattern formation may be difficult for $q$ values
larger than the inverse of the plate spacing (typically 0.2 mm for the CIMA
reaction Ouyang  _et al._ (1992)).

It is worth pointing out the similarities between this type of Turing pattern
formation in reaction-diffusion systems and what occurs in Langevin B systems
immediately after a quench through an equilibrium PT, with initial conditions
being a completely uniform system. Well before formation of any clear fronts
and boundaries between two distinct phases, density undulations at certain
wavevectors would be required to break symmetry. The equations presented here
predict that these undulations would occur at larger spatial frequencies, at
least for short times. These undulations would after long times develop into
fronts separating phases.

Finally, I note that I have now produced two good examples with nonequilibrium
PTs and in both cases the PTs begin first at high spatial frequencies for
modest $X_{DR}$. More examples are needed but this behaviour at high spatial
frequencies is an indicator of universal behaviour Sethna (2006). In these two
examples the dynamics are generally quite slow and inertial effects are not
prominent. Future TI work on faster systems such as ultrasonic propagation and
second sound would require more development. The recent formulation of
Extended Irreversible Thermodynamics is well-suited for studying such fast
phenomena Jou  _et al._ (1988). Incorporating the DR into this formulation
should allow for future TI studies of such fast systems.

###  III.6 Fluctuations with Many Modes and Long Range Order

Any patterns that may be formed near and/or above a critical point will be
determined by a sum over Fourier modes $q$, and such patterns would be
examples of order-by-induction. Some level of long range order would be
required to create a discernible pattern and any function used to describe
long range order should also involve a (weighted) summation over $q$ of plane
waves. The dispersion relation $X_{c}(q)$ will play an important role in the
weighting. For example, if all waves $e^{iqx}$ are weighted equally then there
will be no long range order and one would not expect pattern formation.
Instead one may see an increase in fluctuations at all length scales, a result
resembling hydrodynamic turbulence. Such an effect, termed chemical
turbulence, has been observed in reaction-diffusion systems Ouyang and Swinney
(1991b); Scholz  _et al._ (2015). It may indeed be the case that in systems
exhibiting chemical turbulence, the dispersion relation is very flat. In
contrast, if the dispersion relation has a prominent minimum at $q_{min}$ then
one might expect a clear pattern resembling a standing wave with wavevector
$q_{min}$. However, fluctuations may play an important role in disrupting
pattern formation.

For equilibrium PTs it is well known that thermal fluctuations can disrupt
long range order. The Mermin-Wagner theorem dictates that long range order is
lost in one and two dimensional systems because of fluctuations at long
wavelengths Mermin and Wagner (1966); Gelfert and Nolting (2001). To date the
only way around this theorem is the Kosterlitz-Thouless PT which involves
ordering in topological defects such as magnetic vortices in two dimensions
Kosterlitz and Thouless (1973). Many of the observations of pattern formation
in nonequilibrium systems are essentially two dimensional systems. These
systems may also be interesting exceptions to the Mermin-Wagner theorem.
Indeed it may be easier to achieve long range order in one and two dimensions
near nonequilibrium PTs than it is for equilibrium PTs. For equilibrium PTs
long range order is determined using the Boltzmann factor $\exp{(-H/k_{B}T)}$
in the statistical weighting. The Hamiltonian $H$ typically goes as $q^{2}$ as
it does for systems with spin-spin interactions. The $q^{2}$ dependence makes
the weighting factor soft for long wavelength (Goldstone) modes Plischke and
Bergerson (1986). In contrast the weighting factor for TI2 is
$\exp{(\Psi_{2}(u))}$. The extent of fluctuations is controlled by the term
$-\frac{1}{2}g_{4}u^{2}$ which does not scale as $q^{2}$ and is not expected
to depend on $q$, as the entropy cost of an undulation depends on the
amplitude of the undulation, not the slope. Thus the weighting factor for TI2
remains stiff at large length scales. This means that one does not expect long
wavelength Goldstone modes to disrupt long range order.

For the case where the dispersion relation $X_{c}(q)$ has a minimum at finite
$q_{min}$ then for large $Q$ the PT occurs when $X_{DR}$ is raised up to
$X_{c}(q_{min})$, with the expectation of a clear pattern and long range
order. When $X_{DR}$ is raised slightly higher, modes with $q$ values over the
range from $q_{1}$ to $q_{2}$, where $X_{c}(q_{1})=X_{DR}$ and
$X_{c}(q_{2})=X_{DR}$, will be excited. These excitations are the Goldstone
modes for this case. Following the example for equilibrium PTs, Goldstone
modes are excitations on top of the (nonequilibrium) PT that have a small cost
in $\Psi_{2}$, or the nonequilibrium free energy, $\mathpzc{F}$. If, for the
sake of simplicity, each excited mode is weighted equally, the sum:
$\langle\psi\rangle=\sum_{q}\cos(\bf{q}\cdot\bf{r})$ can show effects of the
disruption of long range order. These sums are similar to the ones used in
elementary optics calculations for coherence in diffracting systems. For
example, in one dimension
$\langle\psi\rangle=q_{2}\text{sinc}(q_{2}x)-q_{1}\text{sinc}(q_{1}x)$. When
$\Delta q=q_{2}-q_{1}$ is small, $\langle\psi(x)\rangle=2\Delta
q\text{sinc}(\Delta qx)\cos(q_{min}x)$, and one identifies the correlation
function as the envelope of $\langle\psi\rangle$, here $2\Delta
q\text{sinc}(\Delta qx)$. As expected then, as $X_{DR}$ is raised higher,
$\Delta q$ grows and the Goldstone modes decrease long range order. Thus if
the quality factor is very high and if $X_{DR}$ is finely tuned enough then
formation of a clear pattern is guaranteed. In some systems $q_{min}$ may be
very large and the pattern not easily observed at small length scales.

For a more general analysis of long range order the correlation function
$\langle\psi(\bf{r})\psi(0)\rangle$ should be investigated. Taking the Fourier
transform and ignoring a prefactor gives

$K(\textbf{r})\equiv\sum_{q}e^{i\textbf{q}\cdot\textbf{r}}\langle\text{u}_{\text{q}}\rangle~{}\,.$
(106)

The average $\langle u_{q}\rangle$ is taken with the weighting factor
$\exp{(\Psi_{2}(u_{q}))}$. The result has already been calculated as the
stationary state from Eq. (28), re-expressed as

$\langle u_{q}\rangle=\langle b_{q}^{2}\rangle_{0}+2Q^{2}\langle
b_{q}^{2}\rangle_{0}\left[z_{q}^{2}-1+\sqrt{(z_{q}^{2}-1)^{2}+z_{q}^{2}/Q^{2}}\right]\,,$
(107)

where $z_{q}=X_{DR}/X_{c}(q)$. The function describing long range order would
be the modulus of $K(\textbf{r})$. When $X_{DR}$ is below the excitation gap,
$K$ is small and then becomes substantial as the excitation threshold is
reached. In the two important examples discussed in Sec. III.4 and Sec. III.5,
the induction was derived from first principles and in both cases $\alpha_{q}$
goes as $q^{2}$ and the dispersion relation $X_{c}(q)$ goes as $q^{-1}$. This
would imply gapless excitation at infinite $q$. However, at microscopic length
scales induction is expected to taper off, and there is a physical cutoff
$q_{cut}$ for the wave vector. For example, in a chemical reaction the mean
free path would likely dictate the value of $q_{cut}$. The excitation gap is
then $X_{c}(q_{cut})$ and this would likely be much smaller than the
excitation gap in a system with a flat dispersion relation. Though, systems
with $\alpha_{q}$ proportional to $q^{2}$ will allow easier excitation of
nonequilibrium PTs, these transitions are more difficult to discover and study
with the very small pattern period.

If $X_{DR}$ is raised further above $X_{c}(q_{cut})$ the period of the pattern
will increase and should become easier to observe. For such $X_{DR}$ the
critical wavevector is $q_{c}<q_{cut}$ such that $X_{c}(q_{c})=X_{DR}$. In the
calculation for $K(\textbf{r})$, $q$ values with magnitudes between $q_{1}$
and $q_{cut}$ will make the largest contributions. For large $q$ the expansion
$\sqrt{1+q^{2}b_{q}^{2}}\approx 1+\frac{1}{2}q^{2}b_{q}^{2}$ used in Sec.
III.4 for the eikonal will break down. This produces a natural cutoff at large
$q$ since $\sqrt{1+q^{2}b_{q}^{2}}\approx qb_{q}$ means that TI2 becomes TI1
which leads to a stationary state $b_{q,ss}=1/q$, and $u_{q,ss}=1/q^{2}$. This
result represents a type of universality since any nonequilibrium front will
show this $b_{q,ss}=1/q$ behaviour for $q>q_{c}$. The critical wavenumber
$q_{c}$ will set the periodicity of any patterns and one would identify this
mode with the PT. Technically, all wavenumbers larger than $q_{c}$ are
involved in individual PTs. Since excitation of these modes tend to suppress
long range order of the mode at $q_{c}$, they may be thought of as Goldstone
modes. For these types of nonequilibrium PTs, Goldstone modes present
themselves at high spatial frequencies and may be generally characterized as
turbulence.

When $u_{q}=1/q^{2}$ the sum in Eq. (106) allows easy calculation of
$K(\textbf{r})=r^{2-d}\cos{(q_{c}r)}$. In one and two dimensions, the long
range order is strong, and somewhat marginal in three dimensions. This trend
is opposite to what happens in equilibrium PTs where long range order is
disrupted by Goldstone modes at low dimensions.

Finally, I note that these results favour the possibility of fluctuations at
large $q$ being enhanced by TI. If so, then the physics at the nanometer scale
becomes even more interesting; At very small length scales systems that are
profoundly influenced by thermal and quantum fluctuations, may also be
influenced by these types of nonequilibrium fluctuations.

###  III.7 Nonequilibrium Thermodynamic Identity

The main result presented as $\mathpzc{S}_{NE}$ being maximized by stationary
states, may be expressed as $d\mathpzc{S}_{NE}=0$. For equilibrium
thermodynamics, the total entropy $S_{T}$ is maximized which leads to the
thermodynamic identity: $dU=TdS-PdV+\sum_{i}\mu_{i}dN_{i}$, where $S$ refers
to the system of interest that is removed from the thermal reservoir, particle
reservoirs, etc..

The generalized entropy $\mathpzc{S}_{NE}$ accounts for the gate which
includes traditional thermodynamic reservoirs. The next step is to take the
gate and separate away the system of interest from such reservoirs including
but not limited to thermal, volume, and particle reservoirs. The variables
$x_{GT}$ and $b$ are adapted to $x_{j}$, $b_{j}$ respectively, where $x_{j}$
can be $U$, $V$, $N_{i}$, etc. This accounts for the Onsager coefficient
$M_{DR}$ depending on internal energy, volume, particle numbers, etc. so that
$\gamma_{j}=\partial M_{DR}/\partial x_{j}$. Equations (38) and (39) are
modified to account for more than one gate variable:

$\mathpzc{S}_{NE}=S_{GT}+\sum_{j}\Xi_{0,j}+\sum_{j}k_{B}\beta_{1,j}b_{j}+\sum_{j}k_{B}\beta_{2,j}u_{j}\,.$
(108)

Separation of the system of interest away from the traditional reservoirs is
straightforward, though care must be taken to distinguish the standard
variables such as $U$, $V$ $N_{i}$, from the mode variable $b_{q}$ that become
excited in TI2. At this point I will explicitly treat the three types
reservoirs, thermal, volume, and particle, where it is understood that there
are least two components present (ex. 2 phases to form a front, or 3
components for chemical reaction, as in the above examples) which allows for
one variable with Fourier modes $b_{q}$ and $u_{q}$:

$\displaystyle\mathpzc{S}_{NE}$ $\displaystyle=$ $\displaystyle
S-\frac{U}{T}-\frac{PV}{T}+\frac{1}{T}\sum_{i}{\mu_{i}N_{i}}+S^{\prime}$ (109)
$\displaystyle+k_{B}\beta_{1,U}(U-U_{eq})+k_{B}\beta_{1,V}(V-V_{eq})+k_{B}\sum_{i}{\beta_{1,N_{i}}(N_{i}-N_{i,eq})}$
$\displaystyle+k_{B}\beta_{2,U}u_{U}+k_{B}\beta_{2,V}u_{V}+k_{B}\sum_{i}\beta_{2,N_{i}}u_{N_{i}}$
$\displaystyle+k_{B}\sum_{q\neq 0}\beta_{2,q}u_{q}\,,$

where $S$ is the entropy of the system of interest, and $S^{\prime}$ is
constant. When applying the condition $d\mathpzc{S}_{NE}=0$ it is understood
that $U$, $V$, $N$, $u_{U}$, $u_{V}$, and $u_{N}$ are varied independently
while $T$, $P$, $\mu$, $\beta_{1,U}$, etc. are held constant. The result is a
revised version of the thermodynamic identity:

$\displaystyle dU$ $\displaystyle=$ $\displaystyle TdS-PdV+\mu dN$ (110)
$\displaystyle+k_{B}T\beta_{1,U}dU+k_{B}T\beta_{1,V}dV+k_{B}T\sum_{i}\beta_{1,N_{i}}dN_{i}$
$\displaystyle+k_{B}T\beta_{2,U}du_{U}+k_{B}T\beta_{2,V}du_{V}+k_{B}T\sum_{i}\beta_{2,N_{i}}du_{N_{i}}$
$\displaystyle+k_{B}T\sum_{q\neq 0}\beta_{2,q}du_{q}\,.$

Equation (110) is illuminating as to the influence of the DR on the system of
interest. When $X_{DR}$ is held constant by some external agent then the
dynamical reservoir has the same type of effect on the system of interest as
the traditional reservoirs. For example, the effect of $M_{DR}$ depending on
$V$ gives an effective pressure $P_{eff}=P-k_{B}T\beta_{1,V}$. Thus, the
classical thermodynamic variables can be altered when the DR is well away from
equilibrium.

Just as working under conditions of constant temperature, pressure, and
chemical potential is often more convenient, so is likely the case for
constant $\beta_{1,U}$, etc. Legendre transformations for the nonequilibrium
variables are just as straightforward as for the equilibrium variables. For
example a generalized Gibbs potential may be defined as

$\mathpzc{G}\equiv U+PV-ST-
k_{B}T\beta_{1,U}U-k_{B}T\beta_{1,V}V-k_{B}T\beta_{2,U}u_{U}-k_{B}T\beta_{2,V}u_{V}\,.$
(111)

Implementing the thermodynamic identity gives for the differential:

$\displaystyle d\mathpzc{G}$ $\displaystyle=$
$\displaystyle\sum_{i}\mu_{i}dN_{i}+VdP-
SdT+k_{B}T\sum_{i}\beta_{1,N_{i}}dN_{i}+k_{B}T\sum_{i}\beta_{2,N_{i}}du_{N_{i}}$
(112) $\displaystyle+k_{B}T\sum_{q\neq 0}\beta_{2,q}du_{q}$ $\displaystyle-
k_{B}Ud(T\beta_{1,U})-k_{B}Vd(T\beta_{1,V})-k_{B}u_{U}d(T\beta_{2,U}u)-k_{B}u_{V}d(T\beta_{2,V})\,.$

Under conditions of constant $T$, $P$, $u_{N_{i}}$, $\beta_{1,U}$,
$\beta_{1,V}$, $\beta_{2,U}$, $\beta_{2,V}$, and $u_{q}$,
$d\mathpzc{G}=\sum_{i}\mu_{i,eff}dN_{i}$, where
$\mu_{i,eff}=\mu_{i}+k_{B}T\beta_{1,N_{i}}$.

New Maxwell relations can be derived from Eq. (112). For example
$\partial\mathpzc{G}/\partial\beta_{1,U}=-k_{B}TU$ and
$\partial\mathpzc{G}/\partial\beta_{1,V}=-k_{B}TV$ and equating the crossing
second order derivatives merely results in a consistency check for
$\partial^{2}M_{DR}/\partial U\partial V=\partial^{2}M_{DR}/\partial V\partial
U$. Another Maxwell relation is created by crossing $P$ and $\beta_{1,V}$,
giving

$\frac{\partial V}{\partial\beta_{1,V}}=-k_{B}T\frac{\partial V}{\partial
P}\,.$ (113)

It is understood that the isothermal compressibility on the right hand side of
Eq. (113) is taken at constant $\beta_{1,V}$ and is not the same as the
traditional isothermal compressibility.

Establishment of the nonequilibrium thermodynamic identity, Eq. (110) shows
that TI can be incorporated into the existing thermodynamic framework by
making use of the DR.

###  III.8 Entropic Coupling Problem With Many Modes

Returning to the important entropic coupling problem, most of the interesting
nonequilibrium systems that exhibit self-organization or pattern formations
would occur with many modes $q$ being excited. A variety of models exist in
the literature that are capable of exhibiting such pattern formation and self-
organization Desai and Kapral (2009); Cross and Greenside (2009). The TI
analysis presented here gives new insights into pattern formation and the key
to gaining these insights is incorporation of the DR into the analysis. With
the DR properly accounted for, one sees that the spontaneous formation of
patterns in the gate do not violate the second law of thermodynamics and that
no patterns can form if the DR is at equilibrium, or even below the excitation
gap. Moreover, the creation of such patterns coincides with an increase in
overall entropy production. In fact, the total rate of entropy production is
maximized. This leads to the somewhat dualistic physical interpretation; 1)
that a large $X_{DR}$ above the excitation gap is what drives the pattern
formation or 2) that the patterns form as the gate facilitates the DR to
create entropy at the greatest rate possible.

Since each mode $q$ acts independently and always adds a positive term to the
total entropy production, there is no limit in principle to how many modes
that can become excited, as long as the $\alpha_{q}$ coefficients are large
enough. With the assumption that all such modes are stationary the entropy
change of the gate due to pattern formation is given by

$\Delta
S_{GT,ss}=-2k_{B}\sum_{q}Q_{q}^{2}\left[z_{q}^{2}-1+\sqrt{(z_{q}^{2}-1)^{2}+z_{q}^{2}/Q_{q}^{2}}\right]\,.$
(114)

With many modes excited this entropy reduction can be substantial. If one
associates this entropy budget with information storage, then it would seem
that the more modes excited the more information that can be stored. If it
assumed that this information would be stored in any pattern that forms then
the situation is not so simple; From the results of Sec. III.6 more excited
modes may actually disrupt pattern formation, because of nonequilibrium
fluctuations of Goldstone modes. There would seem to be a trade-off as far as
the benefit of more excited modes goes. Clearly, more study is warranted on
the topic of information storage in nonequilibrium systems.

##  IV Conclusions

With the general thermodynamics presented here, it is the right time now for
the research community to discuss the establishment of a new set of physical
principles suitable for governing nonequilibrium systems. A new thermodynamic
potential, the generalized entropy, has been constructed such that it is
maximized in nonequilibrium situations when the traditional entropy is not
maximized. The generalized entropy is maximized when the gate is in a
stationary state. Moreover when the entire system is returned to equilibrium
the generalized entropy becomes the traditional entropy. These results have
been summarized in a form very much analogous to the laws of equilibrium
thermodynamics.

The key to arriving at these results was in identifying the dynamical
reservoir and separating this away from the gate, in much the same way as
traditional reservoirs are treated. This separation also answers the
historically important entropic coupling problem. It is now clear that the
second law of thermodynamics is never violated even when patterns self-
organize in the gate. During periods where the entropy of the gate decreases,
the dynamical reservoir creates entropy at a greater rate, ensuring that the
total rate of entropy production is always positive. In two important
examples, front propagation and Turing pattern formation, the dynamical
reservoir is simply the zero wavevector mode. In these examples, pattern
formation with long range order has been shown to be possible, although the
disruptive influence of Goldstone modes has been pointed out. In contrast to
the case of equilibrium second order phase transitions, long range order in
patterns is more robust at lower dimensions.

At the heart of the results presented here is thermodynamic induction.
Developing thermodynamic induction up to second order now allows one to
explain spontaneous symmetry breaking and phase transitions in the
nonequilibrium realm. Real space examples with many wavevector modes have been
discussed here and second order thermodynamic induction in these examples
leads to pattern formation at nonequilibrium phase transitions. Thermodynamic
induction, as a first principles theory, works well at producing the equations
and models such as the Swift-Hohenberg, Edwards-Wilkinson, and Kuramoto-
Sivashinsky equations and that therefore TI provides a good description of all
the systems successfully modeled and reported so far in the literature.
Logically then, the large number of observations of nonequilibrium phenomena
that have been successfully modeled with these equations, becomes evidence for
the existence of thermodynamic induction. For example, the production and
observations of patterns in laminar flame fronts is to be considered as
evidence for second order thermodynamic induction since the first principles
results presented here for second order thermodynamic induction predict the
Kuramoto-Sivashinsky equation and the Kuramoto-Sivashinsky equation has been
developed and used to model the behaviour of these flame fronts.

The approach presented here for obtaining dynamical equations emphasizes
conductance, not energetics. An approach based on energetics, such as a
Hamiltonian-based approach, was destined to be difficult, as evidenced by a
lack of a solution to the entropic coupling problem since the second law of
thermodynamics was established. It is my hope that this work stimulates more
study in the research community of the dynamics of variables coupled to each
other via a conductance, or kinetic coefficient.

##  V Acknowledgments

I thank Cathy J. Meyer for her assistance in editing the manuscript.

##  VI Appendix A

Here is treated the case of two random walk variables, $a$, and $b$, governed
by a multiplicity-type statistical weighting factor
$\Omega[a,b]\equiv\exp(-1/2q_{1}a^{2}-1/2q_{2}b^{2})$. Variable $b$ will have
a constant step length $l_{2}$ while the step length for $a$ is variable, of
the form $l(b)$. If the two variables begin a time step at $(a,b)$ then the
probability of both variables moving to the right (increasing) is proportional