${\bar{\rho}}^{s}$ as before, follows from Eq. (1) and the form (40), as

$\displaystyle\dot{D}\\!\left({\rho}^{s}\parallel{\bar{\rho}}^{s}\right)=\sum_{{{\rm
n}}_{s}}\log\left(\frac{{\rho}^{s}_{{{\rm n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm
n}}_{s}}}\right){\dot{\rho}}^{s}_{{{\rm n}}_{s}}$ $\displaystyle=\sum_{{{\rm
n}}_{s}}\log\left(\frac{{\rho}^{s}_{{{\rm n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm
n}}_{s}}}\right)$ $\displaystyle\phantom{\sum_{{{\rm
n}}_{s}}}\times{\psi}^{sT}\\!\left(e^{-\partial/\partial{{\rm
n}}_{s}}\right){\mathbb{A}}^{s}\operatorname{diag}\left[{\psi}^{s}\\!\left(e^{\partial/\partial{{\rm
n}}_{s}}\right)\right]{\Psi}^{s}\\!\left({{\rm n}}_{s}\right){\rho}^{s}_{{{\rm
n}}_{s}}$ $\displaystyle=\sum_{{{\rm n}}^{\prime}_{s}{{\rm
n}}_{s}}\log\left(\frac{{\rho}^{s}_{{{\rm
n}}^{\prime}_{s}}}{{\bar{\rho}}^{s}_{{{\rm
n}}^{\prime}_{s}}}\right)\cdot{\hat{\mathbb{T}}}^{s}_{{{\rm
n}}^{\prime}_{s},{{\rm n}}_{s}}\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm n}}_{s}}}\right).$ (60)

From the definition (40), $1^{T}{\hat{\mathbb{T}}}^{s}=0$ and
${\hat{\mathbb{T}}}^{s}1=0$, as was the case for _complex-balanced_
${\mathbb{A}}^{s}$. Therefore ${\hat{\mathbb{T}}}^{s}$ can be written as a sum
of adjacency matrices for cycles, weighted by the currents around those cycles
in the steady state. For a general CRN, it would not be assured that these
cycles were all of finite length or that there were only finitely many of them
passing through any given state ${{\rm n}}_{s}$. However, for a CRN in which
both the dimensions of $\mathbb{A}$ and $Y$ and their matrix elements are
finite, the integer hyperflow decomposition under ${\mathbb{A}}^{s}$ and
$Y^{s}$ is in turn finite, and the basis flows can be embedded in the
microstate space to give a decomposition of ${\hat{\mathbb{T}}}^{s}$.

Let $\alpha$ index any basis of integer hyperflows spanning $\ker
Y\mathbb{A}$. The cyclic flows spanning $\ker\mathbb{A}$ embed without
ambiguity in the state space, as images the cycles on the complex network.
Each cycle $\alpha$ defines a sequence of state shifts $\left\\{{\rm
n}\rightarrow{\rm n}+Y_{j}-Y_{i}\right\\}$ when the transitions
$\left\\{ji\mid\alpha\right\\}$ in the cycle are activated in order.

Non-complex-balanced integer flows also create cycles through states, because
by construction the sum of stoichiometric shifts in a balanced flow is zero.
However, there may be a choice in the way a given flow projects into the state
space, depending on the realization, defined as the order in which the
reaction events in the flow are activated. Any order is acceptable, as long as
each state in the cycle can be reached by a reaction that can
execute.222222This condition is called reachability. It can be violated if
there are boundary states with too few individuals to permit the input complex
to some reaction to be formed. States may be grouped into equivalence classes
under reachability, as a means to study network features such as
autocatalysis. Reachability is beyond the scope of this paper. For the large-
population limits to which large-deviation scaling applies, under finitely
generated reactions, all states will have arbitrarily many members and will be
equivalent under reachability, so all embeddings of a finite integer hyperflow
will always complete. We then extend the indexing $\alpha$ from the images of
the complex-balanced cycles to include all the integer flows.

Once a cycle realization has been chosen for each integer hyperflow, an
embedding of these flows in the state space is defined as follows. Pick an
arbitrary starting complex in the cycle, and for each ${{\rm n}}_{s}$, embed
an image of the cycle with the starting complex sampled at ${{\rm n}}_{s}$.
Then every state is passed through by every cycle with each complex in the
cycle sampling from that state exactly once. Every link has a finite number of
cycles passing through it, because the number of integer flows spanning $\ker
Y\mathbb{A}$ and the length of each flow are both finite.

A set of currents $\left\\{{\bar{\jmath}}_{\alpha,{\rm n}}\right\\}$ that sum
to the steady-state currents over each transition is then assigned to the
cycles, indexed by the state ${\rm n}$ where which cycle $\alpha$ samples at
its starting complex. Solving for the values of the
$\left\\{{\bar{\jmath}}_{\alpha,{\rm n}}\right\\}$ is of course equivalent to
solving for $\bar{\rho}$, and is not generally a finite problem.

With these notations in place, Eq. (60) becomes

$\displaystyle\dot{D}\\!\left({\rho}^{s}\parallel{\bar{\rho}}^{s}\right)$
$\displaystyle=-\sum_{{{\rm
n}}_{s}}\sum_{\alpha}{\bar{\jmath}}^{s}_{\alpha,{{\rm n}}_{s}}\sum_{{{\rm
n}}^{\prime}_{s}{{\rm n}}_{s}\mid\alpha}\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm
n}}_{s}}}\right)\log\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}/{\bar{\rho}}^{s}_{{{\rm n}}_{s}}}{{\rho}^{s}_{{{\rm
n}}^{\prime}_{s}}/{\bar{\rho}}^{s}_{{{\rm n}}^{\prime}_{s}}}\right).$ (61)

Eq. (61) is a non-positive sum of KL divergences of probability ratios
$\left({\rho}^{s}_{{{\rm n}}_{s}}/{\bar{\rho}}^{s}_{{{\rm n}}_{s}}\right)$
referenced to their own values advanced around cycles, of the same form as Eq.
(53) for complex activities $e^{\theta Y_{i}}$ for macrostates on the
hypergraph. The state space divergence involves an additional sum over the
reference state ${{\rm n}}_{s}$. Because the sum over $\alpha$ is finite, the
contribution at each ${{\rm n}}_{s}$ is finite and proportional by finite
factors to ${\bar{\rho}}^{s}_{{{\rm n}}^{\prime}_{s}}$ within a finite
neighborhood of ${{\rm n}}_{s}$. Therefore the sum (61) is finite. This proves
that the first line of Eq. (11) is non-decreasing. $\blacksquare$

####  V.2.2 Non-negativity of the housekeeping entropy rate

The cycle decomposition on the state space may also be used to prove non-
negativity of the housekeeping entropy rate (10) introduced by Hatano and Sasa
Hatano:NESS_Langevin:01 . Here rather than work with entropy changes over
extended-time paths, we prove positivity moment by moment from the master
equation.

Unlike the relative entropy within the system, the housekeeping entropy rate
is not only a function of aggregated within-system transition rates
$w^{s}_{{{\rm n}}_{s}{{\rm n}}^{\prime}_{s}}$, but must be disaggregated to
all distinct system-environment interactions. Begin by expressing the sum of
all transition currents both in terms of elementary events and in the previous
cycle decomposition:

$\displaystyle J^{s}_{\rho}$ $\displaystyle\equiv\sum_{{{\rm
n}}_{s}}{\rho}^{s}_{{{\rm n}}_{s}}\sum_{{{\rm n}}^{\prime}_{s}\neq{{\rm
n}}_{s}}\sum_{{\rm n}\mid{{\rm n}}_{s}}\sum_{{{\rm n}}^{\prime}\mid{{\rm
n}}^{\prime}_{s}}\sum_{ji\;\mid\;y_{j}-y_{i}={{\rm n}}^{\prime}-{\rm
n}}w^{\left(ji\right)}_{{{\rm n}}^{\prime}{\rm n}}{\rho}^{e\mid s}_{{\rm n}}$
$\displaystyle\equiv\sum_{{{\rm n}}_{s}}{\rho}^{s}_{{{\rm n}}_{s}}\sum_{{{\rm
n}}^{\prime}_{s}\neq{{\rm n}}_{s}}w^{s}_{{{\rm n}}^{\prime}_{s}{{\rm n}}_{s}}$
$\displaystyle=\sum_{{{\rm
n}}_{s}}\sum_{\alpha}{\bar{\jmath}}^{s}_{\alpha,{{\rm n}}_{s}}\sum_{{{\rm
n}}^{\prime}_{s}{{\rm n}}_{s}\mid\alpha}\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm n}}_{s}}}\right),$ (62)

where $w^{\left(ji\right)}_{{{\rm n}}^{\prime}{\rm n}}$ is labeled by the
particular reaction connecting ${\rm n}$ and ${{\rm n}}^{\prime}$.232323For
notational simplicity, we suppose that each ordered pair of complexes
$\left(ji\right)$ is connected by at most one reaction – this still allows a
common _difference_ vector $Y_{j}-Y_{i}$ to be mediated by several distinct
pairs $\left(ji\right)$ – the generalization to more complex cases is
straightforward.

We follow the construction of correlation functions for a counting process
from Harris:fluct_thms:07 , for a quantity $r$ with event-dependent values
$r^{\left(ji\right)}$ defined as

$\displaystyle r^{\left(ji\right)}_{{{\rm n}}^{\prime}{\rm n}}$
$\displaystyle\equiv\log\left(\frac{w^{\left(ji\right)}_{{{\rm
n}}^{\prime}{\rm n}}{\bar{\rho}}^{s}_{{{\rm n}}_{s}}{\rho}^{e\mid s}_{{{\rm
n}}}}{w^{\left(ji\right)}_{{\rm n}{{\rm n}}^{\prime}}{\bar{\rho}}^{s}_{{{\rm
n}}^{\prime}_{s}}{\rho}^{e\mid s}_{{{\rm n}}^{\prime}}}\right).$ (63)

A total housekeeping entropy rate denoted ${\dot{S}}^{\rm HK}$ is a sum over
pairs of system states, of the quantity in Eq. (10). Written as an expectation
of $r$, it is

$\displaystyle{\dot{S}}^{\rm HK}$ $\displaystyle\equiv\sum_{{{\rm
n}}_{s}}\sum_{{{\rm n}}^{\prime}_{s}\neq{{\rm n}}_{s}}{\sigma}^{\rm HK}_{{{\rm
n}}_{s}{{\rm n}}^{\prime}_{s}}w^{s}_{{{\rm n}}_{s}{{\rm n}}^{\prime}_{s}}$
$\displaystyle=\sum_{{{\rm
n}}_{s}}\sum_{\alpha}{\bar{\jmath}}^{s}_{\alpha,{{\rm n}}_{s}}\sum_{{{\rm
n}}_{s}^{\prime}{{\rm n}}_{s}\mid\alpha}\left(\frac{{\rho}^{s}_{{\rm
n}}}{{\bar{\rho}}^{s}_{{\rm n}}}\right)$
$\displaystyle\mbox{}\times\frac{1}{w^{s}_{{{\rm n}}^{\prime}_{s}{{\rm
n}}_{s}}}\sum_{{\rm n}\mid{{\rm n}}_{s}}\sum_{{{\rm n}}^{\prime}\mid{{\rm
n}}^{\prime}_{s}}\sum_{ji\;\mid\;y_{j}-y_{i}={{\rm n}}^{\prime}-{\rm
n}}w^{\left(ji\right)}_{{{\rm n}}^{\prime}{\rm n}}{\rho}^{e\mid s}_{{\rm
n}}r^{\left(ji\right)}_{{{\rm n}}^{\prime}{\rm n}}.$ (64)

The sign of Eq. (64) can be deduced from convexity of the observable $e^{-r}$,
as in the usual fluctuation theorems. $r$ is nonzero only on transitions, the
terms in the sum in Eq. (62). Let $dt$ be a short time interval. Then the
expectation of any function of $r$ on the interval will be
$\left(1-dt\,J^{s}_{\rho}\right)$ times its value in an unchanging state, plus
a term $\propto dt$ from transitions. Extracting the contribution $\propto dt$
for expectations of $1$ and $e^{-r}$ over the interval, denoted ${\left<\mbox{
}\right>}_{\rm dt}$, gives

$\displaystyle{\left<1\right>}_{\rm dt}-\left(1-dt\,J^{s}_{\rho}\right)$
$\displaystyle=dt\sum_{{{\rm n}}_{s}}\sum_{\alpha}{\bar{\jmath}}_{\alpha,{{\rm
n}}_{s}}\sum_{{{\rm n}}^{\prime}_{s}{{\rm
n}}_{s}\mid\alpha}\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm n}}_{s}}}\right)$
$\displaystyle{\left<e^{-r}\right>}_{\rm dt}-\left(1-dt\,J^{s}_{\rho}\right)$
$\displaystyle=dt\sum_{{{\rm n}}_{s}}\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm n}}_{s}}}\right)\sum_{{{\rm
n}}^{\prime}_{s}\neq{{\rm n}}_{s}}\sum_{{\rm n}\mid{{\rm n}}_{s}}\sum_{{{\rm
n}}^{\prime}\mid{{\rm n}}^{\prime}_{s}}$
$\displaystyle\mbox{}\times\sum_{ji\;\mid\;y_{j}-y_{i}={{\rm n}}^{\prime}-{\rm
n}}w^{\left(ji\right)}_{{{\rm n}}^{\prime}{\rm n}}{\rho}^{e\mid s}_{{\rm
n}}{\bar{\rho}}^{s}_{{{\rm n}}_{s}}e^{-r^{\left(ji\right)}_{{{\rm
n}}^{\prime}{\rm n}}}$ $\displaystyle=dt\sum_{{{\rm
n}}_{s}}\left(\frac{{\rho}^{s}_{{{\rm n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm
n}}_{s}}}\right)\sum_{{{\rm n}}^{\prime}_{s}\neq{{\rm n}}_{s}}\sum_{{\rm
n}\mid{{\rm n}}_{s}}\sum_{{{\rm n}}^{\prime}\mid{{\rm n}}^{\prime}_{s}}$
$\displaystyle\mbox{}\times\sum_{ji\;\mid\;y_{j}-y_{i}={{\rm n}}^{\prime}-{\rm
n}}w^{\left(ji\right)}_{{\rm n}{{\rm n}}^{\prime}}{\rho}^{e\mid s}_{{{\rm
n}}^{\prime}}{\bar{\rho}}^{s}_{{{\rm n}}^{\prime}_{s}}$
$\displaystyle=dt\sum_{{{\rm n}}_{s}}\left(\frac{{\rho}^{s}_{{{\rm
n}}_{s}}}{{\bar{\rho}}^{s}_{{{\rm n}}_{s}}}\right)\sum_{{{\rm
n}}^{\prime}_{s}\neq{{\rm n}}_{s}}w^{s}_{{{\rm n}}_{s}{{\rm
n}}^{\prime}_{s}}{\bar{\rho}}^{s}_{{{\rm n}}^{\prime}_{s}}$
$\displaystyle=dt\sum_{{{\rm n}}_{s}}\sum_{\alpha}{\bar{\jmath}}_{\alpha,{{\rm
n}}_{s}}\sum_{{{\rm n}}^{\prime}_{s}{{\rm
n}}_{s}\mid\alpha}\left(\frac{{\rho}^{s}_{{{\rm
n}}^{\prime}_{s}}}{{\bar{\rho}}^{s}_{{{\rm n}}^{\prime}_{s}}}\right).$ (65)

Between the third and fourth lines in the evaluation of
${\left<e^{-r}\right>}_{\rm dt}$, index labels ${\rm n}$ and ${{\rm
n}}^{\prime}$ are switched. Because the steady state currents decompose into
cycles, whether $\left(\rho/\bar{\rho}\right)$ is summed over the first index
or over the second index along a cycle, the sum is the same. Hence the two
expressions in Eq. (65) are the same. By Jensen’s inequality,
$d{\left<r\right>}_{\rm dt}/dt={\dot{S}}^{\rm HK}\geq 0$, proving that Eq.
(64), which is the second line of Eq. (11), is non-decreasing. $\blacksquare$

Remark: The Hatano-Sasa generating function is constructed to replace the
transition matrix $\hat{\mathbb{T}}$ with its adjoint Harris:fluct_thms:07 .
In Eq. (65), this relation is reflected in the way ${\left<e^{-r}\right>}_{\rm
dt}$ switches $\left(\rho/\bar{\rho}\right)$ to the tail position of links,
whereas in ${\left<1\right>}_{\rm dt}$ it is in the head position. Exactly the
same sum arises if positivity of
$\dot{D}\\!\left({\rho}^{s}\parallel{\bar{\rho}}^{s}\right)$ is proved by a
similar convexity argument, from the expectation of $\left({\rho}^{s}_{{{\rm
n}}^{\prime}_{s}}/{\bar{\rho}}^{s}_{{{\rm
n}}^{\prime}_{s}}\right)/\left({\rho}^{s}_{{{\rm
n}}_{s}}/{\bar{\rho}}^{s}_{{{\rm n}}_{s}}\right)$ over an interval $dt$, which
is the inverse exponential of the log-ratio of Hartley information appearing
in Eq. (61). Thus the two fluctuation theorems are for generating functions
spanning the same chord between two probability measures, though the counting
observables in the two generating functions are distinct.

##  VI Examples

###  VI.1 Dualizing the housekeeping embedding thermodynamics

In the first example the system $s$ is elementary: a 2-state model of
polymerization and hydrolysis. Our interest is in how the embedding of such a
model in a 1-parameter family of environments is captured in the intensive and
extensive thermodynamic parameters, when these produce identical distributions
${\bar{\rho}}^{s}$, but with differing housekeeping entropy rate.

The motivation for the model is a question from astrobiology: in how far can
two environments be considered analogues simply because they produce similar
distributions for a _material_ considered to be of biological interest. For
instance, is Titan an analogue to early Earth if both are believed to support
significant polymerization of small organic molecules Yung:org_Titan:84 ;
Hebrard:Titan_atmos:12 ; Turse:org_Titan:13 , even if polymers on Titan are
stable and near equilibrium at low water activity, whereas Earth produced them
(putatively) through a competition between ligation driven by disequilibrium
leaving groups such as phosphates Westheimer:phosphates:87 ; Pasek:why_P:11
or thioesters Goldford:P_free_metab:17 ; Goldford:bdry_expn:19  and
disequilibrium hydrolysis?

The polymerization/hydrolysis mechanism, though elementary, can be a
foundation for more complex heterogeneous polymerization models
Esposito:copol_eff:10 , in which polymers once formed may be sensitive to
other consequences of water activity, such as propensity to fold or to
aggregate. We ask, to what extent can the rate balance that governs
polymerization, and water activity per se, be decoupled as measures of
environmental similarity. The example will show that in the limit where one
driving buffer goes to zero concentration and a subset of reactions become
strictly irreversible, the two parameters can be made independent.

In this example _all_ influences on what we should call thermodynamic order
come from kinetics in buffered non-equilibrium settings. Total entropy
production is uninformative because the entropy associated with the polymer
distribution sits atop a housekeeping heat that can be varied freely. Even
knowing that housekeeping heat depends on a difference of chemical potentials
gives no information when that difference is taken to infinity in an
irreversible limit. Thus none of the conservation laws linking the model to
microscopically reversible mechanics provide information beyond what is in the
transition matrix. Yet the _thermodynamics_ of the process retains a regular
representation: dualization of the housekeeping heat introduces an intensive
state variable that is just the regular current through the
polymerization/hydrolysis cycle, even in the strictly irreversible limit.

####  VI.1.1 One system, families of environments

##### Elementary reactions:

For simplicity we let environmental distributions ${\rho}^{e\mid s}$ be
Poisson with large capacity as a model of chemostats, and omit them from the
explicit notation. Half-reaction rate constants are denoted ${\hat{k}}_{i}$
and ${\hat{\bar{k}}}_{i}$ where $i$ is a label.

The first reaction type in the model is reversible dehydrating polymerization
and hydrolysis, with schema

$\displaystyle A$
$\displaystyle\overset{{\hat{k}}_{1}}{\underset{{\hat{\bar{k}}}_{1}}{\rightleftharpoons}}A^{\ast}+{\rm
H}_{2}{\rm O}.$ (66)

$A$ are monomers in solution, buffered at unit activity, and $A^{\ast}$ are
monomers attached to the end of a polymer by dehydrating condensation. ${\rm
n}$ denotes the number of $A^{\ast}$, which we will interpret as polymer
_lengths_ , and is the index for the system state. Water is also buffered, at
activity $a_{{\rm H}_{2}{\rm O}}$, which is varied as a control parameter
across a family of environments.

A second process, involving only environment species to define a reference
scale for chemical potentials, is hydrolysis of a phosphoanhydride bond, with
schema

$\displaystyle{\rm P}^{\ast}+{\rm H}_{2}{\rm O}$
$\displaystyle\overset{{\hat{k}}_{2}}{\underset{{\hat{\bar{k}}}_{2}}{\rightleftharpoons}}{\rm
P}.$ (67)

${\rm P}^{\ast}$ is a bound form with activity $a_{{\rm P}^{\ast}}$, and $P$ a
hydrolyzed form, with activity $a_{\rm P}$.242424Eq. (67) is a stand-in for a
variety of phosphoryl group-transfer processes, some linear as shown and some
higher-order in the hydrolyzed species Pasek:why_P:11 . For higher-order
reactions, appropriate powers of the activity would replace $a_{\rm P}$ in the
expressions below, without otherwise changing the results.

Monomer ligation driven by phosphate hydrolysis is the stoichiometrically
coupled reaction

$\displaystyle A+{\rm P}^{\ast}$
$\displaystyle\overset{{\hat{k}}_{3}}{\underset{{\hat{\bar{k}}}_{3}}{\rightleftharpoons}}A^{\ast}+{\rm
P}.$ (68)

Equilibrium constants for the three reactions (66–68) are denoted
${\hat{K}}_{i}={\hat{k}}_{i}/{\hat{\bar{k}}}_{i}$. In actual chemistry,
detailed balance implies the relation
${\hat{K}}_{3}={\hat{K}}_{1}{\hat{K}}_{2}$. To simplify notation we will
choose activity units to set ${\bar{k}}_{3}/{\bar{k}}_{1}=1$. Reaction 2
contributes only to dissipation internal to the environment (the third line in
Eq. (11)), and we omit it, so the scale of ${\bar{k}}_{2}$ never enters.

##### Poisson stationary distribution over polymer lengths:

The stationary distribution ${\bar{\rho}}^{s}$ for polymer length ${\rm n}$
under the joint action of reactions (66,68) is Poisson with parameter $K_{\rm
eff}$, given by

$\displaystyle\frac{{\bar{\rho}}^{s}_{{\rm n}+1}\left({\rm
n}+1\right)}{{\bar{\rho}}^{s}_{{\rm n}}}$ $\displaystyle\equiv K_{\rm
eff}=\frac{{\hat{K}}_{1}+{\hat{K}}_{3}a_{{\rm P}^{\ast}}}{a_{{\rm H}_{2}{\rm
O}}+a_{\rm P}}=\underline{n}.$ (69)

##### The varieties of chemical disequilibrium:

We are interested in families of environments that share the same value of
$K_{\rm eff}$ and so are indistinguishable from within the system. Chemical-
potential coordinates within such a family are

$\displaystyle\log\left(\frac{K_{\rm eff}a_{{\rm H}_{2}{\rm
O}}}{{\hat{K}}_{1}}\right)$ $\displaystyle\equiv{\mu}_{H}$
$\displaystyle\log\left(\frac{{\hat{K}}_{2}a_{{\rm P}^{\ast}}a_{{\rm
H}_{2}{\rm O}}}{a_{\rm P}}\right)$ $\displaystyle\equiv{\mu}_{P}.$ (70)

${\mu}_{H}$ is the chemical potential for hydrolysis in the stationary polymer
distribution. ${\mu}_{P}$ is the chemical potential of the phosphate system
for hydrolysis.

Within surfaces of constant $K_{\rm eff}$ we consider variation of phosphate
and water activities along contours of fixed $a_{\rm P}/a_{{\rm H}_{2}{\rm
O}}$,252525$a_{\rm P^{\ast}}$ is a nonlinear function of the value of $a_{\rm
P}$ along this contour. and vary this ratio later to define routes to
irreversibility. The thermodynamic equilibrium corresponds to
${\mu}_{P}={\mu}_{H}=0$, or

$\displaystyle{\hat{K}}_{2}\underline{a_{{\rm P}^{\ast}}}$
$\displaystyle=\frac{a_{\rm P}}{a_{{\rm H}_{2}{\rm O}}}$
$\displaystyle\frac{\underline{a_{{\rm H}_{2}{\rm O}}}}{{\hat{K}}_{1}}$
$\displaystyle=\frac{1}{K_{\rm eff}}.$ (71)

At equilibrium the sub-system stationary distribution ${\bar{\rho}}^{s}$ is
also the marginal ${\underline{\rho}}^{s}$ of the whole-system equilibrium; to
emphasize that it is fixed as ${\mu}_{H}$ and ${\mu}_{P}$ are varied across a
family of environments, we reference distributions to
${\underline{\rho}}^{s}$.

The activities governing reaction fluxes then depend on the coordinates (70)
as

$\displaystyle a_{{\rm H}_{2}{\rm O}}$
$\displaystyle=\frac{{\hat{K}}_{1}}{K_{\rm eff}}e^{{\mu}_{H}}$ $\displaystyle
a_{\rm P}$ $\displaystyle=\frac{{\hat{K}}_{1}}{K_{\rm
eff}}\frac{e^{{\mu}_{H}}\left(e^{{\mu}_{H}}-1\right)}{\left(e^{{\mu}_{P}}-e^{{\mu}_{H}}\right)}$
$\displaystyle a_{{\rm P}^{\ast}}$
$\displaystyle=\frac{1}{{\hat{K}}_{2}}\frac{e^{{\mu}_{P}}\left(e^{{\mu}_{H}}-1\right)}{\left(e^{{\mu}_{P}}-e^{{\mu}_{H}}\right)}.$
(72)

Fixed $a_{\rm P}/a_{{\rm H}_{2}{\rm O}}$ contours near and far from
equilibrium satisfy

$\displaystyle\frac{{\mu}_{P}-{\mu}_{H}}{{\mu}_{H}}$
$\displaystyle\underset{{\mu}_{P}\rightarrow 0}{\rightarrow}\frac{a_{{\rm
H}_{2}{\rm O}}}{a_{\rm P}}$ $\displaystyle{\mu}_{P}-{\mu}_{H}$
$\displaystyle\underset{{\mu}_{P}\rightarrow\infty}{\rightarrow}\log\left(\frac{a_{{\rm
H}_{2}{\rm O}}}{a_{\rm P}}\right).$ (73)

##### Cycle decomposition of steady-state currents:

If environmental marginals ${\rho}^{e\mid s}$ are chemostats at the indicated
chemical potentials, then currents in the stationary system distribution at a
state ${\rm n}$ are proportional to ${\underline{\rho}}^{s}_{{\rm n}}$ by
factors that do not depend on ${\rm n}$, and can be decomposed into three
“specific currents” around cycles, which are functions of the chemostat
activities

$\displaystyle{\hat{\jmath}}_{1}$
$\displaystyle=\frac{{\hat{K}}_{1}}{2}\left(e^{{\mu}_{H}}+1\right)$
$\displaystyle{\hat{\jmath}}_{3}$
$\displaystyle=\frac{{\hat{K}}_{1}}{2}\left(\frac{e^{{\mu}_{H}}-1}{e^{{\mu}_{P}}-e^{{\mu}_{H}}}\right)\left(e^{{\mu}_{P}}+e^{{\mu}_{H}}\right)$
$\displaystyle{\hat{\jmath}}_{\delta}$
$\displaystyle={\hat{K}}_{1}\left(e^{{\mu}_{H}}-1\right).$ (74)

${\hat{\jmath}}_{1}$ is the average of forward and reverse currents in
reaction (66), and ${\hat{\jmath}}_{3}$ the average of forward and reverse
currents in reaction (68), divided by ${\underline{\rho}}^{s}_{{\rm n}}$.
${\hat{\jmath}}_{\delta}$ is the difference of forward and reverse currents,
which must be equal and opposite in reactions (66) and (68) in stationary
state, also divided by ${\underline{\rho}}^{s}_{{\rm n}}$; it is the only
current in Schnakenberg’s fundamental graph Schnakenberg:ME_graphs:76  for
this CRN.262626${\hat{\jmath}}_{\delta}$ is so named because we have called it
a “$\delta$-flow” in Krishnamurthy:CRN_moments:17 ; Smith:CRN_moments:17 . The
number of possible independent net flows in the stationary state of a CRN
equals Feinberg’s Feinberg:notes:79  topological characteristic termed
deficiency and denoted $\delta$. The CRN of this example has $\delta=1$. The
fundamental graph omits currents ${\hat{\jmath}}_{1}$ and ${\hat{\jmath}}_{3}$
because they do not lead to dissipation in the stationary state, but they do
for more general states.

####  VI.1.2 The housekeeping entropy rate

The housekeeping entropy rate (64), for an arbitrary system distribution
${\rho}^{s}$, evaluates in terms of the specific currents (74) and the density
${\underline{\rho}}^{s}$, to

$\displaystyle{\dot{S}}^{\rm HK}\equiv\sum_{{\rm n}}\sum_{{{\rm
n}}^{\prime}\neq{\rm n}}{\sigma}^{\rm HK}_{{\rm n}{{\rm
n}}^{\prime}}\\!w^{s}_{{\rm n}{{\rm n}}^{\prime}}{\rho}^{s}_{{{\rm
n}}^{\prime}}$ $\displaystyle=\sum_{{\rm
n}}\left\\{{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}+1}\left[\left({\hat{\jmath}}_{1}+\frac{{\hat{\jmath}}_{\delta}}{2}\right){\mu}_{H}-\left({\hat{\jmath}}_{3}-\frac{{\hat{\jmath}}_{\delta}}{2}\right)\left({\mu}_{P}-{\mu}_{H}\right)\right]\right.$
$\displaystyle\mbox{}+\left.{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}}\left[\left({\hat{\jmath}}_{3}+\frac{{\hat{\jmath}}_{\delta}}{2}\right)\left({\mu}_{P}-{\mu}_{H}\right)-\left({\hat{\jmath}}_{1}-\frac{{\hat{\jmath}}_{\delta}}{2}\right){\mu}_{H}\right]\right\\}{\underline{\rho}}^{s}_{{\rm
n}}$ $\displaystyle\sim{\hat{\jmath}}_{\delta}{\mu}_{P}+\int
dn\,{\underline{\rho}}^{s}_{n}\frac{\partial}{\partial
n}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{n}\left[{\hat{\jmath}}_{1}{\mu}_{H}+{\hat{\jmath}}_{3}\left({\mu}_{H}-{\mu}_{P}\right)\right].$
(75)

The second expression is a continuum approximation to first order in
derivatives, with $-\log{\rho}^{s}_{n}$ understood as usual to be approximated
by the appropriate large-deviation function $S_{\rm eff}$.

To check that the exact (discrete) form of Eq. (75) is positive semidefinite
for arbitrary ${\rho}^{s}$, define measures

$\displaystyle p$
$\displaystyle\equiv\frac{{\hat{\jmath}}_{1}-{\hat{\jmath}}_{\delta}/2}{{\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}}$
$\displaystyle 1-p$
$\displaystyle\equiv\frac{{\hat{\jmath}}_{3}+{\hat{\jmath}}_{\delta}/2}{{\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}}$
$\displaystyle q$
$\displaystyle\equiv\frac{{\hat{\jmath}}_{1}+{\hat{\jmath}}_{\delta}/2}{{\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}}$
$\displaystyle 1-q$
$\displaystyle\equiv\frac{{\hat{\jmath}}_{3}-{\hat{\jmath}}_{\delta}/2}{{\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}}.$
(76)

From the formulae (74) it follows that

$\displaystyle\frac{p}{q}$ $\displaystyle=e^{-{\mu}_{H}}$
$\displaystyle\frac{1-p}{1-q}$ $\displaystyle=e^{{\mu}_{P}-{\mu}_{H}},$ (77)

and thus Eq. (75) can be written

$\displaystyle{\dot{S}}^{\rm HK}$ $\displaystyle=\sum_{{\rm
n}}{\underline{\rho}}^{s}_{{\rm
n}}\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right)\left[{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}+1}\\!\\!\\!D\\!\left(q\parallel
p\right)+{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}}D\\!\left(p\parallel q\right)\right],$ (78)

with $D\\!\left(q\parallel p\right)$ a Kullback-Leibler divergence as
elsewhere.

##### Housekeeping entropy rate as an embedding vector:

Eq. (78) can be put in the form

$\displaystyle{\dot{S}}^{\rm HK}$ $\displaystyle=\sum_{{\rm
n}}{\rho}^{s}_{{\rm n}}{\dot{\sigma}}^{\rm HK}_{{\rm n}}.$ (79)

${\dot{\sigma}}^{\rm HK}\equiv\left[{\dot{\sigma}}^{\rm HK}_{{\rm n}}\right]$
is a vector with positive-semidefinite components, which by Eq. (77) equals
zero only at ${\mu}_{P}={\mu}_{H}=0$.

As noted, ${\bar{\rho}}^{s}$ is an extremal vector for the intensive relative
entropy $-D\\!\left({\rho}^{s}\parallel{\bar{\rho}}^{s}\right)$ in the simplex
of distributions ${\rho}^{s}$. By Eq. (75), at this extremal point of $s$,
$\sum_{{\rm n}}{\bar{\rho}}^{s}_{{\rm n}}{\dot{\sigma}}^{\rm HK}_{{\rm
n}}={\hat{\jmath}}_{\delta}{\mu}_{P}$.

####  VI.1.3 Legendre duality for housekeeping entropy rate

The chemical potential ${\mu}_{P}\rightarrow\infty$ if the activity $a_{\rm
P}\rightarrow 0$ at fixed $a_{{\rm P}^{\ast}}$ and $a_{{\rm H}_{2}{\rm O}}$.
In this limit schema (68) becomes an irreversible reaction.
Phenomenologically, all “thermodynamic” characteristics of the system remain
regular, only the energetic accounting breaks down because it is referenced to
the equilibrium state variable $\log a_{\rm P}$, in a system that nowhere
couples to an equilibrium environment.

The divergence of ${\dot{S}}^{\rm HK}$ in this limit is like the divergence of
any extensive thermodynamic potential in the limit that one of the extensive
state variables diverges, except that ${\dot{S}}^{\rm HK}$ is an inherently
_non-equilibrium_ potential, and ${\mu}_{P}$ only behaves like an extensive
variable with respect to _dissipation_.272727Note that ${\dot{S}}^{\rm HK}$
has _no status_ with respect to equilibrium; while entropies are extensive,
${\dot{S}}^{\rm HK}$ depends inherently on a _rate_.

From the view that thermodynamics is about statistical organization and not
fundamentally about energy, it is natural to Legendre transform
${\dot{S}}^{\rm HK}$ to expose the dynamically intensive current that is dual
to ${\mu}_{P}$. For dualization, we work within the tangent space to the
family of constant $K_{\rm eff}$, but now vary ${\mu}_{P}$ independently of
${\mu}_{H}$, rather than varying within the non-linear contours (71) at fixed
$a_{\rm P}/a_{{\rm H}_{2}{\rm O}}$.

The gradient of ${\dot{S}}^{\rm HK}$ with respect to ${\mu}_{P}$ at fixed
${\mu}_{H}$ is

$\displaystyle\frac{\partial{\dot{S}}^{\rm HK}}{\partial{\mu}_{P}}=$
$\displaystyle{\hat{K}}_{1}\sum_{{\rm n}}{\underline{\rho}}^{s}_{{\rm
n}}\left(\frac{e^{{\mu}_{H}}-1}{e^{{\mu}_{P}}-e^{{\mu}_{H}}}\right)\left\\{\left[e^{{\mu}_{P}}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}}-e^{{\mu}_{H}}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}+1}\right]\vphantom{\left(\frac{e^{{\mu}_{P}}e^{{\mu}_{H}}}{e^{{\mu}_{P}}+e^{{\mu}_{H}}}\right)}\right.$
$\displaystyle\mbox{}\phantom{\frac{{\hat{K}}_{1}}{K_{\rm
eff}}}+\left.\left(\frac{e^{{\mu}_{P}}e^{{\mu}_{H}}}{e^{{\mu}_{P}}+e^{{\mu}_{H}}}\right)\left[{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}+1}-{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}+1}\right]\left({\mu}_{H}-{\mu}_{P}\right)\right\\}$
$\displaystyle\sim{\hat{\jmath}}_{\delta}-\int
dn\,{\underline{\rho}}^{s}_{n}\frac{\partial}{\partial
n}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{n}\left[{\hat{\jmath}}_{3}+\frac{\partial{\hat{\jmath}}_{3}}{\partial{\mu}_{P}}\left({\mu}_{P}-{\mu}_{H}\right)\right].$
(80)

By Eq. (74), ${\hat{\jmath}}_{3}\left({\mu}_{P}-{\mu}_{H}\right)$ is convex in
${\mu}_{P}$, so its derivative, the term in square brackets in the final line
of Eq. (80), is invertible to a value for ${\mu}_{P}$. The Legendre dual
potential to ${\dot{S}}^{\rm HK}$ on ${\mu}_{P}$, which we denote
${\dot{F}}_{P}$, is then given by

$\displaystyle{\dot{F}}_{P}\\!\left(\frac{\partial{\dot{S}}^{\rm
HK}}{\partial{\mu}_{P}},{\mu}_{H}\right)\equiv{\mu}_{P}\frac{\partial{\dot{S}}^{\rm
HK}}{\partial{\mu}_{P}}-{\dot{S}}^{\rm HK}$ $\displaystyle\approx-\int
dn\,{\underline{\rho}}^{s}_{n}\frac{\partial}{\partial
n}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{n}\left[\frac{\partial{\hat{\jmath}}_{3}}{\partial{\mu}_{P}}{\mu}_{P}\left({\mu}_{P}-{\mu}_{H}\right)+\left({\hat{\jmath}}_{3}+{\hat{\jmath}}_{1}\right){\mu}_{H}\right].$
(81)

##### Independent variation of control parameters in the irreversible limit:

We may now consider the effects of varying ${\mu}_{H}$, which remains finite,
across the one-parameter family of contours of different $a_{\rm P}/a_{{\rm
H}_{2}{\rm O}}$ as $a_{\rm P}\rightarrow 0$.
$\left({\hat{\jmath}}_{3}+{\hat{\jmath}}_{1}\right)$ approaches a constant
independent of ${\mu}_{P}$, and the only term in Eq. (81) including multiples
of diverging ${\mu}_{P}$ evaluates to

$\displaystyle-\frac{\partial{\hat{\jmath}}_{3}}{\partial{\mu}_{P}}{\mu}_{P}\left({\mu}_{P}-{\mu}_{H}\right)\rightarrow$
$\displaystyle\rightarrow{\hat{K}}_{1}\left(\frac{a_{\rm P}+a_{{\rm H}_{2}{\rm
O}}}{a_{{\rm H}_{2}{\rm
O}}}\right){\mu}_{H}\left[1-\frac{1}{3!}{\left(\frac{a_{{\rm H}_{2}{\rm
O}}}{a_{\rm P}}\right)}^{2}{\mu}_{H}^{2}\right]$
$\displaystyle:\;{\mu}_{H}\rightarrow 0$ $\displaystyle\rightarrow
2{\hat{\jmath}}_{3}e^{-\left({\hat{K}}_{1}/K_{\rm eff}a_{\rm
P}\right){\mu}_{H}}{\mu}_{P}\left({\mu}_{P}-{\mu}_{H}\right)$
$\displaystyle:\;{\mu}_{H}\gtrsim 1.$ (82)

It approaches ${\hat{K}}_{1}{\mu}_{H}$ within a range around ${\mu}_{H}=0$
that becomes vanishingly small as $a_{\rm P}\rightarrow 0$, and converges
exponentially to zero outside that range.

Thus the susceptibility that is the Legendre dual to ${\mu}_{P}$,
$\partial{\dot{S}}^{\rm
HK}/\partial{\mu}_{P}\rightarrow{\hat{\jmath}}_{\delta}$, a function only of
${\mu}_{H}$, almost everywhere. The potential ${\dot{F}}_{P}$ retains only the
${\mu}_{P}$-independent terms in Eq. (81). If, for example, we choose
${\rho}^{s}$ a macrostate with mean $\bar{n}$, then $\int
dn\,{\underline{\rho}}^{s}_{n}\partial{\left(\rho/\underline{\rho}\right)}^{s}_{n}/\partial
n=\left(\bar{n}-\underline{n}\right)$ and
${\dot{F}}_{P}\rightarrow-\left(\bar{n}-\underline{n}\right)\left({\hat{\jmath}}_{3}+{\hat{\jmath}}_{1}\right){\mu}_{H}$.

In the irreversible limit the problem is seen to separate. ${\bar{\rho}}^{s}$
dictates the intensive thermodynamics of the polymer system, while along a
contour that fixes $K_{\rm eff}$, the dual potential ${\dot{F}}_{P}$ describes
the non-trivial dependence of entropy change in the environment on
${\rho}^{s}$. ${\dot{S}}^{\rm HK}$ diverges harmlessly as an extensively-
scaling potential independent of ${\rho}^{s}$ with a regular susceptibility
${\hat{\jmath}}_{\delta}$.

###  VI.2 Metastability and ${\bar{\rho}}^{s}$ as the reference measure for
relative entropy

The second example uses the same buffered driving environment as the first,
but separates phosphate-driven ligation into a distinct, self-catalyzed
channel while leaving the reaction (66) with water uncatalyzed. The model for
system catalysis is a variant on the cubic Schlögl model of Fig. 2. It is
chosen to highlight the difference between the entropy rate (8) typically
assigned as “environmental” and the housekeeping entropy rates (10). A
quantity ${\dot{S}}^{\rm env}$ is a function only of the realized
distributions ${\rho}^{s}$ and ${\rho}^{e\mid s}$. Nowhere does it reflect the
joint participation of uncatalyzed and catalyzed reactions within the same
_system_.

The difference ${\dot{S}}^{\rm HK}-{\dot{S}}^{\rm env}$ may have either sign.
We illustrate the case where phosphate activities are chosen to put the system
in its bistable regime, and ${\mu}_{P}$ is chosen to make the unstable root
$n_{2}$ of the rate equation (47) a fixed point. For equivalent rate constants
a Poisson distribution ${\rho}^{s}$ at mean $n_{2}$ would be the driven
stationary distribution in the previous example, and the ${\dot{S}}^{\rm env}$
_is the same_ in the two models. However, ${\dot{S}}^{\rm HK}-{\dot{S}}^{\rm
env}<0$ because the loss of large-deviation accessibility in the system
results not only from it Shannon entropy change, but from the measure of that
entropy _relative_ to the true, bistable stationary distribution
${\bar{\rho}}^{s}$.

The autocatalytic model replaces the schema (68) with

$\displaystyle 2A^{\ast}+A+{\rm P}^{\ast}$
$\displaystyle\overset{{\hat{k}}_{4}}{\underset{{\hat{\bar{k}}}_{4}}{\rightleftharpoons}}3A^{\ast}+{\rm
P}.$ (83)

Catalysis does not change energetics, so the rate constants must satisfy
${\hat{K}}_{4}={\hat{K}}_{1}{\hat{K}}_{2}$.

In the mass-action rate law (47), the roots $n_{1}<n_{2}<n_{3}$ and $n_{2}$ is
unstable. We set a characteristic scale for transitions between water and
phosphate control by choosing relative rate constants
${\bar{k}}_{4}/{\bar{k}}_{1}=1/n_{2}^{2}$.

The Schlögl model is a birth-death process, so ${\bar{\rho}}^{s}$ is exactly
solvable. In place of $K_{\rm eff}$ from Eq. (69) adjacent indices are related
by a position-dependent effective equilibrium constant

$\displaystyle\frac{{\bar{\rho}}^{s}_{{\rm n}+1}\left({\rm
n}+1\right)}{{\bar{\rho}}^{s}_{{\rm n}}}$ $\displaystyle\equiv K\\!\left({\rm
n}\right)=\frac{{\hat{K}}_{1}+{\hat{K}}_{4}a_{\rm P^{\ast}}\,{\rm n}\left({\rm
n}-1\right)/n_{2}^{2}}{a_{{\rm H}_{2}{\rm O}}+a_{\rm P}\,{\rm n}\left({\rm
n}-1\right)/n_{2}^{2}}.$ (84)

Note that $K\\!\left(n_{2}\right)\rightarrow K_{\rm eff}$ to integer rounding
error, at all activity levels.

In terms of the same roots, choose water activity to set the lower limit of
Eq. (84) at ${\rm n}\leq 1$ to

$\displaystyle\frac{{\hat{K}}_{1}}{a_{{\rm H}_{2}{\rm O}}}$
$\displaystyle=\frac{1}{\sum_{i}1/n_{i}}\equiv\underline{n}.$ (85)

We will vary the activities in the phosphorus system so that
${\bar{\rho}}^{s}$ moves from a lower stable mode near equilibrium, through a
bistable phase, to an upper stable mode driven by phosphorylation farther from
equilibrium.

Equilibrium is again defined as in Eq. (71), and there $K\\!\left({\rm
n}\right)\overset{\rm eq}{=}\underline{n}\;;\;\forall{\rm n}$. The marginal
distribution for $s$ in the whole-system equilibrium,
${\underline{\rho}}^{s}$, is then Poisson with parameter $\underline{n}$.
Recall that, as a solution in detailed balance, it does not and cannot reflect
the fact that $s$ is a CRN capable of bistability. To match rate constants to
the roots $\left\\{n_{1},n_{2},n_{3}\right\\}$ of the Schlögl model, we define
a reference equilibrium for the phosphorus system with activity levels at

$\displaystyle{\hat{K}}_{2}\underline{a_{\rm
P^{\ast}}}=\frac{\underline{a_{\rm P}}}{a_{{\rm H}_{2}{\rm O}}}$
$\displaystyle=1.$ (86)

We now consider a family of driving environments in which the chemical
potential ${\mu}_{P}$ is partitioned between the activities $a_{\rm P^{\ast}}$
and $a_{\rm P}$ in the proportions

$\displaystyle\log\left(\frac{a_{\rm P^{\ast}}}{\underline{a_{\rm
P^{\ast}}}}\right)$
$\displaystyle=\frac{\log\left[\left(\sum_{i}n_{i}\right)n_{2}^{2}/\left(\prod_{i}n_{i}\right)\right]}{\log\left[\left(\sum_{i}n_{i}\right)\left(\sum_{i}1/n_{i}\right)\right]}{\mu}_{P}$
$\displaystyle\log\left(\frac{a_{\rm P}}{\underline{a_{\rm P}}}\right)$
$\displaystyle=-\frac{\log\left[\left(\prod_{i}n_{i}\right)\left(\sum_{i}1/n_{i}\right)/n_{2}^{2}\right]}{\log\left[\left(\sum_{i}n_{i}\right)\left(\sum_{i}1/n_{i}\right)\right]}{\mu}_{P}.$
(87)

The two potentials (87) are coordinates in an exponential family for the
phosphorus system. Where
${\mu}_{P}=\log\left[\left(\sum_{i}n_{i}\right)\left(\sum_{i}1/n_{i}\right)\right]$,
the three roots of the mass action law pass through the values
$\left\\{n_{1},n_{2},n_{3}\right\\}$ from Fig. 2.

The difference in relative dissipations between the driven steady state
${\bar{\rho}}^{s}$and ${\underline{\rho}}^{s}$, at adjacent ${\rm n}$ indices,
is given by the ratio of effective rate constants

$\displaystyle\log\left(\frac{{\bar{\rho}}^{s}_{{\rm
n}+1}/{\underline{\rho}}^{s}_{{\rm n}+1}}{{\bar{\rho}}^{s}_{{\rm
n}}/{\underline{\rho}}^{s}_{{\rm n}}}\right)$
$\displaystyle=\frac{K\\!\left({\rm n}\right)}{\underline{n}}$
$\displaystyle=\frac{1+\left(a_{\rm P^{\ast}}/\underline{a_{\rm
P^{\ast}}}\right){\rm n}\left({\rm n}-1\right)/n_{2}^{2}}{1+\left(a_{\rm
P}/\underline{a_{\rm P}}\right){\rm n}\left({\rm n}-1\right)/n_{2}^{2}}$
$\displaystyle=\frac{1+e^{{\mu}_{P}}\left(a_{\rm P}/\underline{a_{\rm
P}}\right){\rm n}\left({\rm n}-1\right)/n_{2}^{2}}{1+\left(a_{\rm
P}/\underline{a_{\rm P}}\right){\rm n}\left({\rm n}-1\right)/n_{2}^{2}}.$ (88)

Eq. (88) is a sigmoidal function of $\log\left({\rm n}/n_{2}\right)$ in the
range $K\\!\left({\rm n}\right)/\underline{n}\in\left[1,e^{{\mu}_{P}}\right)$,
graphed versus ${\mu}_{P}$ in Fig. 4. The hydrolysis potential ${\mu}_{H}$
from Eq. (70 for the linear system is replaced in the autocatalytic case by a
position-dependent chemical potential

$\displaystyle\mu\\!\left({\rm n}\right)$
$\displaystyle\equiv\log\left(\frac{a_{{\rm H}_{2}{\rm
O}}{\bar{\rho}}^{s}_{{\rm n}+1}\left({\rm
n}+1\right)}{{\hat{K}}_{1}{\bar{\rho}}^{s}_{{\rm
n}}}\right)=\log\left(\frac{K\\!\left({\rm n}\right)}{\underline{n}}\right).$
(89)

Figure 4:  $K\\!\left({\rm n}\right)$ on a sequence of increasing ${\mu}_{P}$
values with proportions (87). Curve at
${\mu}_{P}=\log\left[\left(\sum_{i}n_{i}\right)\left(\sum_{i}1/n_{i}\right)\right]$
is black. Boundaries of bistable range are red. Fixed points $K\\!\left({\rm
n}\right)={\rm n}$ shown with markers.

In the cycle decomposition, the specific currents exiting and entering a site
${\rm n}$, now functions of ${\rm n}$, depend on activities as

$\displaystyle{\hat{\jmath}}_{1}$
$\displaystyle=\frac{1}{2}\left(K\\!\left({\rm n}\right)a_{{\rm H}_{2}{\rm
O}}+{\hat{K}}_{1}\right)$ $\displaystyle{\hat{\jmath}}_{3}$
$\displaystyle=\frac{1}{2}\frac{{\rm n}\left({\rm
n}-1\right)}{n_{2}^{2}}\left({\hat{K}}_{4}a_{\rm P^{\ast}}+K\\!\left({\rm
n}\right)a_{\rm P}\right)$ $\displaystyle{\hat{\jmath}}_{\delta}$
$\displaystyle=K\\!\left({\rm n}\right)a_{{\rm H}_{2}{\rm O}}-{\hat{K}}_{1}$
$\displaystyle={\rm n}\left({\rm n}-1\right)\left({\hat{K}}_{4}a_{\rm
P^{\ast}}-K\\!\left({\rm n}\right)a_{\rm P}\right).$ (90)

In the chemical-potential coordinates, these become

$\displaystyle{\hat{\jmath}}_{1}$
$\displaystyle=\frac{{\hat{K}}_{1}}{2}\left(e^{\mu\left({\rm
n}\right)}+1\right)$ $\displaystyle{\hat{\jmath}}_{3}$
$\displaystyle=\frac{{\hat{K}}_{1}}{2}\frac{{\rm n}\left({\rm
n}-1\right)}{n_{2}^{2}}\left(e^{{\mu}_{P}}+e^{\mu\left({\rm
n}\right)}\right)\frac{a_{\rm P}}{\underline{a_{\rm P}}}$
$\displaystyle{\hat{\jmath}}_{\delta}$
$\displaystyle={\hat{K}}_{1}\left(e^{\mu\left({\rm n}\right)}-1\right)$
$\displaystyle={\hat{K}}_{1}\frac{{\rm n}\left({\rm
n}-1\right)}{n_{2}^{2}}\left(e^{{\mu}_{P}}-e^{\mu\left({\rm
n}\right)}\right)\frac{a_{\rm P}}{\underline{a_{\rm P}}}.$ (91)

Positive-semidefiniteness of ${\dot{S}}^{\rm HK}$ again follows by using the
ratios (76) to produce the form (78), where now instead of Eq. (77), we have

$\displaystyle\frac{p}{q}$ $\displaystyle=e^{-\mu\left({\rm n}\right)}$
$\displaystyle\frac{1-p}{1-q}$ $\displaystyle=e^{{\mu}_{P}-\mu\left({\rm
n}\right)}.$ (92)

For this example we separate out a term ${\dot{S}}^{\rm env}$, which is the
sum over transitions of Eq. (8), from ${\dot{S}}^{\rm HK}$, to give

$\displaystyle{\dot{S}}^{\rm HK}$ $\displaystyle=\sum_{{\rm
n}}\left\\{{\hat{K}}_{1}\frac{{\rm n}\left({\rm
n}-1\right)}{n_{2}^{2}}\left[\frac{a_{\rm P^{\ast}}}{\underline{a_{\rm
P^{\ast}}}}{\rho}^{s}_{{\rm n}}-\frac{a_{\rm P}}{\underline{a_{\rm
P}}}{\rho}^{s}_{{\rm n}+1}\frac{\left({\rm
n}+1\right)}{\underline{n}}\right]{\mu}_{P}\vphantom{\left[{\left(\frac{\rho}{\bar{\rho}}\right)}^{s}_{{\rm
n}}\right]}\right.$ $\displaystyle\phantom{\sum_{{\rm
n}}}\quad+\left.\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right)\left[{\rho}^{s}_{{\rm
n}+1}\frac{\left({\rm n}+1\right)}{K\\!\left({\rm n}\right)}-{\rho}^{s}_{{\rm
n}}\right]\mu\\!\left({\rm n}\right)\right\\}$ $\displaystyle=\sum_{{\rm
n}}\left\\{{\hat{K}}_{1}\frac{{\rm n}\left({\rm
n}-1\right)}{n_{2}^{2}}{\underline{\rho}}^{s}_{{\rm n}}\left[\frac{a_{\rm
P^{\ast}}}{\underline{a_{\rm
P^{\ast}}}}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}}-\frac{a_{\rm P}}{\underline{a_{\rm
P}}}{\left(\frac{\rho}{\underline{\rho}}\right)}^{s}_{{\rm
n}+1}\right]{\mu}_{P}\vphantom{\left[{\left(\frac{\rho}{\bar{\rho}}\right)}^{s}_{{\rm
n}}\right]}\right.$ $\displaystyle\phantom{\sum_{{\rm
n}}}\quad+\left.\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right){\bar{\rho}}^{s}_{{\rm
n}}\left[{\left(\frac{\rho}{\bar{\rho}}\right)}^{s}_{{\rm
n}+1}-{\left(\frac{\rho}{\bar{\rho}}\right)}^{s}_{{\rm
n}}\right]\mu\\!\left({\rm n}\right)\right\\}$ $\displaystyle={\dot{S}}^{\rm
env}+\sum_{{\rm
n}}\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right){\bar{\rho}}^{s}_{{\rm
n}}\left[{\left(\frac{\rho}{\bar{\rho}}\right)}^{s}_{{\rm
n}+1}-{\left(\frac{\rho}{\bar{\rho}}\right)}^{s}_{{\rm
n}}\right]\mu\\!\left({\rm n}\right).$ (93)

${\dot{S}}^{\rm env}$ does not depend on ${\bar{\rho}}^{s}$, and differs from
the sum of terms multiplying ${\mu}_{P}$ in Eq. (75) only by the ${\rm
n}$-dependence now multiplying ${\hat{K}}_{1}$ in Eq. (93).

All dependence of ${\dot{S}}^{\rm HK}$ on ${\bar{\rho}}^{s}$ comes from the
second term involving $\mu\\!\left({\rm n}\right)$. The shape of
${\bar{\rho}}^{s}_{{\rm n}}$ is shown as a function of ${\mu}_{P}$ in Fig. 5,
and the lumped coefficient
$\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right){\bar{\rho}}^{s}_{{\rm
n}}\,\mu\\!\left({\rm n}\right)$ multiplying the difference term in
$\left({\rho}^{s}/{\bar{\rho}}^{s}\right)$ in Eq. (93) is shown in Fig. 6.

Figure 5:  densities ${\bar{\rho}}^{s}$ for a scale factor
$4\times\left\\{1,4,9\right\\}$ for roots  Figure 6:  The combination
$\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right){\bar{\rho}}^{s}_{{\rm
n}}\mu\\!\left({\rm n}\right)$

As an example, consider the value
${\mu}_{P}=\log\left[\left(\sum_{i}n_{i}\right)\left(\sum_{i}1/n_{i}\right)\right]$
that makes $\left\\{n_{1},n_{2},n_{3}\right\\}$ the three fixed points, and
evaluate the embedding entropies for a Poisson distribution ${\rho}^{s}$ with
mean $n_{2}$, the unstable fixed point. The “environmental” component

$\displaystyle{\dot{S}}^{\rm env}$ $\displaystyle={\hat{K}}_{1}\sum_{{\rm
n}}{\rho}^{s}_{{\rm n}}\frac{{\rm n}\left({\rm
n}-1\right)}{n_{2}^{2}}\left(e^{{\mu}_{P}}-\frac{n_{2}}{\underline{n}}\right)\frac{a_{\rm
P}}{\underline{a_{\rm P}}}{\mu}_{P}$
$\displaystyle={\hat{K}}_{1}\left(e^{{\mu}_{P}}-e^{\mu\left(n_{2}\right)}\right)\frac{a_{\rm
P}}{\underline{a_{\rm P}}}{\mu}_{P}$
$\displaystyle={\hat{\jmath}}_{\delta}\\!\left(n_{2}\right){\mu}_{P}.$ (94)

is the same as the value ${\hat{\jmath}}_{\delta}{\mu}_{P}$ following from Eq.
(75) for the non-catalytic CRN, if we set
${\mu}_{H}=\log\left(n_{2}/\underline{n}\right)$ in Eq. (70), where $K_{\rm
eff}\rightarrow n_{2}$ and this ${\rho}^{s}$ is a driven steady state. It is
also the value of ${\dot{S}}^{\rm HK}$ in Eq. (79) at the extremal
distribution ${\rho}^{s}={\bar{\rho}}^{s}$ for the linear model.

For the same ${\rho}^{s}$, recognizing that $\left(\mu\\!\left({\rm
n}\right)-{\mu}_{H}\right)=\log\left(K\\!\left({\rm n}\right)/n_{2}\right)$,
the gradient relative to the bistable reference distribution
${\bar{\rho}}^{s}$ in Eq. (93) expands to ${\bar{\rho}}^{s}_{{\rm
n}}\left[{\left(\rho/\bar{\rho}\right)}^{s}_{{\rm
n}+1}-{\left(\rho/\bar{\rho}\right)}^{s}_{{\rm
n}}\right]\approx-{\rho}^{s}_{{\rm n}}\left[\mu\\!\left({\rm
n}\right)-{\mu}_{H}\right]$. The difference of environment from housekeeping
entropy changes then evaluates to

$\displaystyle{\dot{S}}^{\rm HK}-{\dot{S}}^{\rm env}$
$\displaystyle\approx-\sum_{{\rm
n}}\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right){\rho}^{s}_{{\rm
n}}\mu\\!\left({\rm n}\right)\left[\mu\\!\left({\rm
n}\right)-{\mu}_{H}\right].$ (95)

The combination
$\left({\hat{\jmath}}_{1}+{\hat{\jmath}}_{3}\right){\rho}^{s}_{{\rm
n}}\mu\\!\left({\rm n}\right)$ is increasing through $n_{2}$ (See Fig. 6), and
$\left[\mu\\!\left({\rm n}\right)-{\mu}_{H}\right]$ is antisymmetric, so Eq.
(95) is negative.

Hence not all of the entropy change by ${\dot{S}}^{\rm env}$ reflects a loss
of large-deviation accessibility attributable to the environment. A quantity
equal to ${\dot{S}}^{\rm env}-{\dot{S}}^{\rm HK}$ is assigned in the natural
decomposition (11) to the system, because a Poisson distribution around
$n_{2}$ is unstable and has excess net probability to relax toward the
bistable distribution ${\bar{\rho}}^{s}$.

##  VII Discussion

#### The problem of macroworlds as an alternative central contribution

A suggestion that thermodynamics should not be mainly about the relation
between work and heat flows would have been non-sequitur at the turn of the
20th century. Not only were applications centered around interactions with a
thermal system through mechanical boundaries Carnot:essays:60 ; the nature of
irreversibility and dissipation as scientific problems was framed in their
relation to mechanical principles of time-reversibility Laplace:probs:51  and
conservation of energy Clausius:equivalence:62 ; Joule:sci_papers:44 .

Innovations in scaling methods, however, in context of the unification of
statistical and quantum theories in the second half of the 20th century, have
exposed _understanding the existence of macroworlds_ as one of the central
problems in science. Sec. III explained how large-deviations scaling
formalizes a concept of macrostate in terms of selected distributions over
microstates. When methods derived for phase transitions and critical phenomena
in condensed matter Wilson:RG:74  were merged with the renormalization-group
approach GellMann:RG:54  to vacuum quantum field theory Polchinski:RGEL:84 ,
it became clear that renormalization-group flow, a closely related
operation282828Through the correspondence of Keldysh to 2-field statistical
field methods, the algebras of the two can often be interconverted
Kamenev:DP:02 ; Smith:DP:08 . of dimensional reduction to large-deviations
scaling, provides a concept of effective fields describing the robust
properties of macroworlds through phenomenological Lagrangians
Weinberg:phenom_Lagr:79 . Effective fields are the counterparts under
renormalization to classical states under large-deviations scaling.

In effective field theories nested through phase transitions (as in the models
of this paper) microstates are assimilated back to macrostates. The result has
been a theory of the vacuum and the hierarchy of matter in which all earlier,
partly phenomenological concepts of elementary and composite objects and
interactions has been subsumed within a semantics of distributions. While
these physics examples continue to be organized by symmetries and conservation
laws, similar scaling methods developed independently for reliable coding
Shannon:MTC:49 ; Cover:EIT:91  offer non-mechanical counterparts (see
Smith:geo13:16 , Ch. 7).

The alternative characterization offered, then, is that _thermodynamics is not
fundamentally the theory of the movements of heat, but rather the theory of
the emergence of macroworlds from microworlds._

#### The end of entropy flow, the natural partition, and Hartley information

In the absence of an energy-mediated constraint on the states jointly
realizable by a system and its environment, the notion of “entropy flow”
between subsystems may never arise.292929Even in classical thermodynamics, it
is only well-defined for adiabatic transformations, in which both subsystems
remain in classical states constrained by energy at every moment. The default
description of entropy change ceases to be the metaphor of a fluid and becomes
the literal one: it is the loss of large-deviation accessibility of classical
states due to relaxation of the whole-system distribution. The concept of a
tangent surface delimiting the amount of entropy gain in one component that
can be influenced by the state of another, instead of the adiabatic
transformation, is filled by the the vector of Hartley informations
$-\log{\bar{\rho}}^{s}$ of the stationary state for the ${\mathbb{T}}^{s}$ at
that moment. The natural system-environment partition (11) encodes the
conditional independence of components of the loss of states accessible by
fluctuations.

Hartley information appears as a random variable along trajectories in the
construction of the generating functional for housekeeping heat
Speck:HS_heat_FT:05 ; Seifert:stoch_thermo_rev:12 . In standard treatments,
its interpretation as an entropy is taken to depend on its association with a
dissipated heat.303030The exact statement, referring to Eq. (123) in
Seifert:stoch_thermo_rev:12 , is: “In the absence of a first law for the
master equation dynamics, which would require further physical input not
available at this general stage, this identification is by analogy only.”
Here, $-\log{\bar{\rho}}^{s}$ gains its interpretation as an entropy through
its role as a carrier of information about the interaction of processes within
a system on its own limiting large-deviation function and on its capacity to
limit large-deviations in the environment.

#### Making trajectories first-class citizens

The persistence over 90 years of Onsager’s program Onsager:RRIP1:31 ;
Onsager:RRIP2:31  of making equilibrium Gibbs entropies the foundation of non-
equilibrium thermodynamics has built an asymmetry into this concept even
within physics, at the same time as growth in statistical methods for path
ensembles has reduced the reason for an asymmetry to exist, by defining the
same tools for paths as for states. The equilibrium Gibbs entropy is defined
from the large-deviation function on ensembles of states. The stochastic
effective action (25) Smith:LDP_SEA:11  is the corresponding function _al_ for
trajectories. However, the role that energy conservation plays in stochastic
thermodynamics, as a constraint on which states can be jointly occupied by
components within a system and within its environment, will not generally have
a counterpart for the jointly realizable trajectories involving these
components.313131Consider, as examples, non-Markovian noise sources and error-
correcting codes exploiting time correlations to optimize against those
environments Verdu:Shannon:98 .

Borrowing a term from computer science, the privileged roles of equilibrium
entropy and energy conservation in current non-equilibrium thermodynamics
makes spaces of states “first-class citizens” Abelson:SICP:96 , and admits
thermodynamic interpretations for ensembles only in so far as those derive
from equilibrium heat. Jaynes anticipated Jaynes:caliber:80 ; Jaynes:LOS:03  a
self-contained thermodynamic interpretation for path ensembles, though still
only in a limited form referenced to the calibers of cross-sections of states.
If there is to be a thermodynamics in which trajectories become first-class
citizens on par with states, it will need a foundation in more general
primitives such as large-deviation accessibility and separation of scales, as
in the domain-agnostic framework presented here.

#### Rule-based systems and life

This paper’s formula for a self-contained and substrate-agnostic
thermodynamics is meant to support concept discovery in two domains that
should clearly have such a thermodynamics, and for which energy conservation
should play a limited role or no role at all for many interesting
questions.323232A well-known examples of a constraint that is proved by a
fluctuation theorem, but not binding because a more proximal constraint
exists, is that of total entropy production versus the “excess heat” in a
driven system. Speck:FDT_NESS:06  The former bounds the non-equilibrium
entropy change within a distribution, but is uninformative because it diverges
on long timescales. The tight bound, under some conditions, is the finite
excess heat.

The first domain is rule-based modeling Danos:rule_based_modeling:08  which
recognizes the finitary-to-infinitary mappings exhibited here between CRN
generators and state spaces as a widely generalizable organizing principle.
The algebra and combinatorics of non-commutative rule systems is a very active
area of study Harmer:info_carriers:10 ; Andersen:generic_strat:14 ;
Behr:alg_graph_rewrt:16  which connects to chemistry, systems biology,
theories of algorithms, process calculus, and much more. A thermodynamics of
rule-based systems that expands the conceptual scope of science will not be a
reduction to their consequences for heat generation.

The second domain is the thermodynamic nature of life Smith:geo13:16 ,
encompassing rule-based order in its chemical substrate, the multi-level
control systems and error correction required to maintain complex dynamical
phases, and the nested population dynamics and historical contingency of
evolution. There will be order in such systems that derives, sometimes
cryptically, from microscopic time-reversibility. But it is inconceivable that
there will not be much _more_ order, of novel kinds, that originates from the
abundance of architectures – most of them irreversible – standing between the
microscopic substrate and robust macrophenomena.

##  VIII Concluding remarks

The presentation above is not a review of recent technical innovations, but an
attempt to select core concepts that depend on formalization to be adequately
expressed, and to exhibit them in relation to each other and in a context that
does not presume mechanics. The Hamilton-Jacobi and CRN formalisms are of
course not new, but needed to be reviewed within a coherent framework to make
points about them that are not made elsewhere. Likewise most monotonicity
results are known from fluctuation theorems, though alternative proofs for a
few are given here. It may therefore be useful to summarize the novel
contributions of the paper that are interleaved with standard results above:

* Monotonic change of Kullback-Leibler divergence is obvious and positivity of housekeeping heat was known Speck:HS_heat_FT:05 ; Harris:fluct_thms:07 . An emphasis here on entropy increase as loss of accessibility by large deviations, together with the observation that thermalization in a multi-scale system produces separation in the conditional dependence between system and environment degrees of freedom, leads to the decomposition of $\dot{D}\\!\left(\rho\parallel\underline{\rho}\right)$ in Eq. (11) as the _natural_ partition of irreversibility between system and environment changes, irrespective within each of the degree of irreversibility in the other.

* The “non-equilibrium entropy production” of stochastic thermodynamics is a complicated sum, of an entropy functional of arbitrary distributions (over stochastic microstates) and a Gibbs entropy state-function (for bath macrostates). The designation “violations of the 2nd law” by stochastic events uses that summation to conflate these two entropy summands which have different relations to constraints by boundary conditions. The decomposition in Eq. (34) of $D\\!\left({\rho}^{\left(\bar{n}\right)}\parallel\underline{\rho}\right)$ in terms of the Lyapunov entropy state function $S_{\rm eff}$ and a remainder $D\\!\left(1_{\underline{n}\left(\bar{n}\right)}\parallel\underline{\rho}\right)$ that must be relaxed by instantons, retains an un-violated 2nd law for relative entropy, and also the Lyapunov and large-deviation roles of the classical state-function entropy, in the proper relation.

* The dualization by Baish Baish:DP_duality:15  of observable and response fields in Doi-Peliti theory stands in obvious analogy to the construction of the adjoint transition matrix in the fluctuation theorem for housekeeping heat Speck:HS_heat_FT:05 ; Harris:fluct_thms:07 . The standardization of the CRN generator (35) makes the analogy explicit in the descaling (39) of $\mathbb{T}$ and the descaling (43) of $\mathbb{A}$. Moreover, it shows the analogy as an expression of the general relation of microstates to macrostates, and relates the preservation or loss of information in the classical limit to CRN complexity classes.

* The distinction between relaxation and escape trajectories in terms of formal momenta is widely developed in momentum-space WKB theory Assaf:mom_WKB:17 , as Hamilton-Jacobi theory Bertini:macro_NEThermo:09 , and in relation to information geometry Smith:IGDP:19 . The convexity proof in Eq. (49) relating the Lyapunov to the large-deviation role of the classical-state entropy, and to the structure of the $\mathcal{L}=0$ manifold, is novel.

* The use of cycle decompositions in CRNs to compute heat dissipation in the stationary state is standard since Schnakenberg Schnakenberg:ME_graphs:76 . Here a more complete cycle decomposition (keeping detailed-balanced currents) unifies all proofs of monotonicity of entropies for non-stationary distributions, relating the Lyapunov and large-deviation changes in macrostate entropy on complex-balanced CRNs in Eq. (53), and its counterpart (61) in the state space for all CRNs. It also provides an interesting view of the (known) equivalence of the adjoints constructed in the generating function (64) for housekeeping heat in the environment, and an equivalent function for the relative entropy (61) in the system.

* The Legendre duality of ${\dot{S}}^{\rm HK}$ to the potential ${\dot{F}}_{P}$ in Eq. (81) defines extensivity and intensivity with respect to a different scaling than that for entropy, but a natural one if time-reversal and energy conservation are not the only basis for taking macroscopic limits.

## References

  * (1) Leo Szilard.  Uber die entropieverminderung in einem thermodynamischen system bei eingriffen intelligenter wesen.  Zeitschrift für Physik, 53:840–856, 1929.
  * (2) R. Landauer.  Irreversibility and heat generation in the computing process.  IBM J. Res. Development, 3:183–191, 1961.
  * (3) Charles H. Bennett.  The thermodynamics of computation – a review.  Int. J. Theor. Phys., 21:905–940, 1982.
  * (4) David H. Wolpert.  The stochastic thermodynamics of computation.  J. Phys. A: Math. Theor., 52:193001, 2019.
  * (5) Udo Seifert.  Stochastic thermodynamics, fluctuation theorems, and molecular machines.  Rep. Prog. Phys., 75:126001, 2012.  arXiv:1205.4176v1.
  * (6) Lars Onsager.  Reciprocal Relations in Irreversible Processes. I.  Phys. Rev., 37:405–426, 1931.
  * (7) Lars Onsager.  Reciprocal Relations in Irreversible Processes. II.  Phys. Rev., 38:2265–2279, 1931.
  * (8) P. Glansdorff and I. Prigogine.  Thermodynamic Theory of Structure, Stability, and Fluctuations.  Wiley, New York, 1971.
  * (9) Dilip Kondepudi and Ilya Prigogine.  Modern Thermodynamics: From Heat Engines to Dissipative Structures.  Wiley, New York, 1998.
  * (10) Denis J. Evans and Debra J. Searles.  Fluctuation theorem for stochastic systems.  Phys. Rev. E, 60:159–164, 1999.
  * (11) Denis J. Evans and Debra J. Searles.  The fluctuation theorem.  Adv. Phys., 51:1529–1585, 2002.
  * (12) G. Gallavotti and E. D. G. Cohen.  Dynamical ensembles in non-equilibrium statistical mechanics.  Phys. Rev. Lett., 74:2694–2697, 1995.
  * (13) G. Gallavotti and E. D. G. Cohen.  Dynamical ensembles in stationary states.  J. Stat. Phys., 80:931–970, 1995.
  * (14) C. Jarzynski.  Nonequilibrium work relations: foundations and applications.  Eur. Phys. J. B, 64:331–340, 2008.
  * (15) Raphaël Chetrite and Krzysztof Gawedzki.  Fluctuation relations for diffusion processes.  Commun. Math. Phys., 282:469–518, 2008.
  * (16) Massimiliano Esposito and Christian Van den Broeck.  Three detailed fluctuation theorems.  Phys. Rev. Lett., 104:090601, 2010.
  * (17) Gavin E. Crooks.  Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences.  Phys. Rev. E, 6:2721–2726, 1999.
  * (18) Gavin E. Crooks.  Path-ensemble averages in systems driven far from equilibrium.  Phys. Rev. E, 61:2361–2366, 2000.
  * (19) Jorge Kurchan.  Non-equilibrium work relations.  J. Stat. Mech., 2007:P07005, 2007.
  * (20) Jeremy L. England.  Statistical physics of self-replication.  J. Chem. Phys., 139:121923, 2013.
  * (21) Nikolai Perunov, Robert Marsland, and Jeremy England.  Statistical physics of adaptation.  2015\.  arXiv:1412.1875v1 [physics.bio-ph].
  * (22) R. Duncan Luce.  Individual Choice Behavior.  Wiley, New York, 1959.
  * (23) Daniel McFadden.  Quantal choice analysis: a survey.  Ann. Econ. Soc. Measurement, 5:363–390, 1976.
  * (24) T. M. Lenton, V. N. Livina, V. Dakos, E. H. van Nes, and M. Scheffer.  Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness.  Phil. Trans. R. Soc. A, 370:1185–1204, 2011.
  * (25) J. P. Joule.  The Scientific Papers of James Prescott Joule.  London: Physical Society, Open Library, 1844.  https://openlibrary.org/books/OL239730M/The_scientific_papers_of_James_Prescott_Joule.
  * (26) T. Archer Hirst, editor.  The Mechanical Theory of Heat, London, 1865. John van Voorst.
  * (27) Ludwig Boltzmann.  The second law of thermodynamics.  In Populäre Schriften, pages 25–50. J. A. Barth, Leipzig, 1905.  re-issued Braunschweig: F. Vieweg, 1979.
  * (28) Anders Hald.  A history of probability and statistics and their applications before 1750.  Wiley, New York, 1990.
  * (29) E. T. Jaynes.  Information theory and statistical mechanics.  Phys. Rev., 106:620–630, 1957.  reprinted in Jaynes:Papers:83 .
  * (30) E. T. Jaynes.  Information theory and statistical mechanics. II.  Phys. Rev., 108:171–190, 1957.  reprinted in Jaynes:Papers:83 .
  * (31) Gian-Carlo Rota.  Lectures on Being and Time (1998).  New Yearbook for Phenomenology and Phenomenological Philosophy, 8:225–319, 2008.
  * (32) Murray Gell-Mann and Francis Low.  Quantum electrodynamics at small distances.  Phys. Rev., 95:1300–1312, 1954.
  * (33) K. G. Wilson and J. Kogut.  The renormalization group and the $\varepsilon$ expansion.  Phys. Rep., Phys. Lett., 12C:75–200, 1974.
  * (34) Steven Weinberg.  Phenomenological Lagrangians.  Physica A, 96:327–340, 1979.
  * (35) Joseph G. Polchinski.  Renormalization group and effective lagrangians.  Nuclear Physics B, 231:269–295, 1984.
  * (36) Enrico Fermi.  Thermodynamics.  Dover, New York, 1956.
  * (37) L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim.  Towards a nonequilibrium thermodynamics: a self-contained macroscopic description of driven diffusive systems.  J. Stat. Phys., 135:857–872, 2009.
  * (38) Shun-Ichi Amari.  Methods of Information Geometry.  Amer. Math. Soc., 2001.
  * (39) Nihat Ay, Jürgen Jost, Hông Vân Lê, and Lorentz Schwachhöfer.  Information Geometry.  Schwinger International, Cham, Switzerland, 2017.
  * (40) Richard S. Ellis.  Entropy, Large Deviations, and Statistical Mechanics.  Springer-Verlag, New York, 1985.
  * (41) Hugo Touchette.  The large deviation approach to statistical mechanics.  Phys. Rep., 478:1–69, 2009.  arxiv:0804.0327.
  * (42) Claude Elwood Shannon and Warren Weaver.  The Mathematical Theory of Communication.  U. Illinois Press, Urbana, Ill., 1949.
  * (43) R. V. L. Hartley.  Transmission of information.  Bell system technical journal, July:535–563, 1928.
  * (44) T. Speck and U. Seifert.  Integral fluctuation theorem for the housekeeping heat.  J. Phys. A Math. Gen., 38:L581–L588, 2005.
  * (45) R. J. Harris and G. M. Schütz.  Fluctuation theorems for stochastic dynamics.  J. Stat. Mech., page P07020, 2007.  doi:10.1088/1742-5468/2007/07/P07020.
  * (46) Thomas M. Cover and Joy A. Thomas.  Elements of Information Theory.  Wiley, New York, 1991.
  * (47) Takahiro Hatano and Shin-ichi Sasa.  Steady state thermodynamics of Langevin systems.  Phys. Rev. Lett., 86:3463–3466, 2001.
  * (48) Eric Smith.  Large-deviation principles, stochastic effective actions, path entropies, and the structure and meaning of thermodynamic descriptions.  Rep. Prog. Phys., 74:046601, 2011.  http://arxiv.org/submit/199903.
  * (49) Murray Gell-Mann and Seth Lloyd.  Information measures, effective complexity, and total information.  Complexity, 2:44–52, 1996.
  * (50) Murray Gell-Mann and Seth Lloyd.  Effective complexity.  In Murray Gell-Mann and Constantino Tsallis, editors, Nonextensive Entropy – Interdisciplinary Applications, pages 387–398. Oxford U. Press, New York, 2004.
  * (51) Matteo Polettini and Massimiliano Esposito.  Irreversible thermodynamics of open chemical networks. I. Emergent cycles and broken conservation laws.  J. Chem. Phys., 141:024117, 2014.
  * (52) Matteo Polettini, Gregory Bulnes-Cuetara, and Massimiliano Esposito.  Bridging stochastic and macroscopic thermodynamics.  2016\.  arXiv:1602.06555v1.
  * (53) F. Horn and R. Jackson.  General mass action kinetics.  Arch. Rat. Mech. Anal, 47:81–116, 1972.
  * (54) Martin Feinberg.  Lectures on chemical reaction networks.  lecture notes, 1979.  https://crnt.osu.edu/LecturesOnReactionNetworks.
  * (55) Supriya Krishnamurthy and Eric Smith.  Solving moment hierarchies for chemical reaction networks.  J. Phys. A: Math. Theor., 50:425002, 2017.
  * (56) Eric Smith and Supriya Krishnamurthy.  Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks.  Phys. Rev. E, 96:062102, 2017.
  * (57) Udo Seifert and Thomas Speck.  Fluctuation-dissipation theorem in nonequilibrium steady states.  Europhysics Lett., 89:10007, 2010.
  * (58) J. Schnakenberg.  Network theory of microscopic and macroscopic behavior of master equation systems.  Rev. Mod. Phys, 48:571–585, 1976.
  * (59) Jakob L. Andersen, Christoph Flamm, Daniel Merkle, and Peter F. Stadler.  Generic strategies for chemical space exploration.  Int. J. Comput. Biol. Drug Des., 7:225–258, 2014.
  * (60) Michael Assaf and Baruch Meerson.  Wkb theory of large deviations in stochastic populations.  J. Phys. A, 50:263001, 2017.
  * (61) M. I. Freidlin and A. D. Wentzell.  Random Perturbations in Dynamical Systems.  Springer, New York, second edition, 1998.
  * (62) P. C. Martin, E. D. Siggia, and H. A. Rose.  Statistical dynamics of classical systems.  Phys. Rev. A, 8:423–437, 1973.
  * (63) Alex Kamenev.  Keldysh and doi-peliti techniques for out-of-equilibrium systems.  In I. V. Lerner, B. L. Althsuler, V. I. Fal′ko, and T. Giamarchi, editors, Strongly Correlated Fermions and Bosons in Low-Dimensional Disordered Systems, pages 313–340, Heidelberg, 2002. Springer-Verlag.
  * (64) Eric Smith.  Thermodynamics of natural selection II: Chemical Carnot cycles.  J. Theor. Biol., 252:198–212, 2008.  PMID: 18367209.
  * (65) Eric Smith.  Thermodynamics of natural selection I: Energy and entropy flows through non-equilibrium ensembles.  J. Theor. Biol., 252:185–197, 2008.  PMID: 18367210.
  * (66) Massimiliano Esposito, Katja Lindenberg, and Christian Van den Broeck.  Extracting chemical energy by growing disorder: efficiency at maximum power.  J. Stat. Mech., doi:10.1088/1742-5468/2010/01/P01008:1–11, 2010\.
  * (67) Charles Kittel and Herbert Kroemer.  Thermal Physics.  Freeman, New York, second edition, 1980.
  * (68) D. Siegmund.  Importance sampling in the Monte Carlo study of sequential tests.  Ann. Statist., 4:673–684, 1976.
  * (69) Herbert Goldstein, Charles P. Poole, and John L. Safko.  Classical Mechanics.  Addison Wesley, New York, third edition, 2001.
  * (70) David F. Anderson, George Craciun, and Thomas G. Kurtz.  Product-form stationary distributions for deficiency zero chemical reaction networks.  Bull. Math. Bio., 72:1947–1970, 2010.
  * (71) M. Doi.  Second quantization representation for classical many-particle system.  J. Phys. A, 9:1465–1478, 1976.
  * (72) M. Doi.  Stochastic theory of diffusion-controlled reaction.  J. Phys. A, 9:1479–, 1976.
  * (73) L. Peliti.  Path-integral approach to birth-death processes on a lattice.  J. Physique, 46:1469, 1985.
  * (74) L. Peliti.  Renormalization of fluctuation effects in $a+a\rightarrow a$ reaction.  J. Phys. A, 19:L365, 1986.
  * (75) J. L. Cardy.  Electron localisation in disordered systems and classical solutions in ginzburg-landau field theory.  J. Phys. C, 11:L321 – L328, 1987.
  * (76) Sidney Coleman.  Aspects of Symmetry.  Cambridge, New York, 1985.
  * (77) Eric Smith and Supriya Krishnamurthy.  Eikonal solutions for moment hierarchies of chemical reaction networks in the limits of large particle number.  J. Phys. A Math. Theor., submitted, 2020.
  * (78) Jeremy Gunawardena.  Chemical reaction network theory for in-silico biologists.  lecture notes, June 2003.  vcp.med.harvard.edu/papers/crnt.pdf.
  * (79) Eric Smith and Supriya Krishnamurthy.  Symmetry and Collective Fluctuations in Evolutionary Games.  IOP Press, Bristol, 2015.
  * (80) Jakob L. Andersen, Christoph Flamm, Daniel Merkle, and Peter F. Stadler.  Maximizing output and recognizing autocatalysis in chemical reaction networks is NP-complete.  J. Sys. Chem., 3:1, 2012.
  * (81) Claude Berge.  Graphs and Hypergraphs.  North-Holland, Amsterdam, rev. ed. edition, 1973.
  * (82) John C. Baez and Brendan Fong.  Quantum techniques for studying equilibrium in reaction networks.  J. Compl. Netw., 3:22–34, 2014.  https://academic.oup.com/comnet/article-abstract/3/1/22/490572/Quantum-techniques-for-studying-equilibrium-in?redirectedFrom=fulltext.
  * (83) Andrew James Baish.  Deriving the jarzynski relation from doi-peliti field theory.  Bucknell University Honors Thesis, 2015.
  * (84) Martin Feinberg.  Chemical reaction network structure and the stability of complex isothermal reactors – I. The deficiency zero and deficiency one theorems.  Chem. Enc. Sci., 42:2229–2268, 1987.
  * (85) F. Schlögl.  On thermodynamics near a steady state.  Z. Phys., 248:446–458, 1971.
  * (86) M. I. Dykman, Eugenia Mori, John Ross, and P. M. Hunt.  Large fluctuations and optimal paths in chemical kinetics.  J. Chem. Phys., 100:5735–5750, 1994.
  * (87) N. G. van Kampen.  Stochastic Processes in Physics and Chemistry.  Elsevier, Amsterdam, third edition, 2007.
  * (88) Vincent Danos, Jérôme Feret, Walter Fontana, Russell Harmer, and Jean Krivine.  Rule-based modelling, symmetries, refinements.  Formal methods in systems biology: lecture notes in computer science, 5054:103–122, 2008.
  * (89) John C. Baez.  Quantum techniques for reaction networks.  2013\.  https://arxiv.org/abs/1306.3451.
  * (90) Yuk Yung, M. Allen, and J. P. Pinto.  Photochemistry of the atmosphere of Titan: comparison between model and observations.  Astrophys. J. Suppl. Ser., 55:465–506, 1984.
  * (91) E. Hébrard, M. Dobrijevic, J. C. Loison, A. Bergeat, and K. M. Hickson.  Neutral production of hydrogen isocyanide (HNC) and hydrogen cyanide (HCN) in Titan’s upper atmosphere.  Astron. Astrophys., 541:A21, 2012.
  * (92) Carol Turse, Johannes Leitner, Maria Firneis, and Dirk Schulze-Makuch.  Simulations of prebiotic chemistry under post-imact conditions on Titan.  Life (Basel), 3:538–549, 2013.
  * (93) Frank H. Westheimer.  Why nature chose phosphates.  Science, 235:1173–1178, 1987.
  * (94) Matthew A. Pasek and Terence P. Kee.  On the origin of phosphorylated biomolecules.  In Richard Egel, Dirk-Henner Lankenau, and Armen Y. Mulkidjanian, editors, Origins of Life: The Primal Self-Organization, pages 57–84. Springer-Verlag, Berlin, 2011.
  * (95) Joshua E. Goldford, Hyman Hartman, Temple F. Smith, and Daniel Segrè.  Remnants of an ancient metabolism without phosphate.  Cell, 168:1126–1134, 2017.
  * (96) Joshua E. Goldford, Hyman Hartman, Robert Marsland III, and Daniel Segrè.  Environmental boundary conditions for early life converge to an organo-sulfur metabolism.  Nat. Ecol. Evol., 2019.  doi:10.1038/s41559-019-1018-8.
  * (97) Sadi Carnot.  Reflections on the Motive Power of Fire.  Dover, New York, e. mendoza (ed.) edition, 1960.
  * (98) Pierre Simon Laplace.  A Philosophical Essay on Probabilities.  Dover, New York, original french 6th ed. by truscott, f. w. and emory, f. l. edition, 1951.
  * (99) R. Clausius.  On the application of the theorem of the equivalence of transformations to interior work.  In Hirst Clausius:mech_the_heat:65 , pages 215–250.  Fourth Memoir.
  * (100) Eric Smith.  Quantum-classical correspondence principles for locally non-equilibrium driven systems.  Phys. Rev. E, 77:021109, 2008.  originally as SFI preprint # 06-11-040.
  * (101) Eric Smith and Harold J. Morowitz.  The origin and nature of life on Earth: the emergence of the fourth geosphere.  Cambridge U. Press, London, 2016.  ISBN: 9781316348772.
  * (102) Sergio Verdú.  Fifty years of Shannon theory.  IEEE Trans. Information Theory, 44:2057–2078, 1998.
  * (103) Harold Abelson, Gerald Jay Sussman, and Julie Sussman.  Structure and Interpretation of Computer Programs.  MIT Press, Cambridge, MA, second edition, 1996.
  * (104) E. T. Jaynes.  The minimum entropy production principle.  Annu. Rev. Phys. Chem., 31:579–601, 1980.
  * (105) E. T. Jaynes.  Probability Theory: The Logic of Science.  Cambridge U. Press, New York, 2003.
  * (106) T. Speck and U. Seifert.  Restoring a fluctuation-dissipation theorem in a nonequilibrium steady state.  Europhys. Lett., 74:391–396, 2006.
  * (107) Russ Harmer, Vincent Danos, Jérôme Feret, Jean Krivine, and Walter Fontana.  Intrinsic information carriers in combinatorial dynamical systems.  Chaos, 20:037108, 2010.
  * (108) Nicolas Behr, Vincent Danos, Ilias Garnier, and Tobias Heindel.  The algebras of graph rewriting.  2016\.  arXiv:1612.06240v1 [math-ph] 19 Dec 2016.
  * (109) Eric Smith.  The information geometry of 2-field functional integrals.  submitted, 2019.  http://arxiv.org/abs/1906.09312.
  * (110) R. D. Rosenkrantz, editor.  Jaynes, E. T.: Papers on Probability, Statistics and Statistical Physics.  D. Reidel, Dordrecht, Holland, 1983.