# Revealing hidden structures and symmetries in nonequilibrium transport

David Andrieux

###### Abstract

Recent results have shown how to partition the space of Markov systems into
dynamical equivalence classes. These equivalence classes structure transport
properties in a way that makes, among other features, their responses fully
symmetric. In this note, I illustrate this approach on two representative
systems. First, I derive analytical expressions for the equivalence classes of
a disordered ring model. Second, I verify on a model of ion transport that,
within an equivalence class, the response of coupled currents is symmetric
both near and far from equilibrium.

## The Challenge: Nonequilibrium Response is ill-defined

Nonequilibrium transport is traditionally characterized by quantities such as
currents, affinities or the entropy production of the system. Current
fluctuations and related fluctuation theorems also provide additional
constraints and insights on the behavior of nonequilibrium systems.

However, despite their relevance, these concepts are not enough to build a
well-defined nonequilibrium theory. What do I mean by that?

The core issue is that specifying a system’s currents and affinities (and thus
its entropy production) does not uniquely characterize its nonequilibrium
state. Multiple combinations of parameters can lead to the same affinities and
currents [1]. As a result, you can drive a system out of equilibrium in an
infinite number of ways starting from a given equilibrium state. While in the
linear response regime this ambiguity doesn’t really matter since all response
coefficients are identical regardless of the path you choose, this is no
longer true in the nonlinear regime [2].

Conversely, imagine now that you start with a system out of equilibrium. A
continuum of equilibrium states could serve as reference point and, without a
clearly identified equilibrium state and path to it, the nonequilibrium
response of the system cannot be uniquely defined.

So, how do we remove this arbitrariness and connect equilibrium and
nonequilibrium dynamics in a way that is:

  * •

Well-defined, i.e. the system’s parameters uniquely map all possible
nonequilibrium conditions

  * •

Intrinsic, i.e. the connection emerges from the dynamical characteristics of
the system itself.

To structure the space of possible dynamics while satisfying these
requirements, I previously introduced the concept of dynamical equivalence
classes [3, 4]. These ’hidden structures’ uniquely characterize the
nonquilibrium conditions and parameters of the system, and describe a well-
defined nonequilibrium theory. In addition, using these structures the current
fluctuations can be obtained from an equilibrium dynamics and their nonlinear
responses take an especially simple, symmetric form [5]. These properties go
beyond the traditional current fluctuation theorem [5, 3], offering a new lens
to understand nonequilibrium systems.

To build physical intuition around dynamical equivalence classes, I recently
derived their structure and corresponding transport properties for a molecular
motor toy model [6]. In this note, my objective is to provide additional
examples of how this framework plays out in two specific cases. I first
provide an analytical expression for the equivalence classes of a disordered
ring model. Next, using a model of ion transport in membranes, I illustrate
the symmetries of the response coefficients when multiple currents are
present. I hope these examples will help to build physical intuition and
kickstart the broader application of this framework, e.g. to the design and
control of nonequilibrium systems.

##  I Hidden Structure: Transport on a Disordered Ring example

Let’s consider a $N$-state system characterized by the allowed transitions
$i\rightarrow i\pm 1$ with periodic boundary conditions:

$\displaystyle P$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&p_{1}&&&&1-p_{1}\\\ 1-p_{2}&0&p_{2}&&&\\\
&1-p_{3}&0&p_{3}&&\\\ &&\ddots&0&\ddots&\\\ &&&1-p_{N-1}&0&p_{N-1}\\\
p_{N}&&&&1-p_{N}&0\end{pmatrix}\,.$ (1)

The space of possible dynamics constitues a $N$-dimensional cube
$\Sigma=\\{0<p_{i}\leq 1\\}$ with $i=1,\cdots,N$. Within this space, the set
of equilibrium dynamics forms an $(N-1)$-dimensional hypersurface defined by

$\displaystyle\prod_{i=1}^{N}\frac{p_{i}}{(1-p_{i})}=1\,.$ (2)

Outside of this hypersurface, the affinity

$\displaystyle A=\ln\prod_{i=1}^{N}\frac{p_{i}}{(1-p_{i})}$ (3)

differs from zero and the system is out of equilibrium. We see from this
expression that the nonequilibrium response is not uniquely defined: An
infinite number of paths exist in $\Sigma$ with the same affinities (and
starting from the same equilibrium state) but with different currents [6].

To address this ambiguity, we can partition the space $\Sigma$ into dynamical
equivalent classes as follows [3]. Two dynamics $P$ and $Q$ belong to the same
class if there exists a scalar $\rho\geq 0$ such that:

$\displaystyle P\sim Q\quad{\rm if}\quad P\circ P^{T}=\rho\,\,Q\circ Q^{T}\,,$
(4)

where $\circ$ denotes the Hadamard product.

We can use relation (4) to generate the equivalence classes of the disordered
ring (1). Consider an equilibrium dynamics $\bar{Q}$ with transition
probabilities $\bar{q}_{i}$ satisfying (2). Any dynamics $P$ in $[\bar{Q}]$
then satisfies the system of $N$ polynomial equations

$\displaystyle p_{i}(1-p_{i+1})=\rho\,\bar{q}_{i}(1-\bar{q}_{i+1})$ (5)

for some value of the parameter $\rho$.

This polynomial system can be solved using Gröbner bases. For $N=3$, the
solution of (5) takes a simple form. Introducing the quantities

$\displaystyle a_{i}(\rho)\equiv\rho\,\bar{q}_{i}(1-\bar{q}_{i+1})-1\,,$ (6)

the equivalence class $[\bar{Q}]$ is given by the set of dynamics

$\displaystyle
p^{\pm}_{i}(\rho)=\frac{1}{2a_{i+1}}(\pm\sqrt{\Delta}+a_{i-1}-a_{i}+a_{i+1})$
(7)

with

$\displaystyle\Delta=(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})-2(a_{1}a_{2}+a_{2}a_{3}+a_{1}a_{3})-4\,a_{1}a_{2}a_{3}\,,$
(8)

and where $\rho$ varies between $[0,1]$ and the coefficients $a_{i}$ satisfy
periodic boundary conditions, i.e. $a_{N+1}=a_{1}$ and $a_{0}=a_{N}$. In this
parametrization, equilibrium occurs when $\rho=1$ or equivalently $\Delta=0$
[7]. For $\rho<1$ ($\Delta>0$), both solutions $p^{\pm}_{i}$ belong to
$[\bar{P}]$. They are time-reversed of each other, with opposite affinities
and currents.

Figure 1: The space $\Sigma$ for $N=3$ and its decomposition into equivalence
classes. Each curve corresponds to a different equivalence class $[\bar{Q}$].
Each class crosses the equilibrium hypersurface once and spans the full range
of affinities.

Dynamical equivalence classes thus have a clear physical interpretation. Each
equivalence class determines a curve in $\Sigma$ (more generally a manifold)
that crosses the equilibrium hypersurface only once and whose affinities span
the entire range $[-\infty,\infty]$ as the curves tends to $p_{i}=0$ or
$p_{i}=1$ for all $i$ (Figure 1).

As a result, within an equivalence class the nonequilibrium behavior is well-
defined: there is a unique way to vary the parameters while driving the system
out of equilibrium. Moreover, for each dynamics in $[\bar{Q}]$ its time-
reversed one also belongs to $[\bar{Q}]$ with opposite affinities and
currents. These properties are valid for the equivalence classes of an
arbitrary system [5, 6].

Now that we understand the structure of equivalence classes and their physical
interpretation, let’s use them to study the nonequilibrium behavior of the
disordered ring system.

The current $J(A)$ is illustrated in Fig. 2a for different equivalence classes
(exact expressions for the currents can be obtained using the techniques from
Ref. [8, 2, 9]). It is antisymmetric, $J(-A)=-J(A)$, and reaches the maximal
possible value $J=\pm 1/3$ as $A$ tends to $\pm\infty$.

Figure 2: Current $J(A)$ and diffusion coefficient $K^{(2)}(A)$ for various
equivalent classes. (a) The current reaches the maximal possible value $J=\pm
1/3$ as $A$ tends to $\pm\infty$. (b) The diffusion coefficients are derived
from the currents via Eq. (9). They vanish as the affinity goes to $\pm\infty$
in all equivalence classes. In contrast, varying affinities outside
equivalence classes leads to different behaviors and artifacts related to the
specific choice of the nonequilibrium path [6] (not shown).

The higher-order cumulants $K^{(n)}(A)$ can then be directly obtained from the
average current as

$\displaystyle\frac{1}{2^{n-1}}K^{(n)}(A)=\frac{{\rm d}J^{n}(A)}{{\rm d}A}\,.$
(9)

This relation holds outside the linear regime and at all orders $n$. In
addition, knowledge of the current $J$ fully determines all the cumulants
$K^{(n)}$. In contrast, the traditional fluctuation relations determine only
half of the degrees of freedom and lead to more complex relationships across
cumulants [5, 17].

The diffusion coefficients are symmetric, $K^{(2)}(-A)=K^{(2)}(A)$, and vanish
as the affinity goes to $\pm\infty$ in all equivalence classes (Figure 2b). In
contrast, artifacts can occur when driving a system out of equilibirum through
arbitrary paths in $\Sigma$, such as saturating currents before they reach
their maximal possible value, loss of their anti-symmetry, or the appearances
of additional local structures [6].

The solution (8) of the disordered ring provides insights into the
nonequilibrium dynamics of systems with one independent current. For systems
for more than one currents, the equivalence classes become higher-dimensional
manifolds where additional symmetries emerge [5]. We illustrate these
additional symmetries in the next section.

##  II Hidden symmetries: Ion Transport example

Let’s now turn to a model of ion transport as an archetype of systems with two
independent currents. Following Hill [2], consider a cell surrounded by a
membrane that separates the cell’s interior (In) from its environment (Out). A
complex E, which can exist in two conformations E and E*, has binding sites
for ions L and M. These sites are accessible to inside molecules in
configuration E only, and to outside molecules in configuration E* only. L can
be bound only when M is already bound on its site (Figure 3a).

Figure 3: Ion transport model with two independent currents. (a) Mechanism for
the transport of ions M and L across the membrane. In the normal mode of
operation, molecule M has a larger concentration inside than outside, $[{\rm
M_{i}}]>[{\rm M_{o}}]$, while the opposite holds for molecule L, $[{\rm
L_{o}}]>[{\rm L_{i}}]$. The complex E acts as a free energy transducer and
utilizes the M concentration gradient to drive molecules of L from inside to
outside against its concentration gradient. For example, in the case of the
Na/K-ATPase complex M and L would correspond to ${\rm K}^{+}$ and ${\rm
Na}^{+}$, and transport would be coupled to ATP consumption. (b) Kinetic
diagram. (c) Cycle decomposition. The cycles $a$ and $b$ are chosen as the two
independent cycles. The positive orientation is chosen counterclockwise
(adapted from Hill [2]).

The system has $N=6$ states connected by the kinetic diagram depicted in
Figure 3b. The parameter space $\Sigma$ is $8$-dimensional while the set of
equilibrium dynamics forms an $6$-dimensional manifold in $\Sigma$.

Each equivalence class corresponds to a $2$-dimensional manifold [13, 14]
related to the two independent affinties or currents (here measured along
cycles $a$ and $b$, see Figure 3c). As expected, these currents are coupled,
i.e. a change in $A_{a}$ will drive the current $J_{b}$ and vice versa (Figure
4). They display a nonlinear behavior and, beyond the anti-symmetry
$\boldsymbol{J}(-\boldsymbol{A})=-\boldsymbol{J}(\boldsymbol{A})$
characteristic of equivalence classes, no other symmetry is apparent.

Figure 4: Coupled currents of the ion transport model. Currents $J_{a}$ and
$J_{b}$ and their isolines (solid curves) as a function of the affinities
$A_{a}$ and $A_{b}$ in a given equivalence class. The equivalence class is
defined by the equilibrium dynamics
$p_{12}=0.54,p_{16}=0.46,p_{21}=0.9,p_{23}=0.04,p_{25}=0.06,p_{32}=0.06,p_{34}=0.94,p_{43}=0.65,p_{45}=0.35,p_{52}=0.06,p_{54}=0.38,p_{56}=0.56,p_{61}=0.58,$
and $p_{16}=0.42$. The affinities $\boldsymbol{A}$ take values in the range
$[-10,+10]$.

Despite the nonlinear behavior of the currents, their response displays a
’hidden symmetry’ both near and far from equilibrium [6]:

$\displaystyle\frac{\partial J_{a}}{\partial
A_{b}}(\boldsymbol{A})=\frac{\partial J_{b}}{\partial A_{a}}(\boldsymbol{A})$
(10)

for all values of the affinities $\boldsymbol{A}$ in a given equivalence class
(Figure 5). The symmetry (10) is confirmed for all other equivalence classes
(a Scilab code to generate the response coefficients for an arbitrary
equivalence class is available upon request). Systems with more than two
independent currents display the same symmetries. Higher-order fluctuations
can be obtained by further derivation of the currents and their responses are
also symmetric [6].

Figure 5: The currents response is symmetric both in the linear and nonlinear
regimes. The response coefficients
$\partial\boldsymbol{J}/\partial\boldsymbol{A}$ display a complex structure in
the nonlinear regime. Even so, the cross-response coefficients are identical
both near and far from equilibrium. Parameters take the same values as in Fig.
(4).

##  III Outlook

Properties such as (9) and (10) confirm the importance of dynamical
equivalence classes as a central concept in nonequilibrium thermodynamics.
Where do these remarkable properties come from?

In the traditional way of looking at nonequilibrium systems, affinities are
varied along an arbitrary path in the parameter space (typically by keeping
some parameters constant while varying the others). This makes each dynamics
along the path unrelated to each other and with possibly very different
kinetic properties. In contrast, within an equivalence class, all systems
share similar kinetic properties through relation (4). Their current
generating functions all have the same shapes, modulo a translation [3, 4].
All the fluctuations and response coefficients are thus directly related to
each other, and equilibrium and nonequilibrium fluctuations are essentially
identical.

Beyond their theoretical interests, dynamical equivalence classes bring
intriguing perspectives for the design and control of nonequilibrium systems.
For example, designing mesoscopic devices around the symmetry (10) could make
them easier to control and/ or robust to perturbations.

More generally, suppose you operate a mesoscopic device designed to achieve a
certain output (i.e., currents). The device’s parameters presumably reflect a
trade-off around the desired transport properties and the admissible
fluctuactions and dissipation. Imagine now that you need to achieve a
different output while minimizing any other disruption to the device. How
would you ensure the devive continues to operate as smoothly as possible?

One way to go about it would be to change the device’s outputs while also
staying as close as possible to the starting dynamics to minimize unrelated
deviations or disruptions. This is precisely what dynamical equivalence
classes accomplish! If the device has starting dynamics $G$, its equivalence
class is obtained as [12]

$\displaystyle\min_{P}\left[D(P|G)+\sum_{a}\lambda_{a}J_{a}[P]\right]\,.$ (11)

Here the $\lambda_{a}$s act as control parameters or Lagrange multipliers for
the constrained currents. Meanwhile, the divergence
$D(P|G)=\sum_{i,j}\pi_{i}P_{ij}\ln(P_{ij}/G_{ij})$ measures the statistical
distance between $P$ and $G$ (in information theoric terms, $D$ is the
additional cost in nats required to encode $P$ using measure $G$).

When solving the minimization problem, the Lagrange multipliers relate to the
affinities as $\lambda_{a}=(A_{a}-\bar{A}_{a})/2$, where $\bar{A}_{a}$ are the
affinities of $G$. In other words, the minimization happens across the space
of dynamics $P$ with given affinities $A_{a}=2\lambda_{a}+\bar{A}_{a}$.

Putting everything together, Eq. (11) reveals that equivalence classes select
the dynamics minimizing the statistical distance to $G$ given fixed currents
(or, equivalently, fixed affinities). In other words, using equivalence
classes we can control a device’s output while minimizing its distortions with
respect to a target dynamics $G$. I believe this has practical applications
and I hope these observations will stimulate further exploration of these
concepts.

Disclaimer. This paper is not intended for journal publication.

## References

  * [1] For example, a Markov chain with $N$ states, $E$ edges, and $S$ self-transitions will have $E-N+1$ independent currents or affinities but $2E+S-N>E-N+1$ independent parameters.
  * [2] T. L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics (Dover, 2005).
  * [3] D. Andrieux, arXiv:1208.5699 (2012).
  * [4] D. Andrieux, arXiv:1212.1807 (2012).
  * [5] D. Andrieux, arXiv:2205.10784 (2022).
  * [6] D. Andrieux, arXiv:2306.01445 (2023).
  * [7] The parameter $\rho$ takes values in $[0,1]$ since I choose an equilibrium dynamics $\bar{Q}$ as the reference dynamics. Other choices would lead to a different range of values $\rho$ for which the polynomial system has real solutions [3]. Incidently, $\rho$ is directly related to the current generating function $q=-(1/2)\log\rho$.
  * [8] J. Schnakenberg, Rev. Mod. Phys 48, 571 (1976).
  * [9] D.-Q. Jiang, M. Qian, and M.-P. Qian, Mathematical Theory of Nonequilibrium Steady States (Springer, Berlin, 2004).
  * [10] Using the results from [11, 12], the equilibrium dynamics $\bar{Q}$ of an equivalence class $[P]$ minimizes the Kullback-Leibler divergence between $\bar{Q}$ and $P$, i.e. $\bar{Q}=\min_{G}D_{KL}(G||P)$ where $G$ belongs to the set of compatible equilibrium dynamics.
  * [11] Shi-Feng Shieh, Entropy 13, 2036 (2011).
  * [12] D. Andrieux, arXiv:1201.1232 (2012).
  * [13] For a Markov chain with $N$ states, $E$ edges, and $S$ self-transitions, the number of independent parameters is $2E+S-N$, the equilibrium hyper-surface has dimension $E+S-1$, and an equivalence class forms a $(E-N+1)$-dimensional space.
  * [14] The kinetic diagram (4) has 14 transition probabilities $p_{ij}$. However, since they must satisfy $\sum_{j}p_{ij}=1$ only 8 of them are independent. A neat decomposition of transition matrices into independent parameters can be obtained using the rotational representation of Markov chains [15, 16].
  * [15] S. Alpern, The Annals of Probability 11, 789 (1983).
  * [16] S. Kalpazidou, Cycle Representations of Markov Processes (Springer, 2006).
  * [17] M. Barbier and P. Gaspard, J. Phys. A: Math. Theor. 51 355001 (2018).