Control Strategies for Maintaining Transport Symmetries Far From Equilibrium

TL;DR

The paper introduces two dynamical equivalence-class strategies, $e$ and $m$, to steer transport in mesoscopic Markov devices while preserving symmetry properties far from equilibrium. The $e$-scheme uses an affinity-based representation, yielding anti-symmetric current–affinity relations and a KL-based potential that minimizes distortion relative to the original dynamics. The $m$-scheme builds on a current-based representation, producing linearly parameterized dynamics with symmetric nonlinear response, and a distinct equivalence class that preserves activities. Together, these schemes reveal hidden geometric structures in parameter space and offer practical routes to design and control coupled transport processes in molecular machines and membrane transport, with explicit constructions for representative models. The frameworks bridge information geometry, fluctuation theory, and nonequilibrium thermodynamics, enabling targeted output control under strong driving.

Abstract

We present two strategies for controlling the transport dynamics of mesoscopic devices. In both cases, we manipulate the system's output - such as particle currents and energy flows - while maintaining symmetric transport properties, even under far-from-equilibrium conditions. We provide exact descriptions of each scheme and investigate their characteristics. Notably, one of them minimizes the dissimilarity between the original and modified processes, as quantified by the Kullback-Leibler divergence. These findings can be used to improve the design and control of mesoscopic systems.

Figures (8)

• Figure 1: Schematic representation of the $e$ and $m$ equivalence classes in parameter space.. Both $e$ and $m$ manifolds have dimension $E-N+1$ (here shown as 1-dimensional). They contain the time-reversed dynamics $P^*$, and cross the equilibrium manifold $\Sigma_0$ at $P^e$ and $P^m$, respectively. The equilibrium dynamics $P^e$ and $P^m$ are in general distinct from each other. The $e$-equivalence class typically forms a non-linear manifold while the $m$ class is linear in parameter space.
• Figure 2: e-equivalence classes and corresponding current-affinity curves for the molecular motor. (a) Equivalence classes for different values of $X =[-20, -10, -5, 0, 5, 10, 20]$. Each class traces a non-linear path in parameter space except for the class $X=0$. Each class intersects the equilibrium manifold (dashed line) at a unique point $P_X^e$. (b) Current-affinity curves (\ref{['MM.J.e']}) for different equivalence classes $X$. Each curve is anti-symmetric and invariant under the transformation $X \rightarrow -X$. Each curve spans the entire range of possible current values $\pm 1/2\ell$.
• Figure 3: $m$ equivalence classes and corresponding current-affinity curves for the molecular motor. (a) $m$ equivalence classes for different values of $Y=Y_1 = [0.03, 0.08, 0.12, 0.17, 0.21, 0.26, 0.3]$. Each class forms a straight line in parameter space, and intersects the equilibrium manifold (dashed line) at a unique point $P_Y^m$. (b) Current-affinity curves (\ref{['MM.A.m']}) for different equivalence classes $Y$. Each curve is anti-symmetric and invariant under the transformation $Y \rightarrow 1/\ell-Y$. The currents reach different limiting values depending on the value of $Y$.
• Figure 4: 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 H05).
• Figure 5: Coupled currents of the ion transport model along an $e$ equivalence class. 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 $P^e$ with $p_{12} = 0.62, p_{16} = 0.38,p_{21} = 0.57, p_{23} = 0.13, p_{25} = 0.30, p_{32} = 0.41, p_{34} = 0.59, p_{43} =0.46, p_{45} = 0.54, p_{52} = 0.42, p_{54} = 0.32, p_{56} = 0.26, p_{61} = 0.66,$ and $p_{16} = 0.34$.