Near-optimality of conservative driving in discrete systems

Abstract

Transferring a physical system from an initial to a final state while minimizing energetic losses is an interdisciplinary control problem that bridges stochastic thermodynamics and optimal transport theory. Recent research typically considers problems in which the optimal solution is realized via conservative forces, but whether this situation applies depends on the problem's constraints. In systems with complex topologies like discrete networks, the optimal, dissipation-minimizing protocol involves applying nonconservative forces along cycles if the timescales of the transitions in the network are fixed. We show that although nonconservative driving is optimal in this setting, a conservative protocol exists whose dissipation is at most twice the optimal one. This finding is complemented with an example modeling transport across an energy barrier, which illustrates such improvements of order 1 explicitly. Qualitatively, conservative driving falls short of achieving optimality because direct transport across the barrier is avoided. We conclude with a discussion that the optimality of nonconservative driving might be a generic phenomenon: As fewer degrees of freedom can be optimized, additional degrees of freedom due to adding nonconservative forces become more significant.

Figures (4)

• Figure 1: Optimal versus conservative driving in a three-state Markov network. Colors encode the maximal possible improvement $\sigma^*/\sigma^{\text{cons.}}$ according to the color scale. For given occupation probabilities $p_i$, the configurations that yield a desired current $J = \dot{p}_2 = - \dot{p}_1$ can be parametrized by the cycle affinity $\mathcal{A} = A_{12}+A_{23}+A_{31}$. The minimal thermodynamic cost \ref{['eq:intro:sigma']}$\sigma^*$ after optimizing $\mathcal{A}$ is compared to the cost $\sigma^{\text{cons.}}$ of the conservative protocol satisfying $A_{12}+A_{23}+A_{31} = 0$. The transition rates are parametrized as in Eq. \ref{['eq:intro:k_param']} with fixed ${\kappa_{ij} = 1}$ for $i\neq j$. For $J = 2$, the minimum $\sigma^*/\sigma^{\text{cons.}} = 0.983$ is located at $(p_1, p_2) = (0.03, 0.03)$.
• Figure 2: Transport across an energy barrier in a unicyclic network. a) Set-up: The transitions' timescales are ${\kappa_{12} = e^{-E_b}}$, modeling an energy barrier of height $E_b = N E_0$, and ${\kappa_{ij} = 1}$ otherwise. A desired change in probability $\dot{p}_2 = - \dot{p}_1 = J$ can be accomplished through a local current $j_{12}$ across the energy barrier or a cycle current $j_\mathcal{C} = J - j_{12}$ through the bulk. b) Comparison of conservative and optimal transport with respective costs $\sigma^*$ and $\sigma^{\text{cons.}}$. The theoretical lower bound $\sigma^*/\sigma^{\text{cons.}} \geq 0.5$ (cf. Eq. \ref{['eq:mainres:bounds']}) is compared to numerical calculations for $p_i = 1/N$ and optimized $J$. Increasing $N$ leads to a decrease in $\sigma^*/\sigma^{\text{cons.}}$ but becomes increasingly numerically unstable. A minimal value of $\sigma^*/\sigma^{\text{cons.}} \simeq 0.76$ can be attained for various values of $E_0$. c) Entropy production $\sigma(j_\mathcal{C})$ and $\rho(j_\mathcal{C})$ (cf. Eq. \ref{['eq:mainres:rho']}) as a function of the bulk current $j_\mathcal{C}$ for $N=100$, $E_0 = 0.3$ and the numerically optimized $J \simeq 0.58$. The optimal solution $j_\mathcal{C}^* \simeq 0.3$ balances transport across the energy barrier and through the bulk. In contrast, the conservative solution $j_\mathcal{C}^{\text{cons.}} \simeq 0.55$ minimizes $\rho$ (cf. Eq. \ref{['eq:proof:drho_djc']}) and mostly transports current through the bulk.
• Figure 3: a) A Markov network with two fundamental cycles $\mathcal{C}_1 = (1231)$ and $\mathcal{C}_2 = (1341)$. In this example, $\partial_{j_{\mathcal{C}_1}} = \partial_{j_{12}} + \partial_{j_{23}} - \partial_{j_{13}}$. Note that increasing the cycle currents $j_{12}, j_{13}, j_{31} = - j_{13}$ by the same amount does not affect a given time evolution of the occupation probabilities $p_i(t)$. b) Visualization of the derivation that a relation $\sigma/2 \leq \rho \leq \sigma$ between two cost functions $\sigma$, $\rho$ implies the same chain of inequalities on their minimal values $\sigma^*$, $\rho^*$. We can relate $\sigma^c$, the value of $\sigma$ evaluated at the point at which $\rho$ is minimized, to its minimum $\sigma^*$ via $\sigma^c \leq 2 \sigma^*$. In the main text, the configurations minimizing $\sigma$ and $\rho$ characterize optimal and conservative driving, respectively.
• Figure 4: a) and b) A graphical solution of the transcendental equations \ref{['eq:app_d:f']} and \ref{['eq:app_d:g']}. The left-hand side of each equation is depicted as a black solid curve, whereas the right-hand sides are depicted in shades of blue for different values of $N$, $E_b = N E_0$, $E_0 = 0.3$ and $J = 0.5824$. The abscissa of the intersection of the blue and black curve give the respective value of $j^*_{\mathcal{C}}$ and $j_\mathcal{C}^{\text{cons.}}$ in panel a) and b), respectively. c) The numerically optimized $J(N)$ as a function of $N$ for $E_0 = 0.3$ (parameters identical to the purple curve in Fig. \ref{['fig:fig1']} b)). The conservative and optimal current are also depicted as functions of $N$ for the same parameters. d) Unlike the conservative solution, the optimal solution $j_{\mathcal{C}}^*$ features a nonzero cycle affinity $\mathcal{A}_\mathcal{C}^*$. Its absolute value for the previous set of parameters is compared to the theoretical upper bound $\mathcal{A}_\mathcal{C}^* \leq \mathcal{A}_{\text{max}}(N) = 2(N-2)$ (see Eq. \ref{['eq:app_d:aff']} and Ref. reml21). In addition, the quality factor $\mathcal{Q} = \mathcal{A}^*_{\mathcal{C}}/[2(N-2)]$ is shown, which remains roughly constant across all scales of $N$.