Quantifying dissipation using fluctuating currents

Systems coupled to multiple thermodynamic reservoirs can exhibit nonequilibrium dynamics, breaking detailed balance to generate currents. To power these currents, the entropy of the reservoirs increases. The rate of entropy production, or dissipation, is a measure of the statistical irreversibility of the nonequilibrium process. By measuring this irreversibility in several biological systems, recent experiments have detected that particular systems are not in equilibrium. Here we discuss three strategies to replace binary classification (equilibrium versus nonequilibrium) with a quantification of the entropy production rate. To illustrate, we generate time-series data for the evolution of an analytically tractable bead-spring model. Probability currents can be inferred and utilized to indirectly quantify the entropy production rate, but this approach requires prohibitive amounts of data in high-dimensional systems. This curse of dimensionality can be partially mitigated by using the thermodynamic uncertainty relation to bound the entropy production rate using statistical fluctuations in the probability currents.

. Selecting the bandwidth for kernel density estimation The five-bead model's mean squared error for the temporal estimator (•) and the TUR lower bound ( ) are sensitive to the bandwidth. Data are reported for Tc/T h = 0.1 with the bandwidth scaled by the Bowman and Azzalini rule of thumb value. The MSE is estimated from Eq. (21) by averaging over ten independent trajectories with length τ obs = 1200. Error bars are the standard error, computed by repeating that procedure ten times.
Supplementary Figure 2. Efficiency of TUR bound estimation does not strictly improve with smaller dissipation (a) The dissipation rate, measured in units of kBk2/γ, varies as a function of temperature ratio and spring constant ratio. In generalṠss increases with large temperature differences and spring constant differences. (b) The efficiency of the TUR lower bound calculated numerically using the tilting procedure. Notice that the contour lines of the two plots do not overlap, indicating that the microscopic details are also important to the tightness of the lower bound.

Supplementary Note 1: Bead-spring analytical derivation
Solving for the steady-state behavior of linearly-coupled degrees of freedom is standard, but we review the derivation for completeness. As an ansatz, we insert a Gaussian steady state density, ρ ss ∝ e − 1 2 x T C −1 x , into the Fokker-Planck equation, where C is a symmetric matrix. It is straightforward to confirm from Eq. (3) that for symmetric A. We must then choose C such that the term in parentheses will vanish. The right choice is to set C = lim t→∞ C(t), the long-time limit of the correlation matrix. To see the connection with the correlation matrix, note that the solution to a general Langevin equation of the formẋ = Ax + F ξ can be written as for an arbitrary choice of A and F . Plugging Eq.
(2) into the definition of correlation matrix C ij (t) = x i (0)x j (t) , one easily recovers Eq. (4). Differentiating Eq. (4) with respect to time gives dC(t)/dt = AC(t) + C(t)A T + 2D, where we have used D = F F T /2. The correlation matrix converges to a constant value at long times, so its derivative must vanish, requiring that 0 = AC + CA + 2D. Hence the Gaussian ansatz solves the Fokker-Planck equation with C set by the correlation matrix. The steady-state current in Eq. (5) follows directly by plugging this ρ ss into Eq. (3): The total entropy production rate is an integral of the local entropy production rate, which is the product of the current and the conjugate thermodynamic forcė Inserting the formula for steady-state density and current from Eq. (5) into Supplementary Equation (4) yieldṡ which simplifies to Eq. (7) upon performing the Gaussian integral.
In the main text, we discussed the steady-state properties with two beads, but the model with five beads could be solved following the same procedure. In this case, A and D are 5 × 5 matrices: The total entropy production rate, calculated from the first line of Eq. (7), simplifies tȯ for the five-dimensional case.