Finite-time generalization of the thermodynamic uncertainty relation

Abstract: For fluctuating currents in non-equilibrium steady states, the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving. We show that this relation holds not only for the long-time limit of fluctuations, as described by large deviation theory, but also for fluctuations on arbitrary finite time scales. This generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments, for which infinite timescales are not accessible. Importantly, often this finite-time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long-time limit. We illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle.

• The paper demonstrates that finite-time current fluctuations in Markovian systems obey a refined lower bound relating variance, entropy production, and the Boltzmann constant.
• It derives a universal parabolic bound for the cumulant generating function, validated by both analytical and numerical analyses across various network models.
• Experimental illustrations on colloidal systems show that the finite-time TUR infers significantly tighter entropy production estimates than its infinite-time counterpart.

Finite-Time Generalization of the Thermodynamic Uncertainty Relation

Introduction

The manuscript systematically extends the scope of the thermodynamic uncertainty relation (TUR), which posits a quantitative link between the precision of non-equilibrium steady-state (NESS) currents and the associated entropy production. The original TUR, grounded in large deviation theory, constrains the fluctuation/precision tradeoff only in the infinite time limit. This paper asserts that a finite-time analogue of the TUR holds across arbitrary time scales, thus enhancing its relevance for realistic experiments wherein accessible trajectories are invariably finite.

Main Result

Let $X(t)$ denote a time-integrated current in a Markovian NESS, with average $\langle X(t)\rangle = Jt$ and variance $Var[X(t)]$. The central analytical and numerical result demonstrated is: $Var[X(t)] \frac{\sigma}{J^2 t} \ge 2 k_\mathrm{B}$ where $\sigma$ is the entropy production rate and $k_\mathrm{B}$ is the Boltzmann constant. This relation not only recovers the standard infinite-time TUR as $t \to \infty$—where $Var[X(t)] \sim 2Dt$ for effective diffusion coefficient $D$—but also applies to arbitrary finite $t$. Importantly, in the finite-time regime, the bound may yield a tighter lower limit on $\sigma$ compared to the infinite-time estimate, contingent on the observable chosen and its statistics.

Experimental Illustration

The extended TUR is illustrated through analysis of work fluctuations in a colloidal system: a particle in a one-dimensional, optically trapped potential, where the trap center is stochastically switched according to a Poisson process. Work is defined via two conventions incorporating the force and trap position. For varying switching rates, the authors obtain the full distribution of work over finite intervals up to $1\,\mathrm{s}$. In all regimes, the finite-time TUR is numerically and empirically satisfied. Notably, for particular choices of the switching rate and time horizon, the finite-time lower bound on entropy production inferred from fluctuations is roughly twice as tight as that provided by the steady-state TUR.

This empirical investigation substantiates the finite-time bound's practical utility, especially in single-molecule systems or small-scale circuits where large time statistics are inaccessible and entropy production is otherwise tough to quantify directly.

Theoretical Analysis

The extension of the TUR relies on establishing a universal, parabolic lower bound for the logarithm of the cumulant generating function (CGF) $g(z, t)$ of the considered current,

$$\frac{1}{t}\ln g(z, t) \geq J z (1 + \frac{z J}{\sigma}),$$

holding for all $z\in \mathbb{R}$ and $t \ge 0$. This theoretical statement encapsulates the original, steady-state TUR as a limiting case.

The authors' numerical investigations corroborate the conjectured bound across a broad set of Markovian networks (from unicyclic rings to fully connected graphs), perturbed by random modifications to their transition rates. For unicyclic networks (e.g., asymmetric random walks on a ring), the CGF can be evaluated exactly and is shown to saturate the bound in the linear response regime.

Proofs for short timescales leverage the fluctuation theorem for the joint distribution of entropy production and current, yielding a weaker but rigorous bound,

$Var[X(t)] \ge \frac{2 t J^2}{\sigma},$

which, together with scaling arguments, recovers the finite-time TUR in the linear-response sector.

Implications and Future Directions

This finite-time generalization of the TUR is significant both experimentally and theoretically. Practically, it enables the inference of stringent lower bounds on entropy production from finite-length trajectories, a scenario common in the analysis of nanoscale systems, molecular motors, or synthetic non-equilibrium networks. The result is robust and agnostic to the experimental protocol, applicable provided only the time-integrated current and a Markovian description.

From a theoretical perspective, the global parabolic bound on the CGF suggests a deeper organizing principle in the structure of stochastic non-equilibrium processes, extending beyond known fluctuation theorems and large deviation theory. While the bound is fully proven only in certain limits, its general validity, substantiated by extensive numerics, motivates the search for new mathematical tools to bridge this gap.

Future directions include rigorous extensions beyond Markov jump processes (e.g., to continuous-state Langevin systems), exploring optimal observable selection for tightest entropy bounds, and leveraging the finite-time TUR to reverse-engineer hidden dissipation in biomolecular and soft-matter systems from short-time data. Its adoption as a diagnostic and inferential tool for non-equilibrium thermodynamics at the mesoscopic scale is a natural progression.

Conclusion

The work robustly generalizes the thermodynamic uncertainty relation to finite observation times, solidifying its foundational status in stochastic thermodynamics. The finite-time TUR delivers a stringent, empirically accessible lower bound on entropy production from current fluctuations, relevant for experimentalists probing the energetics and precision of small systems. The parabolic CGF bound encapsulates a universal property of Markovian NESSs, stimulating further theoretical investigation and experimental utility in the quantification of nonequilibrium irreversibility.