Abstract
We introduce a new technique to bound the fluctuations exhibited by a physical system, based on the Euclidean geometry of the space of observables. Through a simple unifying argument, we derive a sweeping generalization of so-called thermodynamic uncertainty relations (TURs). We not only strengthen the bounds but extend their realm of applicability and in many cases prove their optimality, without resorting to large deviation theory or information-theoretic techniques. In particular, we find the best TUR based on entropy production alone. We also derive a periodic uncertainty principle of which previous known bounds for periodic or stationary Markov chains known in the literature appear as limit cases. From it a novel bound for stationary Markov processes is derived, which surpasses previous known bounds. Our results exploit the non-invariance of the system under a symmetry which can be other than time reversal and thus open a wide new spectrum of applications.
Highlights
At several levels of complexity, random processes are successfully employed to model natural phenomena, such as open quantum system [1], soft and active matter [2], biochemical reactions [3], and population ecology [4], just to name a few
Stochastic thermodynamics has emerged as a comprehensive framework to rigorously study the energetics and thermodynamics of stochastic processes [8, 9]
It was first conjectured in [11] that p for a time-integrated current-like observable f is bounded by half the expected entropy σ produced over the interval tf, i.e. pmax σ /2
Summary
At several levels of complexity, random processes are successfully employed to model natural phenomena, such as open quantum system [1], soft and active matter [2], biochemical reactions [3], and population ecology [4], just to name a few. Uncertainty relations appeared as a new powerful tool to investigate dynamical fluctuations They denote a set of inequalities in which the square-mean-to-variance ratio, or precision p(f ), of a generic observable f integrated over a time interval tf is bounded by an f-independent functional, which constitutes an upper estimate on the maximum possible precision pmax:. It was first conjectured in [11] that p for a time-integrated current-like (i.e. odd under time reversal) observable f is bounded by half the expected entropy σ produced over the interval tf , i.e. pmax σ /2 This so-called thermodynamic uncertainty relation, originally proved in the linear response regime and under stationary conditions, triggered an intense activity seeking generalizations or improvements for the largest possible class of out-of-equilibrium conditions. In the case of stationary time-invariant Markov processes, it allows to replace them with simple and tighter bounds, valid over all time intervals, the most revealing being the sum of the absolute values of the currents (tables 1 and 2)
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