Physical insight into the thermodynamic uncertainty relation using Brownian motion in tilted periodic potentials

Abstract

Using Brownian motion in periodic potentials V (x) tilted by a force f , we provide physical insight into the thermodynamic uncertainty relation, a recently conjectured principle for statistical errors and irreversible heat dissipation in nonequilibrium steady states. According to the relation, nonequilibrium output generated from dissipative processes necessarily incurs an energetic cost or heat dissipation q, and in order to limit the output fluctuation within a relative uncertainty , at least 2kBT / 2 of heat must be dissipated. Our model shows that this bound is attained not only at near-equilibrium (f V (x)) but also at far-from-equilibrium (f V (x)), more generally when the dissipated heat is normally distributed. Furthermore, the energetic cost is maximized near the critical force when the barrier separating the potential wells is about to vanish and the fluctuation of Brownian particle is maximized. These findings indicate that the deviation of heat distribution from Gaussianity gives rise to the inequality of the uncertainty relation, further clarifying the meaning of the uncertainty relation. Our derivation of the uncertainty relation also recognizes a new bound of nonequilibrium fluctuations that the variance of dissipated heat (σ 2 q ) increases with its mean (µq) and cannot be smaller than 2kBT µq.Precise determination of an output information from a thermodynamically dissipative process necessarily incurs energetic cost to generate it. Trade-offs between energetic cost and information processing in biochemical and biomolecular processes have been highlighted for the last decades [1][2][3][4][5]. Among others, Barato and Seifert [1] have recently conjectured a fundamental bound in the minimal heat dissipation (q) to generate an output with relative uncertainty ( ). To be specific, when a molecular motor moves along cytoskeletal filament [6][7][8], the chemical free energy transduced into the motor movement, which results in a travel distance of the motor X(t), is eventually dissipated as heat into the surrounding media [9,10], the amount of which increases with the time ( q ∼ t). Because of the inherent stochasticity of chemical processes, the travel distance X(t) has its own variance σ 2 X = (δX(t)) 2 , and defines a time-dependent fluctuation in the output, whose squared quantity decreases with time t, as 2 X ∼ t −1 . The product of the two quantities, Q, is, in fact, independent of t [1, 11], and it was further argued that Q is always greater than 2k B T for any process that can be described as Markov jump process on a suitable network. This notion is concisely written asThe validity of this inequality was claimed for general Markovian networks [1,2], and was partly proved at near equilibrium, linear response regime [1]. This effort has recently been followed by a general proof employing the large deviation theory [12][13][14]. * hyeoncb@kias.re.krHere, while limited to a particular model, we provide a less abstract and physically more tangible proof of the thermodynamic uncertainty relation (Eq....

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