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Register a new userDetailed Jarzynski Equality applied on a Logically Irreversible Procedure
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Authors
Antoine Berut
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A single bit memory system is made with a brownian particle held by an optical tweezer in a double-well potential and the work necessary to erase the memory is measured. We show that the minimum of this work is close to the Landauers bound only for very slow erasure procedure. Instead a detailed Jarzynski equality allows us to retrieve the Landauers bound independently on the speed of this erasure procedure. For the two separated subprocesses, i.e. the transition from state 1 to state 0 and the transition from state 0 to state 0, the Jarzynski equality does not hold but the generalized version links the work done on the system to the probability that it returns to its initial state under the time-reversed procedure.
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We discuss and qualify a previously unnoticed connection between two different phenomena in the physics of nanoscale friction, general in nature and also met in experiments including sliding emu- lations in optical lattices, and protein force spectroscopy. The first is thermolubricity, designating the condition in which a dry nanosized slider can at sufficiently high temperature and low velocity exhibit very small viscous friction f ~ v despite strong corrugations that would commonly imply hard mechanical stick-slip f ~ log(v). The second, apparently unrelated phenomenon present in externally forced nanosystems, is the occurrence of negative work tails (free lunches) in the work probabilty distribution, tails whose presence is necessary to fulfil the celebrated Jarzynski equality of non-equilibrium statistical mechanics. Here we prove analytically and demonstrate numerically in the prototypical classical overdamped one-dimensional point slider (Prandtl-Tomlinson) model that the presence or absence of thermolubricity is exactly equivalent to satisfaction or violation of the Jarzynski equality. The divide between the two regimes, satisfaction of Jarzynski with ther- molubricity, and violation of both, simply coincides with the total frictional work per cycle falling below or above kT respectively. This concept can, with due caution, be extended to more complex sliders, thus inviting crosscheck experiments, such as searching for free lunches in cold ion sliding as well as in forced protein unwinding, and alternatively checking for a thermolubric regime in dragged colloid monolayers. As an important byproduct, we derive a parameter-free formula expressing the linear velocity coefficient of frictional dissipated power in the thermolubric viscous regime, correcting previous empirically parametrized expressions.
We illustrate the Jarzynski equality on the exactly solvable model of a one-dimensional ideal gas in uniform expansion or compression. The analytical results for the probability density $P(W)$ of the work $W$ performed by the gas are compared with the results of molecular dynamics simulations for a two-dimensional dilute gas of hard spheres.
One particle in a classical perfect gas is driven out of equilibrium by changing its mass over a short time interval. The work done on the driven particle depends on its collisions with the other particles in the gas. This model thus provides an example of a non-equilibrium process in a system (the driven particle) coupled to an environment (the rest of the gas). We calculate the work done on the driven particle and compare the results to Jarzynskis equality relating a non-equilibrium work process to an equilibrium free-energy difference. The results for this model are generalised to the case of a system that is driven in one degree of freedom while interacting with the environment through other degrees of freedom.
We have experimentally checked the Jarzynski equality and the Crooks relation on the thermal fluctuations of a macroscopic mechanical oscillator in contact with a heat reservoir. We found that, independently of the time scale and amplitude of the driving force, both relations are satisfied. These results give credit, at least in the case of Gaussian fluctuations, to the use of these relations in biological and chemical systems to estimate the free energy difference between two equilibrium states. An alternative method to estimate of the free nergy difference in isothermal process is proposed too.
We propose a generalized entropy maximization procedure, which takes into account the generalized averaging procedures and information gain definitions underlying the generalized entropies. This novel generalized procedure is then applied to Renyi and Tsallis entropies. The generalized entropy maximization procedure for Renyi entropies results in the exponential stationary distribution asymptotically for q is between [0,1] in contrast to the stationary distribution of the inverse power law obtained through the ordinary entropy maximization procedure. Another result of the generalized entropy maximization procedure is that one can naturally obtain all the possible stationary distributions associated with the Tsallis entropies by employing either ordinary or q-generalized Fourier transforms in the averaging procedure.
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