Detailed Jarzynski equality applied to a logically irreversible procedure

Abstract

Abstract. -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 Landauer's bound only for very slow erasure procedure. Instead a detailed Jarzynski equality allows us to retrieve the Landauer's 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.The connection between thermodynamics and information is nowadays a widely studied problem [1][2][3][4][5]. The main questions concern the amount of energy necessary in order to perform a logical operation in a given time and how the information entropy is related to the free energy difference between the initial and final state of this logical operation. In this context the Landauer's principle [6] is very important as it states that for any irreversible logical operation the minimum amount of entropy production is −k B ln(2) per bit commuted by the logical operation, with k B the Boltzmann constant. Specifically a logically irreversible operation is an operation for which the knowledge of the output does not allow to retrieve the initial state, examples are logical AND, OR and erasure. In a recent paper [7] we have experimentally shown that indeed the mini mum amount of work necessary to erase a bit is actually associated with this Landauer's bound which can be asymptotically reached for quasi-static transformations. The question that arises naturally is whether this work corresponds to the free energy difference between the initial and final state of the system. To answer to this question it seems natural to use the Jarzinsky equality [8] which allows one to compute the free energy difference between two states of a system, in contact with a heat bath at temperature T . When such a system is driven from an equilibrium state A to a state B through any continuous procedure, the Jarzynski equality links the stochastic work W st received by the system during the procedure to the free energy difference ∆F = F B − F A between the two states:Where . denotes the ensemble average over all possible trajectories, and β = 1 kBT (see eq. 2 for the precise definition of the work W st ).In this letter we analyze the question of the application of eq. 1 for estimating the ∆F corresponding to the erasure operation in our experiment, in which a colloidal particle confined in a double well potential is used as a single bit memory. We will show that the classical Jarzynski equality (eq. 1) is not useful here but that a detailed Jarzynski Equality [9] allows us to retrieve the Landauer limit independently of the work done on the system during the memory erasure procedure, and to link this work to the proba...

Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs–Shannon form

“…The final state, 4 after time τ , is in local equilibrium with probability p τ to be in the left well, corresponding to H τ between zero and one bit. Thus, ∆H = H τ − 1 andThis protocol resembles that used in [60,65,66]. However, in those studies, partial erasure was used because the barrier could not be made high enough to ensure full erasure, and correction factors were applied to infer the work required for full erasure of a bit.…”

“…Or, we can selectively stretch one well by a factor η while the other well is unchanged. Such manipulations are not possible in erasure experiments based on optical tweezers [60,65,66], which limits the possible protocols in such cases.The challenge with using feedback traps to measure work values to an accuracy < 0.1 k B T is to calibrate forces accurately and to account for slow drifts in quantities such as the particle's response to an applied voltage. In earlier work, we developed a recursive, real-time calibration technique [73] that allows us to measure accurately the stochastic work done by a changing potential 6 on a particle.…”

Toward Physical Realizations of Thermodynamic Resource Theories

“…We might reconcile the closedness of TRT systems with the openness of experimental systems by modifying the constraint (9). A modification could model the impreciseness with which unitaries can be implemented [1].…”

“…The agent should be able to measure a quasiclassical memory, then reset the memory to some fiducial state by investing work. 9 Schematically,…”

“…I could have advocated for bounding, with idealizations, efficiencies achievable in labs. I should have invoked the connections being developed with fluctuation relations [3][4][5], statistical mechanical theorems that have withstood experimental tests [6][7][8][9][10][11][12][13].…”