Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Collective advantages in finite-time thermodynamics

Alberto Rolandi, Paolo Abiuso, Martí Perarnau-Llobet

TL;DR

The erasure of information in finite time is focused on and a faster convergence to Landauer's bound is proved to prove under realistic levels of control in simple interaction architectures.

Abstract

A central task in finite-time thermodynamics is to minimize the excess or dissipated work $W_{\rm diss}$ when manipulating the state of a system immersed in a thermal bath. We consider this task for an $N$-body system whose constituents are identical and uncorrelated at the beginning and end of the process. In the regime of slow but finite-time processes, we show that $W_{\rm diss}$ can be dramatically reduced by considering collective protocols in which interactions are suitably created along the protocol. This can even lead to a sub-linear growth of $W_{\rm diss}$ with $N$: $W_{\rm diss}\propto N^x$ with $x<1$; to be contrasted to the expected $W_{\rm diss}\propto N$ satisfied in any non-interacting protocol. We derive the fundamental limits to such collective advantages and show that $x=0$ is in principle possible, however it requires long-range interactions. We explore collective processes with spin models featuring two-body interactions and achieve noticeable gains under realistic levels of control in simple interaction architectures. As an application of these results, we focus on the erasure of information in finite time and prove a faster convergence to Landauer's bound.

Collective advantages in finite-time thermodynamics

TL;DR

The erasure of information in finite time is focused on and a faster convergence to Landauer's bound is proved to prove under realistic levels of control in simple interaction architectures.

Abstract

A central task in finite-time thermodynamics is to minimize the excess or dissipated work when manipulating the state of a system immersed in a thermal bath. We consider this task for an -body system whose constituents are identical and uncorrelated at the beginning and end of the process. In the regime of slow but finite-time processes, we show that can be dramatically reduced by considering collective protocols in which interactions are suitably created along the protocol. This can even lead to a sub-linear growth of with : with ; to be contrasted to the expected satisfied in any non-interacting protocol. We derive the fundamental limits to such collective advantages and show that is in principle possible, however it requires long-range interactions. We explore collective processes with spin models featuring two-body interactions and achieve noticeable gains under realistic levels of control in simple interaction architectures. As an application of these results, we focus on the erasure of information in finite time and prove a faster convergence to Landauer's bound.
Paper Structure (14 sections, 1 theorem, 70 equations, 8 figures, 1 table)

This paper contains 14 sections, 1 theorem, 70 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Framework
  3. Fundamental limit of collective advantages
  4. Collective erasure
  5. Collective advantage in driven many-body systems
  6. Conclusions and discussion
  7. Full control scenario
  8. Recovering the quantum Hellinger angle
  9. Geodesics
  10. Fisher metric and minimum Hellinger angle on conditional probabilities
  11. Bounds on the Fisher distance between conditional probabilities
  12. Star Model
  13. Pyramid scheme
  14. Numerics for many-body systems

Key Result

Lemma 1

The geodesic Fisher distance between any $p$ of the form eq:p_cond_3, when fixing $p^{(3)}$ and assuming full control on $p^{(1)}$, is bounded by $3\pi$. When the final point is deterministic on $i_1$ and $i_2$, the bound can be reduced to $2\pi$. That is, for fixed boundary conditions and assuming full control on $p^{(1)}$ only, one has Moreover, if the final point is of the form eq:p_det, the

Figures (8)

  • Figure 1: (a) Minimal dissipation for the erasure of $N$ spins for different control designs analyzed in this work. These are compared with the dissipations that are achievable with no interactions (\ref{['localbound']}, blue-shaded area), and with the dissipations that are not achievable regardless of the protocol (\ref{['boundNqubits']}, red-shaded area). We find $\tau W_{\rm diss}^{\rm *, chain}\approx 1.69 N$, $\tau W_{\rm diss}^{\rm *, all} \approx 2.20 N^{0.857}$, while $\tau W^{\rm *, Star}_{\rm diss} \leq 9\pi^2/4$ (cf. \ref{['B']}). Single points are provided for 2-D and 3-D Pyramids with few layers and an aperture of 8 (cf. \ref{['B']}). (b-e) Depiction of the geometries of the interactions in \ref{['eq:2B_hamiltonian']} (equal colors/labels correspond to equal values of the local fields). (b) all-to-all model with $N=8$, (c) 1-D spin chain with $N=8$, (d) the Star model with $N=9$, (e) 2-D Pyramid model with $4$ layers and an aperture of $1$.
  • Figure 2: Star model: a central spin
  • Figure 3: Pyramid model with $N_i=1+2i$ (cf. \ref{['eq:Npyr_mod']}), in dimension $D=2$.
  • Figure 4: Minimal dissipation for the erasure of $N$ spins in the non-interacting case and full control scenario. The dissipation of the protocol described in \ref{['lem:cond_m']}\ref{['lem:cond_m']} for the erasure of $N$ spins in the star model, and its repeated application for pyramid models in 2-D and 3-D with apertures $a = 2$ and $a=8$. And an extrapolation of the fit of the minimal dissipation in the all-to-all model (to compare the scaling) $W_{\rm diss}^{\rm *, all} = \alpha N^x$ with $x = 0.857$ and $\alpha = 2.20$.
  • Figure 5: Optimal finite-time protocol for approximate erasure ($\varepsilon(\tau) = 4 k_B T$ which has an erasure error of $3\cdot 10^{-4}$) in the all-to-all spin model. This protocol is computed for $N=10$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Lemma 1