Experimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equality

Journal name:
Nature Physics
Year published:
Published online

In 1929, Leó Szilárd invented a feedback protocol1 in which a hypothetical intelligence—dubbed Maxwell’s demon—pumps heat from an isothermal environment and transforms it into work. After a long-lasting and intense controversy it was finally clarified that the demon’s role does not contradict the second law of thermodynamics, implying that we can, in principle, convert information to free energy2, 3, 4, 5, 6. An experimental demonstration of this information-to-energy conversion, however, has been elusive. Here we demonstrate that a non-equilibrium feedback manipulation of a Brownian particle on the basis of information about its location achieves a Szilárd-type information-to-energy conversion. Using real-time feedback control, the particle is made to climb up a spiral-staircase-like potential exerted by an electric field and gains free energy larger than the amount of work done on it. This enables us to verify the generalized Jarzynski equality7, and suggests a new fundamental principle of an ‘information-to-heat engine’ that converts information into energy by feedback control.

At a glance


  1. Schematic illustration of the experiment.
    Figure 1: Schematic illustration of the experiment.

    a, A microscopic particle on a spiral-staircase-like potential with a step height comparable to kBT. The particle stochastically jumps between steps owing to thermal fluctuations. As the downward jumps along the gradient are more frequent than the upward ones, the particle falls down the stairs, on average. b, Feedback control. When an upward jump is observed, a block is placed behind the particle to prevent downward jumps. By repeating this cycle, the particle is expected to climb up the stairs without direct energy injection.

  2. Experimental set-up.
    Figure 2: Experimental set-up29, 30.

    a, The particle was pinned at a single point of the top glass surface and exhibited rotational Brownian motion. To impose a tilted periodic potential on the particle, an elliptically rotating electric field (blue and pink curves) was induced (not to scale; see Methods and Supplementary Information for details). b, Typical potentials with opposite phases to be switched in the feedback control. The particle experienced a tilted periodic potential with a period of 180°. The height and slope were 3.05±0.03kBT and 1.13±0.06kBT/360° (mean±S.E., seven particles), respectively. c, Feedback control. At time t=0, the particle’s angular position is measured. If the particle is observed in the angular region indicated by ‘S’, we switch the potential at t=ε by inverting the phase of the potential (right). Otherwise, we do nothing (left). At t=τ, the next cycle starts. The location of region S is altered by the switching. The potential wells correspond to the steps of the spiral stairs in Fig. 1. The switching of potentials corresponds to the placement of the block.

  3. Trajectories, mean velocities and excess free energy under feedback control.
    Figure 3: Trajectories, mean velocities and excess free energy under feedback control.

    a, Typical trajectories for different values of the feedback delay ε. b, Magnified plot of the region indicated by a rectangle in a. The particle rotates with steps with a size of 90° reflecting the profile of the potential. Red triangles indicate the timings of the switchings. c, Variation of the rotation rate with feedback delay ε. The rotation rate is defined as positive when the particle climbs up the potential. Data of seven particles are averaged. Error bars indicate standard deviations among particles. d, ΔF, the free-energy difference between the initial and final states of the cycle; W, the amount of work done on the particle by the switching calculated as the potential-energy change associated with the switching (see Methods). In cycles without switching, W=0. left fence·right fence denotes the mean per cycle. In the shaded region, we obtain the excess free energy beyond the conventional limitation of the second law of thermodynamics. Error bars indicate standard deviations among particles.

  4. Verification of the generalized Jarzynski equality.
    Figure 4: Verification of the generalized Jarzynski equality.

    a, Circles: left fenceeFW)/kBTright fence. Rectangles: feedback efficacy γ. Data of seven particles are averaged. b, Discrepancy of the equality, defined as [left fenceeFW)/kBTright fenceγ]/left fenceeFW)/kBTright fence, as a percentage. c, Convergence of the estimate of left fenceeFW)/kBTright fence as a function of the number of cycles, plotted for different feedback delays ε. Error bars indicate standard deviations among particles.


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Author information


  1. Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Tokyo 112-8551, Japan

    • Shoichi Toyabe &
    • Eiro Muneyuki
  2. Department of Physics, Graduate School of Science, University of Tokyo, Hongo, Tokyo 113-0033, Japan

    • Takahiro Sagawa,
    • Masahito Ueda &
    • Masaki Sano
  3. ERATO Macroscopic Quantum Control Project, JST, Yayoi, Tokyo 113-8656, Japan

    • Masahito Ueda


S.T. designed and carried out experiments, analysed data and wrote the paper. T.S. and M.S. designed experiments and wrote the paper. T.S. and M.U. supported theoretical aspects. E.M. and M.S. supervised the experiments. All authors discussed the results and implications and commented on the manuscript at all stages.

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