Information Engines Converting Information into Energy

Transcription

1 Information Engines Converting Information into Energy Alfred Hubler Center for Complex Systems Research Department of Physics University of Illinois at Urbana-Champaign Complexity 12 (2), (2006) Energy sources are at the core driving force in most sciences. Photo synthesis drives the growth of plants, nuclear energy sources drive the evolution of stars, and chemical energy drives chemical reactions. Better energy sources may stimulate economic growth and may help to reduce the pollution of the environment. Therefore, finding new sources of energy and efficient energy conversion is an important research topic since a long time. In recent years energy efficient forcing of nonlinear systems was studied systematically 1-4. It was found that nonlinear dynamical systems react most sensitive to perturbations which complement the natural dynamics of the system 1. In this case there is a perfect impedance match 2 and the energy transfer is most effective. In quantum systems as well, the energy transfer is most effective if the forcing function matches the spontaneous radiation of the system 3,5. The most important application in Condensed Matter Physics are quantum dot transistors with extremely low power consumption 6,7. A topic with has received much less attention is the extraction of power from dynamical systems. Power generators such as wind mills, steam engines, and car engines extract mechanical energy from fluids. They can extract a large fraction of the energy stored in coherent patterns, whereas only a much smaller fraction can be extracted from less coherent motion such as a turbulent flow. 1

2 If the wind is steady, wind farms are most efficient in converting energy of the air into electrical energy. The efficiency is much less for wind gusts, and the energy stored in the irregular microscopic motion of the gas molecules is not extracted The energy of a system is due to coherent and irregular motion, ranging from large convection cells to molecular chaos. Heat, or more precisely thermal energy, is the part of energy due to irregular motion. Energy stored in irregular motion is hard to retrieve, since most energy extraction processes require knowledge about the current state of the system and its immediate future. Hence, energy stored in molecular chaos is generally considered to be thermal energy. In the following we divide energy into two parts, thermal energy and the rest, which we call coherent energy or extractable energy. The distinction between thermal energy and extractable energy depends on the given information about the state of the system and the quality of forecasting algorithms. This can be illustrated with a simple thought experiment. We consider a 1cm 3 container with one gas atom. It is assumed that the container walls are movable as in a piston, have a rough, irregular surface, and do not conduct heat. If the position of the atom is known with uncertainty x, for instance x=1mm, the size of container is rapidly reduced around the atom to a volume of size x 3. Finally, the container is slowly expanded to its original volume while the atom is bouncing against the inside of the container wall. From the laws for ideal gases we derive that the work W done by the atom is a fraction of its initial energy K, W=K (1-x 3 /1m 3 )=99.9% of K. Therefore, the amount of extractable energy depends on the amount of information about the systems. As soon as the container expands, the motion is more regular and strongly correlated with the motion of the container wall. Thus only a small portion of the energy is still thermal energy, but most of it is extractable energy. In a system with many atoms the situation is 2

3 similar, but requires forecasting of molecular chaos, since the container can not be reduced instantaneously 12. Data from experiments are always measured with a finite accuracy. For systems governed by deterministic chaos, uncertainty in measurements results in a short-term window of predictability even if an exact model of the dynamics is available 13. This sensitivity to initial conditions means any error will, on average, grow at a rate given by the spectrum of Lyapunov exponents. However, a positive Lyapunov exponent does not rule out increased predictability due to local variation in the effective exponents. Increased predictability due to a local variation of the width of single peaked probability density is often called a return-of-skill effect Unfortunately the effect is small. However, Strelioff and Hubler 16 recently reported that mono-modal course grained probability density functions of chaotic systems often develop pronounced sharp extra peaks. Therefore, the dynamics of the most likely state stays close to the trajectory of the most likely initial state for a certain period of time and jumps to the maximum of one of the extra peaks as soon as the course grained probability density functions becomes multi-modal. A similar algorithm can be used to predict spatial-temporal chaos, including models of irregular fluid flow, with the intent to increase the efficiency of wind farms, hydro-electric power plants, and large and small heat engines, such as gasoline motors or microscopic Brownian motors 17. As an example for such information assisted power generation, we consider a heat engine where a temperature gradient creates irregular convection patterns 18. A power generator such as a small turbine or a propeller 8-11, is placed at a fixed location and extracts energy from the convective flow. The power of the generator depends on the size of the generator. A large 3

4 generator can extract energy from a larger region. However a larger generator does not necessarily produce more power. The power is proportional to the magnitude of the local space and time average of the velocity field. Hence if the flow is rapidly oscillating in time and space, a larger generator may deliver less power than a small generator. In addition, a large generator may divert some of the flow and alter the flow patterns. Power generators are most efficient if they match the size and period of coherent patterns in the flow, i.e. wind mills extract energy from large atmospheric convection patterns, and microscopic Brownian motors extract energy from the coherent motion of atoms. The power delivered by the generator depends on its orientation with respect to the direction of the flow. The efficiency is largest if the generator is perfectly aligned with the direction of the flow at all times. A sensor-actuator network with chaos prediction algorithms can help to improve the alignment. We assume that there are several velocity sensors upstream in the flow. The sensor information is processed by chaos predictors and the forecast is used to adjust the orientation of the generator. The direction of the generator is controlled with a linear feedback control algorithm, where forecast of the predictor sets the target direction of control algorithm of the generator. The efficiency of the power generator is a function of the number and location of the sensors, the response time of the sensor-actuator network, the response time of the converter, and the parameters of the chaos predictor. With a good chaos predictor more energy can be extracted. It is not expected that this simple heat engine has any commercial application, but it is intended to illustrate how sophisticated algorithms can help increase the efficiency of energy sources. 4

5 Information assisted power generation has the potential to multiply the output of some sources of energy. If information can multiply the output of this simple heat engine, then the engine might be considered an information engine as well, which converts information into work. This discussion puts a spot light on an open question: What is the definition of thermal energy and temperature in an open system? Is thermal energy indeed that part of energy with is associated with incoherent irregular motion and thus not extractable? In a turbulent flow, energy stored in large vortexes and convection patterns is coherent and thus no thermal energy. Energy in smaller patterns probably be predicted too, but energy stored in molecular chaos is considered irregular and therefore it is thermal energy. At what scale is the boundary between extractable energy and thermal energy? If there is no clear definition of thermal energy, how can we be sure when the laws of thermodynamics are applicable? With a proper definition of heat and thermal energy it might be possible to design engines which are more efficient than anticipated. 5

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