Fluctuation, Dissipation and Inversion

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Most of us have experienced thermal noise as that undesirable static hiss we get when we turn up the volume of a sound amplifier. Thermal noise is present in any voltage measurement of an electrical circuit and the intensity of the noise increases with both the temperature and the resistance of the circuit elements. One goal of this project is to develop a method to estimate the electrical properties of a circuit from thermal noise measured while heating some circuit elements. One practical application would be to monitor laser welding. We expect that the laser hot spot generates noticeably higher thermal noise with intensity depending mostly on the nearby electric properties. But a defective weld that does not bridge two parts does not conduct electricity well, so one should be able to distinguish a good weld from a bad one based on thermal noise alone. The same principle may also be used in Atomic Force Microscopy to measure the conductivity of a sample. This project will explore these new uses of thermal noise, using mathematical tools to extract valuable information from a usually undesirable andignored quantity. In particular we anticipate to answer questions such as what electric properties can be recovered from thermal noise? How can these be recovered? We will use simulations to illustrate our ideas, with an eye on applications. Another goal of this project is to control heat using small heat pumps called Peltier devices. We use mathematical tools to show that an idealization of these devices can be used to thermally isolate an object, that is, making the temperature surrounding the devices and the object indistinguishable from the ambient temperature. The mathematical framework we will develop could be used for heat control in electronics and to design devices for slow release of drugs. This project provides research training opportunities for graduate and undergraduate students. This project has two main thrusts. The first thrust is about harnessing ubiquitous thermal fluctuations to image the properties of a body and the second thrust concerns active thermal cloaking. An example for the first thrust is Johnson-Nyquist noise, that is, the random currents or voltages that can be measured at the terminals of any resistive electrical component. The voltage variance is proportional to the temperature and the resistance. Thus the resistance can be found from the temperature and the voltage variance. We will consider a similar problem on conducting bodies where the problem is to find the electrical properties (impedance) of a body from the voltage or current covariance measured at a few electrodes (which may be placed either inside the body or on its boundary) all while locally heating the body. A discrete analogue of this problem will be considered for finding the impedance of elements in electric circuits. Johnson-Nyquist noise is a manifestation of the Fluctuation Dissipation Theorem, a fundamental statistical physics result relating the fluctuations about an equilibrium of a system to the dissipation in thesystem. Thus the inverse problems that we plan to study may have applications to other linear systems subject to random externalfluctuations, e.g. thermo-elastic fluctuations. For the second thrust, we will study cloaking strategies for the heat equation that rely on active surfaces to dissimulate objects or heat sources/sinks and are grounded on potential theory (Green identities) for the heat equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.