1. IEEE TRANS ON AUTOMATIC CONTROL, FEBRUARY, 2011 Sandberg, Delvenne, and Doyle http://arxiv.org/abs/1009.2830 today

2. w white, unit intensity (J-N noise) k Boltzmann’s constant Phenomenology R Resistor R Temperature T Capacitor C Voltage v Dissipation  Fluctuation

3. w white, unit intensity k Boltzmann’s constant Dissipation  Fluctuation Origins? Consequences?

4. R y ideal sensor Dissipation  Sensor noise Fluctuation Measurement

5. R y ideal sensor Dissipation  Sensor noise Fluctuation Back action Measurement

6. R y ideal sensor Back action  Sensor noise Measurement

7. R y more ideal sensor Back action  Sensor noise -R Active Assume active device has infinite power supply Measurement

8. Back action  Sensor noise R y -R y Optimal estimator

9. R y -R y Optimal estimator Software Hardware Digital Analog Active Lumped Computers Upside down from other pictures

10. y Optimal estimator

11. y Can compute everything analytically because of special structure.

12. y back-action error

13. back-action error Cold sensors are uniformly easier

14. back-action error

15. “collapse” time t

17. back-action error t small vary R

18. m v m v Heisenberg?

19. Back action  Sensor noise y Optimal estimator

22. error back action Cold sensors (and large masses) are uniformly easier

23. error back action

24. R y more ideal sensor Back action  Sensor noise -R Active Active device has infinite power supply

25. Next steps Estimation to control Efficiency of devices, enzymes Classical to quantum

26. w white, unit intensity k Boltzmann’s constant Dissipation  Fluctuation Origins? Consequences?

27. Resistor R Temperature T Capacitor C Voltage v w white, unit intensity k Boltzmann’s constant T=0 R Temporarily

28. + R step response Caution: this is a visualization of the equations, the “signals” are not physical (“virtual”)

29. + R step response Caution: this is a visualization of the equations, the “signals” are not physical Step response is easier to visualize than impulse…

30. 010 0 0.5 1 1.5 Time (sec) Amplitude dissipative, lossy But the microscope world is lossless (energy is conserved). Where does dissipation come from? + step response

31. + Lossless Approximate + step response dissipative, lossy

32. + step response Lossless Approximate + step response dissipative, lossy

33. Lossless Approximate dissipative, lossy step response Step response Emphasize the differences Cosine series

34. Lossless Approximate dissipative, lossy Step response 00.511.52 0 1 n=10,  =1 Time (sec) n=5,  =1 dissipative, lossy

35. 00.20.40.60.811.2 0 0.2 0.4 0.6 0.8 1 1.2 n=5,  =1 n=10,  =1 n=10,  =2 For n/ ,    step

36.  =1 0 12 0 1 n=10 n/     Lossless Approximate step response age of universe  4e26 nanosecs

37.  =1 n=10 00.20.40.60.81 -1.5 -0.5 0 0.5 1 1.5 00.20.40.60.81 -1.5 -0.5 0 0.5 1 1.5 n=100 Theorem: Linear dissipative (passive) iff linear lossless approximation For n/ ,  

38. Theorem: Linear dissipative (passive) iff linear lossless approximation Corollary: Linear active needs nonlinear lossless approximation Proof: Essentially Fourier series plus elementary control theory. Question: what nonlinearities can be fabricated? For n/ ,  

39. 0246810 0 0.5 1 1.5 Time (sec) + step response v(t)v(t) Lossless Approximate  =10 n=10 n=4

40. 00.511.522.5 -1.5 -0.5 0 0.5 1 1.5 Time (sec) Amplitude n=10 step response Lossless Approximate

41. random initial conditions 00.20.40.60.81 -0.5 0 0.5 n=10

42. random initial conditions 00.511.522.5 -1.5 -0.5 0 0.5 1 1.5 Time (sec) Amplitude n=10 step response 00.20.40.60.81 -0.5 0 0.5

43. 0 11.522.5 -1.5 -0.5 0 0.5 1 1.5 Time (sec) Amplitude n=10 00.20.40.60.81 -0.5 0 0.5 fluctuation dissipation a ( t ) = kT g ( t )(all n ) k = Boltzmann constant, T =temperature Theorem:

44. T=1 n=100 n=10 00.20.40.60.81 0 00.20.40.60.81 0  “white” for n large

45. n=100 00.20.40.60.81 0 T=1 00.20.40.60.81 -1.5 -0.5 0 0.5 1 1.5 Dissipation Fluctuation Theorem: Fluctuation  Dissipation

46. Theorem: Fluctuation  Dissipation Theorem: Linear passive iff linear lossless approximation Corollary: Linear active needs nonlinear lossless approximation “New” “Old”

47. Resistor R Temperature T Capacitor C Voltage v w white, unit intensity k Boltzmann’s constant + T>0

48. Back action  Sensor noise R y -R y Optimal estimator

49. back-action error t small vary R