1. Lee Dong-yun¹, Chulan Kwon², Hyuk Kyu Pak¹ Department of physics, Pusan National University, Korea¹ Department of physics, Myongji University, Korea² Nonequilibrium Statistical Physics of Complex Systems, KIAS, July 8, 2014 Experimental Verification of the Fluctuation Theorem in Expansion/Compression Processes of a Single-Particle Gas

2. Ph. D studnt: Dong Yun Lee Collaborator: Chulan Kwon Page  2

3. Outline  Introduction  Experiment  Results Page  3

4. Crooks Fluctuation Theorem (CFT, G. E. Crooks 1998)  CFT has drawn a lot of attention because of its usefulness in experiment. This theorem makes it possible to experimentally measure the free energy difference of the system during a non- equilibrium process. Page  4

5. Why fluctuation theorem is important? Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies Collin, Bustmante et. al. Nature(2005) Page  5

6. Idea  Consider a particle trapped in a 1D harmonic potential where is a trap strength of the potential.  When the trap strength is either increasing or decreasing isothermally in time, the particle is driven from equilibrium.  Since the size of the system is finite, one can test the fluctuation theorems in this system.  We measure the work distribution and determine the free energy difference of the process by using Crooks fluctuation theorem. Page  6

7. Free Energy Difference of the System  Harmonic oscillator (in 1D) -Hamiltonian is given by -Partition function is -Free energy difference between two equilibrium states at the same temperature is -Forward process :, Backward process: -During these processes: forward backward Page  7

8. 1D Brownian Motion of Single Particle in Heat Bath Thermodynamic 1st Law Page  8

9. 1D Brownian Motion of Single Particle in Heat Bath Thermodynamic 1st Law In equilibrium, In non-equilibrium steady state, Page  9

10. Single Particle Gas under a Harmonic Potential  Quasi-static process (Equilibrium process) x Page  10

11. Single Particle Gas under a Harmonic Potential  Non-equilibrium process(dk/dt=finite)  Forward process (dk/dt>0)  Backward process (dk/dt<0) Page  11

12. Single Particle under a Harmonic Potential  Extreme limit of non-equilibrium process (dk/dt=infinite) Using recent theoretical result, Kwon, Noh, Park, PRE 88 (2013) Page  12

13. Experimental Setup – Optical Tweezers with time dept. trap strength Page  13

14. O Optical Tweezers By strongly focusing a laser beam, one can create a large electric field gradient which can create a force on a colloidal particle with a radial displacement from the center of the trap. For small displacements the force is a Hooke’s law force. Developed by A. Ashkin Potential Energy Page  14

15. Control of the Trap Strength  The optical trap strength is proportional to the laser power.  Therefore, when the laser power is changed linearly in time, the optical strength should be increased or decreased linearly in time.  The laser power is controlled using LCVR(Liquid Crystal Variable Retarders) which allows manipulation of polarization states by applying an electric field to the liquid crystal. Page  15

16. Laser Power Stability  Since the optical trap strength is proportional to the laser power, it is important to have a stable laser power in time.  A feedback control of the laser power is used to reduce the long time fluctuation of the laser power.  During the experiment, the fluctuation of laser power is less than ± 0.5%. Page  16

17. Measurements of Optical Trap Strength  The optical trap strength is calibrated with three different methods -Equi-partition theorem -Boltzmann distribution method -Oscillating optical tweezers method Page  17

18. Passive Method of Measuring Optical Trap Strength  Equi-partition theorem  Boltzmann distribution Page  18

19. Trackingbeam Boltzmann Statistics Objective Potential Energy Condenser QPD Page  19

20. Profile of 1D Harmonic Potential Page  20

21. Measurements of Optical Trap Strength with Controlled Laser Power Page  21

22. Experimental Setup – Optical Tweezers with time dept. trap strength Page  22

23. Experimental Method backwardforward Page  23

24. Laser Power and Trap Strength in Time forward backward EQ Page  24

25. Work Probabilities for Four Different Protocols Page  25

26. Mean Work Value and Expected Free Energy Difference Page  26

27. Verification of Crooks Fluctuation Theorem Page  27

28. Conclusion  We experimentally demonstrated the CFT in an exactly solvable real system.  We also showed that mean works obey in non-equilibrium processes.  Useful to make a micrometer-sized stochastic heat engine. Page  28

29. Thank you for your attentions. Page  29

30. Supplement

31. A piston-cylinder system with ideal gas  Quasi-static Compression (Equilibrium process) -  Thermodynamic work:  dV/dt → 0 ( Quasi-static Compression) - Forward process Page  31

32.  Backward process (Expansion dV/dt>0)  Forward process (Compression dV/dt<0) A piston-cylinder system with ideal gas  Non-equilibrium process (dV/dt= finite) - Page  32

33. O O’ Reference position When a single particle of mass in suspension is forced into an oscillatory motion by optical tweezers, it experiences following two forces; Viscous drag forceSpring-like force The equation of motion for a particle Oscillatory Optical tweezers

34. The equation of motion for a particle trapped by an oscillating trap in a viscous medium, as a function of Neglect the first term ( ) (In extremely overdamped case ), and assume a steady state solution in the form The amplitude and the phase of the displacement of a trapped particle is given by where the amplitude ( D(ω) ) and the phase shift ( δ(ω) ) can be measured directly with a lock-in amplifier and set-up of the oscillatory optical tweezers in the next page,.

35. Active Method of Measuring Optical Trap Strength  Equation of motion  Phase delay Fixed potential well Horizontally oscillating potential well

36. Characteristic equilibration time in this system  In non-equilibrium process, the external parameters have to be changed before the system relaxes to the equilibrium state.  Mean squared displacement: -After the particle loses its initial information then obeys the equi-partition theorem. -In our system, the characteristic equilibration time( ) is about 20ms. (equi-partition theorem) Page  36

37. Calculation of Work Page  37

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