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Lee Dong-yun¹, Chulan Kwon², Hyuk Kyu Pak¹

Experimental Verification of the F luctuation T heorem in Expansion/Compression Processes of a S ingle-Particle Gas. Lee Dong-yun¹, Chulan Kwon², Hyuk Kyu Pak¹ Department of physics, Pusan National University, Korea ¹ Department of physics, Myongji University, Korea ²

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Lee Dong-yun¹, Chulan Kwon², Hyuk Kyu Pak¹

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  1. Experimental Verification of the Fluctuation Theorem in Expansion/Compression Processes of a Single-Particle Gas 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

  2. Bio-soft matter physics lab 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 RNAfolding free energies Collin, Bustmanteet. 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 workdistribution and determine the free energy difference of the process by using Crooks fluctuation theorem. Page 6

  7. Free Energy Difference of the System forward backward • 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: 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, =0 , In non-equilibrium steady state, 0 Work done by external source converts to heat in the heat bath. Page 9

  10. Single Particle Gas under a Harmonic Potential Quasi-static process (Equilibrium process) • Thermodynamic work: Here, (external parameter) • dk/dt→0 : (Quasi-static 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) • The system remembers its previous state. • Consider a sudden change limit(dk/dt→) • The particle is still at initial position. • Position distribution is given by the initial equilibrium Boltzmann distribution • Using Equi-partition theorem • Using recent theoretical result, Kwon, Noh, Park, PRE 88 (2013) Page 12

  13. Experimental Setup – Optical Tweezers with time dept. trap strength **2 PMMA particle in do-decane liquid Temp. of the system is maintained at 27o. Particle position is measured with 1nm resolution. Page 13

  14. Optical Tweezers Developed by A. Ashkin 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. Potential Energy For small displacements the force is a Hooke’s law force. O 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. Boltzmann Statistics QPD Condenser Potential Energy 100 X Oil NA 1.35 Tracking beam Objective Page 19

  20. Profile of 1D Harmonic Potential Page 20

  21. Measurements of Optical Trap Strengthwith Controlled Laser Power Page 21

  22. Experimental Setup – Optical Tweezers with time dept. trap strength **2 PMMA particle in do-decane liquid Temp. of the system is maintained at 27o. Particle position is measured with 1nm resolution. Page 22

  23. Experimental Method backward forward • Using a PMMA particle of 2µm diameter in do-decane solvent • Linearly changing the trap strength in time • From 2.87 to 0.94pN/µm (backward process) • From 0.94 to 2.87pN/µm (forward process) • Theoretical free energy difference : • Data sampling :10kS/s (sampling in every 100sec) • Repetition is over 40000 times • Total number of steps : 360 • Rate of changing trap strength(pN/µm·s) : 0.268, 0.536, 2.68, 5.36 by changing the time difference between the neighboring steps from 1msec to 20msec Page 23

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

  25. Work Probabilities for Four Different Protocols 0.268pN/µm·s 0.536pN/µm·s 2.68pN/µm·s 5.36pN/µm·s Page 25

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

  27. Fast protocol, 2.68pN/µm·s Verification of Crooks Fluctuation Theorem Fastest protocol,5.36pN/µm·s 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 • Partition function • Free energy • Internal Energy : constant. : Fluctuating value • Entropy S

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

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

  34. Oscillatory Optical tweezers 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 Horizontally oscillating potential well Fixed potential well

  36. Characteristic equilibration time in this system (equi-partition theorem) • 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. Page 36

  37. Calculation of Work • Thermodynamic work = • : constant value • Forward work is always positive. f> 0 Therefore, < 0 Page 37

  38. Work Probability : Slowest protocol,0.268pN/µm·s Page 38

  39. Work Probability : Slow protocol,0.536pN/µm·s Page 39

  40. Work Probability : Fast protocol,2.68pN/µm·s Page 40

  41. Work Probability : Fastest protocol,5.36pN/µm·s Page 41

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