Non-Equilibrium Fluctuations in Expansion/Compression Processes of a Single-Particle Gas

Transcription

1 Non-Equilibrium Fluctuations in Expansion/Compression Processes o a Single-Particle Gas Hyu Kyu Pa Department o Physics, UNIST IBS Center or Sot and Living Matter Page 1 November 8, 015, Busan Nonequilibrium Thermodynamics in Quantum and Classical Physics

2 Ulsan National Institute o Science and Technology Page

3 IBS Center or Sot and Living Matter Hyu Kyu Pa What happens at sot matter interaces? How to explain non-equilibrium phenomena? Electrical Properties at Solid-Liquid Interaces Non-Equilibrium Fluctuations in Very Small Systems Colloid and Nano Particles at Interaces Hydrodynamics in Small Systems Wetting and Electro-wetting

4 Electrical Properties at Solid-Liquid Interaces Electric power generation by modulating the metal-water interace. Interacial charge density between solid and liquid. The inluence o ph solution on solid-liquid interaces.

5 Non-Equilibrium Fluctuations in Very Small Systems Realization o a Brownian motor through eedbac control. Optical tweezers studying micro-size systems The inluence o ph solution on solid-liquid interaces.

6 Colloid and Nano Particles at Interaces n-alane NP solution Electrical phenomena across bio-membrane Hydrodynamics o colloid particles near interaces Nano particle adsorption at the liquid-vapor

7 Sot Matter Physics Research Group Contributors: Dr. Dong Yun Lee Dr. Govind Paneru Pro. Chulan Kwon Myongji Univ.

8 Fluctuation theorem Most interesting processes in the nature occur ar rom equilibrium. The second law o thermodynamics predicts that the entropy o an isolated system should tend to increase until it reaches equilibrium. In statistical mechanics, the second law is only a statistical one. There should always be some nonzero probability that the entropy o an isolated system might spontaneously decrease ; the luctuation theorem precisely quantiies this probability. Fluctuation theorem deals with the relative probability that the entropy o a system ar rom equilibrium will increase or decrease over a given amount o time. Page 8

9 Discovery o Fluctuation Theorems (FT) Evans, Cohen, Morris (1993) Evans & Searls (1994), Gallavotti & Cohen (1995) Gallavotti-Cohen Symmetry Detailed FT Integral FT Using Jensen s inequality Thermodynamic nd law is a corollary o G-C symmetry!! Page 9

10 Fluctuations depending on the size o the system Macroscopic system Microscopic system Page 10 Fluctuations in thermodynamic variables are negligible. Fluctuations in thermodynamic variables are very visible.

11 Croos Fluctuation Theorem (CFT, G. E. Croos 1998) F P b ( W ) P (W ) P P b ( W ) ( W ) exp[ ( W F)] CFT has drawn a lot o attention because o its useulness in experiment. This theorem maes it possible to experimentally measure the ree energy dierence o the system during a nonequilibrium process. Page 11

12 Why luctuation theorem is important? Veriication o the Croos luctuation theorem and recovery o RNA olding ree energies P ( W ) P ( W ) b exp[ ( W F)] Collin, Bustmante et. al. Nature(005) Page 1

13 Idea Consider a particle trapped in a 1D harmonic potential V ( x ) x / where is a trap strength o the potential. When the trap strength is either increasing or decreasing isothermally in time, the particle is driven rom equilibrium. Since the size o the system is inite, one can test the luctuation theorems in this system. We measure the wor distribution and determine the ree energy dierence o the process by using Croos luctuation theorem. DY Lee, C. Kwon & H. K. Pa PRL, 114, (015) Page 13

14 Free Energy Dierence o the System Harmonic oscillator (in 1D) - Hamiltonian is given by - Partition unction is Z exp H ( x, p) dxdp / h 1/, / m p 1 H ( x, p) x m - Free energy dierence between two equilibrium states at the same i, temperature is F ( ln Z ) ( ln Zi ) 1/ ln i orward bacward - Forward quasi-static process : Bacward quasi-static process: - During these processes: i i F F 0 ( S 0) 0 ( S 0) Page 14

15 1D Brownian Motion o Single Particle in Heat Bath, ), Thermodynamic 1st Law Page 15

16 1D Brownian Motion o Single Particle in Heat Bath + Thermodynamic 1st Law = In equilibrium, =0, In non-equilibrium steady state, Page 16 Wor done by external source converts to heat in the heat bath.

17 Single Particle Gas under a Harmonic Potential Quasi-static process (Equilibrium process) Thermodynamic wor: W Here, H 1 dt d x (external parameter) Page 17 x d/dt 0 : (Quasi-static process) x eq x dw d ( W ) W W b W b 1 x x b eq T/ B d 1 ln P ( W ) P ( W ) ( W F ) b i 1 BT F d

18 Single Particle Gas under a Harmonic Potential Non-equilibrium process(d/dt=inite) Forward process (d/dt>0) Bacward process (d/dt<0) x d neq eq W dw x eq x b d x neq eq W d W b eq b Page 18 W b W eq F W

19 Single Particle under a Harmonic Potential Extreme limit o non-equilibrium process (d/dt=ininite) The system remembers its previous state. Consider a sudden change limit (d/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, P a 1/ aβw, b(w)= θ( W) ( βw) e a = i π /( i ) W 1/ W neq i d x BT and i i x B T / i Page 19 W b W eq W Kwon, Noh, Par, PRE 88 (013)

20 Experimental Setup Optical Tweezers with time dept. trap strength Page 0 ** PMMA particle in do-decane liquid Temp. o the system is maintained at 7 o o. Particle position is measured with 1nm resolution.

21 Measurements o Optical Trap Strength ot =.87 pn / μm Page 1

22 Measurements o Optical Trap Strength The optical trap strength is calibrated with three dierent methods - Equi-partition theorem 1 x 1 B T - Boltzmann distribution method - Oscillating optical tweezers method 1 U ( x ) ( x ) dx Ce dx, V ( x ) x d x m dt dx dt x Acos( t) x D( )cos( ), tan 1 Page

23 Passive Method o Measuring Optical Trap Strength Equi-partition theorem 1 x 1 B T Boltzmann distribution p 1 V ( x ) ( x) dx Ce dx, V ( x ) x ot =.87 pn / μm Page 3

24 Proile o 1D Harmonic Potential p( x ) dx Ce V ( x ) T B dx in 1Dim V ( x ) T lnp( x ) T lnc B B ot =.87 pn / μm Page 4

25 Measurements o Optical Trap Strength with Controlled Laser Power Page 5

26 Experimental Method Using a PMMA particle o µm diameter in do-decane solvent Linearly changing the trap strength in time - From.87 to 0.94pN/µm (bacward process) - From 0.94 to.87pn/µm (orward process) - Theoretical ree energy dierence : F 1/ ln( / ) Data sampling :10S/s (sampling in every 100 sec) Repetition is over times Total number o steps : 360 i bacward orward Rate o changing trap strength(pn/µm s) : 0.68, 0.536,.68, 5.36 by changing the time dierence between the neighboring steps Page 6 rom 1msec to 0msec

27 Laser Power and Trap Strength in Time EQ EQ EQ bacward orward = ± 0.536pN / μm s Page 7

28 Characteristic equilibration time in this system In non-equilibrium process, the external parameters have to be changed beore the system relaxes to the equilibrium state. Mean squared displacement: xx x( t) x(0) - Ater the particle loses its initial inormation then xx obeys the equi-partition theorem. - In our system, the characteristic equilibration time( ) is about 0ms. t, xx BT / (equi-partition theorem) Page 8

29 Wor Probabilities or Four Dierent Protocols : orward 0.68pN/µm s : bacward 0.536pN/µm s.68pn/µm s 5.36pN/µm s Page 9 P ( W ) P ( W ) b exp[ ( W F)]

30 Mean Wor Value and Expected Free Energy Dierence Page 30 W b W eq F W

31 Veriication o Croos Fluctuation Theorem Fastest protocol, 5.36pN/µm s Fast protocol,.68pn/µm s Page 31 P ( W ) P ( W ) b exp[ ( W F)]

32 Conclusion We experimentally demonstrated the CFT in an exactly solvable real system. P ( W ) exp[ ( W F)] P ( W ) b We also showed that mean wors obey in non-equilibrium processes. W b W eq W Useul to mae a micrometer-sized stochastic heat engine. DY Lee, C. Kwon & H. K. Pa PRL, 114, (015) Page 3