Report

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 BIO-SOFT MATTER PHYSICS LAB Ph. D studnt: Dong Yun Lee Collaborator: Chulan Kwon Page 2 Outline Introduction Experiment Results Page 3 Crooks Fluctuation Theorem (CFT, G. E. Crooks 1998) F Pb ( W ) P f (W ) P f (W ) Pb ( W ) exp[ (W F )] 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 nonequilibrium process. Page 4 Why fluctuation theorem is important? Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies P f (W ) Pb ( W ) Collin, Bustmante et. al. Nature(2005) Page 5 exp[ (W F )] Idea Consider a particle trapped in a 1D harmonic potential V (x ) kx 2 /2 where k 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 Free Energy Difference of the System forward Harmonic oscillator (in 1D) - Hamiltonian is given by H ( x, p ) p 2 2m 1 kx 2 backward 2 - Partition function is Z exp H ( x , p )dxdp / h 1 / , k /m - Free energy difference between two equilibrium states k i , k f at the same temperature is F ( ln Z f ) ( ln Z i ) 1 / 2 ln - Forward process : ki k f ki , F 0 (S 0) k f ki (S 0) Backward process: - During these processes: Page 7 kf F 0 ≡ = 1D Brownian Motion of Single Particle in Heat Bath = = − , ∙ = − ∙ 2 2 2 2 =− + = − + , + − + + = Page 8 ∙ , + − + − ′ , + () + () = 2( − ′ ) = ∙ + Thermodynamic 1st Law − 1D Brownian Motion of Single Particle in Heat Bath 2 2 + = = + − + () Thermodynamic 1st Law − 1 2 1 2 = = = 2 2 In equilibrium, = → =0 , = 0 = 1 2 2 = − In non-equilibrium steady state, , = 0, = , = >0 Page 9 ∆ = ∆ >0 =0 Work done by external source converts to heat in the heat bath. Single Particle Gas under a Harmonic Potential Quasi-static process (Equilibrium process) Thermodynamic work: H 1 W dt 2 dk x 2 Here, = (external parameter) dk/dt→0 : (Quasi-static process) x - eq x x 2 - d (W dW f W W f W k BT / k 2 f - Page 10 x 2 b b ) b 1 x 2 1 2 2 ln eq kf ki dk F 1 k BT 2 k dk Single Particle Gas under a Harmonic Potential Non-equilibrium process(dk/dt=finite) Forward process (dk/dt>0) x x 2 2 Backward process (dk/dt<0) eq x f d W neq f Page 11 d W eq f W x 2 2 eq b d Wb b W eq F W neq f d Wb eq Single Particle under a Harmonic Potential Extreme limit of non-equilibrium process (dk/dt=infinite) The system remembers its previous state. Using recent theoretical result, a Consider a sudden change limit ∞) (dk/dt → P f,b (W) = θ( W) π a = k i / (k ( βW) f 1 / 2 e a βW ki ) - The particle is still at initial position. - Position distribution is given by the initial equilibrium Boltzmann distribution - Using Equi-partition theorem W 1/ 2 kf dk x 2 and x ki W neq Page 12 W k BT / k i k BT k f k i 2 2 b ki W eq W f Kwon, Noh, Park, PRE 88 (2013) Experimental Setup – Optical Tweezers with time dept. trap strength **2 PMMA particle in do-decane liquid Page 13 Temp. of the system is maintained at 27o±0.1o. Particle position is measured with 1nm resolution. 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. F U 6 rV 2 n1 cR k ot re Potential Energy Ftrap 2 n 22 n12 I o 2 2 n 2 n 1 2 2 r / R 2 r2 / R2 e rˆ rˆ For small displacements the force is a Hooke’s law force. Ftrap k ot r O Page 14 where , r k ot 6V 2 n 1 cR 2 n 22 n12 I o 2 2 n 2 2 n1 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 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 Measurements of Optical Trap Strength The optical trap strength is calibrated with three different methods - Equi-partition theorem 1 k x 2 2 1 2 k BT - Boltzmann distribution method (x )dx Ce U (x ) dx , V (x ) 1 2 kx - Oscillating optical tweezers method 2 m d x dt 2 dx kx A cos( t ) dt x D ( ) cos( ), tan Page 17 1 k 2 Passive Method of Measuring Optical Trap Strength Equi-partition theorem Boltzmann distribution 1 2 k x 2 1 2 k BT p (x )dx Ce k ot = 2.87 pN / μm Page 18 V (x ) dx , V (x ) 1 2 kx 2 Boltzmann Statistics p (x )dx Ce V (x ) kBT QPD dx in 1Dim V (x ) k B T ln p (x ) k B T lnC Condenser Potential Energy Tracking beam V (x ) 1 2 k OT x 2 Objective 100 X Oil NA 1.35 Page 19 Profile of 1D Harmonic Potential k ot = 2.87 pN / μm Page 20 Measurements of Optical Trap Strength with Controlled Laser Power Page 21 Experimental Setup – Optical Tweezers with time dept. trap strength **2 PMMA particle in do-decane liquid Page 22 Temp. of the system is maintained at 27o±0.1o. Particle position is measured with 1nm resolution. Experimental Method 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) backward forward - Theoretical free energy difference : F 1 / 2 ln( k f / k i ) 0 . 558 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 Page 23 from 1msec to 20msec Laser Power and Trap Strength in Time EQ EQ backward EQ forward k = ± 0.536 pN / μm s Page 24 Work Probabilities for Four Different Protocols = 0.268pN/µm·s = 0.536pN/µm·s = 2.68pN/µm·s =5.36pN/µm·s P f (W ) Page 25 Pb ( W ) exp[ (W F )] Mean Work Value and Expected Free Energy Difference W Page 26 b W eq F W f Verification of Crooks Fluctuation Theorem Fastest protocol, = 5.36pN/µm·s P f (W ) Page 27 Pb ( W ) Fast protocol, = 2.68pN/µm·s exp[ (W F )] Conclusion We experimentally demonstrated the CFT in an exactly solvable real system. P f (W ) Pb ( W ) exp[ (W F )] We also showed that mean works obey in non-equilibrium processes. W b W eq W Useful to make a micrometer-sized stochastic heat engine. Page 28 f Thank you for your attentions. Page 29 Supplement Partition function Z exp H ( x , p )dxdp / h 1 / , k /m Free energy = − = − = ln ( ℏ) = ln ℏ 1 ℏ 2 = ln + ln Internal Energy ≡ =− ln = = − : constant. : Fluctuating value Entropy S=− = ln + 1 = ln ℏ +1 1 ℏ 2 = − ln − [ ln − 1] A piston-cylinder system with ideal gas Quasi-static Compression (Equilibrium process) - Thermodynamic work: W P (V ) dV dV/dt → 0 (Quasi-static Compression) Forward process Page 31 P f Pb eq - P - dW - F W W f W b f d ( W b ) P W eq dV E S Q E S T S S F S A piston-cylinder system with ideal gas Non-equilibrium process (dV/dt= finite) - Backward process (Expansion dV/dt>0) Pb P Pf eq d Wb Page 32 Forward process (Compression dV/dt<0) d Wb W neq eq b P d W f eq neq W eq F W f d W f eq Oscillatory Optical tweezers When a single particle of mass m in suspension is forced into an oscillatory motion A cos t by optical tweezers, it experiences following two forces; Viscous drag force Fdrag Spring-like force d x ( , t ) a Ftrap k ot A cos t x ( , t ) dt x ( , t ) The equation of motion for a particle A cos t Reference position O O’ m x x k ot A cos t x Oscillatory Optical tweezers The equation of motion for a particle trapped by an oscillating trap in a viscous medium, as a function of x ( , t ) 2 m d x dt 2 dx dt 2 k O T x k O T A cos t d x k OT 2 Neglect the first term ( m dt ) (In extremely overdamped 2 m 4m case ), and assume a steady state solution in the form 2 o 2 x ( , t ) D ( ) co s t ( ) The amplitude and the phase of the displacement of a trapped particle is given by D ( ) A kOT 2 kOT 2 ( ) tan 1 kOT 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, . Active Method of Measuring Optical Trap Strength Fixed potential well Horizontally oscillating potential well Equation of motion m x(t) + γ x (t) + k ot x(t) = A cos ( ωt ) Phase delay tan 1 k 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: xx x (t ) x (0) 2 - After the particle loses its initial information then xx obeys the equi-partition theorem. - In our system, the characteristic equilibration time( ) is about 20ms. t , xx k B T / k (equi-partition theorem) Page 36 Calculation of Work • Thermodynamic work = = 1 2 1 = 2 ∆ 2 • : constant value • Forward work is always positive. f > 0 Therefore, < 0 Page 37 2 Work Probability : Slowest protocol, = 0.268pN/µm·s Page 38 Work Probability : Slow protocol, = 0.536pN/µm·s Page 39 Work Probability : Fast protocol, = 2.68pN/µm·s Page 40 Work Probability : Fastest protocol, =5.36pN/µm·s Page 41