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
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)
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.
)](exp[)()(
FWWPWP
b
f
)( WPb )(WPf
F
Page 4
Why fluctuation theorem is important?
Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies
Collin, Bustmante et. al. Nature(2005)
)](exp[)()(
FWWPWP
b
f
Page 5
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.
2/)( 2kxxV
k
Page 6
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:
mkhdxdppxHZ /,/1/),(exp
22
21
2),( kx
mppxH
i
fif k
kZZF ln2/1)ln()ln(
)0(0 SFkk if
)0(0 SFkk if
fi kk ,
forward
backward
𝑈 ≡ ⟨ 𝐸 ⟩=𝑘𝐵𝑇Page 7
1D Brownian Motion of Single Particle in Heat Bath
�̇�=𝑝𝑚
,
+
)
Thermodynamic 1st Law
,
+ +
𝑑𝑉 (𝑥 ,𝜆 (𝑡 ) )=𝛻𝑉 ∙𝑑𝑟 +𝜕𝑉𝜕𝑡 𝑑𝑡
Page 8
1D Brownian Motion of Single Particle in Heat Bath
+
Thermodynamic 1st Law
In equilibrium, =0 ,𝑑𝐸𝑑𝑡 =−�̇�
�̇�=𝜕𝑉𝜕𝜆 �̇�= 𝜕
𝜕𝑘 ( 12𝑘𝑥2) �̇�=1
2𝑥2 �̇� =
In non-equilibrium steady state, 0 Work done by external source converts to heat in the heat bath.
Page 9
Single Particle Gas under a Harmonic Potential Quasi-static process (Equilibrium process)
Thermodynamic work:
Here, (external parameter)
dk/dt→0 : (Quasi-static process)-
-
-
2
21 xdkHdtW
kTkxxx Bbf
eq/222
dkkTkdkxWddW Beq
bf 21
21)( 2
FkkWWWi
fbf ln2
1
x
Page 10
Single Particle Gas under a Harmonic Potential Non-equilibrium process(dk/dt=finite)
Forward process(dk/dt>0)
eqf
neqf
eq
f
WdWd
xx
22
f
eq
bWFWW
Backward process(dk/dt<0)
eqb
neqb
eq
b
WdWd
xx
22
Page 11
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
i
ifBneq
iB
k
k
kkkTkW
kTkxandxdkW f
i
2
/2/1 22
f
eq
bWWW
βWabf, eβW)(
πaW)θ(=(W)P 2/1
)k(kk=a ifi /
Using recent theoretical result,
Kwon, Noh, Park, PRE 88 (2013)Page 12
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
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.
rrek
rennnnI
cRnrVUF
Rrot
Rro
ˆ
ˆ2
6
22
22
/
/21
22
21
22
212
rkottrapF
r
trapF
Developed by A. Ashkin
Potential Energy
21
22
21
22
212
26,where
nnnnI
cRnVk oot
Page 14
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
- Boltzmann distribution method
- Oscillating optical tweezers method
Tkxk B21
21 2
2)(
21)(,)( kxxVdxCedxx xU
kDx
tAkxdtdx
dtxdm
1
2
2
tan),cos()(
)cos(
Page 17
Passive Method of Measuring Optical Trap Strength
Equi-partition theorem Boltzmann distribution2)(
21)(,)( kxxVdxCedxxp xV
Tkxk B21
21 2
μmpN=kot /2.87Page 18
Tracking beam
Boltzmann Statistics
100 XOil
NA 1.35
Objective
Potential Energy
Condenser
QPD
CTkxpTkxV
DimindxCedxxp
BB
TkxVB
ln)(ln)(
1)()(
221)( xkxV OT
Page 19
Profile of 1D Harmonic Potential
μmpN=kot /2.87
Page 20
Measurements of Optical Trap Strengthwith Controlled Laser Power
Page 21
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
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)- 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
558.0)/ln(2/1 if kkF
backward forward
Page 23
Laser Power and Trap Strength in Time
forwardbackward
EQ EQEQ
sμmpN±=k /0.536
Page 24
Work Probabilities for Four Different Protocols
Page 25
5.36pN/µm·s
0.268pN/µm·s
0.536pN/µm·s
2.68pN/µm·s
)](exp[)()(
FWWPWP
b
f
Mean Work Value and Expected Free Energy Difference
Page 26f
eq
bWFWW
Verification of Crooks Fluctuation Theorem
Fastest protocol, 5.36pN/µm·s Fast protocol, 2.68pN/µm·s
)](exp[)()(
FWWPWP
b
f
Page 27
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
f
eq
bWWW
)](exp[)()(
FWWPWP
b
f
Thank you for your attentions.
Page 29
Supplement
Partition function
Free energy
Internal Energy
: constant. : Fluctuating value Entropy S
mkhdxdppxHZ /,/1/),(exp
A piston-cylinder system with ideal gas Quasi-static Compression (Equilibrium process) -
Thermodynamic work:
dV/dt → 0 (Quasi-static Compression)- - -
Forward process
dVVPW )(
bfeq PPP
dVPWddW eqbf )(
bf WWWF
FSTEQEW
SS
SS
Page 31
Backward process (Expansion dV/dt>0)
Forward process(Compression dV/dt<0)
eqb
neqb
eqb
WdWd
PP
eqf
neqf
eqf
WdWd
PP
f
eq
bWFWW
A piston-cylinder system with ideal gas Non-equilibrium process (dV/dt= finite) -
Page 32
O O’
ttx
d),(dFdrag
),(cosFtrap txtAkot
),( tx
tA cosReference
position
a
When a single particle of mass in suspension is forced into an oscillatory motion by optical tweezers, it experiences following two forces;
tA cos
Viscous drag force Spring-like force
The equation of motion for a particle
xtAkxxm ot cos
m
Oscillatory Optical tweezers
Oscillatory Optical tweezers
2
2 cosOT OTd x dxm k x k A tdt dt
( , ) ( ) cos ( )x t D t
2 2
( ) OT
OT
kDA k
1( ) tan
OTk
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
2
2
d xmdt
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, .
( , )x t
2
22
4mmkOT
o
Active Method of Measuring Optical Trap Strength
Equation of motionPhase delay
k 1tan
Fixed potential well Horizontally oscillating potential well
)(cos ωtA=x(t)k+(t)xγ+(t)xm ot
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.
2)0()( xtxxx
kTkt Bxx /,
xx
(equi-partition theorem)
Page 36
Calculation of Work
• Thermodynamic work
=
• : constant value
• Forward work is always posi-tive.
f > 0
Therefore, < 0
Page 37
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