Transcript Pak - KIAS
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
𝑈 ≡ 𝐸 = 𝑘𝐵 𝑇
1D Brownian Motion of Single Particle in Heat Bath
𝑝
𝑥=
𝑚
𝑝 = −𝛻𝑉 𝑥, 𝜆 𝑡
𝑝∙
𝑝
𝑚
= −𝛻𝑉 ∙
𝑑 𝑝2
𝑑𝑡 2𝑚
𝑑 𝑝2
𝑑𝑡 2𝑚
=−
𝑝
𝑚
𝑑
𝑉
𝑑𝑡
+𝑉 =
−𝛾
𝑝
𝑚
+𝑧 𝑡 ,
+ −𝛾
𝑝
𝑚
+𝑧 𝑡
𝑥 +
𝜕𝑉
𝜆
𝜕𝜆
𝑑𝐸
= 𝑊
𝑑𝑡
Page 8
𝜕𝑉
𝜆
𝜕𝜆
𝑝
𝑚
∙ ,
+ −𝛾
+ −𝛾
𝑧 𝑡 𝑧 𝑡′
𝑝
𝑚
𝑝
𝑚
𝑑𝑉 𝑥, 𝜆 𝑡
+ 𝑧(𝑡)
+ 𝑧(𝑡)
= 2𝐷𝛿(𝑡 − 𝑡 ′ )
𝜕𝑉
= 𝛻𝑉 ∙ 𝑑𝑟 +
𝑑𝑡
𝜕𝑡
𝑝
𝑚
𝑝
𝑚
Thermodynamic 1st Law
−
𝑄
1D Brownian Motion of Single Particle in Heat Bath
𝑑 𝑝2
𝑑𝑡 2𝑚
𝜕𝑉
𝜆
𝜕𝜆
+𝑉 =
𝑑𝐸
= 𝑊
𝑑𝑡
+ −𝛾
𝑝
𝑚
𝑝
𝑚
+ 𝑧(𝑡)
Thermodynamic 1st Law
−
𝑄
𝜕𝑉
𝜕 1 2
1 2
𝑊=
𝜆=
𝑘𝑥 𝑘 = 𝑥 𝑘
𝜕𝜆
𝜕𝑘 2
2
In equilibrium,
𝜆 = 𝑐𝑜𝑛𝑠𝑡
→
𝑊=0 ,
𝑊=
𝜏
𝑊𝑑𝑡
0
=
1 𝑘𝑓 2
𝑥 𝑑𝑘
2 𝑘𝑖
𝑑𝐸
= −𝑄
𝑑𝑡
In non-equilibrium steady state,
𝜌𝑠𝑠 𝑥, 𝑝 = 0, 𝐸 = 𝑐𝑜𝑛𝑠𝑡,
𝑊 = 𝑄 >0
Page 9
∆𝑆𝑒𝑛𝑣 =
∆𝑄
𝑇
>0
𝑑
𝑑𝑡
𝐸 =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, 𝜆 = 𝑘 (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
**2𝜇𝑚 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
**2𝜇𝑚 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 100𝝁sec)
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
𝒌 = 0.268pN/µm·s
𝒌 = 0.536pN/µm·s
𝒌 = 2.68pN/µm·s
𝒌 =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, 𝒌 = 5.36pN/µm·s
P f (W )
Page 27
Pb ( W )
Fast protocol, 𝒌 = 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
𝐹 = 𝑈 − 𝑇𝑆 = −𝑘𝐵 𝑇 𝑙𝑛𝑍 = 𝑘𝐵 𝑇 ln ( 𝛽ℏ𝜔) = 𝑘𝐵 𝑇 ln
ℏ 𝑘
𝑘𝐵 𝑇 𝑚
1
ℏ
2
𝑘𝐵 𝑇 𝑚
= 𝑘𝐵 𝑇 ln 𝑘 + 𝑘𝐵 𝑇 ln
Internal Energy
𝑈≡ 𝐸 =−
𝜕ln𝑍
𝜕𝛽
= 𝑘𝐵 𝑇
𝛥𝐸 = 𝑊 − 𝑄
: constant.
: Fluctuating value
Entropy
S=−
𝜕𝐹
𝜕𝑇
= 𝑘𝐵 ln𝑍 + 1 = 𝑘𝐵 ln
𝑘𝐵 𝑇
𝑚
ℏ
𝑘
+1
1
ℏ
2
𝑘𝐵 𝑇 𝑚
= − 𝑘𝐵 ln 𝑘 − 𝑘𝐵 [ 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
𝑊=
=
𝑑𝑡 𝜆
1
𝑘
2
𝜕𝐻 1
= 𝑘
𝜕𝜆 2
𝑖 ∆𝑡
𝑥2
•
𝑘 : constant value
•
Forward work is always positive.
𝑊f > 0
Therefore, 𝑊𝑏 < 0
Page 37
𝑑𝑡 𝑥 2
Work Probability : Slowest protocol, 𝒌 = 0.268pN/µm·s
Page 38
Work Probability : Slow protocol, 𝒌 = 0.536pN/µm·s
Page 39
Work Probability : Fast protocol, 𝒌 = 2.68pN/µm·s
Page 40
Work Probability : Fastest protocol, 𝒌 =5.36pN/µm·s
Page 41