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