BROWNIAN GYRATOR :
A MINIMAL HEAT ENGINE ON THE NANOSCALE
R. Filliger and P. Reimann, PRL 99, 230602 (2007)
Introduction and summary


The original setup of heat engine consists of two
heat baths in contact with a cyclically working
engine, generating work in the form of a torque.
Smallest and most primitive such engine :
a single particle, gyrating around a generic
potential energy minimum under the influence of
friction and thermal noise forces from two heat
bath
Introduction and summary


The particle generates a systematic average torque
onto the physical object at the origin of the
potential
and an opposite torque of the same magnitude by
way of the dissipation mechanism onto one or both
heat baths.
Model

generic potential energy minimum
1D
xm   U 0xm x  1 U xm x 2  
U x   U 0
2
U x  
xm
1 2
ux
2
Model

generic potential energy minimum
2D
U x  
u1 2 u2 2
y1  y2
2
2
 y1   cos 
   
 y2    sin 
sin   x1 
 
cos   x2 
1
Model

The first heat bath & The second heat bath
The first heat bath along a
preferential direction without
loss of generality
The second heat bath may
either be of isotropic
character of acting only on x2
We focus on the latter case
Model

Langevin equation
mx  U x x  2kBT  t 
m : mass
Potential
T : temperature force
Thermal bath
effects
 : friction coefficient
U x  : potential
 t  : d-correlated Gaussian white noise
Model

1. Neglect inertia effect
2. Consider decoupled heat baths with different T

Overdamped Langevin equations

U xt 
i xi  
 2i k BTi  i t 
xi
dissipation
fluctuation
potential
i  1, 2
2
Solution

Three relevant forces
2
Dissipation : f  t    eii xi t 
i 1
Potential force : fU t   U xt 
2
Fluctuation force : f t    ei 2i k BTi i t 
i 1
Force balance : f t   fU t   f t   0
Solution

Average force
f t   0  fU t    f t 

Average systematic torque
Since f t  x  0 ,
fU t  xt    f t  xt   M t e3
3
Solution

The torque
M t   fU  x   e3
2



  P x, t fU  x  e3d x
Px, t  : probability density to find the particle
at position x at time t
Calculation for eq. (4)

Overdamped Langevin equations
U xt 
i xi  
 2i k BTi  i t 
xi
2k BTi
1 U xt 
t 
i t  t
 xi t  t   xi t  
i xi
i

1

Pz , t  t ; x, t   P  i t  

t

2

1
 1  zi  xi 1 U   

 
exp 

2
2
i xi   
 2 i  t
2i

2k BTi
2
i 
i
Calculation for eq. (4)

Fokker-Planck equation
Px, t 

1  
 
Ai Px, t   
Bij Px, t 
t
i xi
i , j 2 xi x j
1 U
1
2
zi  xi Pzi , t  t; xi , t d z  
Ai x, t   lim lim
 0 t 0 t z  x 
i xi
1
2
2






Bij x, t   lim lim
z

x
z

x
P
z
,
t


t
;
x
,
t
d
z
  i d ij
i
i
j
j
i
i
 0 t 0 t z  x 
Px, t 

 
t
i xi
 1 U 
 2 k BTi
 
 Px, t   2
Px, t 
xi i
 i xi 
Solution

Fokker-Planck equation
Px, t 

 
J i x, t 
t
i xi

4
Probability current density
 1 U k BTi  
J i x, t   

 Px, t 
i xi 
i xi
5
Calculation for P(x,t )

Steady state probability density
Px, t 
0
t
2
2
T

P
Fii  1 1U  kP
k
T

P
B i
B i


i x P  xFP x i x x2 0x 2  0
ii  i i i
ii

ii
i
U
 F  x i
xi
zi 
i
k BTi
X ij 
xi
F ij   OikT uk Okj
  


z
z   1 
  
 z 
 2
k
1
i
Fij
tr X P  z  X   z P   z    z P  0
T
T
T
Calculation for P(x,t )
tr X P  z T  XT   z P   z    z P  0
T

Diagonalize X
Orthonormal matrix TT T  TTT  1
x  T  z
 x'  T   z
T
T X TD
T
T
 D11
D  
 0
P  2 P
i X ii P  Dii xi x  x2  0
i
i
0 

D22 
Calculation for P(x,t )


 1
2
2 
Pst x, t   N exp   x1  2x1 x2  dx2 
 2


1 2
k B2T1T2  q 21 2
A  F11

Ak BT2
C  F22
d
Ck BT1

Bq

Bk BT2

Cq
1 2
1 2
1 2

Bq
2
B  F12  F21
T2  T1
q  kB B
A 2  C1
1
2
or
Bk BT1
1 2

Aq
1
Solution

torque
 U
U 
dx1dx2
Pst x, t  x1
 x2
x1 
 x2
M t   

 
  Ax1 x2  Bx  Bx  Cx2 x1
2
1
x
 d
x
2
2
2
2
 
 1  2 q
2
1
x1 x2  


1 2
AC  B 2
6
Discussion

Average torque of a BG in the steady state
M

k B T1  T2 u1  u2 sin 2
 
u1  u2  1 2 u1  u2  cos 2
1   2
For T1  T2
System will be in the equilibrium state

For u1  u2
Rotationally symmetric

For sin 2  0
symmetric
M 0
M 0
M 0
7 
Discussion

Simple case (equal coupling strengths)
M


k B T1  T2 u1  u2 sin 2
u1  u2
Maximal torque (at  
M max 
1  2

2
 n )
u1  u2
k BT1  k BT2  ~ Q1  Q2
u1  u2
Speaking about efficiencies is no possible as long as one does
not know the resulting rotation speed of the thermal baths
relative to the carrier of the potential U.
Discussion

The direction of rotation of the Brownian particle
If T2  0 , x2 t   0 , fU 2  U x1 ,0 / x2 ~ x1

It is plausible that qualitatively an analogous behavior is
expected also for finite T2 though the details will be more
complicated
Experimental realizations

Block body radiation
vacuum
Charged
particle
Black body
radiation
The dissipation mechanism
is provided by radiation
damping into the vacuum
Experimental realizations

Electrical heat
magnetic
bathsbaths
Helmholtz
coil
plate
Paramagnetic
Charged
particle
Random
Voltage
fluctuation
resistor
Experimental realizations

Usual heat bath and unusual heat bath

identification
2   1   
T   T 
T2  T T1 
 
Anisotropic
Thermal
fluctuations
    T   T

k B T   T u1  u2 sin 2
g
 u1 u2
M
sin 2 g  kBT  


u

u
1
2
u1  u2  cos 2
u1  u2 
2   
Outlook


The same behavior is expected when working in 3
dimensions
It is worthy Abandoning anisotropy condition of the
heat baths



Role of anisotropy condition of the heat baths
1. technical character
2. symmetry breaking
It is possible to consider the potential up to cubic
order
An interesting extension of the present work will be to explore
the collective phenomena due to many interacting Brownian
gyrators