The Fifth KIAS Conference on Statistical Physics
Nonequilibrium Statistical Physics of Complex Systems
INITIAL MEMORY IN
HEAT PRODUCTION
Chulan Kwon, Myongji University
Jaesung Lee, Kias
Kwangmoo Kim, Kias
Hyunggyu Park, Kias
5th KIAS Conference-NSPCS12
I. Introduction
t =t
W, Q
W ', Q'
t=0
W(work), Q(heat emitted)
- path dependent
- accumulated incessantly in time
in nonequilibrium steady state
(NESS)
- Fluctuation theorem
P(W ) / P(-W ) = ebW
Thermodynamic first law
As t ® ¥
W, Q µ t
DE = W -Q
DE = E(t ) - E(0) = finite
P(W ) » P(Q) Þ P(Q) / P(-Q) » ebQ
Steady state FT for entropy production
DSres = Q / T
5th KIAS Conference-NSPCS12
I. Introduction
P(Q)
= e f (Q) ¹ ebQ
P(-Q)
U. Seifert, PRL 95, 040602 (2005)
M. Esposito and V. den Broeck,
PRL 104, 090601 (2010)
P(DStot )
kB-1DStot
=e
P(-DStot )
where DStot = DSsys + Q / T
P(Q)
= h(Q)eQ
P(-Q)
f
Q
R. van Zon and E. Cohen,
PRL 91, 110601 (2003)
“Extended fluctuation theorem”
J.D. Noh and J.-M. Park,
Phys. Rev. Lett. 108, 240603
(2012)
P(Q) should depend on initial distribution.
5th KIAS Conference-NSPCS12
II. Equilibration process
v = -g v + x,
x (t)x (t ') = 2Dd (t - t '), D = g Tr
ted =
Dissipated power
tei =
Injected power
ò
t
0
ò
t
0
dt g v 2
dt x v =
ò
t
0
dt (v + g )v
Q = ted - tei
As t ® ¥, Q ® 0 while ed , ei ® finite
Total heat
Calculate
h1 (e ) : large deviation function, h2 (e ) : correction to ldf
bg - 2bgD v
r (v0 ) =
e
, initial temp T0 = D / bg = b -1Tr
2p D
2
0
5th KIAS Conference-NSPCS12
II. Equilibration Process
gt
P(e d ) =
4p i
gt
ò
i¥
-i¥
dl
e
2
(e d l +1-h )
1+ (1+ l / b )h -1 tanh(hgt )
where e d = e / D, h = 1+ 2 l , l = 2Dl / g
et h( x )
ò -i¥ dx 1+ x
Saddle-point integration near the branch point
Not a Gaussian integral!
i¥
(i) b >1 / 2
gt
(e d -1)2
é
ù
gtb
4e
P(e d ) » ê 2
ú e d
ë pe d (e d +1)[(2 b -1)e d +1] û
1/2
5th KIAS Conference-NSPCS12
II. Equilibration Process
(ii) b <1/ 2 ( b̂ =(1- 2b )-1 )
gt
(e d -1)2
é
ù
gtb
4e
P(e d ) » ê 2
, b̂ - e d >> O(t -1/2 )
ú e d
ë pe d (e d +1)[(2 b -1)e d +1] û
1/2
7/2
é
b
(12
b
)
P(e d ) » ê(gt )3/4
p 2 (1- b )
êë
ò
¥
0
dre-r
4
ù
/4
-gt [( b 2 -b )e d +b ]
ú e
,
úû
e d - b̂ << O(t -1/2 )
é gtb (1- 2 b )2
ù -gt [( b 2 -b )e +b ]
2
d
P(e d ) » ê
p (1- b )ú e
,
ë p (1- b )(e - b̂ )
û
1/2
e d - b̂ >> O(t -1/2 )
5th KIAS Conference-NSPCS12
II. Equilibration Process
Phase transition in the shape of the heat distribution function
at β=1/2
5th KIAS Conference-NSPCS12
III. Dragged Harmonic Potential
x = -(x - vt) + x ,
xi (t)x j (t ') = 2dijd (t - t '), k = g = T =1
Initial distribution
r (x0 ) = (2p )-3/2 e
b
- x02
2
initial temperature b -1
Heat production
Q = DE - W
(x(t ) - vt )2 x02
=
2
2
Probability density
function for heat
ò
b 3/2
P( p) =
2p
t
0
dt (x(t ) - vt ) × (-v)
æ
ö
il 2
÷
wt çç i( p-1) l -l 2 +
2 t ( l -ib ) ÷ø
è
e
ò -¥ d l {(l + i)(l - ib )}3/2
¥
where w = v 2 , p = Q / wt
5th KIAS Conference-NSPCS12
III. Dragged Harmonic Potential
P(p) / P(-p) = ewt f ( p)
FT for heat holds for flat initial distribution, β=0
5th KIAS Conference-NSPCS12
III. Dragged Harmonic Potential
Large deviation form
ì
-wt ( b p-b (1+b ))
e
, p > 2 b +1
ï
ï -wt ( p-1)2 /4
P( p) » í e
,
-1 < p < 2 b +1
ï
wt p
e
,
p < -1
ï
î
LDF is absent in the positive tail for β=0
where there is no exponential decay, but a power-decay
which can be seen from the 1/τ correction in the exponent
5th KIAS Conference-NSPCS12
IV. Summary
1. For the equilibration process of the Brownian motion
the phase transition takes place in the shape of heat
distribution at β=1/2.
2. For the dragged harmonic potential the FT for heat holds
for a flat initial distribution, β=0. The modification of FT
depends for on β.
3. The correction to the LDF needs a nontrivial saddle point
integration near the branch point that is not a conventional
Gaussian integral. Our result may be useful for other cases
in which the generating function, the Fourier transform, of
the distribution function with power-law singularity, as the case
of heat and work distribution.