Fluctuation relations & Arrow of Time - Institut für Physik - Universität

48 Pages·2012·4.03 MB·English

Doing systems: Fluctuation Relations and the Arrow of Time
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Peter Hänggi, Institut für Physik, Universität Augsburg

M. Campisi, P. Hänggi, and P. Talkner M. Campisi and P. Hänggi
Colloquium: Quantum fluctuation relations: Foundations and Fluctuation, Dissipation and the Arrow of Time
applications Entropy 13, 2024–2035 (2012).
Rev. Mod. Phys. 83, 771–791 (2011);
Addendum and Erratum: Quantum fluctuation relations:
Foundations and applications
Rev. Mod. Phys. 83, 1653 (2011).
Acknowledgments
Sekhar Burada
Michele Campisi
Gert Ingold
Eric Lutz
Manolo Morillo
Peter Talkner
SSeeccoonndd LLaaww
Ruddollff Julliius Emanuell Cllausiius
William Thomson alias Lord Kelvin
(1822 – 1888)
(1824 – 1907)
HHeeaatt ggeenneerraallllyy ccaannnnoott  NNoo ccyycclliicc pprroocceessss eexxiissttss wwhhoossee ssoollee 
spontaneously flow from a  effect is to extract heat from a 
material at lower temperature to  single heat bath at temperature T 
a materiiall at hhiighher temperature. andd convert iit entiirelly to workk.  
δQ = TdS (Zu¨rich, 1865)
The famous Laws
Equilibrium Principle -- minus first Law
An isolated, macroscopic system which is placed in an arbitrary
initial state within a finite fixed volume will attain a unique
state of equilibrium.
Second Law (Clausius)
For a non-quasi-static process occurring in a thermally isolated
system, the entropy change between two equilibrium states is
non-negative.
Second Law (Kelvin)
No work can be extracted from a closed equilibrium system
during a cyclic variation of a parameter by an external source.
MINUS FIRST LAW vs. SECOND LAW
-1st Law
2nd Law
Entropy S – content of transformation
„VVerwanddllungswertt““
rreevv iirrrreevv rreevv
dd SS = δδQQ TT ;; δδQQ << δδQQ
V , T
2 2
δQ
Γ
0
rev Γ
irrev
TT ≤
C
II
1
C = Γ + Γ
−rev irrev
V , T
1 1
δQ
S(V , T ) S(V , T ) ∂S
2 2 1 1
− ≥ T
ΓΓ 00 NNOO !!
ZZ iirrrreevv
δQ
∂t ≥≥
S(V , T ) S(V , T ) =
2 2 1 1
− T
Γ
Z rev
Small system – single realizations
11-‐sstt. lalwaw:  : ∆∆E = Q + W
! random quantities ¡
ω ω ω
44 44
... ∆U = QQ ++ W
hh ii ⇒ 4 4
Q = ∆U W
Note:  ∆U =? & W =?
4 − 4 4
weak coupling:  Q = Q
bath
4 − 4
QQ
REV
2‐nd law:  ∆S 44
A B
≥ T
→
IIRRRREEVV
Q
REV
∆S = 4 + ( 0)
A B
T ≥
→
X
IIRRRREEVV
∆U W
= − 4 +
T
X
IRREV
Q
bath ; for weak coupling 
= − 4 +
T
X
What are fluctuation and work theorems
about?
Small systems: fluctuations may become comparable to
average quantities.
Can one infer thermal equilibrium properties from
fluctuations in nonequilibrium processes?
Equilibrium versus nonequilibrium processes
Isothermal quasistatic process:
xx
e−βH(t0)/Z(t0) xx e−βH(tf)/Z(tf)
xxquasi-
BATH weak SYSTEMxxstatic BATH weak SYSTEM
β contact H(t ) β contact H(t )
0 −−−→ f
∆F = w = w
(cid:104) (cid:105)
Non-equilibrium process:
H(t ) HH((tt ))
0 ff
p (w) =? pdf of work
tf,t0
t t t
0 f