Optimizing Cooling Processes - A Dynamic View on the Third Law

Transcription

Optimizing Cooling Processes - A Dynamic View on the Third Law"
Karl Heinz Hoffmann!
TU Chemnitz!
Germany!
A Dynamic View on the Third Law!
IWNET2012, Roros!
August 19-24, 2012!
Professur für Theoretische Physik, insbesondere Computerphysik!
The Third Law"
●  1906 Nernst 1911 Planck!
lim S = 0
T →0
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●  magnetic refrigeration!
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The Third Law"
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–  isotherms!
–  adiabats!
il:in
●  alternate!
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No reversible adiabatic process starting at nonzero temperature can possibly bring a system to zero temperature!
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lim S = 0
T →0
il
●  as !
The Third Law"
●  consequence: absolute zero is unattainable!
●  infinite number of steps would be needed to reach absolute zero!
●  steps are thermodynamic equilibrium steps (infinitely slowly)!
What about non-equilibrium processes?!
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Dynamic View on the Third Law"
●  Processes in finite time!
Finite-Time Thermodynamics Endoreversible Thermodynamics!
●  no longer thermodynamic equilibrium branches!
●  consider a refrigerator!
Are there limits on the cooling rate?!
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Dynamic View on the Third Law"
●  continuous refrigeration process: entropy production!
●  the Second Law:!
σ = −Q̇c /Tc + Q̇h /Th > 0
Rc = Q̇c <
●  What exponent δ does apply in
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Q̇h
Tc
Th
Rc ∝ Tcδ
?!
A Quantum Refrigerator"
●  working medium: an ensemble of quantum harmonic oscillators!
1 2 m ω(t)2 2
Ĥ =
P̂ +
Q̂
2m
2
●  description by density operator!
●  the work parameter: the frequency instead of the volume!
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A Quantum Refrigerator: Average Cooling Rate"
●  for a cyclic refrigerator the cooling rate is the heat up-take per cycle time!
Rc =
Qc
Qc
=
τ
τhc + τc + τch + τh
●  expectation: 3.5
3.0
τch
2.5
For given Tc one needs to choose optimal! ωc , τhc , τc , τch , τh
n 2.0
τh
τc
1.5
τhc
1.0
Qc ( ωc ( Tc ) )
τ ( ωc ( Tc ) )
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0.5
0.5
1.0
1.5
2.0
Ω
2.5
3.0
3.5
The Dynamics of a Quantum Working Fluid"
●  Adiabats:
externally driven Hamiltonian! Ĥ(ω(t))
dÔ(t)
i
∂ Ô(t)
= [Ĥ(t), Ô(t)] +
dt
�
∂t
●  contact to heat bath, fixed frequency: dissipative Lindblad dynamics!
dÔ(t)
i
∗
= L (Ô) = [Ĥ, Ô] + L∗D (Ô)
dt
�
●  Lindblad term establishes thermal equilibrium to a bath!
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Simplified Dynamics"
●  There exists a set of operators which is invariant under the dynamics!
Ĥ, L̂, Ĉ
●  These are the Hamiltonian, the Lagrangian, and the Correlation!
1 2 m ω(t)2 2
Ĥ =
P̂ +
Q̂
2m
2
1 2 m ω(t)2 2
L̂ =
P̂ −
Q̂
2m
2
1
Ĉ = ω(t) (Q̂P̂ + P̂Q̂)
2
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Entropies"
●  von Neumann entropy SvN = −kB Tr(ρ̂ ln ρ̂)
●  energy entropy SE = −kB
with
�
pn ln pn
n
pn = � n |ρ(Ĥ, L̂, Ĉ)| n � expectation values in energy eigen states!
●  in equilibrium!
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SvN = SE
Adiabat Dynamics"
●  frequency changing linear in time [0,10] from 1 to 0.3!
1.0
1.0
0.9
E
L
C
0.8
0.8
Ω
0.7
0.6
0.6
0.4
0.5
0.2
0.4
0.0
0.3
0
2
4
6
t
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8
10
0
2
4
6
8
t
10
Adiabat Dynamics"
●  frequency changing linear in time [0,10] from 1 to 0.3!
1.0
0.99
E
n
0.8
0.98
SvN
SE
0.97
0.6
0.96
0.4
0.2
0.95
0
2
4
6
t
8
10
0.94
0
2
4
6
8
t
●  the mean occupation number n is not constant: quantum friction!
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10
Heat Bath Contact"
●  fixed frequency and coupling to a bath with the contact temperature!
!
!
!
!
!
!W. Muschik, G. Brunk!
1.0 0
0.5
L
0.0
2
E
3.0
4
�0.5
6
�1.0
1.0
E
L
C
2.5
2.0
1.5
1.0
0.5
0.5
0.0
C
�0.5
0.0
�1.0
�0.5
0
2
4
6
8
t
IWNET2012, Roros!
August 19-24, 2012!
10
Heat Bath Contact"
●  fixed frequency and coupling to a bath with the contact temperature!
2.10
2.08
SvN
SE
2.06
2.04
2.02
0
2
4
6
8
10
t
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Heat Bath Contact"
●  fixed frequency and coupling to a bath with a different temperature!
3.0
2.0
E
L
C
2.5
2.0
1.8
SvN
1.5
1.6
1.0
SE
1.4
0.5
0.0
1.2
�0.5
1.0
0
10
20
t
30
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August 19-24, 2012!
40
0
10
20
t
30
40
Heat Bath Contact"
●  fixed frequency and coupling to a bath with a different temperature!
L
0.0
0.5
1.0
1.0
�0.5
0.5
�1.0
1.0
C
0.0
0.5
�0.5
C0.0
�1.0
1.0
0.5
�0.5
0.0L
01
2
E
3
�1.0
4
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�0.5
0
1
2
E
3
4
�1.0
The Cooling Rate"
●  the cooling rate!
Rc
=
1
�ωc (nC − nD )
τhc + τc + τch + τh
●  given the time needed for the adiabats: optimize for relative time allocation on the isotherms (isofrequencies) optimize for time allocation between isotherms and adiabats!
τhc = τch
τh = τc
3.5
eq
Ropt
c
≈
�ωc neq
c
τhc
3.0 nC
τch
2.5 nC
n 2.0
τh
τc
1.5 nD
1.0
0.5
0.5
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τhc
ωc
1.0
1.5
2.0
Ω
2.5
3.0
3.5
Adiabat Dynamics: the 1TURN Process"
●  special frequency protocol!
1.0
1.02
E
0.8
ω̇
µ = 2 = const
ω
n
1.00
SvN
0.6
SE
0.98
0.4
0.96
0.2
0.0
0
5
10
15
20
25
0.94
0
5
10
t
●  the optimal time from hot to cold!
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15
20
t
∗
τhc
=
1 −1
ω
2 c
25
Time needed for IR Adiabats: the 1TURN Process"
●  for given ωc , ωh choose proper speed and time
∗
τhc
1 −1
= ωc
2
!
Ropt
< const. Tc2
c
0.2
C
0.0
0.1
0.2
0.1 L
0.2
0.0
0.1
L
E
0.2
0.4
1.00.80.6
0.0
0.1
0.0
C
0.2
0.2
0.4
0.6
E
0.8
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August 19-24, 2012!
1.0
Adiabat Dynamics: 3JUMP Process"
●  three frequency jumps between given frequencies!
2.0
τ2
E
L
C
2
1.8
1.6
1
1.4
0
Ω
1.2
1.0
0.0
τ1
0.1
0.2
0.3
�1
0.4
t
0.5
IWNET2012, Roros!
August 19-24, 2012!
0.6
0.7
0.0
0.1
0.2
0.3
0.4
t
0.5
0.6
0.7
Adiabat Dynamics: 3JUMP Process"
●  three frequency jumps between given frequencies!
1.0
C
1.0
0.5
0.0
2.5
2.0E
1.5
1.0
1.0
1.5
E
C
2.0
0.5
2.5
�1.5
�1.0
�0.5
L
0.0
IWNET2012, Roros!
August 19-24, 2012!
0.5
0.0
�1.5
�1.0
�0.5
L
0.0
0.5
Shortest Time needed for Adiabats"
●  If frequency is confined between
ωc and
ωh
3JUMP Processes are the adiabats which lead in the shortest time from equilibrium to equilibrium!
●  proof on the basis of control theory, very technical ∗
τhc
1
− 12
= √ ωc
ωh
IWNET2012, Roros!
August 19-24, 2012!
Ropt
c
3
2
< const. Tc
(Dated: July 13, 2012)
In the optimal control problems mentioned above
for these
probl
turned up a[15,
closely
related
invariantofwhich
we herethe
dub
16],
the
constancy
X
reduces
dimension
of
ant of motion for Hamiltonian systems is introduced: the Casimir
thermore the v
the Casimirthe
companion
X
It appears
to be
as simplifies
problem
by [15–18].
oneoscillators)
– a feature
which
greatly
simple dynamical algebras (e.g., coupled
spins,
harmonic
energy
useful for these
problems
as theand
Casimir
is for
the
optimal
control
facilitates
its understanding
andof the sC
problems thatGeneralized
consider adiabatically
varying
the
parameters
in
the operator
Dynamics:
Finite-Dimensional
Lie
Algebra"
the von Neum
problems
traditional,
static
of
interpretation
[17,
18].symmetries. X is calhas proved useful in optimal
controlwith
of changes
in these
parameters.
of X. wh
culated
usingof
the
formula for problem
the
Casimir
but Hterms
allows simple calculation of
the entropy
non-equilibrium
ensembles.
A
Hamiltonian
withoperator
Hamiltonian
is said
inserting
of operators.
ope
have a values
dynamical
algebra
[19]
provided
H can be ex●  There exists
a setexpectation
oftooperators
whichinstead
is invariant
under
the dynamics!
Sv, 05.70.-a, 02.30.Yy
In the optimal
control
problems
mentioned
aboveof the Lie alto
pressed as
a linear
combination
of elements
esses, Entropy, Lie Algebra, Casimir Invariant, Optimal
Control
TH
Ĥ,of
L̂, X
Ĉ reduces the dimension of
[15, 16], thegebra.
constancy
{B
the problem by one – a feature which greatly simplifies
N
!
Consider a s
the
optimal
control
and
facilitates
its
understanding
and
i
and
this
latter
dynamics
follows
from
the
commutation
h (t)Bi . (1)
●  More general setting: there exists a set H
of =
operators
of operators
{
interpretation
[17,
18].
whichrelations
form a Lie algebra with!
i=1
where the Ham
A Hamiltonian problem with Hamiltonian H is said
m mechanics has
N
It then follows
that
theprovided
dynamics
(say)
the energy
can
No
operators.
Th
!
to have a dynamical
algebra
[19]
H of
can
be exk
nvariants simpli[B
cijdynamics
Bkelements
,
(2)
reduced
ofofa the
basis
for
algebra
i, B
j ] =to the
be closedHa
u
pressed as abe
linear
combination
of
Lie
al- the to
these invariants
k=1
{Bi }N
gebra.
i=1 is not
Lie algebra assoN
defining the algebra.
!
htforward invarii
h
(t)B
.
H
=
Optimal
control
involving
only
the iexpectation
values(1)
ants are familiar
i=1handled in the Heisenberg
of (say) the energy, are nicely
omentum.
representation.
The
dynamics
of anyof
operator
A energy
are then
It then follows
that
the dynamics
(say) the
can
Note that thi
ced by Hendrik
be reduced to dA
the dynamics
of ∂A
a basis for the algebra
Hamiltonian ar
d element of the
ı
= [H, A] +
,
(3)Hoffmann!
commutes with
Karl Heinz
IWNET2012, Roros!
Professur für Theoretische Physik, dt
h̄
∂t
August 19-24, 2012!
insbesondere Computerphysik!
ture makes it a
Taking the trace with the (constant) density matrix leads
having the symto ordinary differential equations for the expectation
this means that
The metric
metricg gofofthe
thealgebra
algebra
calculated
directly
The
cancan
be be
calculated
directly
fromthe
thestructure
structureconstants:
constants:
from
Th
N
The osc
N!
!
n n
cincmincm
. Lie
(6) (6)
gikgik= =
.
Generalized Dynamics: Finite-Dimensional
Algebra"
kmckm
m,n=1
m,n=1
Now
Casimir
C [20] is completely deterP, Q
Nowofthe
the
Casimirinvariant
wher
●  Known constant
the motion:
invariant C [20] is completely deter-where
mined
erators,
m
mined
erato
N
lator,lator,
resp
!
N
!
The Casimir Operator!
the finite L
C=
g ik Bikk Bi ,
(7)
the fi
C=
g Bk Bi ,
(7)
i,k=1
i,k=1
with g ikik = [g −1−1
]ki . In addition a second dimensionless
●  There exists invariant,
awith
second
constant
of. the
which
is invariant
under
g the
= [gCasimir
]ki
In motion
addition
second
dimensionless
companion
X,a naturally
exists:
the dynamics:invariant,
the Casimir companion X, naturally exists:
N
!
Ng ik !B "!B " .
!
k
i
ik
g !Bk "!Bi " .
X=
i,k=1
X=
The Casimir Companion!
(8)
Using the
(8)
Using
[H, L] =
[H
Because of the linearity of equation (4), the expectation
values
Bi of
= !B
evolve according
to the(4),
same
Because
the
of equation
theequations
expectationthe metric
i " linearity
●  The von Neumann entropy
– also in non-equilibrium – can always be
of
motion
Bi . The invariance
of X
folvalues
Bthe
= the
!Bi "operators
evolve
according
to the same
equations
the m
i as
expressed in terms
of
Casimir
Companion
!
lows
of motion as the operators Bi . The invariance of X foli,k=1
lows
IWNET2012, Roros!
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Ẋ =
N
!
N
!
i,k=1
Ẋ =
"
#
Professur
für
Theoretische
Physik, g !"B˙k "!Bi " + !Bk "!Ḃi " insbesondere
Computerphysik!
#
N
!
i,k=1
ik
g
ik
!B˙k "!Bi " + !Bk "!Ḃi "
This metr
explicitly
This
above, one
Entropies"
●  von Neumann entropy SvN = kB ln
��
1
X−
4
�
1
E2
−
ω2
4
�
+
√
+
�
X arcsinh
●  energy entropy!
SE = kB ln
��
� √
X
X − 14
 �
�
E2
ω2
E 2 − L2 − C 2
X=
ω2

E2

arcsinh  E 2
2
1
ω
ω2 − 4
SE ≥ SvN
●  one can define "contact temperature"!
1
Tcontact
IWNET2012, Roros!
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∂SE ��
=
�
∂E ω
The Team"
Yair Rezek
Ronnie Kosloff
Fritz Haber Research Center
Hebrew University Jerusalem
IWNET2012, Roros!
August 19-24, 2012!
Peter Salamon
Karl Heinz Hoffmann San Diego
State University
Tech. Universität Chemnitz!
Summary"
●  Dynamic view on the Third Law: Study temperature dependence of the Cooling Rate!
●  "Quantum Refrigerator": parametric harmonic oscillators as working medium!
●  good indicators for non-equilibrium: Von Neumann entropy, energy entropy!
●  Application: Maximum cooling rate of a quantum refrigerator!
●  There exist adiabatic processes in finite time !!!!
●  Generalization: Lie algebra dynamics and the Casimir Companion!
IWNET2012, Roros!
August 19-24, 2012!
Selected References 1"
● 
Optimal control in a quantum cooling problem Salamon, Peter and Hoffmann, Karl Heinz and Tsirlin, Anatoly Applied Mathematics Letters (in press) (2012)!
● 
Time-optimal controls for frictionless cooling in harmonic traps Hoffmann, Karl Heinz and Salamon, Peter and Rezek, Yair and Kosloff, Ronnie Europhysics Letters 96(6): 60015-1--6 (2011)!
● 
Optimal control of the parametric oscillator Andresen, Bjarne and Hoffmann, Karl Heinz and Nulton, James and Tsirlin, Anatoly and Salamon, Peter European Journal of Physics 32(3): 827--843 (2011) ; ISSN 0143-0807!
● 
The quantum refrigerator: The quest for absolute zero Rezek, Y. and Salamon, P. and Hoffmann, K. H. and Kosloff, R. Europhysics Letters 85: 1--5 (2009)!
● 
Maximum work in minimum time from a conservative quantum system Salamon, P. and Hoffmann, K. H. and Rezek, Y. and Kosloff, R. Physical Chemistry Chemical Physics11: 1027--1032 (2009)!
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Selected References 2"
● 
Optimal Process Paths for Endoreversible Systems Hoffmann, K. H. and Burzler, J. and Fischer, A. and Schaller, M. and Schubert, S. Journal of Non-Equilibrium Thermodynamics 28(3): 233--268 (2003)!
● 
Recent Developments in Finite Time Thermodynamics Hoffmann, K. H. Technische Mechanik 22(1): 14--25 (2002)!
● 
Endoreversible Thermodynamics Hoffmann, K. H. and Burzler, J. M. and Schubert, S. Journal of Non-Equilibrium Thermodynamics 22(4): 311--355 (1997)!
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August 19-24, 2012!

Optimizing Cooling Processes - A Dynamic View on the Third Law

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