Thermodynamics of feedback controlled systems Francisco J. Cao - - PowerPoint PPT Presentation
Thermodynamics of feedback controlled systems Francisco J. Cao Open-loop and closed-loop control Open-loop control : Controller the controller actuates on the system Actuation independently of the Evolving system system state.
783 (2005)
Controller Actuation Evolving system Controller Information Actuation Evolving system
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external agent to optimize its actuation on the system, in
performance.
control is incomplete: the role
controlled systems is still not completely understood. In particular, its implications for the entropy of the system.
systems and their limitations is very important form the technological point of view.
Controller Information Actuation Evolving system
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“It is not possible to find any spontaneous process whose
given amount of heat to an equal amount of work through the exchange of heat with only one heat source”.
“It is not possible to find an spontaneous process which its only result is to pass heat from a system to another system with greater temperature”.
Q W T Q T1 T2 Q T1>T2
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a cyclic process we have for each cycle with δQ the infinitesimal amount of work interchanged with the thermal bath at temperature TTB.
process is reversible (in this case also Tsystem=TTB)
TB
V p
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Clausius theorem for reversible cycles tell us that there exist a state function, named entropy, defined by
cycle the change in entropy of the system is zero.
REVERSIBLE
2 1 1 2
V p 1 2
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number of point particles Microstate: position and velocity of each particle at a time t. Macrostate: E, V and N; o p, V and T.
with a great number of constituents experimentally it is only possible to determine the macrostate.
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equilibrium state defined by E, V and N.
has Ω equiprobable compatible microstates
k=1,38 10-23 J/K Boltzmann constant
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pi probability of microstate i n number of microstates compatible with the macrostate
system in equilibrium microcanonical ensemble pi=1/Ω
(particle gas) canonical ensemble.
=
n i i i
1
C.E. Shannon, The Bell System
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=
n i i i
1 2
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equiprobable, the number of bits needed in average to describe the random variable, is simply log2n.
not equiprobable, the average number of bits can be reduced, using a shorter description for the more probable cases.
Example with four values: With this codification the average number of bits needed is ∑pili = 7/4 = 1.75 bits Which coincides with the Shannon “entropy” H = ∑pilog2pi = 7/4 = 1.75 bits While it they were equiprobable it would be log24= 2 bits
8 / 3 3 111 8 / 1 8 / 3 3 110 8 / 1 2 / 1 2 10 4 / 1 2 / 1 1 2 / 1 d c b a l p l chain p values
i i i i
⋅
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the Shannon “entropy” are pi probability or the i microstate n number of microstates compatible with the macrostate
information needed to specify the microstate [The ln(2) factor comes from the change of base.]
=
n i i i
1
=
n i i i
1 2
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system that is coupled to an external agent that uses information of the system to actuate on it.
control is incomplete: the role
controlled systems is still not completely understood. In particular, its implications for the entropy of the system.
from the study of the Maxwell’s demon, and mainly from a computation theory point of view.
Controller Information Actuation Evolving system
J.C.Maxwell, Theory of heat (1871) L.Szilard, Z. Phys. 53, 840 (1929) 19
the middle, and observes where is the particle.
which side is the particle, it attaches a piston in the correct side of the wall to extract a work W. Meanwhile the system is connected to a thermal bath
from it a heat Q=W.
¿¿¿η=W/Q=1??? and with only one thermal bath (¡¡¡2nd principle!!!) Q=W W=kTln2 ΔSs=-kln2 ΔSs=kln2
J.C.Maxwell, Theory of heat (1871) L.Szilard, Z. Phys. 53, 840 (1929) 20
ΔSs=-kln2 ΔSs=kln2 ? W=kTln2 Q=W
183 (1961) 21
second law, therefore it is not a principle.
information produces a growth in the entropy of the enviroment of ΔSe≥kln2 (Szilard engine: one bit is enough to store the information, for example: 0 left, 1 right) Qe≥kTln2 Wd=Qe ΔSe≥kln2
C.H.Bennett, Int.J.Theor.Phys. 21, 905 (1982) 22
ΔSs=-kln2 ΔSs=kln2 W=kTln2 Q=W ΔSd= kln2 ΔSs+ ΔSd+ ΔSe≥0 ΔSe=-kln2 ΔSe≥kln2 ΔSd= -kln2 Wd=Qe Qe≥kTln2
W.H.Zurek, Phys.Rev.A 40, 4731 (1989) 23
nQe
Compressed
nc≤n nWd=nQe ΔSe≥nkln2 ΔSe≥nckln2
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System perspective: For the controller we only need the (deterministic or not) correspondence between the states of the system and the actions of the controller. The entropy of the system before being measured by the system for the first time
1 1
1 1
X x X X b
∈
1 x
left right
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) ( ) 2 ln( : ) ( ) ( ) 2 ln(
1 1 1 1 1
1
C X H k c C X H c p k S
C c C a
= = =
∈
) ( ) 2 ln( : ) ( ln ) (
1 1
1 1 1 1
c C X H k c x p c x p k
X x C X C X
= = − ∑
∈
1 x
1 =
C 1
1 =
C ) (
1 1
x p
C X
) 1 (
1 1
x p
C X
) (
1
C
p
) 1 (
1
C
p
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The average change in a measurement is where its appears the mutual information which is a measurement of the ((dependency)) between two random variables. Thus, we obtain the Landauer “principle” as a consequence
∈ ∈
C c X x C X C X C X
, 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
a d
1 1 1
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For systems with deterministic control after M measurements we obtain H(CM,…,C1) is the average amount of information needed to especify the M actions of the controller on the system This result indicates that only nonredundant information is useful to reduce the entropy of the system (in correspondence with the Zurek idea of compressing the information)
∈
C c M c M C M C M C M C M info
1 1 1
, , 1 1 1
K L L
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For system with NONdeterministic control after M measurements we have where the additional term is nonzero if the present state of the system and the previous history of the controller does not completely determine the action
Example: The entropy reduction in the system due to information after M measurements is with
= − M k k k k M info
1 1 1 1
1
b M info
left right
1-ε 1-ε ε ε
b
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its efficiency can be defined as If the controller does not transfer heat to the system, the maximum efficiency is
have computed the rate of reduction of entropy, the work, and the efficiency, both in the quasistatic and in a nonquasistatic regime.
cont
F W Δ − = η
info cont
S T U W Δ − Δ − = η
1−α α
6 L
1−α α
1 2 3 4 5 7
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