Optimal processes in macro systems (thermodynamics and economics) A.M. Tsirlin and V. Kazakov.
Transcript of Optimal processes in macro systems (thermodynamics and economics) A.M. Tsirlin and V. Kazakov.
Optimal processesin macro systems
(thermodynamics and economics)
A.M. Tsirlin and V. Kazakov
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Macro Systems: thermodynamics, economics, segregated systems
Extensive variables
V, U, …, N0, N
Intensive variables
T, , P, …, p, c
Equation of state
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«Natural processes»
Irreversibility measure,
dissipation
S,
Irreversibility and kinetics
),(//
0)()(lim
2121
21
ppgdtdNdtdN
tptpt
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Structure of MM of the macrosystem
.....,1,0,),,( 21 muuxyfx
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Workout example
thermodynamics microeconomics
Irreversible:
S > 0, A = 0
Reversible:
S = 0, A > 0
Irreversible
> 0, E = 0
Reversible
= 0, E > 0
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Major problems
1. Minimal dissipation processes .
2. Stationary state of an open system that includes intermediary.
3. Intermediary’s limiting possibilities in close, open and non-stationary macro systems.
4. Qualitative measure of irreversibility in microeconomics.
5. Realizability area of macro system.
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Irreversibility measure in microeconomic systems
Wealth function S(N) exists such that
Econom
ic
agent
NRn+1
Resources’ and capital (N0)
endowments
pi(N)Estimate of i-th resource (equilibrium price)
ji
ij
jii
i NNS
ppN
ppNN
Sp
pNS
p
2
0000
01
,,
])()[(
0)(,)()(
1
00
01
00
n
iii
n
iii
NNpNNpS
NpdNNpdNNpdS
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For
capital extraction
voluntariliness principle
000
000
2121 0
NNpS
ppNpNpS
ppgpp
а
ii
iiрез
)(
,const,,
,),(
dSi 0, i=1,2If p1i and p2i have different
signs that it is not less than 2 flows.
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Capital dissipation
00 0NdStptc ),()(
0 00
00 0 .))(,())(,(
pS
dtpcpcgNdtpcpcgpS
– fixed
= g(c,p)(c–p) capital dissipation (trading costs)
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Minimal dissipation processes in thermodynamics
0
1)(
min),(),(tu
dtupXupg
uppp
uppgN
00 0
1
,)(
),,(..
0
1gdtupg ),(
For = ( p )g( p, u )
We get:
const
uX
ugg2
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Minimal dissipation processes in thermodynamics
Heat transfer:
p ~ T1, u ~ T2
12
121
21
11
TTX
Tcq
TT
TTqg
)(),(
),(~
const)(
)(
tTtT
2
1
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Minimal dissipation processes in thermodynamics
0
0
000
0
0
1
0
0
.),(
,)(),,(
,)(),,(
min)()(
gdtpcg
NNpcgdtdN
NNpccgdtdN
Ntc
2
02 g
Nppg
g
cgdNd
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Minimal dissipation processes in thermodynamics
If 00
Np
const2g
cg
ttptcgNpcg const)(),()( *
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Stationary state of open macro system
Thermodynamics n – power, p1i~Ti
q – heat, g – mass, p – intensive variables
for
i i
iii
j j
ijijij
ii
jijiij
jijiij
uq
sgmip
qsg
g
gppgqppq
.,,,
,
,),(,),(
010
0
11
i u
iiiiii upgupqn max),(),(
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If g = 0, qij = ij(Ti – Tj), then
If m = 2, T1 = T+, T2 = T–, then
For g = AX Prigogine’s extremal principle holds for
any u (A – Onsager matrix).
miu
Tu
uT
jijii
ii
i
ii
i ii
i
ii
,,,
;
11
12
221
21
21 1
TTN
TT
TkuTku
max
** ,,,
– limiting power
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Stationary states of open macro systems
Microeconomics ui – prices,
p – estimates
ii
jijiij
i puiiii
ggppg
uupgn
.,),(
max),(,
0
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Analogy of Prigogine extremal principlefor g = A (ij=pi – pj):
A – symmetric.
If gij = ij(pj – pj), gi = i(ui – pj), then
If m = 2, p1 = p+, p2 = p–, then
.,, j
iji
iiiiii upu
50
,,
21
122
21
211 2
2
2
2
ppp
uppp
u
221
21
4 ppn
max
ji i p
iiTiijij
Tij uAA
,
min,50
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Optimal processes
Availability Amax()=?
Control u(t) = (u1, …, um),
h(t) = (h1,…,hm), hi = {0, 1}
k – number of conditions on final state.
StatementsStatements::
1. .u*(t) h – are minimal dissipation processes,
2. For reservoirs {u*(t), h*(t)} are piece-wise constant
function that takes not more than k+1 values.
3. System’s entropy is piece-wise linear function q, g
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If
– exergy
iii
iiiiii TT
cq
TTuq 00 )(,,.
2
2
0
0
00
11
1
1
11
1
)()(exp
)(
)(
,)(
exp
),()(
ln
iiii
i
ii
i iii
ii
i i
iiii
i
iiii
kk
Tk
cT
kkk
TQ
ck
cTQ
kQkQA
TT
TTcA
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Separation systems
m
j i ij
ijj
m
j i i
ijijj
Ax
x
xxRTN
A
00
22
0 00
)(
ln
)(min
00 1
N
N jj ,
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E – analogous of exergy .
– given:
c*(t) obeys conditions of minimal dissipation during all contacts
obeys the conditions
Microeconomics. Profitability =?
.,,),,(),(
;)(),,(
miNNppncN
NNcpnN
iiiiiii
iiiiii
1
0
00
0
.
.
i thtc
ii NNE)(),(
max)()()( 00 0
),(*iii NNc
*iN
i
i
N
N i iiii
iiii NNidN
Nc
NNc0
0 .,*
),(*
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Realizability area
Thermodynamics (heat engine)Thermodynamics (heat engine)
00
00
00DD
Dpp
Dp
~),()(
~),(
TT
K 1
Tp
Tp
Tp
p KK
2
4
1
2
1)(
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Realizability area
Microeconomics (intermediaryMicroeconomics (intermediary))
iiii pcgpp
10
1
1 pp
221
222
21
21
4
ppPmax