optimal processes in macro systems (thermodynamics and economics) a.m. tsirlin and v. kazakov
TRANSCRIPT
Optimal processesin macro systems
(thermodynamics and economics)
A.M. Tsirlin and V. Kazakov
- 2 -
Macro Systems: thermodynamics, economics, segregated systems
Extensive variables
V, U, …, N0, N
Intensive variables
T, , P, …, p, c
Equation of state
- 3 -
«Natural processes»
Irreversibility measure,
dissipation
S,
Irreversibility and kinetics
),(//
0)()(lim
2121
21
ppgdtdNdtdN
tptpt
- 4 -
Structure of MM of the macrosystem
.....,1,0,),,( 21 muuxyfx
- 5 -
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
- 8 -
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.
- 9 -
Capital dissipation
00 0NdStptc ),()(
0 00
00 0 .))(,())(,(
pS
dtpcpcgNdtpcpcgpS
– fixed
= g(c,p)(c–p) capital dissipation (trading costs)
- 10 -
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
- 12 -
Minimal dissipation processes in thermodynamics
0
0
000
0
0
1
0
0
.),(
,)(),,(
,)(),,(
min)()(
gdtpcg
NNpcgdtdN
NNpccgdtdN
Ntc
2
02 g
Nppg
g
cgdNd
- 13 -
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
- 20 -
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
