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

- 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

- 6 -

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.

- 7 -

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

- 11 -

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)(),()( *

- 14 -

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),(),(

- 15 -

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

- 16 -

Stationary states of open macro systems

Microeconomics ui – prices,

p – estimates

ii

jijiij

i puiiii

ggppg

uupgn

.,),(

max),(,

0

- 17 -

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

- 18 -

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

- 19 -

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 ,

- 21 -

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 .,*

),(*

- 22 -

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)(

- 23 -

Realizability area

Microeconomics (intermediaryMicroeconomics (intermediary))

iiii pcgpp

10

1

1 pp

221

222

21

21

4

ppPmax

Optimal processes in macro systems

(thermodynamics and economics)

e-mail: [email protected]

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