# Optimal processes in macro systems (thermodynamics and economics) A.M. Tsirlin and V. Kazakov.

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Optimal processes in macro systems (thermodynamics and economics) A.M. Tsirlin and V. Kazakov

- 2 -- 2 - Macro Systems: thermodynamics, economics, segregated systems Extensive variables V, U, …, N 0, N Intensive variables T, , P, …, p, c Equation of state

- 3 -- 3 - «Natural processes» Irreversibility measure, dissipation S,  Irreversibility and kinetics

- 4 -- 4 - Structure of MM of the macrosystem

- 5 -- 5 - Workout example thermodynamics microeconomics Irreversible:  S > 0, A = 0 Reversible:  S = 0, A > 0 Irreversible  > 0, E = 0 Reversible  = 0, E > 0

- 6 -- 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 -- 7 - Irreversibility measure in microeconomic systems Wealth function S(N) exists such that Economic agent N  R n+1 Resources’ and capital ( N 0 ) endowments pi(N)pi(N) Estimate of i -th resource (equilibrium price)

- 8 -- 8 - For capital extraction voluntariliness principle              000 000 2121 0 NNpS ppNpNpS ppgpp а i i iiрез )(,const,,,),( dS i  0, i=1,2 If p 1i and p 2i have different signs that it is not less than 2 flows.

- 9 -- 9 - Capital dissipation    – fixed  = g (c,p)(c–p) capital dissipation (trading costs)

- 10 - Minimal dissipation processes in thermodynamics For  =  ( p ) g ( p, u ) We get:

- 11 - Minimal dissipation processes in thermodynamics Heat transfer: p ~ T 1, u ~ T 2

- 12 - Minimal dissipation processes in thermodynamics

- 13 - Minimal dissipation processes in thermodynamics If

- 14 - Stationary state of open macro system Thermodynamics n – power, p 1i ~ T i q – heat, g – mass, p – intensive variables for                         i i i ii j j ij i i j iji j iji u q sg mi p q sg g gppgqppq.,,,,,),(,),( 0 10 0 1 1 

- 15 - If g = 0, q ij =  ij (T i – T j ), then If m = 2, T 1 = T +, T 2 = T –, then For g = AX Prigogine’s extremal principle holds for any u ( A – Onsager matrix). – limiting power

- 16 - Stationary states of open macro systems Microeconomics u i – prices, p – estimates

- 17 - Analogy of Prigogine extremal principle for g = A  (  ij =p i – p j ): A – symmetric. If g ij =  ij (p j – p j ), g i =  i (u i – p j ), then If m = 2, p 1 = p +, p 2 = p –, then

- 18 - Optimal processes Availability A max (  )=? Control u(t) = (u 1, …, u m ), h(t) = (h 1,…,h m ), h i = {0, 1} k – number of conditions on final state. Statements: 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

- 20 - Separation systems

- 21 - E  – analogous of exergy.  – given: c*(t) obeys conditions of minimal dissipation during all contacts  obeys the conditions Microeconomics. Profitability =?

- 22 - Realizability area Thermodynamics (heat engine)

- 23 - Realizability area Microeconomics (intermediary)

Optimal processes in macro systems (thermodynamics and economics) e-mail: tsirlin@sarc.botik.ru vladimir.kazakov@uts.edu.au

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