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Optimal processes in 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, …, N 0, N Intensive variables T, , P, …, p, c Equation of state

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«Natural processes» Irreversibility measure, dissipation S, Irreversibility and kinetics

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Structure of MM of the macrosystem

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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 Economic agent N R n+1 Resources’ and capital ( N 0 ) endowments pi(N)pi(N) Estimate of i -th resource (equilibrium price)

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For capital extraction voluntariliness principle 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.

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Capital dissipation – fixed = g (c,p)(c–p) capital dissipation (trading costs)

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

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

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Minimal dissipation processes in thermodynamics

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Minimal dissipation processes in thermodynamics If

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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.,,,,,),(,),(

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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

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

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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

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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

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If – exergy

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Separation systems

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

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Realizability area Thermodynamics (heat engine)

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Realizability area Microeconomics (intermediary)

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Optimal processes in macro systems (thermodynamics and economics)

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