The meaning of the Second Law:Nonlocal effects in continuum physics

Peter VánBUTE, Department of Chemical Physics

– Irreversibility concepts in thermodynamics

– Stability and the Second Law

– How to construct constitutive functions?

– Example:• Ginzburg-Landau equation

– ConceptsConcepts and theories– What kind of thermodynamics?

Discrete(homogeneous)

T(t)

ContinuumT(t,x)

Phenomenological(T is primitive)

Statistical(T is explained)

Equilibrium Irreversible

Second Law

– time dependence – time reversal – variational principles – S exists– S is increasing – local equilibrium

Can be interpreted in a thermodynamic theory

Irreversibility

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Thermodynamic theory

)a(fa Dynamic law:

,...),c,v(a

1 Statics (equilibrium properties)

S

aa

S,,

T

1

e

S

2 Dynamics

0)a(f)a(DSa)a(DS)a(S

1 + 2 + closed system

S is a Ljapunov function of the equilibrium of the dynamic law

Irreversibility trend to equilibrium

Classical Irreversible

Thermodynamics

Local equilibrium (= there is no microstructure)

Violations (nonlocality):

•in time (memory effects)•in space (structure effects)

dynamic variables?

Stability structure Dynamic structure

Example 1

)a(fa 0)( aS for all solutions of the dynamic law

)a(f)a(DS)a(S0 )a(DS)a(l)a(f

?)a(f

relaxational dynamics

Example 2 (Ginzburg-Landau)

gt

0js st

),,( 2

)j,s,g( s

Liu procedure (Farkas’s lemma)

),(s (.)gs),(j(.)j 0s

0g)ss(s

)ss(.)(lgt

state space

constitutive functions

Conclusions

– Second Law is a stability criteria (Irreversibility is a kind of stability)

– Second Law is a tool to construct dynamic laws– Predictive

• Economics (?)