1. Weakly nonlocal non-equilibrium thermodynamics Peter Ván Budapest University of Technology and Economics, Department of Chemical Physics
Non-equilibrium thermodynamics
Beyond local equilibrium and local state
Exploitations of the Second Law - Liu procedure
Examples
Guyer-Krumhansl equation
Ginzburg-Landau equation
One component fluid mechanics - quantum (?) fluids
Conclusions

2. Local equilibrium (~ there is no microstructure)
Classical
Irreversible
Thermodynamics
Local equilibrium (~ there is no microstructure)
Beyond local equilibrium (nonlocality):
in time (memory effects)
in space (structure effects)
dynamic variables ?
Method: local state and beyond …

3. Nonlocalities:
Restrictions from the Second Law.

4. Non-equilibrium thermodynamics
basic balances
(and more)
basic state:
constitutive state:
constitutive functions:
weakly nonlocal
Second law:
(universality)
Constitutive theory
Method: Liu procedure
Speciality: constructive

5. Weakly nonlocal extended thermodynamics
state space
constitutive functions
local state:
Liu procedure (Farkas’s lemma):
solution?

6. extended (Gyarmati) entropy
entropy current (Nyíri)
(B – current multiplier)
balance?
local?

7. Weakly nonlocal internal variables
Ginzburg-Landau (variational):
Second Law
Variational (!)

8. ? Ginzburg-Landau (thermodynamic, relocalized) local state
state space
constitutive functions
local state
Liu procedure (Farkas’s lemma) ?

9. current multiplier
isotropy

10. Ginzburg-Landau (thermodynamic, non relocalizable)
state space
constitutive functions
Liu procedure (Farkas’s lemma)

11. One component weakly nonlocal fluids
and quantum mechanics

12. Schrödinger equation:
Madelung transformation:
de Broglie-Bohm form:
Hydrodynamic form:

13. One component weakly nonlocal fluid
basic state
constitutive state
constitutive functions
Liu procedure (Farkas’s lemma):

14. Schrödinger-Madelung fluid
reversible pressure
(Fisher entropy)
Schrödinger-Madelung fluid

15. Schrödinger equation:
Madelung transformation:
de Broglie-Bohm form:
Hydrodynamic form:

16. One component weakly nonlocal fluid
basic state
constitutive state
constitutive functions
Liu procedure (Farkas’s lemma):

17. Schrödinger-Madelung fluid
reversible pressure
(Fisher entropy)
Schrödinger-Madelung fluid

18. Potential form:
Bernoulli equation
Schrödinger equation
Euler-Lagrange form
Variational origin

19. Why nonequilibrium thermodynamics?
science of temperature
Thermodynamics science of macroscopic
energy changes
general framework of any
Thermodynamics (?) macroscopic (?)
continuum (?)
theories
General framework:
fundamental balances
objectivity - frame indifference
Second Law
reversibility – special limit

20. Conclusions Not everything is a balance
Second order nonlocality is interesting
Constitutive entropy current
Force-current systems: constructivity
Variational principles
Second Law