What is thermodynamics and what is it for? I. Equilibrium and non-equilibrium in discrete systems

Peter Ván HAS, RIPNP, Department of Theoretical Physics

– Introduction– Stability and thermodynamics– Gases– Discussion

Centre of Nonlinear Studies, Tallinn, Estonia, 29/5/2006.

Thermodynamics:

Statics Dynamics

Point-mass Continuum

Mechanics:

Equilibrium Non-equilibrium

Phenomenological Statistical

Discrete (homogeneous)

Continuum

Equilibrium - non-equilibrium?– Mechanics – thermodynamics

• Differential equations?– Equilibrium - time independent– Quasistatic processes – time? (Zeno)– Irreversible processes – something more (internal variables)

II. law?– Heat can flow from a hotter body to the colder.

(Clausius)– There is no perpetuum mobile of the second kind.

(Planck)– In a closed system, in case of spontaneous processes the entropy

increases.– dS = drS + diS és diS0– ….

Discrete systems – equilibrium thermodynamics

Equilibrium thermodynamics

– S entropy

– de = q+w = Tds – pdv Gibbs relation

– There is a tendency to equilibrium

– Thermodynamic stability

What is?

Basic ingredients:

L: E is a Ljapunov function of the equilibrium, if:i) L has a strict maximum at ,

ii) , the derivative of L along the differential equation has a strict minimum at .

Theorem: If there is a Ljapunov function of , then is

asymptotically stable (stable and attractive).

EEfxfx :),()((*) t

Equilibrium of (*): 0xf )( 0

0x

)()()( xfxx DLL

0x 0x

))(())(()())(())(( ttDLttDLtL

dt

dxfxxxx

0x

0x

What is?

Instead of proof:

),()(

),()(

2122

2111

xxftx

xxftx

),,( 21 xxx

0)()()()( 0 xxfxx LDLL

x2

x1

0,, 2121

ffx

L

x

L

What is?

Dynamics without differential equation?

pdvqwqde

fv

vpqe

)( 0TTq

?

.,1

T

p

v

s

Te

s

‘dynamic law?’

A) entropy

B)

T, p T0, p0

as a potential.

q

What is?

Stability structure:

.

.,

0

0

constvvv

consteee

T

T

),(),(),( 000 vesvesvesT

,11

),(0TT

vee

sT

.),(0

0

T

p

T

pve

v

sT fv

vpqe

Ljapunov function

i) )0,0(,),( 00

v

s

e

sveDs TT

T

sDsD T22

0)(11

)(11

),(,),(

00

00

0

0

T

fppq

TTf

T

p

T

ppfq

TT

fpfqv

s

e

sfpfqDss TT

TTii)

convex – thermodynamic stability

Direction of heat: 00 qTT

)(

),()(

0

00

ppv

pppTTe

sD2

0)(11

00

0

T

fppq

TT

convex

E.g. ppf

TTq

0

0

v

RTp

cTe

0,0

.0,0 cR

Dynamic Law

Dynamic material functions(heat exchange, …)

Static material functions (ideal gas)

fv

vpqe

Thermodynamic theory in general

)a(fa Dynamic Law:

1 Statics (properties of equilibrium): existence of entropy

2 Dynamics (properties of interactions): increasing entropy

0)()()()( afaDSaaDSaS

Stability structure Dynamic structure

What for?

Quasistatic processes of a Van der Waals gas:

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

2

3

13

8

3

vv

Tp

veT

5.0

9.0

1

0

0

p

T

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

fv

vpqe

Pitchfork bifurcation of a Van der Waals gas (bifurcation diagram)

v0

T0

p0

Second order equation – internal variables

fv

vpqe

ˆ

ˆˆ

.0)0,,(~),,,(~),(),,(

vep

uvepvepuvep

Non-equilibrium state functions:

T, p T0, p0

State space: q

What is?

),,( vuve

uuvep ),,(~e.g.

Stability structure:

Ljapunov function

0~11

0

T

upq

TTsT

Viscous-damping: 0~ up

)ˆ(ˆ

ˆˆ

0ppfv

vpqe

0

2

000 2),(),(),,(

T

uvesvesuvesTNE

What is?

Entropy of the body:

T

up

T

qs

~ dS = drS + diS és diS0

vpTsvpvpqe )~ˆ(

Irreversible processes of a Van der Waals gas:

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

2

3

13

8

3

vv

Tp

veT

0

5.0

9.0

1

0

0

p

T

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1

5.0

9.0

1

0

0

p

T

Movie-like:

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

1 2 3 4 5 6v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

p

Conclusions

– A thermodynamic structure is a stability structure• Time dependent discrete systems!• Equilibrium – quasistatic – irreversible

– Completing the structure: theory construction• Static: consistency, thermic caloric• Dynamics: Onsager reciprocity, constitutive functions, • Constructive!

– Stability of the theory: stability of the calculations.• Robust numerical codes: numerical viscosity

– Discrete – continuum: the same principles.

What is for what?

Thanks:

To the Hungarian thermodynamic tradition:

Julius Farkas, Imre Fényes, István Gyarmati,...Joe Verhás, Tamás MatolcsiThermodynamic Division

of the Hungarian Physical Society,