Thermodynamic Phase Transitions
and Shock Singularities
Giuseppe De Nittis
Joint work with: Antonio Moro
Proc. R. Soc. A (online) arXiv:1107.0394
Outline
1 Introduction to shock waves
2 Thermodynamic phase transitions vs Shocks
3 Equations of State
4 Singular Sector
Nonlinear Shock Waves: the simplest model
Burgers - Hopf equation
ut +uux =0 u=u(x,t)
The general solutionuis given (locally) by the following algebraic equation
x =ut+f(u) (Hodograph Equation)
Nonlinear Shock Waves: the simplest model
Burgers - Hopf equation
ut +uux =0 u=u(x,t)
The general solutionuis given (locally) by the following algebraic equation
x =ut+f(u) (Hodograph Equation)
Shocks
Burgers-Hopf equation+initial datum
ut +uux =0 u(x,0) =F(x)
t=0
-4 -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2
t»tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
Shocks
Burgers-Hopf equation+initial datum
ut +uux =0 u(x,0) =F(x)
t=0
-4 -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2
t»tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
Shocks
Burgers-Hopf equation+initial datum
ut +uux =0 u(x,0) =F(x)
t=0
-4 -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2
t»tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
Shocks
Burgers-Hopf equation+initial datum
ut +uux =0 u(x,0) =F(x)
t=0
-4 -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2
t»tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
Shocks
Burgers-Hopf equation+initial datum
ut +uux =0 u(x,0) =F(x)
t=0
-4 -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 1.2
t»tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0 1.2
t>tc
-2 0 2 4 6
Critical point
Thecritical point(xc,tc,uc), where thegradient catastropheoccurs is
determined by the following system
xc−uctc−f(uc) =0 tc+f0(uc) =0 f00(uc) =0.
wheref000(uc)<0.
Proof (easy!):Let us define
Ω(x,t,u(x,t)) :=x−ut−f(u) =0.
ComputingΩx =0 we obtain
ux =
1 t+f0(u).
At the gradient catastrophe pointux =∞i.e.
t+f0(u) =0
The solution displays the critical behaviour at theminimum time, such that
Critical point
Thecritical point(xc,tc,uc), where thegradient catastropheoccurs is
determined by the following system
xc−uctc−f(uc) =0 tc+f0(uc) =0 f00(uc) =0.
wheref000(uc)<0.
Proof (easy!):
Let us define
Ω(x,t,u(x,t)) :=x−ut−f(u) =0.
ComputingΩx =0 we obtain
ux =
1 t+f0(u).
At the gradient catastrophe pointux =∞i.e.
t+f0(u) =0
The solution displays the critical behaviour at theminimum time, such that
Critical point
Thecritical point(xc,tc,uc), where thegradient catastropheoccurs is
determined by the following system
xc−uctc−f(uc) =0 tc+f0(uc) =0 f00(uc) =0.
wheref000(uc)<0.
Proof (easy!):Let us define
Ω(x,t,u(x,t)) :=x−ut−f(u) =0.
ComputingΩx =0 we obtain
ux =
1 t+f0(u).
At the gradient catastrophe pointux =∞i.e.
t+f0(u) =0
The solution displays the critical behaviour at theminimum time, such that
Weak solutions
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0
A measurable functionu :R×[0,T]→Ris aweak solutionto the
Burgers-Hopf equation ifu defines a continuous map
[0,T]3t 7−→u(·,t)∈L1loc(R)
such that
Z Z
Ω
(uφt +
u2
2 φx)dx dt =0
Weak solutions
t>tc
-2 0 2 4 6
0.0 0.2 0.4 0.6 0.8 1.0
A measurable functionu :R×[0,T]→Ris aweak solutionto the
Burgers-Hopf equation ifu defines a continuous map
[0,T]3t 7−→u(·,t)∈L1loc(R)
such that
Z Z
Ω
(uφt +
u2
2 φx)dx dt =0
Shock Fitting
In most of the cases a physical solution exists in a weak sense and one should allow discontinuous solutions.
Shock position
Equal Area Principle
Shock speed
Ushock =
u2
r
2 −
u2
l
2
ur −ul
= ul+ur
Shock Fitting
In most of the cases a physical solution exists in a weak sense and one should allow discontinuous solutions.
Shock position
Equal Area Principle
Shock speed
Ushock =
u2
r
2 −
u2
l
2
ur −ul
= ul+ur
Shock Fitting
In most of the cases a physical solution exists in a weak sense and one should allow discontinuous solutions.
Shock position
Equal Area Principle
Shock speed
Ushock =
u2
r
2 −
u2
l
2
ur −ul
= ul+ur
Isothermal Curves vs Nonlinear Waves
A B
T<Tc
V P
T1>T2 T2>T3 T3
V P
Van der Waals gas Ideal Gas
T>Tc T=Tc
T<Tc
P V
Isothermal Curves vs Nonlinear Waves
A B
T<Tc
V P
T1>T2 T2>T3 T3
V P
Van der Waals gas Ideal Gas
T>Tc T=Tc
T<Tc
P V
Isothermal Curves vs Nonlinear Waves
A B
T<Tc
V P
T1>T2 T2>T3 T3
V P
Van der Waals gas Ideal Gas
T>Tc T=Tc
T<Tc
P V
Energy Balance Equation
dE =TdS−PdV +
m
X
j=1 Λjdτj
Introducing the Gibbs free energyΦ =E−TS+VP
dΦ =−SdT +VdP+
m
X
i=1 Λidτi
that is
S =−∂Φ
∂T, V =
∂Φ
∂P, Λi = ∂Φ
∂τi
i=1, . . . ,m
This system of equations is compatible if
∂V ∂T =−
∂S ∂P,
∂V ∂τi
= ∂Λi
Energy Balance Equation
dE =TdS−PdV +
m
X
j=1 Λjdτj
Introducing the Gibbs free energyΦ =E−TS+VP
dΦ =−SdT +VdP+
m
X
i=1 Λidτi
that is
S =−∂Φ
∂T, V =
∂Φ
∂P, Λi = ∂Φ
∂τi
i=1, . . . ,m
This system of equations is compatible if
∂V ∂T =−
∂S ∂P,
∂V ∂τi
= ∂Λi
Let us assume that the entropy functionSis of the form
S(T,P, τ1, . . . , τm) = ˜S V(T,P, τ1, . . . , τm),T, τ1, . . . , τm
Λi(T,P, τ1, . . . , τm) = ˜Λi V(T,P, τ1, . . . , τm),T, τ1, . . . , τm
Then, the system above can be written as follows
∂V
∂P = φ0(V,T, τ1, . . . , τm) ∂V ∂T ∂V
∂τi
= φi(V,T, τ1, . . . , τm)
∂V
∂T i =1, . . . ,m.
where
φ0(V,T, τ1, . . . , τm) :=−
∂S˜ ∂V
!−1
φi(V,T, τ1, . . . , τm) :=φ0(V,T, τ1, . . . , τm)
∂Λ˜i
Let us assume that the entropy functionSis of the form
S(T,P, τ1, . . . , τm) = ˜S V(T,P, τ1, . . . , τm),T, τ1, . . . , τm
Λi(T,P, τ1, . . . , τm) = ˜Λi V(T,P, τ1, . . . , τm),T, τ1, . . . , τm
Then, the system above can be written as follows
∂V
∂P = φ0(V,T, τ1, . . . , τm) ∂V ∂T ∂V
∂τi
= φi(V,T, τ1, . . . , τm)
∂V
∂T i =1, . . . ,m.
where
φ0(V,T, τ1, . . . , τm) :=−
∂S˜ ∂V
!−1
φi(V,T, τ1, . . . , τm) :=φ0(V,T, τ1, . . . , τm)
∂Λ˜i
Under the extra assumptions
˜
S(V,T, τ1, . . . , τm) =σ0(V) +ψ0(T, τ1, . . . , τm),
Λi(V,T, τ1, . . . , τm) =σi(V) +ψi(T, τ1, . . . , τm)
the system takes the so-called hydrodynamic form
∂V
∂P = φ0(V) ∂V ∂T ∂V
∂τi
= φi(V)
∂V
∂T i=1, . . . ,m
The characteristic speedsφ0, . . . , φmdepend on the variableV only!
Remark :
Under the extra assumptions
˜
S(V,T, τ1, . . . , τm) =σ0(V) +ψ0(T, τ1, . . . , τm),
Λi(V,T, τ1, . . . , τm) =σi(V) +ψi(T, τ1, . . . , τm)
the system takes the so-called hydrodynamic form
∂V
∂P = φ0(V) ∂V ∂T ∂V
∂τi
= φi(V)
∂V
∂T i=1, . . . ,m
The characteristic speedsφ0, . . . , φmdepend on the variableV only!
Remark :
Equations of State
The general solution is locally given by the formula
T+φ0(V)P+
m
X
i=1
φi(V)τi =f(V) (Hodograph Equation)
Example 1 : Cauchy problem
Givenφ0js and the “initial datum”V0(P) :=V(T0,P, τ0,1, . . . , τ0,m)
f(V) =T0+φ0(V)P0(V) +
m
X
i=1
φi(V)τ0,i,
Equations of State
The general solution is locally given by the formula
T+φ0(V)P+
m
X
i=1
φi(V)τi =f(V) (Hodograph Equation)
Example 1 : Cauchy problem
Givenφ0js and the “initial datum”V0(P) :=V(T0,P, τ0,1, . . . , τ0,m)
f(V) =T0+φ0(V)P0(V) +
m
X
i=1
φi(V)τ0,i,
Example 2 : allφj(V)’s and f(V)unknown
Givenm+2 distinct isotherms
Vi(P) :=V(Ti,P, τi1, . . . , τim)
compute the set of inverse functionsP0=V0−1, . . . ,Pm=V
−1
m . Then
T0+φ0(V)P0(V) +
m
X
i=1
φi(V)τ0i =f(V)
T1+φ0(V)P1(V) +
m
X
i=1
φi(V)τ1i =f(V)
. . .
Tm+1+φ0(V)Pm(V) + m
X
i=1
φi(V)τmi =f(V)
Ideal Gas
Let us consider a monoatomic gas of constant number of particlesN and equation of state
T +φ0(V)P =f(V).
Assume we are given two particular isobaric curvesV1(T) =V(T,P1)
andV2(T) =V(T,P2)withP16=P2of the form
V1(T) =
T c1
, V2(T) =
T c2
, c1, c2 are consts
We get
φ0(V) =−
V
α f(V) =0, α=const
Then, we obtain thestate equation for the ideal gas
PV =αT (α=nR).
Ideal Gas
Let us consider a monoatomic gas of constant number of particlesN and equation of state
T +φ0(V)P =f(V).
Assume we are given two particular isobaric curvesV1(T) =V(T,P1)
andV2(T) =V(T,P2)withP16=P2of the form
V1(T) =
T c1
, V2(T) =
T c2
, c1, c2 are consts
We get
φ0(V) =−
V
α f(V) =0, α=const Then, we obtain thestate equation for the ideal gas
PV =αT (α=nR).
Real Gas (Virial Expansion)
Assumeτ’s=0 and consider two isotherms (j=1,2) atT16=T2
Pj(V) =
1 V −ν
k
X
n=0
αn(Tj)
Vn , ⇒ αn(Tj) =γnTj+cn
Then thesate equationisT +φ0(V)P =f(V)with
φ0(V) = (ν−V)
k
X
n=0
γn
Vn
!−1
, f(V) =−
Pk
n=0cnVk−n Pk
n=0γnVk−n
.
Particular case: (k=2)
P+n 2a
V2
(V −nb) =nRT (Van der Waals Equation).
This is a particular solution to the equation
∂V ∂P +
V −nb nR
∂V
Real Gas (Virial Expansion)
Assumeτ’s=0 and consider two isotherms (j=1,2) atT16=T2
Pj(V) =
1 V −ν
k
X
n=0
αn(Tj)
Vn , ⇒ αn(Tj) =γnTj+cn
Then thesate equationisT +φ0(V)P=f(V)with
φ0(V) = (ν−V)
k
X
n=0
γn
Vn
!−1
, f(V) =−
Pk
n=0cnVk−n Pk
n=0γnVk−n
.
Particular case: (k=2)
P+n 2a
V2
(V −nb) =nRT (Van der Waals Equation).
This is a particular solution to the equation
∂V ∂P +
V −nb nR
∂V
Real Gas (Virial Expansion)
Assumeτ’s=0 and consider two isotherms (j=1,2) atT16=T2
Pj(V) =
1 V −ν
k
X
n=0
αn(Tj)
Vn , ⇒ αn(Tj) =γnTj+cn
Then thesate equationisT +φ0(V)P=f(V)with
φ0(V) = (ν−V)
k
X
n=0
γn
Vn
!−1
, f(V) =−
Pk
n=0cnVk−n Pk
n=0γnVk−n
.
Particular case: (k=2)
P+n 2a
V2
(V−nb) =nRT (Van der Waals Equation).
This is a particular solution to the equation
∂V ∂P +
V −nb nR
∂V
Singular Sector and Phase Transitions
Let us introduce the function
Ω(V,T,P, τ1, . . . , τm) :=T +φ0(V)P+
m
X
i=1
φi(V)τi−f(V)
and denoteΩ(l)=∂lΩ/∂Vl.
Following Kodama-Konopelchenko (2002), let us introduce thesingular
sectorof codimensions≤m+2
Zs =
(T,P, τ1, . . . , τm)∈Rm+2 | V ∈ Uj such that s = m+2 X
j=1
j |Uj|
Uj :=nV >0|Ω(l)=0 ∀0≤l≤j and Ω(j+1)6=0o.
Singular Sector and Phase Transitions
Let us introduce the function
Ω(V,T,P, τ1, . . . , τm) :=T +φ0(V)P+
m
X
i=1
φi(V)τi−f(V)
and denoteΩ(l)=∂lΩ/∂Vl.
Following Kodama-Konopelchenko (2002), let us introduce thesingular
sectorof codimensions≤m+2
Zs =
(T,P, τ1, . . . , τm)∈Rm+2 | V ∈ Uj such that s = m+2 X
j=1
j |Uj|
Uj :=nV >0|Ω(l)=0 ∀0≤l≤j and Ω(j+1)6=0o.
Singular Sector and Phase Transitions
Let us introduce the function
Ω(V,T,P, τ1, . . . , τm) :=T +φ0(V)P+
m
X
i=1
φi(V)τi−f(V)
and denoteΩ(l)=∂lΩ/∂Vl.
Following Kodama-Konopelchenko (2002), let us introduce thesingular
sectorof codimensions≤m+2
Zs =
(T,P, τ1, . . . , τm)∈Rm+2 | V ∈ Uj such that s = m+2 X
j=1
j |Uj|
Uj :=nV >0|Ω(l)=0 ∀0≤l≤j and Ω(j+1)6=0o.
Singular Sector and Phase Transitions
Let us introduce the function
Ω(V,T,P, τ1, . . . , τm) :=T +φ0(V)P+
m
X
i=1
φi(V)τi−f(V)
and denoteΩ(l)=∂lΩ/∂Vl.
Following Kodama-Konopelchenko (2002), let us introduce thesingular
sectorof codimensions≤m+2
Zs =
(T,P, τ1, . . . , τm)∈Rm+2 | V ∈ Uj such that s = m+2 X
j=1
j |Uj|
Uj :=nV >0|Ω(l)=0 ∀0≤l≤j and Ω(j+1)6=0o.
The critical sector identifies the occurrence of thephase transitionas well as theshock formationof the solution to the hydrodynamic type system.
0 10 20 30 40
0 500 000 1.0´106
1.5´106
T P
0.05 0.10 0.15 0.20 0.25
0 1´106
2´106
3´106
4´106
5´106
V P
Left:Singular sectorZ1for the hydrogen using the Van der Waals model. The
intersection of the two lines is the critical sectorZ2={Tc =33.159K,
Pc =1.29508×106Pasc}.
Right:The bold line shows the critical isotherm. Below the critical isotherm there are two stationary points where the isotherm has a local maximum and minimum. As
The critical sector identifies the occurrence of thephase transitionas well as theshock formationof the solution to the hydrodynamic type system.
0 10 20 30 40
0 500 000 1.0´106
1.5´106
T P
0.05 0.10 0.15 0.20 0.25
0 1´106
2´106
3´106
4´106
5´106
V P
Left:Singular sectorZ1for the hydrogen using the Van der Waals model. The
intersection of the two lines is the critical sectorZ2={Tc =33.159K,
Pc =1.29508×106Pasc}.
Right:The bold line shows the critical isotherm. Below the critical isotherm there are two stationary points where the isotherm has a local maximum and minimum. As
The rôle of the unfolding
Locally, near the critical point, the Van der Waals equation is associated with theuniversal unfoldingof codimension 2 of a Riemann-Hugoniot catastrophe (power 4). More precisely, the universal unfolding gives the Gibbs free energy
Φ =F +PV
whereF =E−TS.
Virial expansions (for "τ-deformed" systems) can be considered
ashigher codimension unfoldingsof the Van der Waals model.
The rôle of the unfolding
Locally, near the critical point, the Van der Waals equation is associated with theuniversal unfoldingof codimension 2 of a Riemann-Hugoniot catastrophe (power 4). More precisely, the universal unfolding gives the Gibbs free energy
Φ =F +PV
whereF =E−TS.
Virial expansions (for "τ-deformed" systems) can be considered
ashigher codimension unfoldingsof the Van der Waals model.
The rôle of the unfolding
Locally, near the critical point, the Van der Waals equation is associated with theuniversal unfoldingof codimension 2 of a Riemann-Hugoniot catastrophe (power 4). More precisely, the universal unfolding gives the Gibbs free energy
Φ =F +PV
whereF =E−TS.
Virial expansions (for "τ-deformed" systems) can be considered
ashigher codimension unfoldingsof the Van der Waals model.
Separable entropy functions
Consider a system the state of which is completely specified by the set of thermodynamic variables(T,P,V).
S(T,P) = ˜S V(T,P),T= ∞
X
j=0
σj V(T,P)
∂ρj
∂T(T)
whereσj(V)andρj(T)are arbitrary functions of their argument.
Examples:
P= nRT
V −
c
V3/2√T (Classical Plasma);
P= nRT
V
Liα+1(z)
ζ(α)
T Tc
α
Separable entropy functions
Consider a system the state of which is completely specified by the set of thermodynamic variables(T,P,V).
S(T,P) = ˜S V(T,P),T= ∞
X
j=0
σj V(T,P)
∂ρj
∂T(T)
whereσj(V)andρj(T)are arbitrary functions of their argument.
Examples:
P = nRT
V −
c
V3/2√T (Classical Plasma);
P= nRT
V
Liα+1(z)
ζ(α)
T Tc
α
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
Summarizing
Equations of state→solutions to nonlinear hyperbolic PDEs;
Isothermal (isobaric) curves⇐⇒nonlinear waves;
Critical point of the phase transition⇐⇒gradient catastrophe.
Further Study
Stability of higher codimensional unfoldings;
Higher component hydrodynamic systems↔Composite thermodynamic systems↔Non-additive entropy;
Higher order effects:
→Phase transition as weak diffusive limit of the Burgers type eq?
! Thank you for your attention !
Ackowledgements:
Grant no. ANR-08-BLAN-0261-01;
Grant of Istituto Nazionale di Alta Matematica - GNFM, Progetti Giovani 2011;
