Post on 29-Jul-2020
Variational Formulation for the KPZ Equation:
Consistency, Galilean-invariance violation, and
fluctuation-dissipation issues in real-space
discretization
Horacio S. Wio
Instituto de Fisica de Cantabria, UC-CSIC, Santander, SPAIN
Electronic address: wio@ifca.unican.es
Workshop on Instabilities and Nonequilibrium Structures,
Viña del Mar, Chile, December 14 to 18, 2009.
COLLABORATORS
• J.A. Revelli (IFCA-Santander)
• R.R. Deza (UNMdP-Mar del Plata)
• C. Escudero (CSIC-Madrid)• C. Escudero (CSIC-Madrid)
• M.S. de la Lama (MPI-Göttingen)
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
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Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:**** * **
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:* Nonequilibrium Potential: Brief Review,* * * * * *
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* * * * *
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,*** *
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,* Discretization & Consistency, * Discretization & Consistency, * *
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,* Discretization & Consistency, * Discretization & Consistency, * Galilean Invariance & Fluctuation-Dissipation, *
Organization of the talk:Organization of the talk:Organization of the talk:Organization of the talk:
* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,* Discretization & Consistency, * Discretization & Consistency, * Galilean Invariance & Fluctuation-Dissipation, * Final Remarks
INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION
R.Graham, in Instabilities and Nonequilibrium Structures, Eds. E. Tirapegui
and D.Villaroel (D.Reidel, Dordrecht,1987)
The main idea behind the notion of nonequilibrium
potential was to extent the usefulness of variational
principles from other areas of physics, particularly principles from other areas of physics, particularly
equilibrium thermodynamics, to nonequilibrium
situations.
It has proved to be very useful, and in particular it has
been extensively exploited to analyze stochastic
resonance in extended or coupled systems, and other
related problems.
INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION
R.Graham, in Instabilities and Nonequilibrium Structures, Eds. E. Tirapegui
and D.Villaroel (D.Reidel, Dordrecht,1987)
Opposing to a claim in a recent paper
“ .. The KPZ equation is in fact a genuine kinetic equation
describing a nonequilibrium process in the sense that the driftdescribing a nonequilibrium process in the sense that the drift
cannot be derived from an
effective free energy...”;
INTRODUCTIONINTRODUCTIONINTRODUCTIONINTRODUCTION
R.Graham, in Instabilities and Nonequilibrium Structures, Eds. E. Tirapegui
and D.Villaroel (D.Reidel, Dordrecht,1987)
Opposing to a claim in a recent paper
“ .. The KPZ equation is in fact a genuine kinetic equation
describing a nonequilibrium process in the sense that the driftdescribing a nonequilibrium process in the sense that the drift
cannot be derived from an
effective free energy...”;
I show here that such a nonequilibrium thermodynamic-like
functional (NEP) for the KPZ equation exists, and also show
its form.
NON-EQUILIBRIUM POTENTIAL:H.S. Wio, in 4th.Granada Lectures in Computational Physics, Eds. J.Marro y P.Garrido
(Springer-Verlag, 1997), pg. 135
Dynamical Systems:
Gradient (Variational):
Relaxational - Non- Gradient:
NON-EQUILIBRIUM POTENTIAL:
Non - Relaxational - Non-Gradient:
NON-EQUILIBRIUM POTENTIAL:
• “No-relaxational”:
NON-EQUILIBRIUM POTENTIAL:
• “No-relaxational”:
The 2nd term will be zero if the orthogonality relation is
fulfilled
NON-EQUILIBRIUM POTENTIAL:
• “No-relaxational”:
The 2nd term will be zero if the orthogonality relation is
fulfilled
NON-EQUILIBRIUM POTENTIAL:
Dynamical System + Stochastic Terms:
NON-EQUILIBRIUM POTENTIAL:
Dynamical System + Stochastic Terms:
and the associated Fokker-Planck Equation:
NON-EQUILIBRIUM POTENTIAL:
Dynamical System + Stochastic Terms:
and the associated Fokker-Planck Equation:
If: and
NON-EQUILIBRIUM POTENTIAL:
• Definition of the non-equilibrium potential:
NON-EQUILIBRIUM POTENTIAL:
• Definition of the non-equilibrium potential:
• General form: • General form:
NON-EQUILIBRIUM POTENTIAL:
• Definition of the non-equilibrium potential:
• General form: • General form:
• solution of a Hamilton-Jacobi (like) Eq.
Independent of (which is a solution of a 1st
order pdf)
NEP FOR THE KPZ EQUATIONVariational formulation for the KPZ and related kinetic equations, H. S. Wio,
Int. J. Bif. Chaos 19 (8), 2813 (2009)
KPZ (Kardar-Parisi-Zhang) equation: a description of the
surface roughness during the evolution and grow of surfaces
and intefaces
NEP FOR THE KPZ EQUATIONVariational formulation for the KPZ and related kinetic equations, H. S. Wio,
Int. J. Bif. Chaos 19 (8), 2813 (2009)
Edward-Wilkinson
NEP FOR THE KPZ EQUATIONVariational formulation for the KPZ and related kinetic equations, H. S. Wio,
Int. J. Bif. Chaos 19 (8), 2813 (2009)
Edward-Wilkinson + curvature KPZ
NEP FOR THE KPZ EQUATION
NEP FOR THE KPZ EQUATION
NEP FOR THE KPZ EQUATION
NEP FOR THE KPZ EQUATION
NEP FOR THE KPZ EQUATION
How could we derive the previous form of NEP?
We start with the scalar reaction-diffusion equation
where is a general nonlinear function where is a general nonlinear function
NEP FOR THE KPZ EQUATION
How could we derive the previous form of NEP?
We start with the scalar reaction-diffusion equation
where is a general nonlinear function where is a general nonlinear function
NEP FOR THE KPZ EQUATION
How could we derive the previous form of NEP?
We start with the scalar reaction-diffusion equation
where is a general nonlinear function where is a general nonlinear function
NEP FOR THE KPZ EQUATION
The Hopf-Cole transformation
with the inverse yields
NEP FOR THE KPZ EQUATION
The Hopf-Cole transformation
with the inverse yields
In order to obtain the “usual” KPZ equation we reduce the
nonlinear function to
with and
NEP FOR THE KPZ EQUATION
Some properties (shift invariance, etc):
NEP FOR THE KPZ EQUATION
Definition
NEP FOR THE KPZ EQUATION
Definition
NEP FOR THE KPZ EQUATION
Definition
NEP FOR THE KPZ EQUATION
Fokker-Planck Eq.
NEP FOR THE KPZ EQUATION
Fokker-Planck Eq.
with the SPD
NEP FOR THE KPZ EQUATION
Fokker-Planck Eq.
with the SPD
NEP FOR THE KPZ EQUATION
Fokker-Planck Eq.
with the SPD
NEP FOR THE KPZ EQUATION
Fokker-Planck Eq.
with the SPD
NEP FOR THE KPZ EQUATION
Fokker-Planck Eq.
with the SPD
NEP FOR THE KPZ EQUATION
Other kinetic equations related to KPZ and Non-Locality
Consider the following, nonlocal, reaction-diffusion
equation
NEP FOR THE KPZ EQUATION
Other kinetic equations related to KPZ and Non-Locality
Consider the following, nonlocal, reaction-diffusion
equation
NEP FOR THE KPZ EQUATION
Applying the Hopf-Cole transformation we obtain the
KPZ-like equation
NEP FOR THE KPZ EQUATION
Applying the Hopf-Cole transformation we obtain the
KPZ-like equation
and the following form of NEP
NEP FOR THE KPZ EQUATION
If we assume that the kernel is symmetric and of “short
range”, we can expand it as
with with
NEP FOR THE KPZ EQUATION
If we assume that the kernel is symmetric and of “short
range”, we can expand it as
with with
Reeplacing into the kinetic equation we obtain several
contributions leading to several kown (obtained in a
“phenomenological” or “geometrical” form) contributions
(Kuramoto-Shivashinski, Sun-Guo-Grant, etc).
NEP FOR THE KPZ EQUATION
• Even the case of density dependent diffusion could be also
treated
with
NEP FOR THE KPZ EQUATION
• Even the case of density dependent diffusion could be also
treated
with
via Hopf-Cole transformation leads to
NEP FOR THE KPZ EQUATION
• Even the case of density dependent diffusion could be also
treated
with
via Hopf-Cole transformation leads to
with a NEP
Discretization & Consistency KPZ equation: Galilean-invariance violation, Consistency, and fluctuation--
dissipation issues in real-space discretization, H.S.Wio, J.A.Revelli, R.R.Deza,
C.Escudero & M.S. de La Lama, submitted to Europhys.Lett.
Let us return to the RD equation with multiplicative noise
Discretization & Consistency KPZ equation: Galilean-invariance violation, Consistency, and fluctuation--
dissipation issues in real-space discretization, H.S.Wio, J.A.Revelli, R.R.Deza,
C.Escudero & M.S. de La Lama, submitted to Europhys.Lett.
Let us return to the RD equation with multiplicative noise
Discretizing it and applying the
“discrete” Hopf-Cole transformation we find the
following form of the KPZ eq.
Discretization & Consistency
Expanding the previous equation up to 2nd order in
we obtain
Discretization & Consistency
Starting from the LF for deterministic part of the RD eq.
with the form
Discretization & Consistency
Starting from the LF for deterministic part of the RD eq.
with the form
and assuming the discrete form
where F=0, we obtain for the functional
Discretization & Consistency
Starting from the LF for deterministic part of the RD eq.
with the form
and assuming the discrete form
where F=0, we obtain for the functional
with
Discretization & Consistency
Starting from the LF for deterministic part of the RD eq.
with the form
and assuming the discrete form
where F=0, we obtain for the functional
with
Discretization & Consistency
Another possible form to discretize the Laplacian is
that corresponds to a more accurate approximation for
the Laplacian.
Discretization & Consistency
Another possible form to discretize the Laplacian is
that corresponds to a more accurate approximation for
the Laplacian. The associated KPZ term results
Discretization & Consistency
Similar results are obtained if we start with the discrete
form of the NEP for KPZ
Discretization & Consistency
Similar results are obtained if we start with the discrete
form of the NEP for KPZ
resulting again in
Galilean Invariance & Fluctuation-Dissipation
The Fokker-Planck equation associated to the KPZ is
and has the following form of SPD (in 1d)
The fluctuation-dissipation relation for KPZ in 1d reflects the fact
that the SPD shows that the nonlinear term is not operative at long
times.
Galilean Invariance & Fluctuation-Dissipation
The SPD is
that in a discrete representation reads
Galilean Invariance & Fluctuation-Dissipation
The SPD is
that in a discrete representation reads
Inserting this expression into the FPE we found that the only surviving
contribution is
Galilean Invariance & Fluctuation-Dissipation
The SPD is
that in a discrete representation reads
Inserting this expression into the FPE we found that the only surviving
contribution is
with a continuous limit
Galilean Invariance & Fluctuation-Dissipation
KPZ is Galilean invariant: it fullils the following transformations
Galilean Invariance & Fluctuation-Dissipation
KPZ is Galilean invariant: it fullils the following transformations
If we consider a gradient discretization
it fulfills the discrete (Galilean) transformations
Galilean Invariance & Fluctuation-Dissipation
KPZ is Galilean invariant: it fullils the following transformations
If we consider a gradient discretization
it fulfills the discrete (Galilean) transformations
while other discretizations do not.
Galilean Invariance & Fluctuation-Dissipation
Galilean invariance has been always associated with the
exactness of the one-dimensional KPZ exponents, and with
a relation that connects the critical exponents in higher
dimensions.
If the numerical solution obtained from a finite-differenceIf the numerical solution obtained from a finite-difference
scheme as the ones shown before, some which are not Galilean
invariant, yields the well known critical exponents, that would
strongly suggest that Galilean invariance is not a fundamental
symmetry as usually considered.
Galilean Invariance & Fluctuation-Dissipation
Galilean invariance => exponent relation:
Where roughness exponent , dynamic exponent and
growth exponent
Family-Vicsek Ansatz:Family-Vicsek Ansatz:
Galilean Invariance & Fluctuation-Dissipation
Standard:
Galilean Invariance & Fluctuation-Dissipation
Standard:
Consistent:
Galilean Invariance & Fluctuation-Dissipation
Standard:
Consistent:
High Accuracy:
Galilean Invariance & Fluctuation-Dissipation
Galilean Invariance & Fluctuation-Dissipation
Final Remarks:
Final Remarks:
The moral from the present analysis is clear:
Final Remarks:
The moral from the present analysis is clear:
* A NEP for the KPZ equation exists! and for any general KPZ-like eq.
we can obtain the NEP “à la carte”
Final Remarks:
The moral from the present analysis is clear:
* A NEP for the KPZ equation exists! and for any general KPZ-like eq.
we can obtain the NEP “à la carte”
* Due to the locality of the Hopf--Cole transformation the discrete forms
of the Laplacian and the nonlinear (KPZ) term cannot be chosen
independently; moreover, the prescriptions should be the same,
regardless of the fields they are applied to.
Final Remarks:
The moral from the present analysis is clear:
* A NEP for the KPZ equation exists! and for any general KPZ-like eq.
we can obtain the NEP “à la carte”
* Due to the locality of the Hopf--Cole transformation the discrete forms
of the Laplacian and the nonlinear (KPZ) term cannot be chosen
independently; moreover, the prescriptions should be the same,
regardless of the fields they are applied to.
* A consistent discretization scheme, accurate up to
corrections, and whose prescription is not more complex than other
known proposals, was presented,
Final Remarks:
The moral from the present analysis is clear:
* A NEP for the KPZ equation exists! and for any general KPZ-like eq.
we can obtain the NEP “à la carte”
* Due to the locality of the Hopf--Cole transformation the discrete forms
of the Laplacian and the nonlinear (KPZ) term cannot be chosen
independently; moreover, the prescriptions should be the same,
regardless of the fields they are applied to.
* A consistent discretization scheme, accurate up to
corrections, and whose prescription is not more complex than other
known proposals, was presented,
* Whereas the fluctuation--dissipation relation essentially tells that the
nonlinearity is not operative for long times in 1D, our analysis indicates
that the problem with the fluctuation--dissipation theorem in 1+1, can be
circumvented by improving the numerical accuracy.
Final Remarks:
• Galilean invariance has been always associated with the exactness of the one-dimensional KPZ exponents, and with a relation that connects the critical exponents in higher dimensions. However,it is worth remarking that this interpretation has been recently criticized. One of the main results of our analysis indicates that if the numerical solution obtained with an on-Galilean-invariant finite-difference scheme yields the well known critical exponents, that difference scheme yields the well known critical exponents, that would strongly suggest that Galilean invariance is not a fundamental symmetry for KPZ as usually considered.
Final Remarks:
• Galilean invariance has been always associated with the exactness of the one-dimensional KPZ exponents, and with a relation that connects the critical exponents in higher dimensions. However,it is worth remarking that this interpretation has been recently criticized. One of the main results of our analysis indicates that if the numerical solution obtained with an on-Galilean-invariant finite-difference scheme yields the well known critical exponents, that difference scheme yields the well known critical exponents, that would strongly suggest that Galilean invariance is not a fundamental symmetry for KPZ as usually considered.
• Work towards exploiting the known form of the KPZ NEP in order to obtain expressions for the critical exponents, etc, is in progress.
• Variational formulation for the KPZ and related kinetic equations,
H. S. Wio, Int. J. Bif. Chaos 19 (8), 2813 (2009)
• KPZ equation: Galilean-invariance violation, Consistency, and
fluctuation--dissipation issues in real-space discretization, H.S.Wio,
J.A.Revelli, R.R.Deza, C.Escudero & M.S. de La Lama, submitted
to Europhys.Lett.
• Discretization-related issues in the KPZ equation:Consistency,
Galilean-invariance violation,and fluctuation--dissipation relation,
H.S.Wio, J.A.Revelli, R.R.Deza, C.Escudero & M.S. de La Lama,
submitted to Phys.Rev.E
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