Variational Formulation for the KPZ Equation: Consistency, … · 2016-06-04 · R.Graham, in...

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: [email protected]

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:

* **** * **


* Introduction:**** * **


* Introduction:* Nonequilibrium Potential: Brief Review,* * * * * *


* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* * * * *


* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,*** *


* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,* Discretization & Consistency, * Discretization & Consistency, * *


* Introduction:* Nonequilibrium Potential: Brief Review,* Nonequilibrium Potential for KPZ* NEP for KPZ-related equations,* Discretization & Consistency, * Discretization & Consistency, * Galilean Invariance & Fluctuation-Dissipation, *


* 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.




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...”;




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:


• “No-relaxational”:


• “No-relaxational”:

The 2nd term will be zero if the orthogonality relation is

fulfilled


Dynamical System + Stochastic Terms:



and the associated Fokker-Planck Equation:



and the associated Fokker-Planck Equation:

If: and


• Definition of the non-equilibrium potential:



• General form: • General form:



• 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


Int. J. Bif. Chaos 19 (8), 2813 (2009)

Edward-Wilkinson


Int. J. Bif. Chaos 19 (8), 2813 (2009)

Edward-Wilkinson + curvature KPZ

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


The Hopf-Cole transformation

with the inverse yields


The Hopf-Cole transformation

with the inverse yields

In order to obtain the “usual” KPZ equation we reduce the

nonlinear function to

with and


Some properties (shift invariance, etc):


Definition


Fokker-Planck Eq.


Fokker-Planck Eq.

with the SPD


Other kinetic equations related to KPZ and Non-Locality

Consider the following, nonlocal, reaction-diffusion

equation


Applying the Hopf-Cole transformation we obtain the

KPZ-like equation


Applying the Hopf-Cole transformation we obtain the

KPZ-like equation

and the following form of NEP


If we assume that the kernel is symmetric and of “short

range”, we can expand it as

with with


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).


• Even the case of density dependent diffusion could be also

treated

with



treated

with

via Hopf-Cole transformation leads to



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


Starting from the LF for deterministic part of the RD eq.

with the form



with the form

and assuming the discrete form

where F=0, we obtain for the functional



with the form

and assuming the discrete form

where F=0, we obtain for the functional

with


Another possible form to discretize the Laplacian is

that corresponds to a more accurate approximation for

the Laplacian.


Another possible form to discretize the Laplacian is

that corresponds to a more accurate approximation for

the Laplacian. The associated KPZ term results


Similar results are obtained if we start with the discrete

form of the NEP for KPZ


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.


The SPD is

that in a discrete representation reads


The SPD is


Inserting this expression into the FPE we found that the only surviving

contribution is


The SPD is


Inserting this expression into the FPE we found that the only surviving

contribution is

with a continuous limit


KPZ is Galilean invariant: it fullils the following transformations



If we consider a gradient discretization

it fulfills the discrete (Galilean) transformations



If we consider a gradient discretization

it fulfills the discrete (Galilean) transformations

while other discretizations do not.


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 => exponent relation:

Where roughness exponent , dynamic exponent and

growth exponent

Family-Vicsek Ansatz:Family-Vicsek Ansatz:


Standard:


Standard:

Consistent:


Standard:

Consistent:

High Accuracy:

Final Remarks:

Final Remarks:

The moral from the present analysis is clear:

Final Remarks:


* A NEP for the KPZ equation exists! and for any general KPZ-like eq.

we can obtain the NEP “à la carte”

Final Remarks:




* 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:








* A consistent discretization scheme, accurate up to

corrections, and whose prescription is not more complex than other

known proposals, was presented,

Final Remarks:








* 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

THE ENDTHE ENDTHE ENDTHE END

Variational Formulation for the KPZ Equation: Consistency, … · 2016-06-04 · R.Graham, in...

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