1. Thermal fluctuations in multiphase lattice Boltzmann methods
Fathollah Varnik, Markus Gross
ICAMS, Ruhr-University Bochum, Germany
ERC-Workshop “Multiphase physics at the micro- and nano-scales”
University of Rome Tor Vergata, May 2013

2. Introduction Newtonian mechanics Lattice Gas averaging
Thermal fluctuations inherently present
averaging
Boltzmann eq.
discretization
Lattice Boltzmann
Thermal fluctuations fully eliminated
Navier-Stokes equations

3. Need for fluctuations in LB
deterministic B.eq.: distribution function f = avg. # of particles in a phase space cell (r…r+dr , c…c+dc)
but particles are permanently scattered in and out of each phase space cell
 “instantaneous” distribution function fluctuates
 Fluctuating Boltzmann equation:
noise
Kadomtsev (Sov. Phys. JETP 1957) , Bixon, Zwanzig (Phys. Rev. 1969)
Fox, Uhlenbeck (Phys. Fluids 1970)

4. Examples
Some examples for the importance of thermal fluctuations in LB:
Brownian particles in a solvent [Ladd, PRL 1993]
Hybrid MD-LB approaches
Polymer dynamics in a solvent [Ahlrich, Dünweg,J Chem Phys 1999]
Nano-confined polymers [Usta, Butler, Ladd, PRL 2007]
DNA translocation [Farahpour, Maleknejad, FV, Ejtehadi, Soft Matter 2013]

5. (small mean-free path, sub-sonic flows)
Ladd’s approach
stochastic Boltzmann:
stochastic Navier-Stokes:
Chapman- Enskog
noise xi
fluct. stress R
(small mean-free path, sub-sonic flows)
[Ladd, PRL 1993]
Stress fluctuations uncorrelated in space and time
This ensures equipartition on hydrodynamic (large) length scale
Further improvement by thermalizing the kinetic (ghost) modes
 Equipartition at all length scales
[Adhikari, Stratford, Cates, Wagner, EPL2004]

6. Test: Decay of velocity autocorrelation
[Ladd, PRL 1993]
Symbols: Fluctuating LB
Solid lines: Theory

7. Why fluctuations in non-ideal fluid LB?
Capillary fluctuations
vapor
interfacial fluctuations induced by thermal motion in the bulk
liquid
Critical dynamics T
liquid-vapor coexistence
metastable region
spinodal r

8. Need for fluctuations…
Colloids in a drop
competition between
Brownian motion
capillary flow
colloidal crystallization at the substrate
multitude of residual colloid patterns observed on the substrate (after evaporation)

9. Non-ideal fluid LB
force-based model [Lee/Fischer (PRE 2006), He/Shan/Doolen (PRE 1998)]:
ideal gas equilibrium distribution
interaction force: ( LB force Fi)
modified-equilibrium model [Swift/Osborn/Yeomans (PRL 1995)]:
pressure tensor constraint:

10. What is needed? Covariance matrix of fluctuations: Previous studies:
Ideal gas 
[Ladd, JFM 1994, Adhikari et al., EPL (2005)]
[Dünweg, Schiller, Ladd, PRE (2007)]
[Gross, Adhikari, Cates, FV, PRE (2010)]
[Ollila, Denniston, Karttunen, Ala-Nissila, PRE (2011)]
Non-ideal gas:
Within forced-based non-ideal LB, we encountered difficulties in handling non-linear advection term and force term
Route via Discrete Boltzmann Equation

11. Procedure: Start with Fluctuating Discrete Boltzmann Equation (FDBE)
Bring the equations to a form similar to generalized Langevin equation
Derive expression for the covariance matrix of fluctuations
Translate it to the Lattice Boltzmann Framework

12. Generalized Langevin equation
A set of coupled Langevin equations
Onsager, Machlup (1953) / Fox, Uhlenbeck (1970)
Apply to the one dimensional case
friction random force equipartition

13. Fluctuating DBE Thermal noise in Discrete Boltzmann Equation:
Linearize DBE, express it in terms of moments and Fourier transform:
This is a set of coupled Langevin equations, thus: Onsager, Machlup (1953) / Fox, Uhlenbeck (1970)
Gross, Cates, Varnik, Adhikari (J. Stat. Mech. 2011)
time-evolution operator:
equilibrium correlation matrix:

14. Elements of the G-matrix
How to choose the equilibrium correlations of the distribution function?
ideal gas
[Kadomtsev (Sov. Phys. JETP 1957) / Bixon, Zwanzig (Phys. Rev. 1969) / Fox, Uhlenbeck (Phys. Fluids 1970) / Adhikari et al (EPL 2005)]
non-ideal gas [Klimontovich (1973) / Gross, Adhikari, Cates, Varnik (Phys. Rev. E 2010)]
Note: this guarantees

15. Noise covariance matrix
Force based LB:
Using linearized force, one obtains the same result as for an ideal gas
Non-ideal fluid interactions are reversible and do not contribute to dissipation
The same non-correlated noise also for non-linear forces are expected (and found!)
on a lattice, however, microscopic reversibility is often violated
Modified equilibrium LB:
Spatially correlated noise!
[Gross, Cates, FV, Adhikari (J. Stat. Mech. 2011)]
[Gross, Adhikari, Cates, FV (Phys. Rev. E, 2010)]

16. Hydrodynamic limit Fluctuation-Dissipation Relation
for a continuum square-gradient, two-phase fluid: FDT identical to the single-phase fluid case [Kim/Mazenko (J. Stat. Phys. 1991)]
This is violated by the modified equilibrium non-ideal LB!
bulk viscosity
R: random stress tensor
shear viscosity

17. Hydrodynamics of the LB fluid
Fluctuation-Dissipation Relation (LB fluid)
modified-equilibrium (Swift/Yeomans) LBM:
modified bulk viscosity:
random stress tensor spatially correlated
force-based (Lee/Fischer) LBM:
standard (‘ideal gas’) expressions for the viscosities (indep. of k)
random stress tensor same as for a single-phase fluid (uncorrelated)
[Kim/Mazenko (J. Stat. Phys. 1991)]
shear viscosity bulk viscosity

18. Benchmark Momentum correlations: Density correlations:
Sufficient to consider

19. Benchmark Capillary fluctuations: vap. liq. h(r) vap.
deviation from straight line due to Fourier-transform of discrete Laplacian
[Gross, Cates, FV, Adhikari (J. Stat. Mech. 2011)]

20. Stochastic lubrication equation [Davidovitch, Moro, Stone (PRL 2005)]
Spreading of nanodroplets
Stochastic lubrication equation [Davidovitch, Moro, Stone (PRL 2005)]
h(t)
r(t)
Noise dominated spreading
r  t1/6
log r
perfectly wetting substrate
r  t1/10
Tanner’s law
(no noise)
prediction: thermal fluctuations enhance late-time spreading rate
log t

21. Spreading of nanodroplets
 cross-section through a 3D simulation

22. Thank you for listening
Summary
Analysis of fluctuating Discrete Boltzmann Equation:
Ideal gas noise within force-based non-ideal fluid LB
Spatially correlated noise for modified equilibrium LB
Use of ideal gas noise works well for a certain parameter range
[Gross, Cates, FV, Adhikari (J. Stat. Mech. 2011)]
[Gross, Adhikari, Cates, FV (Phys. Rev. E, 2010)]
[Gross, Adhikari, Cates, FV (Phil. Trans. R. Soc. A 2011)]
Thank you for listening