Navigator 2007 – Centerfold – NAVO MSRC CAP II – Version 1.2.3.3 – Cleared for public release: VS07.005-129 Case No. ESC 07-0689.

Lattice model of fluid turbulence

Je!rey Yepez1, George Vahala2, Linda Vahala3, Min Soe4, Sean Ziegeler51Air Force Research Laboratory, Hanscom Air Force Base, MA 017312Department of Physics, William & Mary, Williamsburg, VA 23185

3Department of Electrical & Computer Engineering, Old Dominion University, Norfolk, VA 235294Department of Mathematics and Science, Rogers State University, Claremore, OK 74017

5High Performance Computing Modernization Program, Mississippi State University, MS 38677

The kinetic lattice gas is an optimal model of a vast many-particle system (such as a fluid) withcomplicated particle-particle interactions and irregular boundary conditions. With it the fluid dy-namicist can achieve higher nonlinearity (measured by Reynolds number), unconditional stabilityand accuracy, with less memory and processor time than with other models of turbulence for sit-uations with tortuous boundaries. For an engineer, it is simple to code, runs perfectly on parallelsupercomputers, and is suited to a plethora of computational physics applications. As demonstratedhere, it is a competitive alternative to large eddy simulations with Smagorinsky sub-grid closure.To theorists and experimentalist in quantum information science, its kinetic transport equation isa special case of the quantum dynamics, particularly governing a parallel array of quantum pro-cessors, a type-II quantum computer architecture. Presented are turbulent fluid simulations usingthe kinetic lattice gas model carried out on the supercomputer Babbage. As an illustration of thee!ciency of the lattice model, presented is a discovery of a universal range in the morphologicalevolution of the laminar-to-turbulent flow transition: the breaking subrange.

Keywords: kinetic lattice gas model, type II quantum computing, entropic lattice Boltzmann equation, fluidturbulence

I. INTRODUCTION

In the defense community there are a plethora ofneutral-fluid flows problems, particularly flows by regionswith non-trivial spatial boundaries, such around an air-craft fuselage or a ship hull, through a jet engine com-partment, or over the Earth’s surface. While analyti-cal solutions to such problems remain elusive and gen-erally intractable, today they are within the reach ofthe computational physicist. To tackle such problems,practitioners usually resort to direct numerical simula-tion methods, yet here the amount of computer memoryand processing time grows faster that the number of de-sired computed field points. Even in cases with simpleor periodic boundaries, where more e"cient numericalrepresentations are available (such as a psuedo spectralapproach pioneered by G.I. Taylor in the 1930’s and S.A.Orszag et al. in the 1970’s [1]), the scaling of computerresources with grid size is daunting. This computationalcomplexity has translated into significant annual costs todefense department high performance computing o"cespurchasing large numbers of processing elements (typi-cally thousands or ten of thousands), configured in mas-sive parallel computing arrays. But this cost, increasingyear after year, has been borne so engineers can solve mis-sion critical fluid problems, perpetually requiring higherresolution grids and more accurate and faithful simula-tions. For example, high resolution flow simulations arevital to aeronautical engineers designing the shape of ad-vanced fighter jets or unmanned aerial vehicles or sub-marines to economize on fuel consumption and minimizemaneuvering instabilities and wakes, to propulsion engi-neers designing nozzle and flow control orifices to max-imize thrust, or to meteorologists trying to understand

intermittent turbulence induced in the upper atmosphereunder the jet stream to maximize laser propagation fromairborne platforms. Yet to the theoretical physicist, thesituation is even more dire: the prediction of any aspectsof turbulence (beyond Kolmogorov’s 1941 universalityhypothesis), using advanced statistical methods and per-turbation methods, borrowed from triumphant quantumfield theory and statistical mechanics, remains the oldestand most prominent of classical grand challenge prob-lems, open now for over 150 years.

This dire situation arises because, even in the macro-scopic limit, strong correlations and feedback mecha-nisms between large scale and small scale flow struc-tures, over many decades of spatial separation, dominatethe overall flow evolution. The clearest high level pic-ture capturing the essential physics of this problem, withrestrictioned attention to divergence free and low Machnumber flows, are the incompressible Navier-Stokes equa-tions. The strong correlation between disparate scales iscaptured by the extremely simple non-local convectivederivative (the second order nonlinearity in the velocityfield).

II. LAMINAR TO TURBULENT FLOWTRANSITION

The lattice model now a!ords a deep insight into theorigin and essential inner workings of free shear turbu-lence. This is a kinetic lattice gas model, the clearestlow level picture correctly capturing the essential physicsand hydrodynamics of the problem. As the well knownIsing lattice gas model is fundamental to a statistical me-chanics understanding of the essential physics of ferro-

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FIG. 1: Supercomputer simulation of Navier-Stokes turbulence.Surface of constant vorticity in turbulent flow, showing entangledvortex tubes (top) and anisotropic flow in the breaking subrange(bottom). The dot product of the velocity and vorticity fields aredisplayed in the red-blue color coding. The bottom image is azoomed view of at t = 5K!t time step shown if Fig. 2, where thevertical white line is one edge of the cubical simulation grid. Longvortex tendrils are easily seen. Fig. 1 shows the fastest executingcode for computational fluid dynamics simulations of turbulenceto date. This was computed on the newest defense departmentsupercomputer, Babbage, located at the Naval Oceanographic Of-fice MSRC at the Stennis Space Center in Mississippi, through aCapability Application Program (CAP) II grant. Also tested werenew sub-grid models of turbulence, using lattice Boltzmann equa-tion techniques, an entropic method and a Smagorinsky closuremethod. These codes are close cousins to quantum algorithms foraerodynamics. Our CAP II simulations used over one million hours,using a dedicated block of thousands of high performance proces-sors over the period of four weeks, generating terabytes of data perday. These large-scale simulations o"er a better understanding ofthe morphological evolution and structural development of turbu-lence in fluids, but they also give a preview of the kind of numericaloutput to be available from future quantum computers.

FIG. 2: Surfaces of constant enstrophy (E = 12

RdV |!"|2 where

!" = ! " !u) illustrating an incompressible fluid’s morphologicalevolution from t = 0 up to t = 7, 000!t iterations, in time stepsof 1, 000!t on a cubical cartesian grid of size L = 512!x. Surfacecoloring uses !u · !" (red equals -1 and blue 1). This 3+1 dimen-sional turbulent neutral fluid simulation run was on the newestdefense department supercomputer, Babbage, using the entropiclattice Boltzmann equation with 15-body particle-particle collisions(ELB-Q15 model) computed at every lattice site at each time step.At each site, local relaxation of the single-particle probability dis-tribution a desired equilibrium function, represented as a low Machnumber polynomial expansion. The initial flow is a Kida andMurakami profile [2] with a super cell size set to L! = 512!x,the total grid size. So the flow configurations within all 8 oc-tants of the large grid are initially identical. The characteristicflow speed is u! = 0.07071!x

!t . The collisional inversion parame-ter is set to # = 0.99592, corresponding to a kinematic viscosity

of $! = 6.8 " 10"4 !x2

!t , for % = 2. The Reynolds number is

Re = L!u!!!

= 53, 024. The resulting turbulence is not fully re-

solved down to the dissipation scale, which in the model is the cell

size !x. Do do this, set L = Re34 # 2, 078!x. So the flow is under

resolved by about a factor of 4.

3

magnetism and the order-disorder phase transition, thekinetic lattice gas model is fundamental to a dynamicalmechanics understanding of fluids and the laminar-to-turbulent flow transition. See Appendix A for a briefmathematical overview of the model.

2000 4000 6000 8000 10000 12000 14000time

-20

-10

0

10

20

30

40

50

enstrophy

12

3

4

56 7

-0.06 -0.02 0.02 0.06velocity

0

0.001

0.002

0.003

0.004

probability

-0.1 -0.05 0 0.05 0.1vorticity

00.0020.0040.0060.0080.010.0120.014

probability

STAGE 1 STAGE 2 STAGE 3

0

FIG. 3: Plot of enstrophy versus time (smooth black curve) show-ing three stages in the morphological evolution: (1) vortex stretch-ing range (t < 3, 200!t), (2) breaking subrange (3, 200 < t <9, 000!t), and (3) inertial subrange (t > 9, 000!t). The enstrophyis normalized so that at t = 0 it is unity. The isovalues used tovisualize the 8 images in Fig. 2 are shown (black squares). Stage1: The initial exponential increase (blue curve) of enstrophy des-ignates the initial vortex stretching range with characteristic lam-inar flow. There is excellent agreement between the analytical fit(blue curve) and the enstrophy data (black curve). Stage 2: Thetime derivative of the enstrophy curve (jagged black curve) is alsoplotted. The time period of generally negative slope of the enstro-phy derivative (gray shaded region) is here termed the breakingsubrange, where large scale anisotropic ordering of turbulence oc-curs and intermittently breaks down over time. The first majorbreaking point occurs at about t = 3, 200!t (vertical red line) andsubsequent intermediate avalanches occur at about t = 5, 100!tand t = 6, 750!t (thin vertical red lines), respectively. Stage3: The final exponential decrease of enstrophy (red curve) desig-nates the inertial subrange where the homogeneous and isotropic“small scale” turbulent flow morphology, with characteristic entan-gled vortex tubes, is organized in a spatially self-similar way. Herethe energy spectral density obeys the Kolmogorov universality hy-

pothesis, the famous k"53 power law for energy cascade downward

to smaller scales. The onset of the inertial subrange occurs close tot = 9, 000!t (vertical blue line). Here the velocity probability dis-tribution functions, for each component, are Gaussian (top inset)and the vorticity probability distribution function approaches anexponential (bottom inset). There is excellent agreement betweenthe analytical fit (blue curve) and the enstrophy data (black curve).There exists a fourth stage of the morphology of turbulence at latetimes (t > 14, 000!t), not shown here, called the viscous subrange.

A new universal feature of the laminar-to-turbulenttransition becomes clear in high Reynolds number simu-lations. Following an initial period of laminar flow withvortex sheet stretching, and preceding the final inertialsubrange period of isotropic and homogeneous turbulentflow with self-similar vortex tubes, there exists a well-defined interim period, here termed the breaking sub-range. The hypothesis is that this subrange manifestsself-organized criticality. The breaking subrange is dom-

inated by anisotropic and non-homogeneous turbulentflow. Avalanches occur intermittently, where large coher-ent spatial structures grow, become unstable under max-imal shear, and subsequently break into isotropic andhomogeneous turbulence. These avalanches occur pro-gressively in time across the entire space. See the bottomimage of Fig. 1 showing the kind of anisotropic turbulentflow that occurs during an avalanche event, with youngvortex tendrils. The time rate of change of the enstrophy,!E!t , has a generally negatively sloped avalanche cascade,with marked peaks, indicating the successive breakingpoints, as shown in Fig. 3.

By carefully comparing the renderings in Fig. 2 to theenstrophy plots in in Fig. 3, it is possible to see three dis-tinct morphological stages of the flow: vortex stretchingrange, breaking subrange, and the inertial subrange. Themorphological evolution transitions from (1) large vortexsheets to (2) convoluted vortex sheets with virgin vortextendrils to (3) small entangled vortex tubes.

III. Q VERSUS S MODELS: CLOSURE OFSUB-GRID EFFECTS

The continuum hydrodynamical equations are projec-tions of the entropic lattice Boltzmann (ELB) equation,a projection down from the Q! LD-dimensional kineticphase space on to a 4 ! LD-dimensional hydrodynamicnull space 1. This projection recovers the Navier-Stokesequations in the Chapman-Enskog limit. This has animportant consequence: there exist many “qubit” mod-els (or Q models for short) with a di!erent local sten-cils (i.e. lattice vectors sets with di!erent finite pointgroup symmetries and coordination number), which willalso recover the Navier-Stokes equations asymptotically(continuous rotational symmetry).

ELB is ideal for large eddy simulation (LES) closuressince in LES one typically deals with mean strain ratesfor modeling the eddy viscosity. These nonlocal fluidcalculations are immediately recovered from simple localmoments in ELB.

At this stage, the ELB runs are a validation of themethod and this is important because ELB is a crucialprecursor to a viable quantum lattice Boltzmann method[3]. The good numerical agreement between the LES-LB(lattice Boltzmann equation with sub-grid Smagorinskyclosure here referred to as an S model) and the basic ELBQ models is encouraging. Now fully convinced of the va-lidity of ELB, work to speed it up for present day super-computer implementations is underway. But the most

1 A subset of all the eigenvectors in the Q-dimensional kineticspace, have zero eigenvalues. These eigenvectors span the hy-drodynamic null space. It is 4 dimensional because the kineticlattice gas model conserves probability (mass) and probabilityflux along the spatially orthogonal directions (three componentsof the momentum vector).

4

promising opportunity is to simply build a type-II quan-tum computer [4, 5], which could far outstrip any classi-cal supercomputers, for turbulence fluid simulations. Inthe fullness of time, the ELB will outpace the LES-LB,even without quantum computers.

There are a few reasons for this view. First, the kineticlattice gases obey detail balance while sub-grid closuremethods, such as the LES-LB, do not. And another ad-vantage of the ELB over LES-LB is that ELB obeys thesecond law of thermodynamics while LES-LB and otherLES methods do not necessarily obey the second law.Third, in the LES-LB the strain tensor must be com-puted at every site. With no-slip boundaries, computingthe strain tensor becomes problematic. In contrast, ELBis purely local, so grid sites near boundaries are handledas easily as sites far away from boundaries. All these areimportant di!erences when the model is used for practi-cal engineering grade applications.

IV. TIMINGS AND SCALING

Turbulent dynamics are easily solved since the under-lying kinetic equation (A1) has only local algebraic non-linearities in the macroscopic variables and simple linearadvection. At this mesoscopic level there are various ki-netic lattices (Q=15, 19, or 27) with di!erent lattice vec-tors on a cubic lattice, which model the Navier-Stokesequation to leading order in the Chapman-Enskog per-turbative asymptotics.

With the CAP data obtained on Babbage, the e!ectsof the underlying lattice symmetry on the fluid turbu-lence statistics (through autocorrelation tensors of veloc-ity, vorticity, pdfs of vorticity, and the like) can be deter-mined, but there is not have su"cient space to presentdetails here. The Q15 model seems to be the most e"-cient model. An example output of this model is shownin Fig. 2. Even on a relatively modest size 5123 grid, wecan achieve such a high degree of resolved nonlinearity(Re=26,512) that the consequent isotropic turbulence atthe onset of the inertial subrange (at about t " 9, 000#t)outstrips the ability to visualize the myriad vortex tubes.Fig. 2 is just a test case. The sustained floating pointper seconds (MFLOPS/PE) we achieved are the best ofall scientific codes run, for example on the Earth Simu-lator, attaining over 67% of peak performance on 4,800processors. Achieving the world’s highest Reynolds num-ber in the field of computational fluid dynamics shouldalso occur in the near future. The current tested codeon Babbage, on 1, 6003 grid with 2,048 processing ele-ments, can achieve a Reynold’s number of Re=565,667.Modeling atmospheric scale turbulence, in the range ofRe " 106 is possible today, for the first time in the halfcentury long history of numerical digital computers ap-plied to aerodynamics.

Some runs with these models–in particular the Q15,Q19, Q27 and S27 models–have been completed. Theadvantages of these lower Q models are reduced wall-

FIG. 4: TFlops/sec scaling of ELB-Q27 code on Babbage withnumber of CPUs.

clock times with less memory demands. The nonlinearconvective derivatives of in the Navier-Stokes equationare recovered from purely local moments of kinetic spacedistribution function. This is the basic reason why ELBscales so well with PEs: the algorithm consists of simplelocal computations and streaming of information only tonear neighbor grid sites.

No. PEs GRID MODEL WALLCLOCK (s) GFlops/s per PE2912 19503 ELB-Q27 7,554.7 2.172912 19503 ELB-Q19 5,602.7 2.242912 19503 ELB-Q15 4,798.4 2.052912 19503 LES-LB-S27 4,451.2 1.05

TABLE I: The gigaflops per second per processor element for2912 CPU runs on a 1952! 1946! 1950 grid for four latticeBoltzmann codes variants. The wallclock time is for 2,000"t(lattice time steps). A full turbulence simulation takes about54,000 time steps.

During Phase I, investigating the scaling propertiesof the Q27 code, over 6.3 tera flops per second on thefull 2912 processor elements available on Babbage wasachieved, see Fig. 4.

The LES-LB-S27 code, which no longer needs the so-lution of an entropy constraint equation, has also beentested in Phase I. It is less computationally intensive (dueto the avoidance of log-calls and the need for a Newton-Raphson root finder at each spatial node and time itera-tion) and shorter wallclock time than the ELB-Q27 code,see Table I.

V. QUANTUM INFORMATION PROCESSING

Quantum information processing (QIP) and quantumcommunications will be integral to the 21st century. Formany years, QIP has been included in the Developing

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FIG. 5: Shown here is a newsworthy achievement of a first quan-tum information processor circuit to embody a qubit (left), a basicdevice for storing quantum information, designed by Terry Orlandoof MIT and build at MIT Lincoln Laboratory at Hanscom AFB in2000 [6, 7]. The circuit must be placed in a dilution refrigera-tor. Air Force O#ce of Scientific Research (AFOSR) supportednovel quantum computing technology based on superconductiveelectronics under the Quantum Computation for Physical Model-ing (QCPM) theme, and this research has recently found follow-onuse. A superconducting wire loop with multiple Josephson junc-tions forms a qubit and such qubits are coupled together to makequantum logic gates. AFOSR funded this new solid-state tech-nology, fabricated at the superconductive electronics foundry atMIT Lincoln laboratory. This helped establish the basic fabrica-tion techniques to build scalable quantum computers and mappedthe quantum control methodology, proven with NMR spectroscopy[4, 5], onto the field of superconductive electronics for quantum in-formation processing, mapping pulse protocols to allow qubit-qubitlogical operations, constituting a basic 2-qubit quantum processor.Going from a 2-qubit processor to a 16-qubit processor necessarilyentails a significant applied research e"ort recently announced byD-Wave, a Canadian start-up company. D-Wave’s 16-qubit proces-sor, fabricated by NASA, is shown (right).

Science and Technologies List, in the section of criticalinformation systems technology. There now exists thepossibility of the development of large quantum computerarrays, potentially outstripping any supercomputers nowused for defense department computations.

On Babbage Q27, Q19, Q15 lattice Boltzmann codeswere tested, like the one shown in Figure 1. Using 2,048processors it took two days to complete a single job. A1, 0243 grid takes about 44 hours for Q27 on 512 proces-sors, while the corresponding run for Q15 takes about28 hours. The cost of the largest supercomputer parallelarrays annually adds to a significant fraction of a billiondollars for new government-owned systems in the UnitedStates (e.g. 19,000 processor Franklin at DOE/NERSCcost about $50 million and occupies the space of a gym-nasium). Remarkably, exploiting quantum mechanicalcomplexity new quantum device technology can be usedto e"ciently compute the collision operator of the Q15lattice Boltzmann code, the basic engine of the code.

Therefore, a relatively low cost and small type-II quan-tum computer with a few thousand 16-qubit quantumprocessor chips, perhaps cooled with dilution refrigera-tion, could handle the same supercomputing job. It couldcost a few million dollars to assemble, a couple orders ofmagnitude less expensive than classical digital electronics

based supercomputers, such as Babbage, and physicallysmaller by many orders of magnitude as well. Further-more, future quantum processor arrays with more qubitsper node, if available in the near future, could outstripany traditional parallel supercomputer purchased underthe department of defense high performance computingmodernization program.

The kinetic lattice gas model has proven to be a state-of-the-art tool for understanding the morphological evo-lution of turbulence.

VI. ACKNOWLEDGEMENTS

This work was supported in part by a grant of com-puter time from the defense department High Perfor-mance Computing Modernization Program, with compu-tational support from the Major Shared Resource Cen-ters: the Naval Oceanographic O"ce and the US ArmyEngineer Research and Development Center. JY wouldlike to acknowledge the leaders in the Air Force O"ceof Scientific Research. The AFOSR Novel Strategiesfor Parallel Computing Initiative launched in 1992, andtwo subsequent AFOSR quantum computing researchprojects launched in 1996 and 2000, allowed us to laya firm foundation theoretically, experimentally, and nu-merically for this lattice model of turbulence suited forboth parallel supercomputers and quantum computers.

Appendix A: Mathematical formulation

In the lattice model, dynamics is projected into a dis-crete kinetic phase space. The logical “1” state (the ex-cited state) of a qubit |q# associated with the spacetimepoint (!x, t) encodes the probability fq of the existence ofa particle at that point moving with velocity !cq = !"xq

!t ,where #!xq are lattice vectors, for q = 1, 2, . . . , Q. Afundamental property of the lattice model is that parti-cle motions in momentum space and position space oc-cur independently [8]. Particle momentum and positionspace motions are generated by the combination of anengineered qubit-qubit interaction Hamiltonian H ! anda free Hamiltonian $i!

!q !cq ·%, respectively.

All the particle-particle interactions, the 2-body up toand including all the (Q$2)-body interactions, generatedby H ! are mapped to a local collision function H ! &' $q

that depends on all the fq’s at the lattice site [3].In the type-II quantum computing case, quantum en-

tanglement is localized among qubits associated with thesame (!x, t) [9], so:

f !q(!x, t) = fq(!x, t) + $q(f1, f2, . . . fQ) (A1a)fq(!x, t) = f !q(!x$#!xq, t$#t), (A1b)

where fq and f !q are called the incoming and outgoingprobabilities, respectively. In the classical limit, there

6

exists a fundamental entropy function

H (f1, . . . , fQ) =Q"

q=1

fq ln("qfq), (A2)

where the "q are self-consistently determined positiveweights (

!q "q = 1). $q in (A1a) is determined by the

constant entropy condition:

H (f !1, . . . , f!Q) = H (f1, . . . , fQ). (A3)

It is (A1) through (A3) that constitutes the lattice modelof fluid turbulence suited for parallel supercomputers.

In the Q-dimensional kinetic space, the equilibriumdistribution f eq

q , taken as a vector, is the bisector of thedi!erence of the incoming and outgoing kinetic vectors:

fq = f eqq $ 1

#$$q f !q = f eq

q +#

1$ 1#$

$$q,(A4a)

when lim#"0 #$ = 2 and where f eqq is analytically deter-

mined by extremizing (A3) subject to the two constraintsof conversation of probability and probability flux. Elim-inating $q from (A4) yields an operative collision equa-tion simpler than (A1a)

f !q = fq + #$%f eq

q $ fq

&. (A5)

Finally, eliminating f !q in (A1b) and (A5), yields the lat-tice Boltzmann equation

fq(!x + #!xq, t + #t) = fq(!x, t) + #$'f eq

q (!x, t)$ fq(!x, t)(,

(A6)

in the BGK collisional limit [10]. Since #$ and f eqq are

determined by (A2), (A6) is called the entropic latticeBoltzmann equation [11].

The kinetic lattice gas model becomes equivalent tothe Navier Stokes equations:

%t!u+!u ·%!u = $%P +&%2!u and % ·!u = 0, (A7)

where !u = !u(!x, t) is the vectorial flow field, where thepressure is proportional to the field density (P = 'c2

s,with spatial dimension D and sound speed cs = 1#

D!r!t ),

and where the shear viscosity is the measure of dissipa-tion (a renormalized transport coe"cient for momentumdi!usion). In the limit when #!r ' 0 and #t ' 0 andthe hydrodynamic variables are cast as moments of theprobability distribution, ('c, !u) =

!q(c,!cq)fq [12]. The

hydrodynamic variables are independently evaluated ateach spacetime point (!x, t). The shear viscosity is ana-lytically determined, and the result is

& = 'c2s#t

#1

#$$ 1

2

$, (A8)

so the model approaches the inviscid limit where & ' 0as #$ ' 2 [13].

[1] S. A. Orszag and G. S. Patterson. Statistical Models andTurbulence. Springer-Verlag, New York, 1972. editors M.Rosenblatt and C. Van Atta.

[2] S. Kida and Y. Murakami. Kolmogorov similarity infreely decaying turbulence. Physics of Fluids, 30(7):2030–2039, 1987.

[3] Je#rey Yepez. Quantum lattice-gas model for computa-tional fluid dynamics. Physical Review E, 63(4):046702,Mar 2001.

[4] Marco Pravia, Zhiying Chen, Je#rey Yepez, andDavid G. Cory. Towards an NMR implementation of aquantum lattice gas algorithm. Computer Physics Com-munications, 146(3):339–344, 2002.

[5] Zhiying Chen, Je#rey Yepez, and David G. Cory. Sim-ulation of the Burgers equation by NMR quantum-information processing. Physical Review A, 74(4):042321,Oct 2006.

[6] J.E. Mooij, Terry P. Orlando, L. Levitov, Lin Tian,Casper H. van der Wal, and Seth Lloyd. Josephsonpersistent-current qubit. Science, 285:1036–1039, 1999.

[7] T.P. Orlando, J.E. Mooij, Lin Tian, Caspar H. van derWal, L.S. Levitov, Seth Lloyd, and J.J. Mazo. Super-conducting persistent-current qubit. Physical Review B,60(22):15398–15413, 1999.

[8] Je#rey Yepez. Relativistic path integral as a lattice-basedquantum algorithm. Quantum Information Processing,4(6):471–509, Dec 2005.

[9] Je#rey Yepez. Open quantum system model of the one-dimensional Burgers equation with tunable shear viscos-ity. Physical Review A, 74(4):042322, Oct 2006.

[10] P. L. Bhatnagar, E. P. Gross, and M. Krook. A model forcollision processes in gases. i. small amplitude processesin charged and neutral one-component systems. Phys.Rev., 94(3):511–525, May 1954.

[11] Bruce M. Boghosian, Je#rey Yepez, Peter Coveney, andAlexander Wagner. Entropic lattice Boltzmann methods.Proceedings of the Royal Society A: Mathematical, Phys-ical and Engineering Sciences, 457(2007):717–766, Mar2001.

[12] Bruce M. Boghosian, Peter J. Love, Peter V. Coveney,Iliya V. Karlin, Sauro Succi, and Je#rey Yepez. Galilean-invariant lattice-Boltzmann models with H theorem.Physical Review E, 68(2):025103, Aug 2003.

[13] Brian Keating, George Vahala, Je#rey Yepez, Min Soe,and Linda Vahala. Entropic lattice Boltzmann represen-tations required to recover navier-stokes flows. PhysicalReview E, 75(3):036712, 2007.