text: Large-scale Optimization-based Non-negative Computational Framework for Diﬀusion Equations: Parallel Implementation and Performance Studies J. Chang, S. Karra and K. B. Nakshatrala Correspondence to: e-mail: knakshatrala@uh.edu, phone:+1-713-743-4418 Abstract. It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the ﬁnite element method for solving self-adjoint diﬀusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diﬀusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diﬀusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computa- tional performance of current algorithms available in these scientiﬁc libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the eﬃciency of the solvers. Our studies indicate that the proposed non-negative computa- tional framework for diﬀusion-type equations exhibits excellent strong scaling for real-world large-scale problems. 1. INTRODUCTION The modeling of ﬂow and transport in subsurface is vital for energy, climate and environmental ap- plications. Examples include CO migration in carbon-dioxide sequestration, enhanced geothermal sys- 2 tems, oil and gas production, radio-nuclide transport in a nuclear waste repository, groundwater contam- ination, and thermo-hydrology in the Arctic permafrost due to the recent climate change [10,15,17,23]. Several numerical codes (e.g., FEHM [40], TOUGH [37], PFLOTRAN [22]) have been developed to model ﬂow and transport in subsurface at reservoir-scale. These codes typically solve unsteady Darcy equations for ﬂow and advection-diﬀusion equation for transport. The predictive capability of a nu- merical simulator depends on the robustness of the underlying numerical methods. A necessary and essential requirement is to satisfy important mathematical principles and physical constraints. One such property in transport and reactive-transport problems is that the concentration of a chemical species cannot be negative. Mathematically, this translates to the satisfaction of the discrete maximum princi- ple (DMP) for diﬀusion-type equations. Subsurface ﬂow and transport applications typically encounter geological media that are highly heterogeneous and anisotropic in nature, and it is well-known that Key words and phrases. high performance computing; anisotropic diﬀusion; maximum principles; non-negative con- straint; large-scale optimization. 1 ##LTLine##

text: Large-scale

text: Optimization-based

text: Non-negative

text: Computational

text: Framework

text: for

text: Diﬀusion

text: Equations:

text: Parallel

text: Implementation

text: and

text: Performance

text: Studies

text: J.

text: Chang,

text: S.

text: Karra

text: and

text: K.

text: B.

text: Nakshatrala

text: Correspondence

text: to:

text: e-mail:

text: knakshatrala@uh.edu,

text: Abstract.

text: It

text: is

text: well-known

text: that

text: the

text: standard

text: Galerkin

text: formulation,

text: which

text: is

text: often

text: the

text: formulation

text: of

text: choice

text: under

text: the

text: ﬁnite

text: element

text: method

text: for

text: solving

text: self-adjoint

text: diﬀusion

text: equations,

text: does

text: not

text: meet

text: maximum

text: principles

text: and

text: the

text: non-negative

text: constraint

text: for

text: anisotropic

text: diﬀusion

text: equations.

text: Recently,

text: optimization-based

text: methodologies

text: that

text: satisfy

text: maximum

text: principles

text: and

text: the

text: non-negative

text: constraint

text: for

text: steady-state

text: and

text: transient

text: diﬀusion-type

text: equations

text: have

text: been

text: proposed.

text: To

text: date,

text: these

text: methodologies

text: have

text: been

text: tested

text: only

text: on

text: small-scale

text: academic

text: problems.

text: The

text: purpose

text: of

text: this

text: paper

text: is

text: to

text: systematically

text: study

text: the

text: performance

text: of

text: the

text: non-negative

text: methodology

text: in

text: the

text: context

text: of

text: high

text: performance

text: computing

text: (HPC).

text: PETSc

text: and

text: TAO

text: libraries

text: are,

text: respectively,

text: used

text: for

text: the

text: parallel

text: environment

text: and

text: optimization

text: solvers.

text: For

text: large-scale

text: problems,

text: it

text: is

text: important

text: for

text: computational

text: scientists

text: to

text: understand

text: the

text: computa-

text: tional

text: performance

text: of

text: current

text: algorithms

text: available

text: in

text: these

text: scientiﬁc

text: libraries.

text: The

text: numerical

text: experiments

text: are

text: conducted

text: on

text: the

text: state-of-the-art

text: HPC

text: systems,

text: and

text: a

text: single-core

text: performance

text: model

text: is

text: used

text: to

text: better

text: characterize

text: the

text: eﬃciency

text: of

text: the

text: solvers.

text: Our

text: studies

text: indicate

text: that

text: the

text: proposed

text: non-negative

text: computa-

text: tional

text: framework

text: for

text: diﬀusion-type

text: equations

text: exhibits

text: excellent

text: strong

text: scaling

text: for

text: real-world

text: large-scale

text: problems.

text: INTRODUCTION

text: The

text: modeling

text: of

text: ﬂow

text: and

text: transport

text: in

text: subsurface

text: is

text: vital

text: for

text: energy,

text: climate

text: and

text: environmental

text: ap-

text: plications.

text: Examples

text: include

text: CO

text: migration

text: in

text: carbon-dioxide

text: sequestration,

text: enhanced

text: geothermal

text: sys-

text: tems,

text: oil

text: and

text: gas

text: production,

text: radio-nuclide

text: transport

text: in

text: a

text: nuclear

text: waste

text: repository,

text: groundwater

text: contam-

text: ination,

text: and

text: thermo-hydrology

text: in

text: the

text: Arctic

text: permafrost

text: due

text: to

text: the

text: recent

text: climate

text: change

text: Several

text: numerical

text: codes

text: (e.g.,

text: FEHM

text: TOUGH

text: PFLOTRAN

text: have

text: been

text: developed

text: to

text: model

text: ﬂow

text: and

text: transport

text: in

text: subsurface

text: at

text: reservoir-scale.

text: These

text: codes

text: typically

text: solve

text: unsteady

text: Darcy

text: equations

text: for

text: ﬂow

text: and

text: advection-diﬀusion

text: equation

text: for

text: transport.

text: The

text: predictive

text: capability

text: of

text: a

text: nu-

text: merical

text: simulator

text: depends

text: on

text: the

text: robustness

text: of

text: the

text: underlying

text: numerical

text: methods.

text: A

text: necessary

text: and

text: essential

text: requirement

text: is

text: to

text: satisfy

text: important

text: mathematical

text: principles

text: and

text: physical

text: constraints.

text: One

text: such

text: property

text: in

text: transport

text: and

text: reactive-transport

text: problems

text: is

text: that

text: the

text: concentration

text: of

text: a

text: chemical

text: species

text: cannot

text: be

text: negative.

text: Mathematically,

text: this

text: translates

text: to

text: the

text: satisfaction

text: of

text: the

text: discrete

text: maximum

text: princi-

text: ple

text: (DMP)

text: for

text: diﬀusion-type

text: equations.

text: Subsurface

text: ﬂow

text: and

text: transport

text: applications

text: typically

text: encounter

text: geological

text: media

text: that

text: are

text: highly

text: heterogeneous

text: and

text: anisotropic

text: in

text: nature,

text: and

text: it

text: is

text: well-known

text: that

text: Key

text: words

text: and

text: phrases.

text: high

text: performance

text: computing;

text: anisotropic

text: diﬀusion;

text: maximum

text: principles;

text: non-negative

text: con-

text: straint;

text: large-scale

text: optimization.

text: ##LTLine##


text: Table 2. Commonly used PETSc operations and their respective Total Bytes Trans- · · · ferred. Here we note X, Y, Z as vectors with i = 1, , N entries, a is a scalar value, and nz denotes the total number of non-zeros. We assume that the sizes of integers and doubles are 4 and 8 bytes respectively. (cid:113) (cid:80) PETSc function (cid:80)Operation Total Bytes Transferred VecNorm() a = N X(i)2 8(N + 1) i ∗ VecDot() a = N X(i) Y (i) 8(2N + 1) i ← VecCopy() Y X 8(2N) VecSet() Y (i) = a 8(2N) ∗ VecScale() Y = a Y 8(2N) ∗ VecAXPY() Y = a X + Y 8(3N) ∗ VecAYPX() Y = X + a Y 8(3N) ∗ VecPointwiseMult() Z(i) = X(i) Y (i) 8(3N) MatMult() SpMV 4(N + nz) + 8(2N + nz) To this end, we propose a rooﬂine-like performance model where the TBT assumes a “perfect cache” – each byte of the data needs to be fetched from DRAM only once. This assumption enables us to predict a slightly more realistic upper bound of the peak performance than by simply comparing to the TPP. Table 2 lists the key PETSc functions used for the solvers and their respective estimates of TBT based on the perfect cache assumption. The formula for SpMV follows the procedure outlined in [9]. We assume that the TBT formula for operations also involving a sparse matrix and vector like the incomplete lower-upper (ILU) factorization to be the same as MatMult(). Estimating the TBT for other important operations like the sparse matrix-matrix and triple matrix products (which are important for multi-grid methods) is an area of future work. In short, our AI formulation relies on the following four key assumptions: (i) All ﬂoating-point operations (add, multiply, square roots, etc.) are treated equally and equate to one FLOP count. (ii) There are no conﬂict misses. That is, each matrix and vector element is loaded into cache only once. (iii) Processor never waits on a memory reference. That is, any number of loads and stores are satisﬁed in a single cycle. (iv) Compilers are capable of storing scalar multipliers in the register only for pure streaming compu- tations. Therefore, the eﬃciency based on this new rooﬂi(cid:26)ne-like performance model as: Measured FLOPS/s × Eﬃciency (%) = 100 (4.2) TPP min × AI STREAMS where the numerator is reported by the PETSc program and the denominator is the ideal performance upper-bounded by both the TPP and the product of AI and STREAMS bandwidth. STREAMS Triad [14] is one of the most popular benchmarks for determining the achievable memory performance of a given machine. Figure 2 denotes the estimated memorybandwidth as a function of number of cores on a single Mustang and Wolf node. It is interesting to note that although the Wolf node has a greater 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

label: paragraph

text: Commonly

text: used

text: PETSc

text: operations

text: and

text: their

text: respective

text: Total

text: Bytes

text: Trans-

text: ferred.

text: Here

text: we

text: note

text: X,

text: Y,

text: Z

text: as

text: vectors

text: with

text: i

text: N

text: entries,

text: a

text: is

text: a

text: scalar

text: value,

text: and

text: nz

text: denotes

text: the

text: total

text: number

text: of

text: non-zeros.

text: We

text: assume

text: that

text: the

text: sizes

text: of

text: integers

text: and

text: doubles

text: are

text: and

text: bytes

text: respectively.

text: PETSc

text: function

text: Total

text: Bytes

text: Transferred

text: VecNorm()

text: a

text: N

text: i

text: VecDot()

text: a

text: N

text: X(i)

text: Y

text: (i)

text: i

text: VecCopy()

text: Y

text: X

text: VecSet()

text: Y

text: (i)

text: a

text: VecScale()

text: Y

text: a

text: Y

text: VecAXPY()

text: Y

text: a

text: X

text: Y

text: VecAYPX()

text: Y

text: X

text: a

text: Y

text: VecPointwiseMult()

text: Z(i)

text: X(i)

text: Y

text: (i)

text: MatMult()

text: SpMV

text: nz)

text: nz)

text: To

text: this

text: end,

text: we

text: propose

text: a

text: rooﬂine-like

text: performance

text: model

text: where

text: the

text: TBT

text: assumes

text: a

text: “perfect

text: cache”

text: each

text: byte

text: of

text: the

text: data

text: needs

text: to

text: be

text: fetched

text: from

text: DRAM

text: only

text: once.

text: This

text: assumption

text: enables

text: us

text: to

text: predict

text: a

text: slightly

text: more

text: realistic

text: upper

text: bound

text: of

text: the

text: peak

text: performance

text: than

text: by

text: simply

text: comparing

text: to

text: the

text: TPP.

text: Table

text: lists

text: the

text: key

text: PETSc

text: functions

text: used

text: for

text: the

text: solvers

text: and

text: their

text: respective

text: estimates

text: of

text: TBT

text: based

text: on

text: the

text: perfect

text: cache

text: assumption.

text: The

text: formula

text: for

text: SpMV

text: follows

text: the

text: procedure

text: outlined

text: in

text: We

text: assume

text: that

text: the

text: TBT

text: formula

text: for

text: operations

text: also

text: involving

text: a

text: sparse

text: matrix

text: and

text: vector

text: like

text: the

text: incomplete

text: lower-upper

text: (ILU)

text: factorization

text: to

text: be

text: the

text: same

text: as

text: MatMult().

text: Estimating

text: the

text: TBT

text: for

text: other

text: important

text: operations

text: like

text: the

text: sparse

text: matrix-matrix

text: and

text: triple

text: matrix

text: products

text: (which

text: are

text: important

text: for

text: multi-grid

text: methods)

text: is

text: an

text: area

text: of

text: future

text: work.

text: In

text: short,

text: our

text: AI

text: formulation

text: relies

text: on

text: the

text: following

text: four

text: key

text: assumptions:

text: (i)

text: All

text: ﬂoating-point

text: operations

text: (add,

text: multiply,

text: square

text: roots,

text: etc.)

text: are

text: treated

text: equally

text: and

text: equate

text: to

text: one

text: FLOP

text: count.

text: (ii)

text: There

text: are

text: no

text: conﬂict

text: misses.

text: That

text: is,

text: each

text: matrix

text: and

text: vector

text: element

text: is

text: loaded

text: into

text: cache

text: only

text: once.

text: (iii)

text: Processor

text: never

text: waits

text: on

text: a

text: memory

text: reference.

text: That

text: is,

text: any

text: number

text: of

text: loads

text: and

text: stores

text: are

text: satisﬁed

text: in

text: a

text: single

text: cycle.

text: (iv)

text: Compilers

text: are

text: capable

text: of

text: storing

text: scalar

text: multipliers

text: in

text: the

text: register

text: only

text: for

text: pure

text: streaming

text: compu-

text: tations.

text: Therefore,

text: the

text: eﬃciency

text: based

text: on

text: this

text: new

text: performance

text: model

text: as:

text: Measured

text: FLOPS/s

text: Eﬃciency

text: TPP

text: min

text: AI

text: STREAMS

text: where

text: the

text: numerator

text: is

text: reported

text: by

text: the

text: PETSc

text: program

text: and

text: the

text: denominator

text: is

text: the

text: ideal

text: performance

text: upper-bounded

text: by

text: both

text: the

text: TPP

text: and

text: the

text: product

text: of

text: AI

text: and

text: STREAMS

text: bandwidth.

text: STREAMS

text: Triad

text: is

text: one

text: of

text: the

text: most

text: popular

text: benchmarks

text: for

text: determining

text: the

text: achievable

text: memory

text: performance

text: of

text: a

text: given

text: machine.

text: Figure

text: denotes

text: the

text: estimated

text: memorybandwidth

text: as

text: a

text: function

text: of

text: number

text: of

text: cores

text: on

text: a

text: single

text: Mustang

text: and

text: Wolf

text: node.

text: It

text: is

text: interesting

text: to

text: note

text: that

text: although

text: the

text: Wolf

text: node

text: has

text: a

text: greater

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: s/SPOLFG s/SPOLFG 0.5 Galerkin TRON1 TRON2 TRON3 BLMVM 0.4 0.3 0.2 0.1 0 A1 B1 C1 A2 B2 C2 A3 B3 C3 Cases (a) Mustang 1.5 Galerkin TRON1 TRON2 TRON3 BLMVM 1 0.5 0 A1 B1 C1 A2 B2 C2 A3 B3 C3 Cases (b) Wolf Figure 7. Cube with a hole: measured ﬂoating-point rate (FLOPS/s) on a single core. times the amount of wall-clock time as the standard Galerkin method. TRON on the other hand, does not consume nearly as much time but tightening the KSP tolerances will gradually increase the amount of time. We are interested in determine why these optimization solvers consume more wall-clock time, whether it be mostly due to additional workload associated with optimization-based techniques or due to the presence of relatively more complicated and expensive data structures compared to the standard solvers used for the Galerkin formulation. The ﬁrst step is noting the total KSP and TAO iterations needed and how they vary with respect to problem size. Figure 5 depicts the KSP and TAO iterations for the Galerkin and BLMVM methods respectively. It is well-known that block Jacobi (also known as ILU(0)) requires more iterates as the size of the problem increases. In other words, the solver may exhibit poor scaling for extremely large problems, but we see that the BLMVM algorithm has an even poorer scaling rate of the solver iterates. For the TRON solvers, we document both the KSP and TAO iterates as shown in Figure 6. We see that tightening the KSP tolerance increases the number of KSP iterates but requires slightly fewer TAO iterates. This behavior indicates that the more accurate the computed gradient projection is, the fewer optimization loops the solver has to perform. 17 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: s/SPOLFG

font: default

text: s/SPOLFG

text: Galerkin

text: BLMVM

text: Cases

text: (a)

text: Mustang

text: Galerkin

text: BLMVM

text: Cases

text: (b)

text: Wolf

text: Figure

text: Cube

text: with

text: a

text: hole:

text: measured

text: ﬂoating-point

text: rate

text: (FLOPS/s)

text: on

text: a

text: single

text: core.

text: times

text: the

text: amount

text: of

text: wall-clock

text: time

text: as

text: the

text: standard

text: Galerkin

text: method.

text: TRON

text: on

text: the

text: other

text: hand,

text: does

text: not

text: consume

text: nearly

text: as

text: much

text: time

text: but

text: tightening

text: the

text: KSP

text: tolerances

text: will

text: gradually

text: increase

text: the

text: amount

text: of

text: time.

text: We

text: are

text: interested

text: in

text: determine

text: why

text: these

text: optimization

text: solvers

text: consume

text: more

text: wall-clock

text: time,

text: whether

text: it

text: be

text: mostly

text: due

text: to

text: additional

text: workload

text: associated

text: with

text: optimization-based

text: techniques

text: or

text: due

text: to

text: the

text: presence

text: of

text: relatively

text: more

text: complicated

text: and

text: expensive

text: data

text: structures

text: compared

text: to

text: the

text: standard

text: solvers

text: used

text: for

text: the

text: Galerkin

text: formulation.

text: The

text: ﬁrst

text: step

text: is

text: noting

text: the

text: total

text: KSP

text: and

text: TAO

text: iterations

text: needed

text: and

text: how

text: they

text: vary

text: with

text: respect

text: to

text: problem

text: size.

text: Figure

text: depicts

text: the

text: KSP

text: and

text: TAO

text: iterations

text: for

text: the

text: Galerkin

text: and

text: BLMVM

text: methods

text: respectively.

text: It

text: is

text: well-known

text: that

text: block

text: Jacobi

text: (also

text: known

text: as

text: requires

text: more

text: iterates

text: as

text: the

text: size

text: of

text: the

text: problem

text: increases.

text: In

text: other

text: words,

text: the

text: solver

text: may

text: exhibit

text: poor

text: scaling

text: for

text: extremely

text: large

text: problems,

text: but

text: we

text: see

text: that

text: the

text: BLMVM

text: algorithm

text: has

text: an

text: even

text: poorer

text: scaling

text: rate

text: of

text: the

text: solver

text: iterates.

text: For

text: the

text: TRON

text: solvers,

text: we

text: document

text: both

text: the

text: KSP

text: and

text: TAO

text: iterates

text: as

text: shown

text: in

text: Figure

text: We

text: see

text: that

text: tightening

text: the

text: KSP

text: tolerance

text: increases

text: the

text: number

text: of

text: KSP

text: iterates

text: but

text: requires

text: slightly

text: fewer

text: TAO

text: iterates.

text: This

text: behavior

text: indicates

text: that

text: the

text: more

text: accurate

text: the

text: computed

text: gradient

text: projection

text: is,

text: the

text: fewer

text: optimization

text: loops

text: the

text: solver

text: has

text: to

text: perform.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:90) (cid:90) (cid:88) − NΘ T − (cid:88) αj(cid:90) (cid:90) (cid:16) (cid:90)( δf(x, t)(cid:17))Θj(s)p(x, t)ds dx dt 0 Ω Ω j=1 NΘ T = α Θ (s)ds δf(x, t)p(x, t) dx dt. (14) (cid:80) (cid:82) j(cid:82) (cid:0) (cid:82) j (cid:1) 0 Ω Ω j=1 (cid:80) (cid:82) (cid:82) (cid:0) (cid:82) − (cid:1) T For the term α δf(x+s, t)Θ (s)ds p(x, t) dx dt we apply the substitu- j j 0 Ω ↔Ω j − − T tion y = x+s, then exchange x y, so that it recasts to α p(y, t)Θ (x − j j 0 Ω Ω j y)dy δf(x, t)dx dt. Then, again, we substitute s = x y and, by inserting also the former (cid:90) (cid:90) (cid:90) term, we get (cid:104) (cid:88) (cid:105) − T NΘ − − α (p(x s, t) p(x))Θ (s)ds δf(x, t)dx dt. (15) j j 0 Ω Ω j=1 (cid:90) (cid:90) For the variation of the time derivative, integrating by parts, one obtains | − T δf(x, t) p(x, t) T dx ∂ p(x, t) δf(x, t)dxdt. (16) 0 t Ω 0 (cid:82) (cid:82) The variation δf(x, 0) = 0 holds because of the Cauchy initial condition, while the variation in T can be deﬁned in some points X . Next, we integrate by parts the term l | − with the ﬁrst order derivative in x and obtain b T (δf(x, t)p(x, t)) Ωb dt b ∂ p δfdx dt. 0 − Ωa Ω x Due to periodicity in the ﬁrst term, δf(Ω , t)(p(Ω , t) p(Ω , t)) has to be zero, hence a b a p(Ω , t) = p(Ω , t). b a From(cid:90)the(cid:90) diﬀusive term we get (cid:90) (cid:90) −σ2 T −σ2 T − ∂2δfp(x, t)dxdt = [∂ δfp δf∂ p]Ωb dt + ∂2p(x, t)dxdt. (17) 2 x 2 x x Ωa x 0 Ω 0 Ω The ﬁrst boundary term is zero because of the periodic condition of the variation of the derivative of f at the boundaries, and because of the previous periodic condition on p. The second is analogous and has to vanish, we therefore get the continuity condition (cid:82) ∂ p(Ω , t) = ∂ p(Ω , t). x a x b By collecting all the terms under double integral, we get the adjoint equation. The remaining boundary term δf(x, T )p(x, T )dx will be considered below. Ω To calculate the variation on f in the functional J we perform an additional integration in space, so that 11 ##LTLine## ##LTLine##

label: equation

text: NΘ

text: T

text: δf(x,

text: t)ds

text: dx

text: dt

text: Ω

text: Ω

text: NΘ

text: T

text: α

text: Θ

text: (s)ds

text: δf(x,

text: t)p(x,

text: t)

text: dx

text: dt.

text: j

text: Ω

text: Ω

text: T

text: For

text: the

text: term

text: α

text: δf(x+s,

text: t)Θ

text: (s)ds

text: p(x,

text: t)

text: dx

text: dt

text: we

text: apply

text: the

text: substitu-

text: j

text: j

text: Ω

text: ↔Ω

text: j

text: T

text: tion

text: y

text: x+s,

text: then

text: exchange

text: x

text: y,

text: so

text: that

text: it

text: recasts

text: to

text: α

text: p(y,

text: t)Θ

text: (x

text: j

text: j

text: Ω

text: Ω

text: j

text: y)dy

text: δf(x,

text: t)dx

text: dt.

text: Then,

text: again,

text: we

text: substitute

text: s

text: x

text: y

text: and,

text: by

text: inserting

text: also

text: the

text: former

text: term,

text: we

text: get

text: T

text: NΘ

text: α

text: (p(x

text: s,

text: t)

text: p(x))Θ

text: (s)ds

text: δf(x,

text: t)dx

text: dt.

text: j

text: j

text: Ω

text: Ω

text: For

text: the

text: variation

text: of

text: the

text: time

text: derivative,

text: integrating

text: by

text: parts,

text: one

text: obtains

text: T

text: δf(x,

text: t)

text: p(x,

text: t)

text: T

text: dx

text: p(x,

text: t)

text: δf(x,

text: t)dxdt.

text: t

text: Ω

text: The

text: variation

text: δf(x,

text: holds

text: because

text: of

text: the

text: Cauchy

text: initial

text: condition,

text: while

text: the

text: variation

text: in

text: T

text: can

text: be

text: deﬁned

text: in

text: some

text: points

text: X

text: Next,

text: we

text: integrate

text: by

text: parts

text: the

text: term

text: l

text: with

text: the

text: ﬁrst

text: order

text: derivative

text: in

text: x

text: and

text: obtain

text: b

text: T

text: (δf(x,

text: t)p(x,

text: t))

text: Ωb

text: dt

text: b

text: p

text: δfdx

text: dt.

text: Ωa

text: Ω

text: x

text: Due

text: to

text: periodicity

text: in

text: the

text: ﬁrst

text: term,

text: δf(Ω

text: t)(p(Ω

text: t)

text: p(Ω

text: t))

text: has

text: to

text: be

text: zero,

text: hence

text: a

text: b

text: a

text: p(Ω

text: t)

text: p(Ω

text: t).

text: b

text: a

text: diﬀusive

text: term

text: we

text: get

text: −σ2

text: T

text: −σ2

text: T

text: t)dxdt

text: δfp

text: δf∂

text: p]Ωb

text: dt

text: t)dxdt.

text: x

text: x

text: x

text: Ωa

text: x

text: Ω

text: Ω

text: The

text: ﬁrst

text: boundary

text: term

text: is

text: zero

text: because

text: of

text: the

text: periodic

text: condition

text: of

text: the

text: variation

text: of

text: the

text: derivative

text: of

text: f

text: at

text: the

text: boundaries,

text: and

text: because

text: of

text: the

text: previous

text: periodic

text: condition

text: on

text: p.

text: The

text: second

text: is

text: analogous

text: and

text: has

text: to

text: vanish,

text: we

text: therefore

text: get

text: the

text: continuity

text: condition

text: p(Ω

text: t)

text: p(Ω

text: t).

text: x

text: a

text: x

text: b

text: By

text: collecting

text: all

text: the

text: terms

text: under

text: double

text: integral,

text: we

text: get

text: the

text: adjoint

text: equation.

text: The

text: remaining

text: boundary

text: term

text: δf(x,

text: T

text: )p(x,

text: T

text: )dx

text: will

text: be

text: considered

text: below.

text: Ω

text: To

text: calculate

text: the

text: variation

text: on

text: f

text: in

text: the

text: functional

text: J

text: we

text: perform

text: an

text: additional

text: integration

text: in

text: space,

text: so

text: that

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:88) 1 L − log(f(x, T ))δ(x X )dx, (18) l L Ω l=1 − where δ(.) is the δ Dirac me(cid:90)asure, then variate f(x, T ) + δf(x, T ), hence (cid:90) − log(f(x, T ) + δf(x, T ))δ(x X )dx l Ω − = (log(f(x, T )) + δf(x, T )/f(x, T ))δ(x X )dx, (19) l Ω (cid:90) (cid:90) so that the ﬁrst (cid:88)order terms plus the remaining boundary, give 1 L δf(x, T ) − δ(x X )dx + p(x, T )δf(x, T )dx. (20) l L f(x, T ) Ω Ω l=1 This expression have to be zero for each δf(x, T ). It represents the terminal con- (cid:80) − dition for the adjoint equation: that is p(X , T ) = 1/(Lf(x , T )), and p(x, T ) = 0, (cid:54) { } l l if x = X , . . . , X . In case of multiplicity of X the condition becomes p(X , T ) = − 1 L (cid:48) l l 1/(L (cid:48) f(X (cid:48), T )), with l running on the multiplicity value. l l  Summarizing, the adjoint equation (Kolmogorov’s backward equation) is as follows:  (cid:82) (cid:80)       − − − σ2 − − − ∂ p(x, t) b(cid:80)∂ p(x, t) ∂2p(x, t) NΘ α (p(x s, t) p(x, t))Θ (s)ds = 0  t x 2 x Ω j=1 j j      − −  p(x, T ) = 1 L δ(x X )/f(X , T ) L l=1 l l p(Ω , t) = p(Ω , t), ∂ p(Ω , t) = ∂ p(Ω , t). a b x a x b (21) We note that by reverting the sign of the time we get the same PIDE as the forward equation (up to a reﬂection of the drift and jump direction), hence this equation has a unique solution, also for the non regular ﬁnal value problem [8]. L − L Second, we variate in Eq.(13) the ﬁtting parameters (α + δα ) (α ), from which j j j we found(cid:90)the(cid:90)op(cid:90)timality equations: − − T − π (cid:48) (f(x + s, t) f(x, t))p(x, t)Θ (s) ds dx dt = 0, j = 1, . . . , N , (22) j j Θ 0 Ω Ω 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L

text: log(f(x,

text: T

text: ))δ(x

text: X

text: )dx,

text: l

text: L

text: Ω

text: where

text: δ(.)

text: is

text: the

text: δ

text: Dirac

text: then

text: variate

text: f(x,

text: T

text: δf(x,

text: T

text: hence

text: log(f(x,

text: T

text: δf(x,

text: T

text: ))δ(x

text: X

text: )dx

text: l

text: Ω

text: (log(f(x,

text: T

text: δf(x,

text: T

text: )/f(x,

text: T

text: ))δ(x

text: X

text: )dx,

text: l

text: Ω

text: so

text: that

text: the

text: ﬁrst

text: terms

text: plus

text: the

text: remaining

text: boundary,

text: give

text: L

text: δf(x,

text: T

text: δ(x

text: X

text: )dx

text: p(x,

text: T

text: )δf(x,

text: T

text: )dx.

text: l

text: L

text: f(x,

text: T

text: Ω

text: Ω

text: This

text: expression

text: have

text: to

text: be

text: zero

text: for

text: each

text: δf(x,

text: T

text: It

text: represents

text: the

text: terminal

text: con-

text: dition

text: for

text: the

text: adjoint

text: equation:

text: that

text: is

text: p(X

text: T

text: T

text: and

text: p(x,

text: T

text: l

text: l

text: if

text: x

text: X

text: X

text: In

text: case

text: of

text: multiplicity

text: of

text: X

text: the

text: condition

text: becomes

text: p(X

text: T

text: L

text: l

text: l

text: f(X

text: T

text: with

text: l

text: running

text: on

text: the

text: multiplicity

text: value.

text: l

text: l

text: Summarizing,

text: the

text: adjoint

text: equation

text: (Kolmogorov’s

text: backward

text: equation)

text: is

text: as

text: follows:

text: σ2

text: p(x,

text: t)

text: p(x,

text: t)

text: t)

text: NΘ

text: α

text: (p(x

text: s,

text: t)

text: p(x,

text: t))Θ

text: (s)ds

text: t

text: x

text: x

text: Ω

text: j

text: j

text: p(x,

text: T

text: L

text: δ(x

text: X

text: )/f(X

text: T

text: L

text: l

text: l

text: p(Ω

text: t)

text: p(Ω

text: t),

text: p(Ω

text: t)

text: p(Ω

text: t).

text: a

text: b

text: x

text: a

text: x

text: b

text: We

text: note

text: that

text: by

text: reverting

text: the

text: sign

text: of

text: the

text: time

text: we

text: get

text: the

text: same

text: PIDE

text: as

text: the

text: forward

text: equation

text: (up

text: to

text: a

text: reﬂection

text: of

text: the

text: drift

text: and

text: jump

text: direction),

text: hence

text: this

text: equation

text: has

text: a

text: unique

text: solution,

text: also

text: for

text: the

text: non

text: regular

text: ﬁnal

text: value

text: problem

text: L

text: L

text: Second,

text: we

text: variate

text: in

text: the

text: ﬁtting

text: parameters

text: (α

text: δα

text: (α

text: from

text: which

text: j

text: j

text: j

text: we

text: equations:

text: T

text: π

text: (f(x

text: s,

text: t)

text: f(x,

text: t))p(x,

text: t)Θ

text: (s)

text: ds

text: dx

text: dt

text: j

text: N

text: j

text: j

text: Θ

text: Ω

text: Ω

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − − ≥ ≤ where A˜ = A diag(A) is a positive matrix. Provided that fm 1 0 and δt 1/a the r.h.s. is a non negative vector. We observe that the matrix on the l.h.s is always diagonal dominant, hence it has a convergent regular splitting and consequently is an M-matrix − − [27]. Therefore, (I δtA) 1 is non negative and fm will be too. In order to prove the positivity of the BDF2 numerical scheme (35), we need of the following Lemma that gives a lower bound to the velocity of decreasing of the solution. ∈ R Lemma 1. Let the vector fm N be given non negative. Take a number ξ > 1, then the solution fm+q calculated with the Euler scheme of Eq. (37) after q time steps satisﬁes the following inequality ≥ fm+q fm/ξq, − ξ 1 provided that δt < , with parameters deﬁned in Eq. (34). aξ + β(1 + ω)/h Proof. A proof is given for a particular case in [11] (see also Refs. therein). Here we prove − it as follows. Given fm and fm+1 calculated with (37), let deﬁne v = ξfm+1 fm. By − − − − applying the operator I δtA, we get (I δtA)v = (ξI (I δtA))fm + δtξQ(fm; α¯), i.e. − − − (I δtA)v = ((ξ 1 δt(β(1 + ω)/h + aξ))I + δtA˜ + δtξR)fm, − ˜ where A = A diag(A) is a positive matrix. Now provided the bound for δt, then the ≥ r.h.s. is positive and from Th. 1 we get that v 0. By iterating that inequality q times, we get the thesis. Remark 3. The upper bound on δt in Lemma 1 results to be δt < 1/a for ξ > 1, hence the condition on the Lemma is stricter than those on non negativity of Thm. 1. Now we show a Lemma similar to Lemma 1 valid for the BDF2 scheme. ≤ − − Lemma 2. Let 1 < ξ < 3 and δt h(ξ 1)(3 ξ)/(aξh + 2β(1 + ω)) be the time step size of the numerical scheme of Eq. (35) that generates the sequence of vectors fm for ∗ ∗ − ∗ ≥ m = 2, 3, ... from the starting vectors f0, f1. If there exists m such that ξfm +1 fm 0 ∗ ≥ − ≥ ∗ and fm 0, then ξfm+1 fm 0 for all m > m . − − Proof. We apply the operator (3I 2δtA) to v = ξfm+2 fm+1, − − − − (3I 2δtA)v = ξ(3I 2δtA)fm+2 (3I 2δtA)fm+1 and use Eq. (35) to the ﬁrst term on the r.h.s. to get − − − − (3I 2δtA)v = [4ξ 3 δt(aξ + 2β(1 + ω)/h)]fm+1 ξfm + δt(2A˜ + ξR)fm+1, 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: A˜

text: A

text: diag(A)

text: is

text: a

text: positive

text: matrix.

text: Provided

text: that

text: fm

text: and

text: δt

text: the

text: r.h.s.

text: is

text: a

text: non

text: negative

text: vector.

text: We

text: observe

text: that

text: the

text: matrix

text: on

text: the

text: l.h.s

text: is

text: always

text: diagonal

text: dominant,

text: hence

text: it

text: has

text: a

text: convergent

text: regular

text: splitting

text: and

text: consequently

text: is

text: an

text: M-matrix

text: Therefore,

text: (I

text: δtA)

text: is

text: non

text: negative

text: and

text: fm

text: will

text: be

text: too.

text: In

text: order

text: to

text: prove

text: the

text: positivity

text: of

text: the

text: numerical

text: scheme

text: we

text: need

text: of

text: the

text: following

text: Lemma

text: that

text: gives

text: a

text: lower

text: bound

text: to

text: the

text: velocity

text: of

text: decreasing

text: of

text: the

text: solution.

text: R

text: Lemma

text: Let

text: the

text: vector

text: fm

text: N

text: be

text: given

text: non

text: negative.

text: Take

text: a

text: number

text: ξ

text: then

text: the

text: solution

text: fm+q

text: calculated

text: with

text: the

text: Euler

text: scheme

text: of

text: Eq.

text: after

text: q

text: time

text: steps

text: satisﬁes

text: the

text: following

text: inequality

text: fm+q

text: fm/ξq,

text: ξ

text: provided

text: that

text: δt

text: with

text: parameters

text: deﬁned

text: in

text: Eq.

text: aξ

text: ω)/h

text: Proof.

text: A

text: proof

text: is

text: given

text: for

text: a

text: particular

text: case

text: in

text: (see

text: also

text: Refs.

text: therein).

text: Here

text: we

text: prove

text: it

text: as

text: follows.

text: Given

text: fm

text: and

text: calculated

text: with

text: let

text: deﬁne

text: v

text: fm.

text: By

text: applying

text: the

text: operator

text: I

text: δtA,

text: we

text: get

text: (I

text: δtA)v

text: (ξI

text: (I

text: δtA))fm

text: δtξQ(fm;

text: α¯),

text: i.e.

text: (I

text: δtA)v

text: ((ξ

text: ω)/h

text: aξ))I

text: δtA˜

text: δtξR)fm,

text: where

text: A

text: A

text: diag(A)

text: is

text: a

text: positive

text: matrix.

text: Now

text: provided

text: the

text: bound

text: for

text: δt,

text: then

text: the

text: r.h.s.

text: is

text: positive

text: and

text: from

text: Th.

text: we

text: get

text: that

text: v

text: By

text: iterating

text: that

text: inequality

text: q

text: times,

text: we

text: get

text: the

text: thesis.

text: Remark

text: The

text: upper

text: bound

text: on

text: δt

text: in

text: Lemma

text: results

text: to

text: be

text: δt

text: for

text: ξ

text: hence

text: the

text: condition

text: on

text: the

text: Lemma

text: is

text: stricter

text: than

text: those

text: on

text: non

text: negativity

text: of

text: Thm.

text: Now

text: we

text: show

text: a

text: Lemma

text: similar

text: to

text: Lemma

text: valid

text: for

text: the

text: scheme.

text: Lemma

text: Let

text: ξ

text: and

text: δt

text: h(ξ

text: ξ)/(aξh

text: ω))

text: be

text: the

text: time

text: step

text: size

text: of

text: the

text: numerical

text: scheme

text: of

text: Eq.

text: that

text: generates

text: the

text: sequence

text: of

text: vectors

text: fm

text: for

text: m

text: from

text: the

text: starting

text: vectors

text: If

text: there

text: exists

text: m

text: such

text: that

text: ξfm

text: fm

text: and

text: fm

text: then

text: fm

text: for

text: all

text: m

text: m

text: Proof.

text: We

text: apply

text: the

text: operator

text: to

text: v

text: and

text: use

text: Eq.

text: to

text: the

text: ﬁrst

text: term

text: on

text: the

text: r.h.s.

text: to

text: get

text: δt(aξ

text: ξfm

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: R → φ : ( , +) (Ω, + ) (6) K be the group homomorphism deﬁned by φ(x) = (x) mod (K). Let X(t) = φ(Y (t)) be R a stochastic process on Ω. If Y (t) is a L´evy process on the group ( , +), the same applies to X(t) with respect to the group (Ω, + ). Note that in the deﬁnition of L´evy processes K R only the group structure of ( , +) enters. L´evy processes are naturally deﬁned on locally R compact Abelian groups like ( , +) or also (Ω, + ), see [8]. Let us now consider the K characteristic function of the Ω-valued process X(t) at time t. By the periodicity of Ω, Z only k values from 2π are needed. To derive a L´evy - Kinchine formula (2) for X(t) on K R Ω from that of Y (t) on , we consider for such values of k (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) E E eiX(t)k = eiφ(Y (t))k E = eiY (t)k = etψ(k). (7) Inserting (5), we obtain (cid:90) (cid:0) (cid:1) − σ2 (cid:90) − ψ(k) = ibk k2 + (cid:0)eisk 1 d(cid:1)ν(s) 2 R\{ } 0 − σ2 (cid:90) − = ibk k2 + (cid:0)eiφ(s)k (cid:1)1 dν(s) (8) 2 R\{ } 0 − σ2 − = ibk k2 + eisk 1 dφ∗ν(s), 2 \{ } Ω 0 with φ∗ν the image measure on ν under φ. Note that under the hypothesis that ν has − − → R support in [Ω , Ω ], ν can be reconstructed from φ∗ν as ν = φ∗ 1φ∗ν, where φ 1 : Ω a b is the natural embedding. Using this, we identify ν and φ∗ν in the following. Summing up, we consider the maximum likelihood parameter estimation problem with ﬁxed b, σ2, a piecewise linear density function with N grid points for the ﬁnite L´evy Θ measure and periodic boundary conditions on Ω. By reﬁning the grid for the linear inter- polation, enlarging the size of the torus and letting the ﬁxed diﬀusion go to zero, we can approximate the distribution of any L´evy process with one of our candidate processes in the topology set by weak convergence in law. This constitutes the hierarchy of maximum likelihood estimation procedures. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: R

text: φ

text: (Ω,

text: K

text: be

text: the

text: group

text: homomorphism

text: deﬁned

text: by

text: φ(x)

text: (x)

text: mod

text: (K).

text: Let

text: X(t)

text: φ(Y

text: (t))

text: be

text: R

text: a

text: stochastic

text: process

text: on

text: Ω.

text: If

text: Y

text: (t)

text: is

text: a

text: L´evy

text: process

text: on

text: the

text: group

text: the

text: same

text: applies

text: to

text: X(t)

text: with

text: respect

text: to

text: the

text: group

text: (Ω,

text: Note

text: that

text: in

text: the

text: deﬁnition

text: of

text: L´evy

text: processes

text: K

text: R

text: only

text: the

text: group

text: structure

text: of

text: enters.

text: L´evy

text: processes

text: are

text: naturally

text: deﬁned

text: on

text: locally

text: R

text: compact

text: Abelian

text: groups

text: like

text: or

text: also

text: (Ω,

text: see

text: Let

text: us

text: now

text: consider

text: the

text: K

text: characteristic

text: function

text: of

text: the

text: Ω-valued

text: process

text: X(t)

text: at

text: time

text: t.

text: By

text: the

text: periodicity

text: of

text: Ω,

text: Z

text: only

text: k

text: values

text: from

text: 2π

text: are

text: needed.

text: To

text: derive

text: a

text: L´evy

text: Kinchine

text: formula

text: for

text: X(t)

text: on

text: K

text: R

text: Ω

text: from

text: that

text: of

text: Y

text: (t)

text: on

text: we

text: consider

text: for

text: such

text: values

text: of

text: k

text: E

text: E

text: eiX(t)k

text: eiφ(Y

text: (t))k

text: E

text: eiY

text: (t)k

text: etψ(k).

text: Inserting

text: we

text: obtain

text: σ2

text: ψ(k)

text: ibk

text: R\{

text: σ2

text: ibk

text: dν(s)

text: R\{

text: σ2

text: ibk

text: eisk

text: dφ∗ν(s),

text: Ω

text: with

text: φ∗ν

text: the

text: image

text: measure

text: on

text: ν

text: under

text: φ.

text: Note

text: that

text: under

text: the

text: hypothesis

text: that

text: ν

text: has

text: R

text: support

text: in

text: [Ω

text: Ω

text: ν

text: can

text: be

text: reconstructed

text: from

text: φ∗ν

text: as

text: ν

text: φ∗

text: 1φ∗ν,

text: where

text: φ

text: Ω

text: a

text: b

text: is

text: the

text: natural

text: embedding.

text: Using

text: this,

text: we

text: identify

text: ν

text: and

text: φ∗ν

text: in

text: the

text: following.

text: Summing

text: up,

text: we

text: consider

text: the

text: maximum

text: likelihood

text: parameter

text: estimation

text: problem

text: with

text: ﬁxed

text: b,

text: a

text: piecewise

text: linear

text: density

text: function

text: with

text: N

text: grid

text: points

text: for

text: the

text: ﬁnite

text: L´evy

text: Θ

text: measure

text: and

text: periodic

text: boundary

text: conditions

text: on

text: Ω.

text: By

text: reﬁning

text: the

text: grid

text: for

text: the

text: linear

text: inter-

text: polation,

text: enlarging

text: the

text: size

text: of

text: the

text: torus

text: and

text: letting

text: the

text: ﬁxed

text: diﬀusion

text: go

text: to

text: zero,

text: we

text: can

text: approximate

text: the

text: distribution

text: of

text: any

text: L´evy

text: process

text: with

text: one

text: of

text: our

text: candidate

text: processes

text: in

text: the

text: topology

text: set

text: by

text: weak

text: convergence

text: in

text: law.

text: This

text: constitutes

text: the

text: hierarchy

text: of

text: maximum

text: likelihood

text: estimation

text: procedures.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Electromagnetic currents induced by color ﬁelds Naoto Tanji∗ Institut fu¨r Theoretische Physik, Universit¨at Heidelberg Philosophenweg 16, 69120 Heidelberg, Germany Abstract The quark production in classical color ﬁelds is investigated with a focus on the induction of an electromagnetic current by produced quarks. We show that the color SU(2) and the SU(3) theories lead signiﬁcantly diﬀerent results for the electromag- netic current. In uniform SU(2) color ﬁelds, the net electromagnetic current is not generated, while in SU(3) color ﬁelds the net current is induced depending on the color direction of background ﬁelds. Also the numerical study of the quark production in inhomogeneous color ﬁelds is done. Motivated by gauge ﬁeld conﬁgurations provided by the color glass condensate framework, we introduce an ensemble of randomly dis- tributed color electric ﬂuxtubes. The spectrum of photons emitted from the quarks by a classical process is shown. 1 Introduction Photons are one of the most important observables to explore the properties of high- energy matter produced in relativistic heavy-ion collisions, since they are penetrating probes which are free from strong interactions. In recent years, direct photon measurement by the PHENIX [1, 2] and the ALICE Collaborations [3, 4] posed a puzzle to theoretical understanding of photon production. The direct photon puzzle consists of two issues. (i) State-of-the-art theoretical calculations [5, 6, 7, 8] underestimate direct photon excess measured in experiments [1, 3]. (ii) A large elliptic ﬂow of direct photons comparable in magnitude to that of pions are observed [2, 4]. It is diﬃcult to explain these two issues simultaneously because a large direct photon yield favors photon emission at early time when temperature of the system is high, while a large elliptic ﬂow indicates that photons are emitted at later time when the ﬂow of matter is largely developed. This puzzle naturally drives people to think about a novel mechanism of photon production, e.g. [9, 10, 11, 12, 13]. In Ref. [14], it has been shown that the photon spectra measured in p+p, d+Au, Au+Au collisions at RHIC and Pb+Pb collisions at LHC satisfy geometric scaling. This result indicates that the photons are emitted in early time when the saturation momentum Q is the only dominant scale of the system. In the leading order description of the s color glass condensate, the system right after a heavy-ion collision consists of longitudinal color electric and magnetic ﬁelds [15, 16, 17]. This state is called Glasma. The photon production in the Glasma has been studied in Refs. [9, 10]. Although promising results are presented, these studies are based on a simpliﬁed model for the gluon distribution function ∗ E-mail address: tanji@thphys.uni-heidelberg.de 1 ##LTLine##

text: Electromagnetic

label: title

text: currents

text: induced

text: by

text: color

text: ﬁelds

text: Naoto

label: author

text: Tanji∗

text: Institut

label: date

text: fu¨r

text: Theoretische

text: Physik,

text: Universit¨at

text: Heidelberg

text: Philosophenweg

text: Heidelberg,

text: Germany

text: Abstract

text: The

label: abstract

text: quark

text: production

text: in

text: classical

text: color

text: ﬁelds

text: is

text: investigated

text: with

text: a

text: focus

text: on

text: the

text: induction

text: of

text: an

text: electromagnetic

text: current

text: by

text: produced

text: quarks.

text: We

text: show

text: that

text: the

text: color

text: and

text: the

text: theories

text: lead

text: signiﬁcantly

text: diﬀerent

text: results

text: for

text: the

text: electromag-

text: netic

text: current.

text: In

text: uniform

text: color

text: ﬁelds,

text: the

text: net

text: electromagnetic

text: current

text: is

text: not

text: generated,

text: while

text: in

text: color

text: ﬁelds

text: the

text: net

text: current

text: is

text: induced

text: depending

text: on

text: the

text: color

text: direction

text: of

text: background

text: ﬁelds.

text: Also

text: the

text: numerical

text: study

text: of

text: the

text: quark

text: production

text: in

text: inhomogeneous

text: color

text: ﬁelds

text: is

text: done.

text: Motivated

text: by

text: gauge

text: ﬁeld

text: conﬁgurations

text: provided

text: by

text: the

text: color

text: glass

text: condensate

text: framework,

text: we

text: introduce

text: an

text: ensemble

text: of

text: randomly

text: dis-

text: tributed

text: color

text: electric

text: ﬂuxtubes.

text: The

text: spectrum

text: of

text: photons

text: emitted

text: from

text: the

text: quarks

text: by

text: a

text: classical

text: process

text: is

text: shown.

text: Introduction

text: Photons

text: are

text: one

text: of

text: the

text: most

text: important

text: observables

text: to

text: explore

text: the

text: properties

text: of

text: high-

text: energy

text: matter

text: produced

text: in

text: relativistic

text: heavy-ion

text: collisions,

text: since

text: they

text: are

text: penetrating

text: probes

text: which

text: are

text: free

text: from

text: strong

text: interactions.

text: In

text: recent

text: years,

text: direct

text: photon

text: measurement

text: by

text: the

text: PHENIX

text: and

text: the

text: ALICE

text: Collaborations

text: posed

text: a

text: puzzle

text: to

text: theoretical

text: understanding

text: of

text: photon

text: production.

text: The

text: direct

text: photon

text: puzzle

text: consists

text: of

text: two

text: issues.

text: (i)

text: State-of-the-art

text: theoretical

text: calculations

text: underestimate

text: direct

text: photon

text: excess

text: measured

text: in

text: experiments

text: (ii)

text: A

text: large

text: elliptic

text: ﬂow

text: of

text: direct

text: photons

text: comparable

text: in

text: magnitude

text: to

text: that

text: of

text: pions

text: are

text: observed

text: It

text: is

text: diﬃcult

text: to

text: explain

text: these

text: two

text: issues

text: simultaneously

text: because

text: a

text: large

text: direct

text: photon

text: yield

text: favors

text: photon

text: emission

text: at

text: early

text: time

text: when

text: temperature

text: of

text: the

text: system

text: is

text: high,

text: while

text: a

text: large

text: elliptic

text: ﬂow

text: indicates

text: that

text: photons

text: are

text: emitted

text: at

text: later

text: time

text: when

text: the

text: ﬂow

text: of

text: matter

text: is

text: largely

text: developed.

text: This

text: puzzle

text: naturally

text: drives

text: people

text: to

text: think

text: about

text: a

text: novel

text: mechanism

text: of

text: photon

text: production,

text: e.g.

text: In

text: Ref.

text: it

text: has

text: been

text: shown

text: that

text: the

text: photon

text: spectra

text: measured

text: in

text: p+p,

text: d+Au,

text: Au+Au

text: collisions

text: at

text: RHIC

text: and

text: Pb+Pb

text: collisions

text: at

text: LHC

text: satisfy

text: geometric

text: scaling.

text: This

text: result

text: indicates

text: that

text: the

text: photons

text: are

text: emitted

text: in

text: early

text: time

text: when

text: the

text: saturation

text: momentum

text: Q

text: is

text: the

text: only

text: dominant

text: scale

text: of

text: the

text: system.

text: In

text: the

text: leading

text: order

text: description

text: of

text: the

text: s

text: color

text: glass

text: condensate,

text: the

text: system

text: right

text: after

text: a

text: heavy-ion

text: collision

text: consists

text: of

text: longitudinal

text: color

text: electric

text: and

text: magnetic

text: ﬁelds

text: This

text: state

text: is

text: called

text: Glasma.

text: The

text: photon

text: production

text: in

text: the

text: Glasma

text: has

text: been

text: studied

text: in

text: Refs.

text: Although

text: promising

text: results

text: are

text: presented,

text: these

text: studies

text: are

text: based

text: on

text: a

text: simpliﬁed

text: model

text: for

text: the

text: gluon

text: distribution

text: function

text: E-mail

text: address:

text: tanji@thphys.uni-heidelberg.de

text: ##LTLine##


text: [4] D. Lohner, J.Phys.Conf.Ser. 446, 012028 (2013). [5] H. van Hees, C. Gale, and R. Rapp, Phys.Rev. C84, 054906 (2011). [6] O. Linnyk, W. Cassing, and E. Bratkovskaya, Phys.Rev. C89, 034908 (2014). [7] C. Shen, U. W. Heinz, J.-F. Paquet, and C. Gale, Phys.Rev. C89, 044910 (2014). [8] C. Shen, U. W. Heinz, J.-F. Paquet, I. Kozlov, and C. Gale, Phys.Rev. C91, 024908 (2015). [9] M. Chiu et al., Nucl.Phys. A900, 16 (2013). [10] L. McLerran and B. Schenke, Nucl.Phys. A929, 71 (2014). [11] G. Basar, D. Kharzeev, and V. Skokov, Phys.Rev.Lett. 109, 202303 (2012). [12] K. Tuchin, Phys.Rev. C91, 014902 (2015). [13] C. Gale et al., Phys.Rev.Lett. 114, 072301 (2015). [14] C. Klein-B¨osing and L. McLerran, Phys.Lett. B734, 282 (2014). [15] A. Kovner, L. D. McLerran, and H. Weigert, Phys.Rev. D52, 6231 (1995). [16] A. Kovner, L. D. McLerran, and H. Weigert, Phys.Rev. D52, 3809 (1995). [17] T. Lappi and L. McLerran, Nucl.Phys. A772, 200 (2006). [18] E. Iancu, A. Leonidov, and L. McLerran, The Color glass condensate: An Introduc- tion, in QCD perspectives on hot and dense matter. Proceedings, NATO Advanced Study Institute, Summer School, Cargese, France, August 6-18, 2001, pages 73–145, 2002. [19] H. Weigert, Prog. Part. Nucl. Phys. 55, 461 (2005). [20] F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010). [21] F. Gelis and J. Jalilian-Marian, Phys. Rev. D66, 014021 (2002). [22] F. Gelis and J. Jalilian-Marian, Phys. Rev. D66, 094014 (2002). [23] R. Baier, A. H. Mueller, and D. Schiﬀ, Nucl. Phys. A741, 358 (2004). [24] A. Kovner and A. H. Rezaeian, Phys. Rev. D90, 014031 (2014). [25] F. Gelis, Int. J. Mod. Phys. A28, 1330001 (2013). [26] G. Aarts and J. Smit, Nucl.Phys. B555, 355 (1999). [27] S. Borsanyi and M. Hindmarsh, Phys.Rev. D79, 065010 (2009). [28] J. Berges, D. Gelfand, and D. Sexty, Phys.Rev. D89, 025001 (2014). [29] V. Kasper, F. Hebenstreit, and J. Berges, Phys.Rev. D90, 025016 (2014). [30] F. Gelis and N. Tanji, (2015). 19

label: reference

text: D.

text: Lohner,

text: J.Phys.Conf.Ser.

text: H.

text: van

text: Hees,

text: C.

text: Gale,

text: and

text: R.

text: Rapp,

text: Phys.Rev.

text: O.

text: Linnyk,

text: W.

text: Cassing,

text: and

text: E.

text: Bratkovskaya,

text: Phys.Rev.

text: C.

text: Shen,

text: U.

text: W.

text: Heinz,

text: J.-F.

text: Paquet,

text: and

text: C.

text: Gale,

text: Phys.Rev.

text: C.

text: Shen,

text: U.

text: W.

text: Heinz,

text: J.-F.

text: Paquet,

text: I.

text: Kozlov,

text: and

text: C.

text: Gale,

text: Phys.Rev.

text: M.

text: Chiu

text: et

text: al.,

text: Nucl.Phys.

text: L.

text: McLerran

text: and

text: B.

text: Schenke,

text: Nucl.Phys.

text: G.

text: Basar,

text: D.

text: Kharzeev,

text: and

text: V.

text: Skokov,

text: Phys.Rev.Lett.

text: K.

text: Tuchin,

text: Phys.Rev.

text: C.

text: Gale

text: et

text: al.,

text: Phys.Rev.Lett.

text: C.

text: Klein-B¨osing

text: and

text: L.

text: McLerran,

text: Phys.Lett.

text: A.

text: Kovner,

text: L.

text: D.

text: McLerran,

text: and

text: H.

text: Weigert,

text: Phys.Rev.

text: A.

text: Kovner,

text: L.

text: D.

text: McLerran,

text: and

text: H.

text: Weigert,

text: Phys.Rev.

text: T.

text: Lappi

text: and

text: L.

text: McLerran,

text: Nucl.Phys.

text: E.

text: Iancu,

text: A.

text: Leonidov,

text: and

text: L.

text: McLerran,

text: The

text: Color

text: glass

text: condensate:

text: An

text: Introduc-

text: tion,

text: in

text: QCD

text: perspectives

text: on

text: hot

text: and

text: dense

text: matter.

text: Proceedings,

text: NATO

text: Advanced

text: Study

text: Institute,

text: Summer

text: School,

text: Cargese,

text: France,

text: August

text: pages

text: H.

text: Weigert,

text: Prog.

text: Part.

text: Nucl.

text: Phys.

text: F.

text: Gelis,

text: E.

text: Iancu,

text: J.

text: Jalilian-Marian,

text: and

text: R.

text: Venugopalan,

text: Ann.

text: Rev.

text: Nucl.

text: Part.

text: Sci.

text: F.

text: Gelis

text: and

text: J.

text: Jalilian-Marian,

text: Phys.

text: Rev.

text: F.

text: Gelis

text: and

text: J.

text: Jalilian-Marian,

text: Phys.

text: Rev.

text: R.

text: Baier,

text: A.

text: H.

text: Mueller,

text: and

text: D.

text: Schiﬀ,

text: Nucl.

text: Phys.

text: A.

text: Kovner

text: and

text: A.

text: H.

text: Rezaeian,

text: Phys.

text: Rev.

text: F.

text: Gelis,

text: Int.

text: J.

text: Mod.

text: Phys.

text: G.

text: Aarts

text: and

text: J.

text: Smit,

text: Nucl.Phys.

text: S.

text: Borsanyi

text: and

text: M.

text: Hindmarsh,

text: Phys.Rev.

text: J.

text: Berges,

text: D.

text: Gelfand,

text: and

text: D.

text: Sexty,

text: Phys.Rev.

text: V.

text: Kasper,

text: F.

text: Hebenstreit,

text: and

text: J.

text: Berges,

text: Phys.Rev.

text: F.

text: Gelis

text: and

text: N.

text: Tanji,


text: with useful information to understand more complicated and realistic situation, which is discussed in the next section. Furthermore, the uniform ﬁeld conﬁgurations can be used as a benchmark test for the real-time lattice numerical computations. By comparing numerical results with known analytic results, we can check the validity of our numerical method on lattice before applying it to ﬁeld conﬁgurations that analytical solutions are not available. See Appendix for details of the lattice numerical method. In this paper, we consider only one family of quark ﬂavor which has an EM charge e and mass m, for simplicity. We will brieﬂy comment on the eﬀects of multi-ﬂavors in Sec. . 2.1 Abelianization We brieﬂy review a method to simplify the color structure of the Dirac equation under uniform color electromagnetic ﬁelds [33, 34]. Thanks to this simpliﬁcation, the induction of EM currents in uniform color ﬁelds can be understood intuitively. If the ﬁeld strength is spatially uniform, its color dependence can be factorized as F a = F na , (1) µν µν where na is a constant vector in color space. This kind of ﬁelds are called covariant constant ﬁelds [35, 36]. This ﬁeld strength is given by the gauge ﬁeld Aa(x) = A (x)na , (2) µ µ where A (x) is related with F by F = ∂ A − ∂ A . The gauge ﬁeld (2) is a µ µν µν µ ν ν µ solution of the Yang–Mills equation. Because the color vector na is constant, the struc- ture of the covariant derivative D = ∂ − igA (x)naT a can be reduced from SU(N ) to µ µ µ c [U(1)]Nc. Since naT a is an N × N hermitian matrix, one can diagonalize it by a unitary c c transformation as UnaT aU† = diag (w , · · · , w ) . (3) 1 Nc The color-rotated quark ﬁeld ψ(cid:48) = (Uψ) (i = 1, · · · , N ) obeys i i c [iγµ (∂ − iw gA ) − m] ψ(cid:48) = 0 , (4) µ i µ i which has the same structure as the U(1) theory but a coupling e is replaced by the eﬀective coupling w g. i The SU(2) color space is isotropic since the (cid:18)rank of(cid:19)SU(2) is one. For any unit color vector na, the color matrix naT a takes the same form after diagonalization: 1 1 UnaT aU† = diag , − . (5) 2 2 Therefore, the eﬀective couplings are always +g/2 and −g/2. For N ≥ 3, the color space is anisotropic in a sense that the eﬀective couplings depend c on the direction of the color vector na in a gauge-invariant way. For SU(3) , there are two c independent diagonal generators. If we take the generators T a = 1λa with the Gell-Mann 2 3 ##LTLine## ##LTLine## ##LTLine##

text: with

text: useful

text: information

text: to

text: understand

text: more

text: complicated

text: and

text: realistic

text: situation,

text: which

text: is

text: discussed

text: in

text: the

text: next

text: section.

text: Furthermore,

text: the

text: uniform

text: ﬁeld

text: conﬁgurations

text: can

text: be

text: used

text: as

text: a

text: benchmark

text: test

text: for

text: the

text: real-time

text: lattice

text: numerical

text: computations.

text: By

text: comparing

text: numerical

text: results

text: with

text: known

text: analytic

text: results,

text: we

text: can

text: check

text: the

text: validity

text: of

text: our

text: numerical

text: method

text: on

text: lattice

text: before

text: applying

text: it

text: to

text: ﬁeld

text: conﬁgurations

text: that

text: analytical

text: solutions

text: are

text: not

text: available.

text: See

text: Appendix

text: for

text: details

text: of

text: the

text: lattice

text: numerical

text: method.

text: In

text: this

text: paper,

text: we

text: consider

text: only

text: one

text: family

text: of

text: quark

text: ﬂavor

text: which

text: has

text: an

text: EM

text: charge

text: e

text: and

text: mass

text: m,

text: for

text: simplicity.

text: We

text: will

text: brieﬂy

text: comment

text: on

text: the

text: eﬀects

text: of

text: multi-ﬂavors

text: in

text: Sec.

text: Abelianization

text: We

text: brieﬂy

text: review

text: a

text: method

text: to

text: simplify

text: the

text: color

text: structure

text: of

text: the

text: Dirac

text: equation

text: under

text: uniform

text: color

text: electromagnetic

text: ﬁelds

text: Thanks

text: to

text: this

text: simpliﬁcation,

text: the

text: induction

text: of

text: EM

text: currents

text: in

text: uniform

text: color

text: ﬁelds

text: can

text: be

text: understood

text: intuitively.

text: If

text: the

text: ﬁeld

text: strength

text: is

text: spatially

text: uniform,

text: its

text: color

text: dependence

text: can

text: be

text: factorized

text: as

text: F

text: a

text: F

text: µν

text: µν

text: where

text: is

text: a

text: constant

text: vector

text: in

text: color

text: space.

text: This

text: kind

text: of

text: ﬁelds

text: are

text: called

text: covariant

text: constant

text: ﬁelds

text: This

text: ﬁeld

text: strength

text: is

text: given

text: by

text: the

text: gauge

text: ﬁeld

text: Aa(x)

text: A

text: (x)na

text: µ

text: µ

text: where

text: A

text: (x)

text: is

text: related

text: with

text: F

text: by

text: F

text: A

text: A

text: The

text: gauge

text: ﬁeld

text: is

text: a

text: µ

text: µν

text: µν

text: µ

text: ν

text: ν

text: µ

text: solution

text: of

text: the

text: Yang–Mills

text: equation.

text: Because

text: the

text: color

text: vector

text: is

text: constant,

text: the

text: struc-

text: ture

text: of

text: the

text: covariant

text: derivative

text: D

text: igA

text: (x)naT

text: a

text: can

text: be

text: reduced

text: from

text: SU(N

text: to

text: µ

text: µ

text: µ

text: c

text: Since

text: naT

text: a

text: is

text: an

text: N

text: N

text: hermitian

text: matrix,

text: one

text: can

text: diagonalize

text: it

text: by

text: a

text: unitary

text: c

text: c

text: transformation

text: as

text: UnaT

text: aU†

text: diag

text: (w

text: w

text: Nc

text: The

text: color-rotated

text: quark

text: ﬁeld

text: (Uψ)

text: (i

text: N

text: obeys

text: i

text: i

text: c

text: [iγµ

text: iw

text: gA

text: m]

text: µ

text: i

text: µ

text: i

text: which

text: has

text: the

text: same

text: structure

text: as

text: the

text: theory

text: but

text: a

text: coupling

text: e

text: is

text: replaced

text: by

text: the

text: eﬀective

text: coupling

text: w

text: g.

text: i

text: The

text: color

text: space

text: is

text: isotropic

text: since

text: the

text: is

text: one.

text: For

text: any

text: unit

text: color

text: vector

text: na,

text: the

text: color

text: matrix

text: naT

text: a

text: takes

text: the

text: same

text: form

text: after

text: diagonalization:

text: UnaT

text: aU†

text: diag

text: Therefore,

text: the

text: eﬀective

text: couplings

text: are

text: always

text: and

text: For

text: N

text: the

text: color

text: space

text: is

text: anisotropic

text: in

text: a

text: sense

text: that

text: the

text: eﬀective

text: couplings

text: depend

text: c

text: on

text: the

text: direction

text: of

text: the

text: color

text: vector

text: in

text: a

text: gauge-invariant

text: way.

text: For

text: there

text: are

text: two

text: c

text: independent

text: diagonal

text: generators.

text: If

text: we

text: take

text: the

text: generators

text: T

text: a

text: with

text: the

text: Gell-Mann

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: matrix λa, the diagonal generators are T 3 and T 8, by which the diagonalized color matrix naT a can be expanded: UnaT aU† = T 3 cos θ − T 8 sin θ. (6) (cid:2) (cid:3) The angle θ is related with the color vector na through the second Casimir invariant C = dabcnanbnc 2 as [34] 2 sin2 3θ = 3C . (7) 2 By this equation, the angle θ characterizes the color direction of the vector na in a gauge-invariant way. The eigenvalues of the m(cid:16)atrix n(cid:17)aT a are parametrized by θ as (cid:18) (cid:19) 1 π w = √ cos θ + , 1 3 6 (cid:18) (cid:19) 1 5π w = √ cos θ + , (8) 2 3 6 1 3π w = √ cos θ + . 3 3 2 Unlike the SU(2) case, these eﬀective couplings have the color direction dependence. c 2.2 EM currents in uniform color electric ﬁelds Having discussed the diagonalization (Abelianization) of the color matrix, one can easily prove the cancellation of EM currents in SU(2) uniform electric ﬁelds (see Fig. 1). c Let us suppose that the electric ﬁeld is applied to the z-direction with the strength E 0 (> 0). In the case of SU(2) , the eﬀective couplings are independent of the color direction c of the gauge ﬁeld. A quark with color 1 couples to the electric ﬁeld with the eﬀective coupling of +1g, and it also has an EM charge +e (we assume e > 0). This quark is 2 accelerated to the +z-direction by the electric ﬁeld and causes a positive EM current. An anti-quark is accelerated to the opposite direction but it has a negative EM charge, so that it also induces a positive EM current. On the other hand, a quark with color 2 has the eﬀective coupling −1g. Therefore, this quark is accelerated to −z-direction. 2 Although the color charges are opposite in sign between color 1 and 2, the EM charge is independent of the color components. Consequently, the quark with color 2 induces a color 1 color2 Figure 1: SU(2) quarks in a constant color electric ﬁeld. c 4 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: matrix

text: λa,

text: the

text: diagonal

text: generators

text: are

text: T

text: and

text: T

text: by

text: which

text: the

text: diagonalized

text: color

text: matrix

text: naT

text: a

text: can

text: be

text: expanded:

text: UnaT

text: aU†

text: T

text: cos

text: θ

text: T

text: sin

text: θ.

text: The

text: angle

text: θ

text: is

text: related

text: with

text: the

text: color

text: vector

text: through

text: the

text: second

text: Casimir

text: invariant

text: C

text: dabcnanbnc

text: as

text: 3θ

text: By

text: this

text: equation,

text: the

text: angle

text: θ

text: characterizes

text: the

text: color

text: direction

text: of

text: the

text: vector

text: in

text: a

text: gauge-invariant

text: way.

text: The

text: eigenvalues

text: of

text: the

text: a

text: are

text: parametrized

text: by

text: θ

text: as

text: π

text: w

text: cos

text: θ

text: 5π

text: w

text: cos

text: θ

text: 3π

text: w

text: cos

text: θ

text: Unlike

text: the

text: case,

text: these

text: eﬀective

text: couplings

text: have

text: the

text: color

text: direction

text: dependence.

text: c

text: EM

text: currents

text: in

text: uniform

text: color

text: electric

text: ﬁelds

text: Having

text: discussed

text: the

text: diagonalization

text: (Abelianization)

text: of

text: the

text: color

text: matrix,

text: one

text: can

text: easily

text: prove

text: the

text: cancellation

text: of

text: EM

text: currents

text: in

text: uniform

text: electric

text: ﬁelds

text: (see

text: Fig.

text: c

text: Let

text: us

text: suppose

text: that

text: the

text: electric

text: ﬁeld

text: is

text: applied

text: to

text: the

text: z-direction

text: with

text: the

text: strength

text: E

text: In

text: the

text: case

text: of

text: the

text: eﬀective

text: couplings

text: are

text: independent

text: of

text: the

text: color

text: direction

text: c

text: of

text: the

text: gauge

text: ﬁeld.

text: A

text: quark

text: with

text: color

text: couples

text: to

text: the

text: electric

text: ﬁeld

text: with

text: the

text: eﬀective

text: coupling

text: of

text: and

text: it

text: also

text: has

text: an

text: EM

text: charge

text: +e

text: (we

text: assume

text: e

text: This

text: quark

text: is

text: accelerated

text: to

text: the

text: +z-direction

text: by

text: the

text: electric

text: ﬁeld

text: and

text: causes

text: a

text: positive

text: EM

text: current.

text: An

text: anti-quark

text: is

text: accelerated

text: to

text: the

text: opposite

text: direction

text: but

text: it

text: has

text: a

text: negative

text: EM

text: charge,

text: so

text: that

text: it

text: also

text: induces

text: a

text: positive

text: EM

text: current.

text: On

text: the

text: other

text: hand,

text: a

text: quark

text: with

text: color

text: has

text: the

text: eﬀective

text: coupling

text: Therefore,

text: this

text: quark

text: is

text: accelerated

text: to

text: −z-direction.

text: Although

text: the

text: color

text: charges

text: are

text: opposite

text: in

text: sign

text: between

text: color

text: and

text: the

text: EM

text: charge

text: is

text: independent

text: of

text: the

text: color

text: components.

text: Consequently,

text: the

text: quark

text: with

text: color

text: induces

text: a

text: color

text: Figure

text: quarks

text: in

text: a

text: constant

text: color

text: electric

text: ﬁeld.

text: c

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Draft version May 27, 2020 Preprint typeset using LATEX style emulateapj v. 12/16/11 “SLIMPLECTIC” INTEGRATORS: VARIATIONAL INTEGRATORS FOR GENERAL NONCONSERVATIVE SYSTEMS † David Tsang 1, , Chad R. Galley2, Leo C. Stein3,4, Alec Turner1 Draft version May 27, 2020 ABSTRACT Symplectic integrators are widely used for long-term integration of conservative astrophysical problems due to their ability to preserve the constants of motion; however, they cannot in general be applied in the presence of nonconservative interactions. In this Letter, we develop the “slimplectic” integrator, a new type of numerical integrator that shares many of the beneﬁts of traditional symplectic integrators yet is applicable to general nonconservative systems. We utilize a ﬁxed-time-step variational integrator formalism applied to the principle of stationary nonconservative action developed in Galley (2013); Galley, Tsang, & Stein (2014). As a result, the generalized momenta and energy (Noether current) evolutions are well-tracked. We discuss several example systems, including damped harmonic oscil- lators, Poynting-Robertson drag, and gravitational radiation reaction, by utilizing our new publicly available code to demonstrate the slimplectic integrator algorithm. Slimplectic integrators are well-suited for integrations of systems where nonconservative eﬀects play an important role in the long-term dynamical evolution. As such they are particularly appropriate for cosmological or celestial N-body dynamics problems where nonconservative interactions, e.g. gas interactions or dissipative tides, can play an important role. Keywords: 1. INTRODUCTION proach (Marsden & West 2001; Lew et al. 2004). Here, we utilize the more general nonconservative action princi- Symplectic integrators are a class of mappings that al- ple, recently developed by Galley (2013); Galley, Tsang, low for numerical integration of conservative dynamical & Stein (2014). This formalism was developed to accom- systems and which, up to round-oﬀ, exactly preserve cer- modate generically the causal dynamics of untracked or tain constants of motion (e.g. the symplectic form). As inaccessible degrees of freedom that might result from a result the integrations do not suﬀer from numerical an integrating-out or coarse-graining procedure at the “dissipation” which would cause an unphysical drift over level of an action/Lagrangian/Hamiltonian. many dynamical times. Due to these properties, sym- In this Letter, we develop variational integrators from plectic integrators are widely used in the long-term inte- the nonconservative action principle. The resulting map- gration of many physical systems, particularly in celes- pings are no longer symplectic, as the symplectic form tial dynamics (Wisdom & Holman 1991; Gladman et al. (and momenta) are no longer conserved, but evolve ac- 1991; Levison & Duncan 1994; Rein & Tamayo 2015). cording to the nonconservative dynamics. We instead Conservative variational integrators (see e.g. Marsden refer this type of numerical integrator as “slimplectic” & West 2001) are a subclass of symplectic integrators since phase space volumes tend to slim down for dissipa- where the mappings are determined by the extremiza- tion of a discretized action.5 The discretized action can tive systems. We show that our method inherits many of the same performance features of the symplectic in- inherit the symmetries of the full action such that, by tegrator. Previous works have demonstrated some suc- Noether’s theorem, the discrete equations of motion ex- cess by including weakly dissipative forces in 2nd-order actly conserve the symplectic form and the momenta. symplectic integrators (see e.g. Malhotra 1994; Cordeiro Since discretizing the time coordinate breaks the con- et al. 1996; Mikkola 1997; Hamilton et al. 1999; Zhang tinuous time-shift symmetry, ﬁxed-time-step variational & Hamilton 2007); in fact, these can be shown to be integrators that preserve the symplectic form and the particular cases of our more general (arbitrary order) momenta cannot also conserve energy (Ge & Marsden method. For brevity, we focus on a basic ﬁxed-time- 1988). However, the energy error tends to be bounded step slimplectic integrator leaving further developments, by a constant, even over long integration times (see e.g. such as adaptive time-stepping and detailed discussion of Lew et al. 2004, and references therein), in contrast with Noether current evolution, to a longer follow-up paper. traditional integration methods where the error tends to grow with time. Variational integrators can be applied to some dis- sipative problems using the Lagrange-d’Alembert ap- 1 Department of Physics, McGill University, Montreal, QC, H3A 2T8, Canada † dtsang@physics.mcgill.ca 2 Theoretical Astrophysics, Walter Burke Institute for Theo- retical Physics, California Institute of Technology, Pasadena, CA 91125, USA 3 Center for Radiophysics and Space Research, Cornell Uni- versity, Ithaca, NY 14853, USA 4 Einstein Fellow 5 Most symplectic integrators can be written as (local) varia- tional integrators.

text: Draft

label: section

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: “SLIMPLECTIC”

text: INTEGRATORS:

text: VARIATIONAL

text: INTEGRATORS

text: FOR

text: GENERAL

text: NONCONSERVATIVE

text: SYSTEMS

text: David

text: Tsang

text: Chad

text: R.

text: Leo

text: C.

text: Alec

text: Draft

text: version

text: May

text: ABSTRACT

text: Symplectic

text: integrators

text: are

text: widely

text: used

text: for

text: long-term

text: integration

text: of

text: conservative

text: astrophysical

text: problems

text: due

text: to

text: their

text: ability

text: to

text: preserve

text: the

text: constants

text: of

text: motion;

text: however,

text: they

text: cannot

text: in

text: general

text: be

text: applied

text: in

text: the

text: presence

text: of

text: nonconservative

text: interactions.

text: In

text: this

text: Letter,

text: we

text: develop

text: the

text: “slimplectic”

text: integrator,

text: a

text: new

text: type

text: of

text: numerical

text: integrator

text: that

text: shares

text: many

text: of

text: the

text: beneﬁts

text: of

text: traditional

text: symplectic

text: integrators

text: yet

text: is

text: applicable

text: to

text: general

text: nonconservative

text: systems.

text: We

text: utilize

text: a

text: ﬁxed-time-step

text: variational

text: integrator

text: formalism

text: applied

text: to

text: the

text: principle

text: of

text: stationary

text: nonconservative

text: action

text: developed

text: in

text: Galley

text: Galley,

text: Tsang,

text: Stein

text: As

text: a

text: result,

text: the

text: generalized

text: momenta

text: and

text: energy

text: (Noether

text: current)

text: evolutions

text: are

text: well-tracked.

text: We

text: discuss

text: several

text: example

text: systems,

text: including

text: damped

text: harmonic

text: oscil-

text: lators,

text: Poynting-Robertson

text: drag,

text: and

text: gravitational

text: radiation

text: reaction,

text: by

text: utilizing

text: our

text: new

text: publicly

text: available

text: code

text: to

text: demonstrate

text: the

text: slimplectic

text: integrator

text: algorithm.

text: Slimplectic

text: integrators

text: are

text: well-suited

text: for

text: integrations

text: of

text: systems

text: where

text: nonconservative

text: eﬀects

text: play

text: an

text: important

text: role

text: in

text: the

text: long-term

text: dynamical

text: evolution.

text: As

text: such

text: they

text: are

text: particularly

text: appropriate

text: for

text: cosmological

text: or

text: celestial

text: N-body

text: dynamics

text: problems

text: where

text: nonconservative

text: interactions,

text: e.g.

text: gas

text: interactions

text: or

text: dissipative

text: tides,

text: can

text: play

text: an

text: important

text: role.

text: Keywords:

text: INTRODUCTION

text: proach

text: (Marsden

text: West

text: Lew

text: et

text: al.

text: Here,

text: we

text: utilize

text: the

text: more

text: general

text: nonconservative

text: action

text: princi-

text: Symplectic

text: integrators

text: are

text: a

text: class

text: of

text: mappings

text: that

text: al-

text: ple,

text: recently

text: developed

text: by

text: Galley

text: Galley,

text: Tsang,

text: low

text: for

text: numerical

text: integration

text: of

text: conservative

text: dynamical

text: Stein

text: This

text: formalism

text: was

text: developed

text: to

text: accom-

text: systems

text: and

text: which,

text: up

text: to

text: round-oﬀ,

text: exactly

text: preserve

text: cer-

text: modate

text: generically

text: the

text: causal

text: dynamics

text: of

text: untracked

text: or

text: tain

text: constants

text: of

text: motion

text: (e.g.

text: the

text: symplectic

text: form).

text: As

text: inaccessible

text: degrees

text: of

text: freedom

text: that

text: might

text: result

text: from

text: a

text: result

text: the

text: integrations

text: do

text: not

text: suﬀer

text: from

text: numerical

text: an

text: integrating-out

text: or

text: coarse-graining

text: procedure

text: at

text: the

text: “dissipation”

text: which

text: would

text: cause

text: an

text: unphysical

text: drift

text: over

text: level

text: of

text: an

text: action/Lagrangian/Hamiltonian.

text: many

text: dynamical

text: times.

text: Due

text: to

text: these

text: properties,

text: sym-

text: In

text: this

text: Letter,

text: we

text: develop

text: variational

text: integrators

text: from

text: plectic

text: integrators

text: are

text: widely

text: used

text: in

text: the

text: long-term

text: inte-

text: the

text: nonconservative

text: action

text: principle.

text: The

text: resulting

text: map-

text: gration

text: of

text: many

text: physical

text: systems,

text: particularly

text: in

text: celes-

text: pings

text: are

text: no

text: longer

text: symplectic,

text: as

text: the

text: symplectic

text: form

text: tial

text: dynamics

text: (Wisdom

text: Holman

text: Gladman

text: et

text: al.

text: (and

text: momenta)

text: are

text: no

text: longer

text: conserved,

text: but

text: evolve

text: ac-

text: Levison

text: Duncan

text: Rein

text: Tamayo

text: cording

text: to

text: the

text: nonconservative

text: dynamics.

text: We

text: instead

text: Conservative

text: variational

text: integrators

text: (see

text: e.g.

text: Marsden

text: refer

text: this

text: type

text: of

text: numerical

text: integrator

text: as

text: “slimplectic”

text: West

text: are

text: a

text: subclass

text: of

text: symplectic

text: integrators

text: since

text: phase

text: space

text: volumes

text: tend

text: to

text: slim

text: down

text: for

text: dissipa-

text: where

text: the

text: mappings

text: are

text: determined

text: by

text: the

text: extremiza-

text: tion

text: of

text: a

text: discretized

text: The

text: discretized

text: action

text: can

text: tive

text: systems.

text: We

text: show

text: that

text: our

text: method

text: inherits

text: many

text: of

text: the

text: same

text: performance

text: features

text: of

text: the

text: symplectic

text: in-

text: inherit

text: the

text: symmetries

text: of

text: the

text: full

text: action

text: such

text: that,

text: by

text: tegrator.

text: Previous

text: works

text: have

text: demonstrated

text: some

text: suc-

text: Noether’s

text: theorem,

text: the

text: discrete

text: equations

text: of

text: motion

text: ex-

text: cess

text: by

text: including

text: weakly

text: dissipative

text: forces

text: in

text: actly

text: conserve

text: the

text: symplectic

text: form

text: and

text: the

text: momenta.

text: symplectic

text: integrators

text: (see

text: e.g.

text: Malhotra

text: Cordeiro

text: Since

text: discretizing

text: the

text: time

text: coordinate

text: breaks

text: the

text: con-

text: et

text: al.

text: Mikkola

text: Hamilton

text: et

text: al.

text: Zhang

text: tinuous

text: time-shift

text: symmetry,

text: ﬁxed-time-step

text: variational

text: Hamilton

text: in

text: fact,

text: these

text: can

text: be

text: shown

text: to

text: be

text: integrators

text: that

text: preserve

text: the

text: symplectic

text: form

text: and

text: the

text: particular

text: cases

text: of

text: our

text: more

text: general

text: (arbitrary

text: order)

text: momenta

text: cannot

text: also

text: conserve

text: energy

text: (Ge

text: Marsden

text: method.

text: For

text: brevity,

text: we

text: focus

text: on

text: a

text: basic

text: ﬁxed-time-

text: However,

text: the

text: energy

text: error

text: tends

text: to

text: be

text: bounded

text: step

text: slimplectic

text: integrator

text: leaving

text: further

text: developments,

text: by

text: a

text: constant,

text: even

text: over

text: long

text: integration

text: times

text: (see

text: e.g.

text: such

text: as

text: adaptive

text: time-stepping

text: and

text: detailed

text: discussion

text: of

text: Lew

text: et

text: al.

text: and

text: references

text: therein),

text: in

text: contrast

text: with

text: Noether

text: current

text: evolution,

text: to

text: a

text: longer

text: follow-up

text: paper.

text: traditional

text: integration

text: methods

text: where

text: the

text: error

text: tends

text: to

text: grow

text: with

text: time.

text: Variational

text: integrators

text: can

text: be

text: applied

text: to

text: some

text: dis-

text: sipative

text: problems

text: using

text: the

text: Lagrange-d’Alembert

text: ap-

text: Department

text: of

text: Physics,

text: McGill

text: University,

text: Montreal,

text: QC,

text: Canada

text: dtsang@physics.mcgill.ca

text: Theoretical

text: Astrophysics,

text: Walter

text: Burke

text: Institute

text: for

text: Theo-

text: retical

text: Physics,

text: California

text: Institute

text: of

text: Technology,

text: Pasadena,

text: CA

text: USA

text: Center

text: for

text: Radiophysics

text: and

text: Space

text: Research,

text: Cornell

text: Uni-

text: versity,

text: Ithaca,

text: NY

text: USA

text: Einstein

text: Fellow

text: Most

text: symplectic

text: integrators

text: can

text: be

text: written

text: as

text: (local)

text: varia-

text: tional

text: integrators.


text: 3 eﬀective modulus. Considering the DMT adhesive force F = 2πwR acting normal to the contact surface − − DMT c mQ¨ = K (Q u + Rθ ) [19, 20] (where w is the work of adhesion between two j S j 0,j j − surfaces), the net normal force is given by + G (Q 2Q + Q − ) N j+1 j j 1 − − mZ¨ = K (Z w ) j N j 0,j − − − − + G [Z 2Z + Z − R(θ θ − )] F = F F . (2) S j+1 j j 1 j+1 j 1 N,net N DMT − − Iθ¨ = K R(Q u + Rθ ) j S j 0,j j − G R[R(θ + 2θ + θ − ) S j+1 j j 1 − − To describe the shear contact, we utilize the Mindlin (Z Z − )], j+1 j 1 model [21], which assumes small relative displacements (6) and no slip at the contact surface. For two elastic bod- ies with shear moduli G and G , the restoring elastic 1 2 where u and w are horizontal and ver- force F to displacement δ of the particle center in the 0,j 0,j S S tical displacements of the substrate sur- direction transverse to the contact normal is given by [21] face at the point of contact, respectively. ∗ C. Eﬀective Medium Approximation F = 8G R1/2δ δ1/2, (3) S c S N Considering wavelengths much longer than the sphere ∗ − − − diameter, we treat the monolayer as an eﬀective contin- where G = [(2 ν )/G +(2 ν )/G ] 1 is the eﬀective 1 1 2 2 uous medium. By substituting the center diﬀerence for- · − · ≈ · · − shear modulus. Here, the factor of δ1/2 arises from the mulas [( ) ( ) − ]/(2D) ∂( )/∂x and [( ) N · · j+1 j ≈1 · j+1 Hertzian relation between the contact radius and FN−. 2( )j + ( )j−1]/(D2) ∂2( )/∂x2 into Eq. (6), we arrive By substituting the relative displacements δ = Z w at the equations of motion of the monolayer in eﬀective − N 0 and δ = Q u + Rθ into Eqs. 2 and 3, and lin- medium form: S 0 earizing about the equilibrium conﬁguration of δ = N,0 ∗ [3FDMT /(4E Rc1/2)]2/3 and δS,0 = 0, we derive linearized ∂2Q − − normal and shear contact stiﬀnesses m = K (Q u + Rθ) (cid:16) (cid:17) ∂t2 S 0 ∂2Q + 4G R2 (cid:32) ∗ 1/(cid:33)3 N ∂x2 2 KN = 6E RcFDMT ∂2Z − − m = K (Z w ) ∗ 1/3 (4) ∂t2 N 0 3 G 3 (7) KS = 8 4 E∗ RcFDMT , + 4GSR2(∂∂2xZ2 − ∂∂xθ ) ∂2θ − − I = K R(Q u + Rθ) with G and G given by equations of the same form, ∂t2 S 0 N ∗ S∗ but with E , G , R , and F adjusted for sphere- − ∂2θ ∂Z c DMT 4G R2(R2 + θ + ). sphere contacts. In the special case where the spheres S ∂x2 ∂x and substrate are composed of the same material, the relative magnitudes of the stiﬀness constants are deter- The coupling between the monolayer and substrate is mined exclusively by Poisson’s ratio ν of the material, described by the following boundary conditions at the surface z = 0, which describe the average force acting on the surface due to the motion of the spheres: − G = 2 2/3K N N ∗ KGS == ν2−K2/N3ν∗K , (5) σzx = KAS (Q − u0 + Rθ) S N (8) K − σ = N (Z w ), ∗ − − zz A 0 where ν = 2(1 ν)/(2 ν). where σ and σ are components of the elastic zx zz B. Equations of Motion of the Spheres stress tensor [17] and A = D2 is the area of a primitive unit cell in our square-packed monolayer. Assuming small displacements (i.e. Q , Z , and Rθ The combination of Eq. (7) and the linear elas- j j j are much less than D), the jth sphere obeys the equations tic wave equations describing waves in the substrate of motion [17], coupled by the boundary conditions of Eq. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: eﬀective

text: modulus.

text: Considering

text: the

text: DMT

text: adhesive

text: force

text: F

text: acting

text: normal

text: to

text: the

text: contact

text: surface

text: DMT

text: c

text: mQ¨

text: K

text: (Q

text: u

text: Rθ

text: (where

text: w

text: is

text: the

text: work

text: of

text: adhesion

text: between

text: two

text: j

text: S

text: j

text: j

text: surfaces),

text: the

text: net

text: normal

text: force

text: is

text: given

text: by

text: G

text: (Q

text: Q

text: N

text: j

text: j

text: mZ¨

text: K

text: (Z

text: w

text: j

text: N

text: j

text: G

text: [Z

text: Z

text: R(θ

text: θ

text: F

text: F

text: F

text: S

text: j

text: j

text: j

text: N,net

text: N

text: DMT

text: Iθ¨

text: K

text: R(Q

text: u

text: Rθ

text: j

text: S

text: j

text: j

text: G

text: R[R(θ

text: 2θ

text: θ

text: S

text: j

text: j

text: To

text: describe

text: the

text: shear

text: contact,

text: we

text: utilize

text: the

text: Mindlin

text: (Z

text: Z

text: j

text: model

text: which

text: assumes

text: small

text: relative

text: displacements

text: and

text: no

text: slip

text: at

text: the

text: contact

text: surface.

text: For

text: two

text: elastic

text: bod-

text: ies

text: with

text: shear

text: moduli

text: G

text: and

text: G

text: the

text: restoring

text: elastic

text: where

text: u

text: and

text: w

text: are

text: horizontal

text: and

text: ver-

text: force

text: F

text: to

text: displacement

text: δ

text: of

text: the

text: particle

text: center

text: in

text: the

text: S

text: S

text: tical

text: displacements

text: of

text: the

text: substrate

text: sur-

text: direction

text: transverse

text: to

text: the

text: contact

text: normal

text: is

text: given

text: by

text: face

text: at

text: the

text: point

text: of

text: contact,

text: respectively.

text: C.

text: Eﬀective

text: Medium

text: Approximation

text: F

text: S

text: c

text: S

text: N

text: Considering

text: wavelengths

text: much

text: longer

text: than

text: the

text: sphere

text: diameter,

text: we

text: treat

text: the

text: monolayer

text: as

text: an

text: eﬀective

text: contin-

text: where

text: G

text: ν

text: )/G

text: ν

text: )/G

text: is

text: the

text: eﬀective

text: uous

text: medium.

text: By

text: substituting

text: the

text: center

text: diﬀerence

text: for-

text: shear

text: modulus.

text: Here,

text: the

text: factor

text: of

text: arises

text: from

text: the

text: mulas

text: )/∂x

text: and

text: N

text: j

text: ≈1

text: Hertzian

text: relation

text: between

text: the

text: contact

text: radius

text: and

text: FN−.

text: )j

text: into

text: Eq.

text: we

text: arrive

text: By

text: substituting

text: the

text: relative

text: displacements

text: δ

text: Z

text: w

text: at

text: the

text: equations

text: of

text: motion

text: of

text: the

text: monolayer

text: in

text: eﬀective

text: N

text: and

text: δ

text: Q

text: u

text: Rθ

text: into

text: Eqs.

text: and

text: and

text: lin-

text: medium

text: form:

text: S

text: earizing

text: about

text: the

text: equilibrium

text: conﬁguration

text: of

text: δ

text: and

text: we

text: derive

text: linearized

text: normal

text: and

text: shear

text: contact

text: stiﬀnesses

text: m

text: K

text: (Q

text: u

text: Rθ)

text: S

text: N

text: KN

text: RcFDMT

text: m

text: K

text: (Z

text: w

text: N

text: G

text: KS

text: E∗

text: RcFDMT

text: ∂∂xθ

text: ∂2θ

text: I

text: K

text: R(Q

text: u

text: Rθ)

text: with

text: G

text: and

text: G

text: given

text: by

text: equations

text: of

text: the

text: same

text: form,

text: S

text: N

text: S∗

text: but

text: with

text: E

text: G

text: R

text: and

text: F

text: adjusted

text: for

text: sphere-

text: ∂2θ

text: ∂Z

text: c

text: DMT

text: θ

text: sphere

text: contacts.

text: In

text: the

text: special

text: case

text: where

text: the

text: spheres

text: S

text: ∂x

text: and

text: substrate

text: are

text: composed

text: of

text: the

text: same

text: material,

text: the

text: relative

text: magnitudes

text: of

text: the

text: stiﬀness

text: constants

text: are

text: deter-

text: The

text: coupling

text: between

text: the

text: monolayer

text: and

text: substrate

text: is

text: mined

text: exclusively

text: by

text: Poisson’s

text: ratio

text: ν

text: of

text: the

text: material,

text: described

text: by

text: the

text: following

text: boundary

text: conditions

text: at

text: the

text: surface

text: z

text: which

text: describe

text: the

text: average

text: force

text: acting

text: on

text: the

text: surface

text: due

text: to

text: the

text: motion

text: of

text: the

text: spheres:

text: G

text: N

text: N

text: KGS

text: σzx

text: KAS

text: (Q

text: Rθ)

text: S

text: N

text: K

text: σ

text: N

text: (Z

text: w

text: zz

text: A

text: where

text: ν

text: ν).

text: where

text: σ

text: and

text: σ

text: are

text: components

text: of

text: the

text: elastic

text: zx

text: zz

text: B.

text: Equations

text: of

text: Motion

text: of

text: the

text: Spheres

text: stress

text: tensor

text: and

text: A

text: is

text: the

text: area

text: of

text: a

text: primitive

text: unit

text: cell

text: in

text: our

text: square-packed

text: monolayer.

text: Assuming

text: small

text: displacements

text: (i.e.

text: Q

text: Z

text: and

text: Rθ

text: The

text: combination

text: of

text: Eq.

text: and

text: the

text: linear

text: elas-

text: j

text: j

text: j

text: are

text: much

text: less

text: than

text: D),

text: the

text: jth

text: sphere

text: obeys

text: the

text: equations

text: tic

text: wave

text: equations

text: describing

text: waves

text: in

text: the

text: substrate

text: of

text: motion

text: coupled

text: by

text: the

text: boundary

text: conditions

text: of

text: Eq.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##